# AQM: A Refresh of the Abstract Qubit Model for Quantum Co-design

Chenxu Liu  Email<EMAIL_ADDRESS>Physical and Computational Sciences,
Pacific Northwest National Laboratory, Richland, WA, 99354, USA  Samuel A.
Stein  Physical and Computational Sciences, Pacific Northwest National
Laboratory, Richland, WA, 99354, USA  Muqing Zheng  Physical and Computational
Sciences, Pacific Northwest National Laboratory, Richland, WA, 99354, USA
James Ang  Physical and Computational Sciences, Pacific Northwest National
Laboratory, Richland, WA, 99354, USA  Ang Li  Email<EMAIL_ADDRESS>Physical
and Computational Sciences, Pacific Northwest National Laboratory, Richland,
WA, 99354, USA

###### Abstract

Qubits are the fundamental building blocks of quantum information science and
applications, whose concept is widely utilized in both quantum physics and
quantum computation. While the significance of qubits and their implementation
in physical devices have been extensively examined, we posit that now is the
right time to revisit this understanding. In this paper, we introduce an
abstract qubit model (AQM), offering a mathematical framework for higher-level
algorithms and applications, and setting forth criteria for lower-level
physical devices to enable quantum computation. We first provide a
comprehensive definition of “qubits”, regarded as the foundational premise for
quantum algorithms (bottom-up support), and examine their requisites for
devices (top-down demand). We then investigate the feasibility of moderating
specific requirements, thereby broadening device support while considering
techniques that tradeoff extra costs to counterbalance this moderation.
Lastly, we delve into the quantum applications that only require incomplete
qubits, and discuss the physical systems having restricted AQM support but are
still useful in quantum applications. AQM may serve as an intermediate
interface between quantum algorithms and devices, facilitating quantum
algorithm-device co-design.

##  I Introduction

Quantum computing (QC) holds the potential for significant acceleration of
classically challenging problems Grover (1996); Shor (1994, 1997); Simon
(1997), while quantum information processing (QIP) facilitates secure
communication underpinned by quantum mechanics Bennett and Brassard (1984); Xu
_et al._ (2020); Portmann and Renner (2022). Along with the rapid advancement
of quantum technology is the growing selection of physical systems promising
for QC and QIP. Despite these devices being rooted in diverse materials and
fabrication processes, certain essential functionalities exist, such as
initialization, control, and readout of quantum information, that every device
is deemed to offer. These functionalities represent the commonality, or the
criteria, necessary for a device to be capable of engaging in QC and QIP.

In order to examine whether a physical system can be a promising candidate for
QC and QIP, DiVincenzo in his seminal work DiVincenzo (2000) proposed the
well-known criteria. One of the key concepts, which will also be the main
focus of our paper, is the concept of qubit.

As the fundamental building blocks of gate-based quantum computers, the
abstraction of qubits is necessary to build a model for quantum computers and
support the upper-stack design. As described by Nielsen and Chuang, a qubit is
defined as a mathematical object, each of which has a state that is a unit
vector in a two-dimensional complex vector space Nielsen and Chuang (2010).
Looking from the bottom of the QC design stack, this criteria implies also the
requirements placed on the physical platforms. The physical platform should
support such a system, abstracted as a qubit. To better understand such a
qubit in realistic physical systems, especially when the physical degrees of
freedom are more than a qubit, in Ref. Viola  _et al._ (2001), Viola et al.
proposed that: in addition to the operator algebra, the physical system should
also support (1) universal control, (2) initialization, and (3) read-out
capabilities to build an operational qubit.

Despite classical, it has been more than 20 years since DiVincenzo’s qubit
criteria. During the recent development of QC and QIP, new opportunities and
challenges have arisen. Firstly, although a large integration of physical
qubits has been demonstrated Gambetta (2022); Bluvstein  _et al._ (2024);
Gambetta (2023), a perfect logical qubit has not been built yet. We still need
to take advantage of the hardware features to maximize the benefits in QC.
Second, some recent quantum algorithms do not require a complete AQM to
support. The rigorous AQM has the potential to be released. Beyond the
commonly used quantum circuit model, various universal computation models
exist, such as measurement-based quantum computation (MBQC) Raussendorf and
Briegel (2001); Raussendorf  _et al._ (2003); Browne and Rudolph (2005);
Briegel  _et al._ (2009) and quantum random walk Kadian  _et al._ (2021);
Venegas-Andraca (2012); Childs (2009); Singh  _et al._ (2021). These models
have unique requirements for physical systems. Lastly, with the recent
discussion of quantum co-design Shi  _et al._ (2020); Tomesh and Martonosi
(2021), partially supporting AQM can benefit both quantum algorithm designs
and hardware development. How to leverage the AQM for algorithm-device co-
design is still unclear. These emerging topics call for a reconsideration of
the AQM.

In this paper, we address these challenges by revisiting the AQM. We begin by
presenting a comprehensive definition of qubits, essential for quantum
algorithms (bottom-up support), and analyze their device requirements (top-
down demand). We explore the opportunities of relaxing specific requirements
to expand device support, considering techniques that trade off extra costs to
compensate for the relaxation. Finally, we examine quantum applications that
only need incomplete qubits and discuss physical systems with limited AQM
support yet remain valuable in quantum applications.

This paper is organized as follows. In Sec. II, we define the abstracted qubit
model. In Sec. III, we discuss how the physical requirements can be released
and compensated. In Sec. IV, we focus on releasing the complete AQM from both
the upper and lower stacks opportunities. Finally, we conclude in Sec. V.

Figure 1: The abstracted qubit model in the design stacks of quantum
information processing. The abstracted qubit model contains the mathematical
description of qubits, which can be used in the upper stacks, while it places
physical requirements that need to be fulfilled by the lower stacks.

##  II Abstract model of qubits

We present the definition of qubits and construct the AQM. The AQM design
stack is shown in Fig. 1. Specifically, we consider how qubits can support the
upper stacks of QC and QIP systems by defining the mathematical description of
the states and the operations on qubits in Sec. II.1. Then, we discuss the
requirements of AQM placed on the physical systems in Sec. II.2.

###  II.1 Mathematical description of qubits

We first extract the essential properties of qubits. We point out how a qubit
should be viewed in quantum algorithms and applications.

  * •

State of qubits: The state of a qubit can be represented as a complex two-
dimensional vector with the unit norm, same as Ref. Nielsen and Chuang (2010).
The state of $n$ qubits is a complex vector with dimension $2^{n}$. Assuming
the state of qubit A is $\ket{A}$, while the state of qubit B is $\ket{B}$,
the state of both qubits is $\ket{A}\otimes\ket{B}$, where $\otimes$ represent
the Kronecker product of two vectors.

  * •

Operations on qubits: The quantum manipulation on qubits can be represented as
a unitary matrix $U$, i.e., $U\,U^{\dagger}=U^{\dagger}\,U=I$, where $I$ is an
identity matrix. The dimension of the unitary matrix is $2^{n}\times 2^{n}$ if
the operation is acting on $n$ qubits. Any unitary matrices ($U$) with the
given dimension should be obtainable by qubit operations ($\tilde{U}$) with a
global phase factor, $U=e^{i\phi}\tilde{U}$.

  * •

Measurements on qubits: A measurement on $n$ qubits is represented by a set of
$2^{n}\times 2^{n}$ matrices, noted as $\mathbf{M}=\\{m_{1},m_{2},...\\}$,
where the matrices satisfies $\sum_{\forall m\in\mathbf{M}}m^{\dagger}m=I$.
After performing the measurement, the state of the qubits is transformed into
$\ket{\psi_{j}}=m_{j}\ket{\psi_{0}}/\sqrt{p_{j}}$, where
$p_{j}=\bra{\psi_{0}}m_{j}^{\dagger}m_{j}\ket{\psi_{0}}$ is the probability of
getting $j$-th measurement outcome ($j=0,\,1$ for a projective measurement on
a single qubit). In other words, the state of the qubit after the measurement
can be treated as a probabilistic (classical) superposition of all possible
outcomes.

Qubit vs Qudit: With the possibility of data compression and the advantages of
simulating the dynamics with other symmetry groups, qudits become an
attractive generalization and alternative to qubits Wang  _et al._ (2020); Chi
_et al._ (2022); Liu  _et al._ (2023a); González-Cuadra  _et al._ (2022). A
qudit is a system with $d$ states, where $d$ can be more than $2$. The
mathematical definition discussed above can be easily generalized to qudit
systems by redefining the dimensionality of a single qudit to $d$, while the
operation matrix dimensions are $d\times d$. In the following of our
discussion, we mainly focus on qubits, while the generalization to qudit
systems is straightforward as a simple linear mapping.

###  II.2 Physical requirements for constructing qubits

Recall DiVincenzo’s five criteria DiVincenzo (2000) for examining physical
systems for QC:

  1. 1.

A scalable physical system with well-characterized qubits.

  2. 2.

The ability to initialize the state of the qubits to a simple fiducial state.

  3. 3.

Long relevant decoherence times, much longer than the gate operation time.

  4. 4.

A “universal” set of quantum gates.

  5. 5.

A qubit-specific measurement capability.

and two extra criteria for quantum communication,

  * 6.

The ability to interconvert stationary and flying qubits.

  * 7.

The ability to faithfully transmit flying qubits between specified locations.

Subsequent works refine qubit requirements and explore building qubits for QC
and QIP Viola  _et al._ (2001); Duckering (2022). Here we focus on
DiVincenzo’s criteria because of the wide adoption in QC. The ideal qubit
model contains:

  * •

Rule 1: States. A qubit should have $2$ quantum states which can be addressed
at any time of interest.

  * •

Rule 2: Operations. The capability of performing operations on qubits requires
the physical system that makes these $n$ qubits to be completely controllable
Dong and Petersen (2010); D’Alessandro (2021). Obviously, with the complete
controllability of the physical systems, universal quantum computing can be
supported.

  * •

Rule 3: Connectivity. The capability of performing any unitary operations on
qubits requires the qubits to have all-to-all connectivity.

  * •

Rule 4: Coherence. The qubit should have infinite long coherent time and
experiences no other incoherent errors, which ensures that the state of the
qubits at the time of interest can always be represented as a $2$-dimensional
complex vector with unit length.

  * •

Rule 5: Readout. In addition, as projective measurements and post-selection
results can be well described by the measurement formalism Nielsen and Chuang
(2010), with the above measurement operation support, the qubit states can be
readout into classical information. We also implicitly require that the
measurement is error-free, meaning that the extracted classical information is
accurate. The initialization of the qubits to a known state (noted as
$\ket{\psi_{0}}$) can also be realized through this measurement operation by
defining the measurement operator set
$\mathbf{M}=\\{\outerproduct{\psi_{0}}{j}\\}$ for all basis states of the
qubit systems $\ket{j}$.

##  III Incomplete support of the AQM from physical systems

Table 1:  Comparison of the ideal AQM and the incomplete AQM. We also consider the techniques to compensate for the imperfection and their corresponding costs. Ideal model |  Restricted model |  Compensation techniques |  Cost
---|---|---|---
Two quantum states |  Two quantum states |  |
Infinitely long coherence time |  Finite but long coherence time |  Quantum Error Detection and Correction |  More physical qubits and more gate and measurement operations
|  |  Quantum error mitigation |  Extra quantum algorithm implementations, and post-processing loads.
|  |  Dynamical decoupling |  Extra control complexities
No quantum operation errors |  gate imperfection are allowed |  Quantum Error Detection and Correction |  More physical qubits and more gate and measurement operations
Directly support any unitary operations. |  A discrete universal gate set is available. |  Gate decomposition |  More gate operations, which require more operation time. Needs more qubits if the gate set is computationally universal,
All-to-all connectivity. |  All qubits can be connected. |  Gate routing |  More physical operations and extra times.
General measurements are accessible. |  Projective measurements can be performed, even indirectly. |  Measurements with auxiliary qubits |  Extra qubits and quantum gates to construct a general measurement.
|  |  Routing for measurements |  Swapping the quantum states to the qubits requires extra gate operations and time.
Figure 2: The incomplete support of the AQM from the physical systems. With
compensation techniques, the physical systems can support the complete AQM
with extra costs on time and resources.

The complete AQM is demanding. Most present physical systems can only support
the AQM partially, as shown in Fig. 2. Nevertheless, full support of a
computation qubit in the AQM is still possible by using the relaxation with
compensation techniques. However, these techniques use extra physical
resources, time, and complexity cost. These compensation techniques answer the
question mark shown in Fig. 2. In the following, we break down the
requirements to construct a qubit from physical systems and present the
available techniques to compensate for the “imperfection” of the physical
platforms.

#####  Relaxing qubit states: The qubit should have two quantum states.

The existence of two quantum states for a qubit is necessary. One commonly
seen example is in trapped ion or neutral atom systems, where the quantum
levels of the ions and atoms are used to encode quantum information. Another
example is to use the presence or absence of a particle (photons or electrons,
etc.) in a single mode as the two quantum states of the qubits. We will
discuss it in more detail in Sec. IV.2.3. There may be more than two quantum
states in a certain physical system Viola  _et al._ (2001). Other than simply
picking up two distinct states from all available quantum states, the quantum
states need to be controllable (see the following discussion about Operation).

On the other side, some physical platforms may not have enough quantum states
that fulfill even the minimum requirements of a single qubit, e.g., the space
formed by the two states can be addressed, and the operations on the qubits
should map the state space back to itself. Similar constraints apply to qudit
constructions. When a single physical system is not sufficient to provide such
a state space, however, the composite system of multiple physical systems can
be utilized to form a qubit, e.g., the presence or absence of a single
electron in a quantum dot cannot be encoded as qubit state due to the lack of
coherent manipulation techniques in the Hilbert space spanned by these two
states DiVincenzo (2000). In contrast, the presence or absence of a microwave
photon can be encoded as a qubit with the help of a superconducting qubit
Mariantoni  _et al._ (2011a, b); Besse  _et al._ (2020); Reuer  _et al._
(2022); Gao  _et al._ (2019).

We note that this requirement is weaker than the requirements of single-qubit
completely controllable (Rule 2) Dong and Petersen (2010); D’Alessandro
(2021). Despite incomplete control over the physical system, if a set of
computational universal gates can be executed, and the qubit state can be
manipulated to a specific state (e.g., $\ket{0}$), the construction of a qubit
is possible. The arbitrary state of the qubit Hilbert space can be
approximately approached with the extra cost of quantum gate operations (see
the discussion of Operations).

The quantum information encoded in qubit Hilbert space can be transmitted via
various physical systems. In measurement-based quantum computing (MBQC)
Raussendorf and Briegel (2001); Raussendorf  _et al._ (2003); Raussendorf and
Harrington (2007); Briegel  _et al._ (2009), for instance, information is
”teleported” to connected physical qubits in resource states, with the
physical qubits being destroyed during computation, especially in photonic
systems.

#####  Relaxing operations: The qubit should support a set of computational
universal gates.

It is quite costly to require any unitary matrices on an arbitrary number of
qubits to be directly implementable as different operations. What is the
minimum requirement on the set of supported operations to allow us, at least
approximate a given unitary matrix on a number of qubits? This question
motivates the research of the universal gate set of QC. It turns out that a
discrete set of gates (unitary matrices) on a small number of qubits is enough
($n\leq 3$), e.g., the set of Hadamard gate, CNOT gate, phase gate, and
$\pi/8$ gate is universal Nielsen and Chuang (2010).

However, releasing this requirement also brings extra cost, which has two
aspects. One is from the gate decomposition, which decomposes the required
unitary matrix into the directly supported gate operations in the gate set.
The gate decomposition requires an increasing number of quantum gates applied
to the qubits. For example, the Solovay-Kitaev theorem showed that
approximating an arbitrary unitary matrix in $SU(d)$ using a discrete set of
gates that generate a dense subgroup of $SU(d)$ requires
$O(log^{c}(1/\epsilon))$ gates, where $c$ is a constant and $\epsilon$ is the
accuracy Nielsen and Chuang (2010); Dawson and Nielsen (2005). The second
aspect is that the different universal gate sets supported by the physical
system can cost extra quantum gates and qubits for the same computation, as
highlighted by the difference between ‘computational universal’ and ‘strictly
universal’ Aharonov (2003). One example is the gate set that consists of the
Toffoli gate and the Hadamard gate. This gate set is universal Aharonov
(2003); Shi (2002), but a general unitary on a $n$-qubit system cannot be
decomposed with arbitrary accuracy using just Toffoli and Hadamard gates.
Instead, the computation process of that unitary operation can be simulated
within an arbitrarily small error Aharonov (2003). However, it costs
polynomially many more qubits and gates. Similar discussions about
universality can be found in qudit systems Brylinski and Brylinski (2001);
Wang  _et al._ (2020). We stress that the cost of decomposing target unitaries
on the qubit system should be manageable, e.g., not using exponentially more
time and physical qubits.

Nevertheless, there are other QC models other than the quantum circuit models.
For example, in the MBQC scheme Raussendorf and Briegel (2001); Raussendorf
_et al._ (2003); Raussendorf and Harrington (2007); Briegel  _et al._ (2009),
universal computing is driven by the measurements on a prepared entangled
state. In this model, the computation can still be mapped back to the circuit
model, where the flow of quantum information can be treated as qubits that
carry the quantum information. The physical systems that support MBQC can
still be treated as supporting full universal control, though indirectly with
the help of measurements and pre-existed quantum entanglements.

#####  Relaxing connectivity: The qubits needs to be connected.

For a QC system supporting a universal gate set applicable up to $n$ qubits, a
prerequisite for using these qubits in general QC algorithms is that any
subset of $n$ qubits must be capable of performing gates from the universal
set. Note that all-to-all connectivity is not necessary. For example, if the
universal gate set contains CNOT, Hadamard, phase and $\pi/8$ gate Nielsen and
Chuang (2010), this is equivalent to requiring that all qubits can perform the
Hadamard, phase and $\pi/8$ gates, and any two qubits can find a path between
in their connection graph. Even if two qubits are not directly connected, a
CNOT gate between them can be implemented using SWAP gates along the path,
decomposable into three CNOTs. This less stringent connectivity requirement
incurs additional costs compared to all-to-all connectivity.

In the new paradigm of distributed QC models, the “connection” between the two
qubits we discussed above is not necessarily made of physical contacts, or
direct coupling between two physical qubits. Instead, quantum communication
between two physically remote qubits enables remote gate operations Cuomo  _et
al._ (2020); Cacciapuoti  _et al._ (2020), which relies on a reliable channel
to transmit flying qubits from one end to the other, or a reliable source of
entangled resource states (like Bell states) to be used for gate teleportation
Gottesman and Chuang (1999); Huang  _et al._ (2004); Wan  _et al._ (2019);
Chou  _et al._ (2018); Nielsen and Chuang (2010). This should also comply with
DiVincezo’s criteria for quantum communication (criteria 6 and 7, see Sec.
II.2).

#####  Relaxing coherence: The qubit should have finite but long enough
coherence time.

1\. relaxing to the coherence time enough for gate operation, 2. using QEC to
extend logical qubit coherence.

Quantum coherence is critical for quantum computing. However, physical systems
can hardly be completely isolated. The interaction with the external
environment causes decoherence of the quantum systems. The coherence time
gives a time scale for preserving the quantum coherence. These unwanted
interactions also introduce decoherence errors to the quantum gates applied to
the systems. Therefore, the coherence time should be long enough for the gate
operations time to perform a meaningful QC on a physical system. This
motivates the development of noisy intermediate-scale quantum (NISQ)
algorithms, e.g., variational quantum algorithms Cerezo  _et al._ (2021)

Another strategy is to suppress the noise and extend the coherence time of the
logical qubits by quantum error correction (QEC) codes Nielsen and Chuang
(2010); Calderbank and Shor (1996). By combining multiple physical qubits into
a logical qubit, QEC codes use entangled states as qubit states to detect and
correct errors. Increasing the code distance, achieved by using more physical
qubits per logical qubit, allows for suppressing logical error rates as long
as noise-induced error rates remain below the code threshold. What’s more, QEC
not only extends the coherence time of logical qubits but also suppresses
noises from other sources, e.g., imperfect control during gate implementation
and cross-talk between adjacent qubits, etc.

#####  Relaxing readout: The qubit should have a way of performing
measurements on qubit states.

Compared to the mathematical description of the general measurements on ideal
qubits, a quantum system to build a qubit may only support projective
measurements along the computational basis. It is possible to construct
general measurements using auxiliary qubits, entangling gates, projective
measurements on the auxiliary qubits, and quantum gates conditioned on the
projective measurement outcomes.

Furthermore, even the direct projective measurements on the physical systems
can be released if we consider heterogeneous QC architectures, where the
qubits used in different functional units inside the QC units Stein  _et al._
(2023a, b); Liu  _et al._ (2023b) can be different. The role of the qubit used
in the quantum processing units and quantum memory units is to process and
store quantum information, which means direct measurements on them are not
necessary. We can think of using another physical system, that supports qubit
measurement functionalities and can interact with the qubits in the quantum
memory modules to extract the stored quantum states. Therefore, when the
measurement of the quantum information is needed, it is performed on the
second type of qubits by swapping the information out from the computing
qubits. A more detailed discussion is in Sec. IV.1.5.

Table 1 summarizes our discussion on releasing requirements for an ideal AQM.
We also cover compensation techniques and their associated costs. If a
physical system meets the requirements of a restricted AQM and supports the
compensation techniques, it’s suitable for constructing a large-scale general-
purpose quantum computer. Such a system can serve as a candidate for building
a qubit sufficient to support any quantum algorithms.

Figure 3: The algorithm-device co-design. Quantum algorithms and applications
may only have partial demand on the AQM, while the physical systems can only
have partial support of the AQM. The co-design of quantum algorithms and
physical systems can be beneficial.

##  IV AQM for Algorithm-Device codesign

In this section, we discuss the algorithm-device co-design opportunities
demonstrated by Fig. 3. Specifically, a quantum application may not require
the complete support of the AQM, and hence a physical system that fulfills its
requirements can still be useful, even without building a qubit, and vice
versa. Therefore, the algorithm-device codesign opportunities by partially
breaking the abstraction layer of qubits can be beneficial for the current
stage of QC system design. Specifically, in Sec. IV.1, we discuss the
algorithm and applications that can be supported without the complete AQM,
while in Sec. IV.2, we discuss different physical systems supporting the AQM
or quantum applications. In Sec. IV.3, we discuss co-design opportunities of
the physical systems.

Table 2:  Comparison of the demand of quantum algorithms and applications on the AQM. We focus on quantum annealing (QA), quantum random walk (QRW), measurement-based quantum computing (MBQC), analogue quantum simulation (AQS), and quantum memory (QM). $\checkmark$ means the demand is similar to the AQM. |  States |  Operation |  Connectivity |  Coherence |  Readout
---|---|---|---|---|---
QA |  $\checkmark$ |  Released |  Critical to maintain high connectivity |  $\checkmark$ |  Computational basis
QRW |  Encoding methods can be different. |  Released |  Based on the problem, can be released. |  $\checkmark$ |  Computational basis
MBQC |  Physical qubits are not always alive. |  Released (between physical qubits) |  Resource state generation. |  $\checkmark$ |  Arbitrary basis
AQS |  $\checkmark$ |  Released |  Based on the problem. |  $\checkmark$ |  Computational basis (depending on problems).
QM |  $\checkmark$ |  Released |  Connectivity to computing component is necessary. |  Long coherence time. |  Released

###  IV.1 Applications without the complete AQM

In this section, we consider three computation models, including quantum
annealing, MBQC, and quantum random walk for computation. In addition, we also
focus on the application of analog quantum simulation and quantum memory,
which do not necessitate the complete AQM. The demand for these quantum
applications is summarized in Table. 2.

####  IV.1.1 Quantum Annealing

Analogous to the classical annealing method, quantum annealing is a QC
paradigm to solve combinatorial optimization problems Das and Chakrabarti
(2008); Hauke  _et al._ (2020); Albash and Lidar (2018). Unlike the quantum
circuit computing model, quantum annealing is an analog quantum computation
framework. The optimization problem is encoded into the device Hamiltonian,
whose ground state corresponds to the global optimum of the problem. Solving
the problem corresponds to annealing the system states from an initial state
to the ground state of the problem Hamiltonian.

The device Hamiltonian can be described by

$H=\lambda_{0}(t)H_{0}+\lambda_{1}(t)H_{1},$ (1)

where $\lambda_{0,1}(t)$ are two tunable parameters, $H_{1}$ is the
Hamiltonian that encodes the optimization problem, $H_{0}$ is another
Hamiltonian which does not commute with the terms in $H_{1}$. In the
beginning, $\lambda_{0}=1$ and $\lambda_{1}=0$, and the system is initialized
into the ground state of $H_{0}$. During the annealing process, $\lambda_{0}$
is slowly decreased to $0$, while $\lambda_{1}$ is increased to $1$. As long
as the change of the system Hamiltonian is slow enough, based on the adiabatic
theorem, the state of the system will be tuned to the ground state of $H_{1}$,
which solves the optimization problem.

Therefore, quantum annealing does not require the support for Rule 2 of the
AQM. As the computation is not driven by quantum gate operation, there is no
need to support a full universal gate set. Instead, as long as the physical
device can realize the Hamiltonian $H_{0}$ and $H_{1}$ that encodes the
optimization problem, e.g., the nearest-neighbor spin-spin coupling and
single-qubit Pauli-X rotations Johnson  _et al._ (2011), and can slowly tune
the system Hamiltonian and readout the final state of the system, the system
can support the quantum annealing algorithm to solve the specific type of
optimization problems.

For example, one type of physical system commonly considered in quantum
annealing is the Ising spin glass, where $H_{0}$ contains single-site
$\sigma^{(x)}$ terms, while $H_{1}$ contains two-site $Z$ couplings, i.e.,
$\sigma^{(z)}_{i}\sigma^{(z)}_{j}$. Although the coupling terms are restricted
to two-body interactions, NP-hard problems can be cast into quadratic forms
using ancilla Hauke  _et al._ (2020); Lucas (2014). A survey of NP problems
mapping to spin systems can be found in Ref. Lucas (2014). Note that realizing
the Hamiltonian terms does not lead to universal control of individual spin in
the system.

In order to keep the resource cost low when encoding the optimization problems
into a quantum annealing setup, the physical systems need to have high
connectivity, which is Rule 3 of the AQM. However, there are embedding
techniques that allow mapping a quantum annealing problem onto physical
systems with limited connectivity Djidjev  _et al._ (2018); Zaribafiyan  _et
al._ (2017); Lechner  _et al._ (2015).

In conclusion, the quantum annealing applications require Rule 1 and 4 of the
complete AQM, while the Rule 2 and 5 can be released. In addition, the Rule 3
is also critical to keep the resource cost low.

####  IV.1.2 Quantum random walk.

Quantum random walk is a computation paradigm for QC, especially for solving
graph problems. There are broad categories of quantum walk computations. In
this section, we do not aim to give a thorough review of quantum random walk.
For a review of quantum random walk for QC, we refer to Refs. Kadian  _et al._
(2021); Venegas-Andraca (2012). Instead, we focus on the coined discrete-time
quantum walk as an example.

Compared to the classical random walk model, where the walker stochastically
chooses the movement, in the coined discrete-time quantum walk, there are two
quantum systems, one is used as a coin, while the other system is the walker
system. The walker’s movement is determined by the coin state. The walker’s
trajectory is entangled with the coin states and different trajectories are
coherently superposed.

For instance, when a quantum walker working on a 1D line, who can only walk to
the right or left. The coin system can be a two-level system, while the
walker’s state encodes the position of the walker on the line. The walk
operation can be represented as a unitary acting on both systems,

$U=\outerproduct{0}{0}^{c}\otimes U_{0}^{w}+\outerproduct{1}{1}^{c}\otimes
U_{1}^{w},$ (2)

where $U_{0}^{w}$ and $U_{1}^{w}$ are unitary operations acting on the walker
system representing the movements. To encode different probability
distributions of the walker, the coin system should be able to apply a general
unitary in the Hilbert space (a general $SU(2)$ operation if it is a qubit).
In addition, it is necessary to perform measurements on the coin and walker
systems to extract the computation results.

The quantum walk model can be used to implement universal QC Childs (2009);
Singh  _et al._ (2021). The quantum random walk model needs a different set of
requirements compared to the AQM. For example, quantum information can be
encoded not into the internal state of the qubit, but instead, into the
spacial modes that can hold a physical particle Tang  _et al._ (2018). Other
than supporting a universal gate set on qubits, quantum walk computation only
requires that the physical qubit can hop between different spatial modes that
encode the quantum information to carry out the computation.

In summary, quantum walk applications require Rule 4 of the AQM. However, they
have distinct requirements from the general AQM. Adaptable to problem demands,
the quantum walk can operate on systems with more than two quantum states by
encoding the coin system into a multi-level system based on walking choices,
which differs from the Rule 1 of the AQM. Rule 2 of the AQM is released, as
universal controllability and a universal gate set are not essential while
controlled-walk operation suffices. Rule 3 can also be partially released,
where the unique connectivity involves linking the systems encoding coin and
space degrees of freedom (DOFs). Full connectivity is not required for
unreachable space DOFs. For readout, the quantum walk model primarily
emphasizes distribution in space DOFs Gong  _et al._ (2021a); Esposito  _et
al._ (2022), making complete quantum information extraction (e.g., state
tomography) unnecessary. Therefore, Rule 5 is also relaxed.

####  IV.1.3 Measurement-based quantum computation.

MBQC is a computation model different from the quantum circuit computation
model Raussendorf and Briegel (2001); Raussendorf  _et al._ (2003); Browne and
Rudolph (2005); Briegel  _et al._ (2009). In MBQC, the computation is driven
by measurements of qubits in an entangled state of qubits. This entangled
state is the resource of MBQC. A two-dimensional and higher-dimensional
cluster state can be used as the resource state of MBQC Raussendorf  _et al._
(2003); Van den Nest  _et al._ (2006); Brown and Roberts (2020).

As demonstrated in Ref. Raussendorf  _et al._ (2003), a set of universal gates
can be equivalently performed in MBQC. Therefore, MBQC is equivalent to the
quantum circuit computation model and it can support universal QC. However,
compared to the requirements placed on the AQM, MBQC releases a few
requirements. The physical qubits used in the MBQC are not accessible at all
times, and they do not need to support a universal gate set for computation.
Furthermore, the connectivity requirements are also different from the AQM, as
it only requires that the physical qubits need to be able to prepare a
specific resource state for MBQC. During the computation, connectivity between
different physical qubits is not necessary.

MBQC requires fast and high-fidelity single-qubit measurements on the qubits,
and the measurement basis can be easily adjusted. Additionally, although on-
demand entangling operations aren’t essential during computation, preparing an
entangled state of all qubits can be demanding. Moreover, due to MBQC’s
nature, performing the same algorithm requires a larger number of physical
qubits compared to the circuit computation model.

In summary, MBQC requires Rule 4 and 5 of the AOM. In contrast, MBQC relaxes
Rule 1 of the AQM, where the physical qubits do not need to be alive
throughout the computation. Rule 2 of the AQM is also released, as the
physical qubits do not need to perform entangling gates. Rule 3 is supported
in a different manner, which refers to the connectivity of the resource state.

####  IV.1.4 Analogue Quantum Simulation

Quantum simulation utilizes the power of QC systems to model the behavior of a
desired target system. By measuring the quantum system, we can extract
physical properties of the target systems that are difficult to compute using
classical methods.

There are two main quantum simulation categories based on the type of quantum
systems used. One involves digital quantum computers equipped with algorithms
to compute physical properties, such as the VQE algorithm designed for
determining molecular ground states and energies Cerezo  _et al._ (2021). The
other method focuses on employing quantum devices for analogue simulations
Monroe  _et al._ (2021); Lv  _et al._ (2018); Ebadi  _et al._ (2021); Daley
_et al._ (2022), which is the emphasis of this subsection.

Analogue quantum simulations can address low-energy steady states or dynamics
of physical systems. Solving the ground state of a physical system can use
adiabatic evolution, akin to the discussion of the quantum annealing method
(see Sec. IV.1.1). Alternatively, when the problem directly maps to a quantum
system, i.e., the quantum simulator can realize the target Hamiltonian,
manipulating the simulator can simulate the dynamics of the target system.
Therefore, the measurement of the quantum simulator can help to reveal the
physical properties of the target problem. For example, the Ising model can
naturally map to a neutral atom lattice Ebadi  _et al._ (2021). The quantum
many-body phase transition can be directly observed by manipulating the atom
interactions Ebadi  _et al._ (2021); Bloch  _et al._ (2008).

In summary, analogue quantum simulation still demands Rule 1 and 4 of the AQM.
As the simulator does not need to support a universal gate set for computing,
or achieving universal control on each qubit, as long as the target
Hamiltonian can be realized, Rule 2 of the AQM is released. As the
connectivity and the readout capability depend on the physical problems being
simulated, Rule 3 and 5 of the AQM have the potential to be released.

####  IV.1.5 Quantum Memory

Quantum memory is an application of quantum systems, which can preserve
quantum information for an extended period of time. Unlike the other quantum
computation paradigms and quantum simulation schemes, quantum memory does not
aim to perform computation. Therefore, the quantum memory application has
different demands on the AQM Liu  _et al._ (2023b).

To support quantum memory applications, Rule 1 and 4 of the AQM are required.
The AQM Rule 2 can be released, as a quantum system does not need to support
universal control of individual qubits and a universal gate set for quantum
computation. Rule 5 can also be released as when the quantum memory acts as a
computing module in the future QPU architecture design, its readout capability
is not necessary. Rule 3 is differently demanded, as the connectivity of the
qubits inside the quantum memory modules is not essential, while connectivity
to computing qubits is necessary Liu  _et al._ (2023b).

###  IV.2 Physical systems supporting the AQM

In this section, we examine the physical system from the lower stacks of the
AQM. We do not aim to provide a thorough review of useful physical systems for
QC and QIP. Instead, we mainly focus on three types of physical systems: (1)
physical systems with strong support of the AQM, e.g., superconducting
circuits, trapped ions, and neutral atoms, (2) systems with unique strength
and weakness in supporting the AQM, e.g., nitrogen-vacancy centers in diamond
crystals, and photonic systems, (3) systems with incomplete support of the
AQM, e.g, the quantum memory systems for photons.

####  IV.2.1 Physical systems for AQM support

Superconducting circuits Devoret  _et al._ (2004); Girvin  _et al._ (2009);
Kjaergaard  _et al._ (2020); Kockum and Nori (2019); Blais  _et al._ (2021),
trapped ions Häffner  _et al._ (2008); Monroe and Kim (2013); Bruzewicz  _et
al._ (2019); Brown  _et al._ (2021), and neutral atom systems Wu  _et al._
(2021); Saffman  _et al._ (2010); Bluvstein  _et al._ (2022, 2024) are widely
adopted in QC. The superconducting circuit system is known for its fast and
reliable gate operations. The trapped ions system is famous for its
connectivity and long coherence while the neutral atom system is good for
parallel operations and integration. Although there are other physical systems
also supporting the AQM well, e.g., quantum dot and solid-state spin qubits,
in the interest of conciseness, we mainly focus on the superconducting circuit
system to discuss the support of the AQM from the physical stack. Detailed
discussions of trapped ions and neutral atoms can be found in the Appendix.

##### Qubit states.

The superconducting qubits use the modes of plasma oscillation as quantum
states to encode quantum information. Depending on the ratio of Josephson
energy ($E_{J}$) to the charging energy ($E_{C}$). When $E_{J}\ll E_{C}$, the
qubit states have definite Cooper pair numbers, while in the opposite case,
the qubit states have a more definite superconducting phase.

Both transmon and fluxonium qubits have higher excited states Kjaergaard  _et
al._ (2020); Blais  _et al._ (2021); Manucharyan  _et al._ (2009); Koch  _et
al._ (2007), while the ions and atoms have other electronic levels. In
transmon qubits, due to the anharmonicity provided by Josephson junctions, the
transitions to higher excited states can be energetically distinguished from
the transition between ground and excited states Koch  _et al._ (2007);
Manucharyan  _et al._ (2009). Although it is still likely to populate higher
excited states, especially for transmon qubits, the leakage can be reduced
using quantum control methods, e.g., DRAG pulses Chow  _et al._ (2010).

In the trapped ion and neutral atoms systems, qubits are made of individual
ions and atoms that are trapped in electromagnetic/optical traps. The qubit
states can be the electronic levels Bermudez  _et al._ (2017); Pogorelov  _et
al._ (2021); Levine  _et al._ (2018); Madjarov  _et al._ (2020); Picken  _et
al._ (2018); Morgado and Whitlock (2021) or the hyperfine levels of the
ion/atom Harty  _et al._ (2014); Gaebler  _et al._ (2016); Ballance  _et al._
(2016); Wang  _et al._ (2017, 2021); Sheng  _et al._ (2018); Graham  _et al._
(2019); Levine  _et al._ (2019); Bluvstein  _et al._ (2022). Ions and atoms
also have other levels, which can be distinguished by the energy difference
and by the transition rules.

The Hilbert space spanned by the qubit states of transmon, fluxonium, ions,
and neutral atoms can be completely addressed using microwave and optical
control methods, which will be discussed below in more detail.

##### Operations.

The universal control of a single superconducting qubit can be obtained
through external microwave drives Chow  _et al._ (2010); Koch  _et al._
(2007); Manucharyan  _et al._ (2009); Houck  _et al._ (2008); Blais  _et al._
(2021). Specifically, a superconducting qubit can be driven by microwave
fields to apply arbitrary angle Pauli-X and Pauli-Y gates. The Pauli-Z
rotations can be implemented virtually. The universal control over a single-
qubit Hilbert space can be achieved.

Two-qubit gates between superconducting qubits can also be implemented. For
example, control-phase gates can be implemented by tuning the frequency of one
transmon through flux drives DiCarlo  _et al._ (2010); Barends  _et al._
(2013), while cross-resonance gates between two fixed-frequency transmons or
fluxoniums can be implemented using microwave drives Rigetti and Devoret
(2010); Chow  _et al._ (2011); Nesterov  _et al._ (2022); Dogan  _et al._
(2023). Furthermore, the recent development of tunable couplers enables
tunable couplings between transmon qubits, which enables CZ gates with
fidelity reaching $99.8\%$ with about $40$ ns Stehlik  _et al._ (2021); Sung
_et al._ (2021); Marxer  _et al._ (2023). The high-fidelity two-qubit gates
can also be realized. An iSWAP gate with gate time $50$ ns can reach fidelity
$99.72\%$ Bao  _et al._ (2022), while microwave-activated CZ gates can be
expected to be realized with fidelity $99.9\%$ within $100$ ns Nesterov  _et
al._ (2018); Nguyen  _et al._ (2022). With all these single-qubit and two-
qubit gates, a universal gate set is supported.

The universal control of the qubits made of ions or atoms can also be realized
using microwave and optical methods Hilder  _et al._ (2022); Ospelkaus  _et
al._ (2011); Harty  _et al._ (2014); Yavuz  _et al._ (2006); Xia  _et al._
(2015); Bluvstein  _et al._ (2022); Levine  _et al._ (2022). The complete
Hilbert space spaced by the qubit states are addressable. Fast and high-
fidelity two-qubit gate operations are achieved Gaebler  _et al._ (2016);
Ballance  _et al._ (2016); Saner  _et al._ (2023); Levine  _et al._ (2019);
Bluvstein  _et al._ (2022); Evered  _et al._ (2023); Bluvstein  _et al._
(2023, 2024).

Although the gate operations still suffer from errors, there are active
explorations on building quantum error correction codes to mitigate the
imperfection Krinner  _et al._ (2022); Acharya  _et al._ (2023a); Bermudez
_et al._ (2017); Bluvstein  _et al._ (2024); Stricker  _et al._ (2020).

##### Connectivity.

However, the coupling between superconducting qubits is limited, where a
single superconducting qubit can only couple to the qubits nearby (mostly
physically connected). Therefore, the connectivity is usually limited to the
nearest neighbors Blais  _et al._ (2021).

Compared to the superconducting qubits, the trapped ions and neutral atoms can
maintain higher connectivity. Due to the nature of the mechanics of two-qubit
gates between ions, it is possible to perform long-range two-qubit gates
between ions trapped in a single trap Sørensen and Mølmer (1999). In the
neutral atom system, the coherent transport of atoms enables shuttling atoms
to another site to perform gate operations Bluvstein  _et al._ (2022, 2024).

##### Long coherence time.

Transmon and fluxonium qubits can preserve a long coherence time compared to
the gate time. With the current improvement of material choices and
fabrication techniques, the lifetime of transmon qubits has been improved from
$\sim 1~{}\mu$s Schreier  _et al._ (2008); Houck  _et al._ (2008) to $100$
Barends  _et al._ (2013); Chang  _et al._ (2013); Jin  _et al._ (2015) to
$500~{}\mu$s Place  _et al._ (2021); Wang  _et al._ (2022a), while the
coherence time $T_{2}^{*}$ can reach $0.3$ ms Place  _et al._ (2021); Wang
_et al._ (2022a). The coherence time can be further improved using dynamical
decoupling to $0.557$ ms Wang  _et al._ (2022a). the fluxonium qubit is less
sensitive to charge noise, which extends its coherence time $100$ to
$300~{}\mu$s. Recently, a fluxonium qubit with coherence time $T_{2}^{*}$
reaches $1.48$ ms has been reported Somoroff  _et al._ (2023)

Compared to superconducting qubit systems, the trapped ions and neutral atoms
can have longer coherence time, especially when the qubit is encoded into the
hyperfine levels of the ions/atoms. For example, the coherence time of an ion
qubit can reach $5500$ s Wang  _et al._ (2021).

##### Qubit readout.

The state of transmon or fluxonium qubits can be read out by dispersive
coupling to a microwave field Wallraff  _et al._ (2004). The phase accumulated
by the microwave field when the qubit is in its ground or excited state is
different. Using the phase information of the microwave field, quantum
nondemolition measurement on qubit state can be achieved in 40 ns to 100 ns
with fidelity 99.0% to 99.7% Walter  _et al._ (2017); Sunada  _et al._ (2022);
Swiadek  _et al._ (2023); Sunada  _et al._ (2024). Although the measurements
via this method are only projective measurements along the computational
basis, more general measurements, e.g, the stabilizer measurements and error
corrections used in quantum error correction codes, can be realized using
projective measurements on an auxiliary qubit, which is widely used in quantum
error correction Krinner  _et al._ (2022); Acharya  _et al._ (2023b).

Although superconducting qubits fully satisfy the requirements of the released
AQM, they still suffer from finite gate errors. In order to reduce the gate
errors, several attempts of using superconducting qubits to build error
correction codes have been demonstrated Andersen  _et al._ (2020); Krinner
_et al._ (2022); Acharya  _et al._ (2023a); Gong  _et al._ (2021b). We
conclude that superconducting qubits meet all the requirements from the
released AQM, and have nearly all mitigation techniques available, which makes
superconducting qubits one of the most promising systems for QC. But as the
current construction of superconducting qubit QC systems still suffers from
finite gate errors and the connectivity in a large integration of qubits,
there are opportunities to use superconducting qubits as a small but fast
quantum arithmetic unit in the future QC design Liu  _et al._ (2023b).

####  IV.2.2 Nitrogen-Vacancy centers

Nitrogen-vacancy centers Wrachtrup and Jelezko (2006); Doherty  _et al._
(2011, 2013) combined with other solid-state defect centers have emerged as
promising candidates for QIP. Defect color centers inside the solid state
systems can be nicely fabricated and implanted inside the solid crystal, while
they can have long coherence spin states that can be manipulated using
microwave and optically readout.

##### Qubit states.

The negatively charged NV centers have six electrons localized around the
defect. The electronic ground state manifold consists of three spin-1 states.
When there is no magnetic field applied, the state with $S_{1}=0$ and two
states with $S_{z}=\pm 1$ splits by $2.87$ GHz Doherty  _et al._ (2011, 2013).
When the NV center electronic spin states are utilized to encode quantum
information, a magnetic field is applied to further break the degeneracy of
the two states with $S_{z}=\pm 1$, and use one of them with the $S_{z}=0$
state to form a qubit space Fuchs  _et al._ (2009, 2010); Abobeih  _et al._
(2018). Another way to encode quantum information is to encode into the
nuclear spin states of the nearby atoms (Carbon or Nitrogen atoms) Maurer  _et
al._ (2012); Bradley  _et al._ (2019); Pompili  _et al._ (2021); Bartling  _et
al._ (2022); Hermans  _et al._ (2022); Xie  _et al._ (2023).

##### Operations.

The universal control of a qubit encoded into NV center electronic states can
be achieved by coherent microwave drives Fuchs  _et al._ (2009, 2010). The
universal control on the nuclear spin states can be achieved by addressing the
nuclear spin states by manipulating the electronic states of the NV center via
the hyperfine interaction Jelezko  _et al._ (2004); Abobeih  _et al._ (2022).

However, implementing entangling gates between NV centers to support a full
universal gate set for QC is challenging. Entanglement between two electronic
spin qubits of NV centers can be entangled by photon heralded entanglement
generation process Bernien  _et al._ (2013); Pfaff  _et al._ (2014); Hensen
_et al._ (2015); Pompili  _et al._ (2021). Entanglement gates between the
electronic spin qubit and the nearby nuclear spin qubits can be performed by
microwave drive on electronic spin degree of freedom Cramer  _et al._ (2016);
Taminiau  _et al._ (2014, 2012); Dong  _et al._ (2020); Abobeih  _et al._
(2022).

With the heralded entanglement between the electronic states of two NV
centers, the entangled state can be used to enable gate operations between the
nearby nuclear spin states. It can potentially be used to obtain a remote
entangling gate for nuclear spin qubits, which can be used to build a quantum
network Pompili  _et al._ (2021).

##### Connectivity.

Despite the possibility of heralded entanglement generation, two-qubit
entanglement occurs relatively slowly, with a reported rate of $9$ Hz for
remote entanglement between two NV centers Pompili  _et al._ (2021). Although
improvements in photon emission, collection, and detection efficiency could
enhance generation speed, the speed of generating entanglement of NV qubits
remains slower compared to trapped ion and superconducting systems.

##### Long coherence time.

The electronic qubit of NV centers can preserve a long lifetime. Even at room
temperature, the NV electronic states can preserve $1.8$ ms coherence time in
an isotopically pure diamond crystal Balasubramanian  _et al._ (2009). The
nuclear spin states can have even longer lifetime Maurer  _et al._ (2012);
Bradley  _et al._ (2019); Pompili  _et al._ (2021); Bartling  _et al._ (2022);
Hermans  _et al._ (2022); Xie  _et al._ (2023). One challenge of NV center
electronic states and nuclear spin states is the relatively short coherence
time, which is due to the spin bath of the surrounding nuclear spins in the
diamond crystal. However, the effect of the spin bath can be mitigated by
dynamical decoupling Maurer  _et al._ (2012); Bar-Gill  _et al._ (2013);
Abobeih  _et al._ (2018, 2022); Bradley  _et al._ (2019). With the help of
dynamical decoupling, the coherence time of the electronic spin state can
reach $1.58$ s Abobeih  _et al._ (2018), while the nuclear spin states can be
extended to $63$ s Bradley  _et al._ (2019); Abobeih  _et al._ (2022).

##### Qubit Readout.

The measurement of NV center electronic states is implemented by optically
pumping the states to the excited state manifold, and detecting the emitted
photons. Due to the existence of a non-radiative relaxation path from the
excited spin $0$ states, the state of the NV electronic qubit can be
distinguished by the photon counts Doherty  _et al._ (2011, 2013). As the
nuclear spin states are well isolated from the environment, the nuclear spin
states can be read out by swapping the nuclear state to NV electronic states.
Then the state can be detected using the above method Abobeih  _et al._
(2022).

Overall, the NV center systems have well-defined and controllable qubit states
(Rule 1), and long coherence time (Rule4) and easy single-qubit measurement
capabilities (Rule 5). However, the connectivity and two-NV entangling gates
(Rule 2 and 3) are relatively weakly supported.

####  IV.2.3 Photonic qubits for quantum computing

Optical photons are widely used in quantum communication and computing.
However, their weak nonlinearity in optical nonlinear materials limits fast
gate operations compared to microwave photons Byer (1974); Chen and Liu
(1986), which benefit from Josephson junctions. Another distinct feature of
optical photons used in quantum communication and quantum computing is that
they are considered ‘flying qubits’, i.e., itinerant photons rather than
stationary photons stored in an optical cavity mode. To avoid repeating the
discussion covered in the previous sub-sections, we focus on itinerant photons
used as qubits in quantum communication and quantum computing, especially in
the MBQC model Raussendorf and Briegel (2001); Raussendorf  _et al._ (2003);
Browne and Rudolph (2005); Briegel  _et al._ (2009), offering an alternative
approach to supporting the AQM.

##### Qubit states.

There are multiple ways of encoding quantum information into optical photons.
For example, the quantum information can be encoded into the polarization of
photons (polarization encoding) Turchette  _et al._ (1995); Knill  _et al._
(2001); Duan and Kimble (2004); O’Brien  _et al._ (2003); Basset  _et al._
(2021), the presence or absence of a photon (Fock encoding) Chuang and
Yamamoto (1995), the presence of a photon in early or late time bins (time
bin) Brendel  _et al._ (1999); Jayakumar  _et al._ (2014); Sun  _et al._
(2022); Lee  _et al._ (2019); Ortu  _et al._ (2022), the presence of a photon
in two modes with different frequencies (frequency encoding) Olislager  _et
al._ (2010); Lu  _et al._ (2019); Sabattoli  _et al._ (2022); Clementi  _et
al._ (2023), etc.

##### Operations.

Universal control of a single qubit state varies based on encoding methods.
Polarization-encoded photonic qubits can be controlled using passive optical
elements like wave plates, beam splitters, phase shifters, and polarizers Kok
_et al._ (2007). However, other encoding methods may require active elements
and operations. For example, time-bin encoded qubits require delay lines and
active optical switches to separate spatial modes, coherently convert photons
between modes, and recombine them into time bins Kok  _et al._ (2007).

Supporting a universal gate set in photonic systems involves two approaches.
One method is implementing an entangling gate, akin to other matter qubit
systems. This gate can be executed probabilistically using photon measurements
inducing nonlinearity Knill  _et al._ (2001); O’Brien  _et al._ (2003); Ralph
_et al._ (2001); Zou  _et al._ (2002); Pittman  _et al._ (2002, 2003), or
deterministically utilizing nonlinearity from strongly-coupled cavity-QED
systems Duan and Kimble (2004); Kimble (2008); Zhan and Sun (2020). However,
quantum gates suffer from imperfections. Probabilistic photon gates can
achieve high fidelity through heralding but require more resources to
implement Kok  _et al._ (2007); O’Brien  _et al._ (2003). Photonic gates using
cavity-QED systems also face challenges due to imprecise control and photon
absorption, making gate-based photonic QC difficult at the current stage.

On the other hand, universal computing in photonic systems can be implemented
by MBQC model Raussendorf and Briegel (2001); Raussendorf  _et al._ (2003);
Browne and Rudolph (2005); Briegel  _et al._ (2009). In this model, universal
computation is driven by measurements of selected qubits in a certain order on
an existing entangled state. The entangled photonic states can be generated by
time-delayed feedback Pichler  _et al._ (2017); Sun  _et al._ (2016); Zhan and
Sun (2020); Shi and Waks (2021), or fusing small pieces of entangled states
together Bartolucci  _et al._ (2023), or by photon emission from entangled
photonic emitters Lindner and Rudolph (2009); Economou  _et al._ (2010);
Buterakos  _et al._ (2017); Russo  _et al._ (2019).

The physical support of a universal gate set can be disparate from other
matter qubit systems discussed in the rest section. However, a universal gate
used in the gate model, e.g., a CNOT gate, can be mapped to a sequence of
measurements on a resource state Raussendorf and Briegel (2001); Raussendorf
_et al._ (2003). Therefore, unlike gate-based models, after a gate operation,
the quantum information is not carried by the same set of physical photonic
qubits before the gate.

##### Connectivity.

The physical connectivity of optical photons can be easily achieved by guiding
the photons together. However, the computation connectivity of photonic qubits
is restricted by the capability of performing entangling gates, or the
entanglement structure of the resource states for MBQC.

##### Long coherence time.

As photons do not interact with each other naturally, the photonic qubits can
have long coherence times when they propagate in the vacuum. However, when
they propagate in optical media, the material absorption causes photon loss,
which is the leading error in photonic-based quantum systems. Another source
of decoherence is the fluctuations of the optical paths, inducing phase noises
to the photonic qubits. Stabilizing optical paths is essential in optical-
based quantum systems Bernien  _et al._ (2013); Pfaff  _et al._ (2014); Hensen
_et al._ (2015).

To mitigate photon loss, dual-rail-type encoding methods, such as using two
time, frequency, or spatial modes, can enable error detection. Photon
detectors can flag errors when no detection event occurs during qubit
measurement.

Another way to mitigate the error is to enable error correction on photonic-
based QC systems, which have been theoretically proposed and investigated
Varnava  _et al._ (2006, 2007); Buterakos  _et al._ (2017); Zhan and Sun
(2020); Hilaire  _et al._ (2021); Zhan  _et al._ (2023); Bartolucci  _et al._
(2023). Generating tree graph states for photon loss error has been
demonstrated in experiments Zhan and Sun (2020). However, due to the resource
demand, it is still challenging to demonstrate the error correction
functionality in photonic systems Li  _et al._ (2015); Bartolucci  _et al._
(2023).

##### Qubit readout.

Photon detection involves capturing photons by detectors, converting them into
electric signals (e.g., current) Scully and Zubairy (1997). Utilizing photon
detectors and linear optical devices like wave plates and polarizers, quantum
information encoded in photonic qubits can be measured. For example,
polarization-encoded photon qubits can be measured using polarizers followed
by photon detectors, while the time-bin encoded photon qubits can be detected
using optical switches to split them into two spatial modes, and then measured
separately.

A notable distinction between itinerant photonic qubits and matter qubits is
that measurement destroys photonic qubits. Unlike matter qubits, where the
qubit itself remains alive even after destructive measurement, photonic qubits
are absorbed by detectors. This unique feature affects the support for the AQM
by photonic systems. Resetting photonic qubits through measurements is not
physically feasible, but they can be replenished once the measurement result
is known. In quantum algorithms using photonic systems, the number of photons
qubits can exceed the number of logical qubits, especially when there are mid-
circuit measurements in the algorithm. Multiple photonic qubits can represent
a single computational qubit in certain quantum algorithms.

To summarize, although the physical photons may not remain alive during the
computation, as the photons can potentially be replenished, Rule 1 of the AQM
can still be supported. As the photonic two-qubit gates are still challenging
to perform, Rule 2 of the AQM can only be weakly supported. Rule 3 of the AQM
is hard to be supported by photonic qubits in terms of gate operations,
however, can be supported with the aid of photon emitters. Rule 4 and 5 of the
AQM are well supported by photonic systems.

####  IV.2.4 Quantum memory systems for photons

At the end of this section, we consider a type of physical system well-suited
for buffering optical lights, which are used as optical quantum memories.
These systems cannot support the AQM, however, still useful for QIP
applications.

##### Qubit states.

There are different constructions of optical memories. Quantum memories using
single atoms, or defect centers in solid-state systems can also be utilized as
a computational qubit, as we discussed in Sec. IV.2.2. In this section, we
examine two main types: atomic clouds and rare-earth-ion-doped crystals. Both
types store quantum information in collective excitations of atoms or ions,
rather than individual ones Lvovsky  _et al._ (2009); Heshami  _et al._
(2016); Fleischhauer  _et al._ (2005); Lukin (2003); Ledingham  _et al._
(2012); Ferguson  _et al._ (2016); Alexander  _et al._ (2006); Gorshkov  _et
al._ (2007).

The qubit states in these systems can be defined by the presence or absence of
such collective excitations. However, universal control over the Hilbert space
encompassing these states is challenging. Therefore, while these systems offer
two distinguishable states for quantum information storage, they may not
support two states for the AQM.

##### Operations.

Realizing universal control of the Hilbert space spanned by the two states
requires the collective control of all the atoms/ions inside the systems,
which hinders the arbitrary transformations in the Hilbert space. However, by
applying external optical light or electronic voltage to control the
absorption of the atoms/ions, the incoming photons can be absorbed and
converted into the collective excitation of the material, and then the
absorbed photon can be re-emitted. Several techniques have been developed for
these two operations, including electromagnetically-induced transparency (EIT)
Fleischhauer  _et al._ (2005); Lukin (2003); Hosseini  _et al._ (2011); Dudin
_et al._ (2013); Ding  _et al._ (2013); Nicolas  _et al._ (2014); Ding  _et
al._ (2015a, b); Parigi  _et al._ (2015); Saunders  _et al._ (2016); Katz and
Firstenberg (2018); Hsiao  _et al._ (2018); Jiang  _et al._ (2019); Wang  _et
al._ (2019); Li  _et al._ (2020); Dideriksen  _et al._ (2021); Wang  _et al._
(2022b); Meßner  _et al._ (2023); Buser  _et al._ (2022), controlled
reversible inhomogeneous broadening (CRIB) Moiseev and Kröll (2001); Nilsson
and Kröll (2005); Kraus  _et al._ (2006); Alexander  _et al._ (2006); Hétet
_et al._ (2008), atomic frequency combs (AFC) McAuslan  _et al._ (2012);
Saglamyurek  _et al._ (2011); Clausen  _et al._ (2011); Ledingham  _et al._
(2012); Zhou  _et al._ (2012); Ferguson  _et al._ (2016); Jin  _et al._
(2022); Ma  _et al._ (2021a); Seri  _et al._ (2017); Kutluer  _et al._ (2017);
Laplane  _et al._ (2017); Holzäpfel  _et al._ (2020); Businger  _et al._
(2020); Askarani  _et al._ (2021); Ma  _et al._ (2021b), and rephased
amplified spontaneous emission (RASE) Ledingham  _et al._ (2012, 2010).

##### Connectivity.

The multi-mode feature of the collective excitation used in atomic clouds and
doped ions enables the integration of multiple memory cells into a single
physical system Vernaz-Gris  _et al._ (2018); Meßner  _et al._ (2023).
However, due to the control complexity, entangling different modes can be
hardly implemented. Therefore, the connectivity of different memory modes is
limited.

In contrast, two memory systems can be physically connected through optical
paths, where the optical light stored in one memory can be emitted and
absorbed by the connected memory system. Routing the optical photons can be
achieved by active optical elements, which can potentially configure the
memory system connectivity.

##### Long coherence time.

The collective quantum state of the ensemble of atoms and ions can preserve
the quantum information for a long time. For example, the $1/e$ decay lifetime
of the stored light can reach $16$ s with dynamical decoupling in a cold
atomic cloud using the EIT technique Dudin  _et al._ (2013), while an AFC-
based ion-doped crystal system an obtain storage lifetime $52.9$ min with
dynamical decoupling Ma  _et al._ (2021b).

##### Qubit readout.

Direct measurement of the collective quantum states is intimidating to realize
due to the difficulty of controlling the quantum states of all the atoms/ions
simultaneously. However, the measurement can be possible by detecting the
state of the emitted photons.

Overall, the quantum memory systems have good support of Rule 1 and 4 of the
AQM, as they have good coherence time while they have quantum states that
encode quantum information. However, Rule 2 is not supported, as universal
gate sets are not available. The AQM Rule 3 and Rule 5 are partially
supported, as the direct entangling operations and measurements on these
quantum memory systems are challenging.

###  IV.3 Physical systems with co-design opportunities

The AQM facilitates efficient application-device co-design by aligning quantum
algorithm demand with physical system support. When the requirements of
quantum algorithms and the capabilities of the physical system supporting the
AQM coincide, the application can be performed on this physical system. We
illustrate a few examples.

As the photonic system is challenging to perform two-qubit gates between
physical qubits, Rule 2 and 3 of the AQM is not well supported between
physical qubits. According to the unique support of the AQM by photonic qubit
systems, MBQC is well-suited Bartolucci  _et al._ (2023); Kok  _et al._
(2007). In addition, although the entangling gates between photons are
challenging, the photon hopping between coupled modes can be relatively easy
to achieve, which provides the necessary connectivity (Rule 3) requirements
for the application, such as quantum random walk models Tang  _et al._ (2018)
and sampling problems Zhong  _et al._ (2020); Deng  _et al._ (2023).

In contrast, NV center systems have long coherence times (Rule 4) and good
support of the AQM Rule 1 and 5. Although the support of Rule 2 and 3 are
limited, they are promising candidates for quantum memory and communication
applications where a universal gate set isn’t necessary Bernien  _et al._
(2013); Pfaff  _et al._ (2014); Hensen  _et al._ (2015); Borregaard  _et al._
(2020); Pompili  _et al._ (2021); Jing and Razavi (2022); Ruf  _et al._
(2021); Liu  _et al._ (2023b). Similar to the quantum memory systems for
photons, where Rule 2 and 3 are not supported. In addition, the quantum memory
systems for photons also have limited support to the readout capability (Rule
5). However, referring to Table. 2, they can be well suited to be used as
quantum memory, which can not only be used in quantum communication and
quantum networks Askarani  _et al._ (2021); Jobez  _et al._ (2016), but also
in quantum memory of a heterogeneous QC architecture Liu  _et al._ (2023b).

Superconducting, trapped ions, and neutral atom systems offer support of the
complete AQM, making them promising candidates for not only universal quantum
computing but also applications with stronger partial AQM demands on specific
aspects. Compared to the superconducting qubit systems, trapped ions and
neutral atoms have higher connectivity and longer coherence time, which also
enable them to be used as quantum interconnect modules or quantum memory
modules in future heterogenous QC systems with high fidelity coupling Stein
_et al._ (2023a); Liu  _et al._ (2023b). In addition, the high connectivity
also makes them suitable for quantum annealing and quantum simulation
applications Daley  _et al._ (2022); Monroe  _et al._ (2021); Ebadi  _et al._
(2021).

##  V Conclusion

In conclusion, we refine the abstract qubit model, addressing the demands
posed by quantum algorithms and applications, as well as the requirements on
physical systems. However, achieving the ideal AQM is challenging for state-
of-the-art physical systems. Instead, these systems can meet a less stringent
set of requirements, utilizing techniques such as unitary decomposition,
quantum error mitigation, and error correction methods, to compensate for the
incomplete support of the AQM, albeit at the cost of more time and resources.

Furthermore, we observe that physical systems can offer unique advantages in
supporting the AQM and quantum applications, especially for quantum algorithms
and applications that don’t need the complete AQM. This presents co-design
opportunities to utilize the strengths of current physical systems by breaking
the AQM abstraction layer. We discuss examples such as quantum annealing,
quantum random walk, measurement-based quantum computing, analogue quantum
simulation, and quantum memory applications, focusing on their specific
demands on the AQM. Subsequently, we discuss how physical systems like
superconducting qubit systems, NV centers, photonics, and quantum memory
materials can support the AQM and their potential roles in future QC system
designs.

As we have shown, the AQM can be a flexible model, which can be adjusted to
better support various quantum applications. Therefore, we believe that the
AQM can be a useful tool for future quantum algorithm-device co-design. The
AQM discussion can guide QC and QIP researchers in developing new algorithms
and physical devices, and take benefits of quantum co-design to enhance the
capabilities of existing QC devices.

##  VI Acknowledgement

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, National Quantum Information Science Research Centers, Co-
design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704,
(Basic Energy Sciences, PNNL FWP 76274). The Pacific Northwest National
Laboratory is operated by Battelle for the U.S. Department of Energy under
Contract DE-AC05-76RL01830.

##  Appendix A Supporting the AQM by trapped ions

The trapped ion systems support well on the AQM. For a thorough review of
trapped ion systems for QC and QIP, we suggest referring to Refs. Häffner  _et
al._ (2008); Monroe and Kim (2013); Bruzewicz  _et al._ (2019); Brown  _et
al._ (2021).

##### Qubit states.

In a trapped ion system, each ion inside the trap can be used as a qubit. The
quantum information can either be encoded into the hyperfine levels Harty  _et
al._ (2014); Gaebler  _et al._ (2016); Ballance  _et al._ (2016); Wang  _et
al._ (2017, 2021) or Zeeman sublevels of a same orbital Keselman  _et al._
(2011); Ruster  _et al._ (2016); Hilder  _et al._ (2022), or other quantum
states in the specific ion level structures Bermudez  _et al._ (2017); Toyoda
_et al._ (2010); Pogorelov  _et al._ (2021).

##### Universal control.

Depending on the type of ion qubits, the quantum manipulation schemes for
qubit universal control of the ion qubits also vary. Specifically, for the
hyperfine ion qubits, single-qubit gates can be implemented using either
optical Raman transitions Hilder  _et al._ (2022) or microwave drives
Ospelkaus  _et al._ (2011); Harty  _et al._ (2014). The microwave Pauli-X and
Pauli-Y rotations with fidelity $\sim 99.9999\%$ have been reported Harty  _et
al._ (2014). The microwave gates duration has been improved to $\sim 1~{}\mu$s
without sacrificing much gate fidelity Leu  _et al._ (2023). Raman transition-
based single-qubit gates achieve fidelity $99.993\%$ in $7.5~{}\mu$s Ballance
_et al._ (2016). Using ultra-fast laser pulses to strongly drive Raman
transition can achieve a $\pi$-pulse in $\sim 50$ ps with fidelity $\sim 99\%$
Campbell  _et al._ (2010). These gates provide the necessary tools for
universal control of single qubits.

The coupling between two ions can be induced using the Coulomb interactions
between them. There are several schemes to perform two-qubit gates between
trapped ions, e.g., the Cirac-Zoller gate Cirac and Zoller (1995), the Mølmer-
Sørensen (MS) gate Sørensen and Mølmer (1999), and the Leibfried geometric
phase gate Leibfried  _et al._ (2003). Specifically, MS gate utilizes the
phonon modes of the trapped ion chain to mediate the coupling between
different trapped ions, which is widely used in trapped ion systems Gaebler
_et al._ (2016); Blümel  _et al._ (2021); Moses  _et al._ (2023); Jeon  _et
al._ (2023); Saner  _et al._ (2023), while the fidelity of two-qubit gates can
reach $99.9\%$ Ballance  _et al._ (2016). The two-qubit gates can be performed
in $10$ to $500~{}\mu$s Gaebler  _et al._ (2016); Ballance  _et al._ (2016);
Saner  _et al._ (2023). The two-qubit entangling gates combined with the
single-qubit universal control provide universal gate sets for QC.

##### Long coherence time.

Depending on the type of encoding, the coherence time of the qubits can be
different. For instance, in the Zeeman qubits, the coherence time can reach
$300$ ms Ruster  _et al._ (2016), while the hyperfine states are more
coherent, and hence the coherence time of a hyperfine qubit can reach several
minutes to an hour Harty  _et al._ (2014); Wang  _et al._ (2017, 2021) ($5500$
s reported in Ref. Wang  _et al._ (2021)).

##### Qubit measurements.

The qubit measurement is achieved using optical approaches, where one of the
ion qubit states is excited to an optical active state, such that the
relaxation of the ion can emit optical photons for detection. Therefore, by
distinguishing the collected emitted photons, the state of the ion can be
determined. This physical process is a projective measurement of the qubit
computation states. The measurements on ion states can be obtained with
fidelity $>99.9\%$ Burrell  _et al._ (2010); Christensen  _et al._ (2020);
Todaro  _et al._ (2021).

##### Connectivity.

The ions are trapped using radio-frequency Paul traps Neuhauser  _et al._
(1980) and other types of electromagnetic traps Dehmelt (1990); Chiaverini
_et al._ (2005); Seidelin  _et al._ (2006); Labaziewicz  _et al._ (2008). The
trapped ion systems can have all-to-all connectivity between the ions trapped
in a single trap. This is due to the mechanism of two-qubit gate operations.
The Mølmer-Sørensen gate requires using the collective motion (phonon) modes
to mediate the coupling, which enables this all-to-all connectivity. However,
limited by the size of the ion trap and the distinguishability between
different phonon modes, it is impossible to have millions of ions trapped in a
single trap and have all-to-all connectivity. Furthermore, due to the 1D
nature of the ion traps, it is hard to maintain a higher-dimensional ion array
to improve integration and connectivity.

With the techniques of ion shuttling, it is possible to select the ion qubits
that are necessary to be coupled in the quantum algorithm, and shuttling the
ion qubits into the same ion trap to enable the coupling Moses  _et al._
(2023). This technique potentially increases the connectivity of the ion
qubits. However, shuttling operations can increase the decoherence noise and
time consumption.

In summary, trapped ion systems also support the AQM well. Similar to the
superconducting qubit systems, the imperfection of the gate operations and
decoherence limit the support to an ideal AQM. With the quantum error
correction codes, it is possible to overcome the errors in the quantum
operations. Compared with superconducting qubits, trapped ions can have longer
coherence time, and all-to-all connectivity between ions in the same trap,
which enables potential application to quantum memory and quantum simulation
Zhang  _et al._ (2017); Lv  _et al._ (2018); Monroe  _et al._ (2021);
Kiesenhofer  _et al._ (2023).

##  Appendix B Supporting the AQM by neutral atoms

The trapped neutral atom system is becoming increasingly popular with the new
progress of showing the potential of enabling error correction
functionalities. Neutral atoms can also have good support for the AQM.

##### Qubit states.

Neutral atom systems use the spin-electronic states of the trapped atoms to
encode quantum information, similar to the ion qubits in the trapped ion
system. There are a few strategies to encode quantum information into states
of Rydberg atoms. For instance, the ground state of the atom and its Rydberg
excited state can be used to encode the qubit state $\ket{0}$ and $\ket{1}$
states Levine  _et al._ (2018); Madjarov  _et al._ (2020), which is referred
to as GR qubits in Ref. Picken  _et al._ (2018); Morgado and Whitlock (2021),
while two hyperfine ground states of the Rydberg atoms can also be used Sheng
_et al._ (2018); Graham  _et al._ (2019); Levine  _et al._ (2019); Bluvstein
_et al._ (2022), which is called GG qubits in Ref. Morgado and Whitlock
(2021)). The Hilbert space spanned by the states in different selections can
also be fully addressed.

##### Universal control.

The single-qubit universal control can be achieved using optical Raman
transitions or using microwave drives with optically activated Stack shifts
Yavuz  _et al._ (2006); Xia  _et al._ (2015); Bluvstein  _et al._ (2022);
Levine  _et al._ (2022). The Rabi rate of $2$ MHz has been realized for
single-qubit gate operations Levine  _et al._ (2022). With these control
methods, the Hilbert space can be fully addressed.

To support a universal gate set for QC, entangling gates between two qubits
are necessary. The Rydberg interactions can be leveraged to perform entangling
gates. When the atom is excited to a highly excited state (Rydberg state), the
radius of the Rydberg state is much larger than the radius of the atom in the
ground state, which activates a strong dipole-dipole interaction between
Rydberg atoms. Specifically, the dipole interaction makes an excited Rydberg
atom strongly shift the energy level of the other Rydberg atoms, which blocks
the excitation of the nearby atoms. This effect is named “Rydberg blockade”
Lukin  _et al._ (2001); Jaksch  _et al._ (2000); Urban  _et al._ (2009), which
enables fast two-qubit gates. The two-qubit gates with fidelity $97.4\%$
Levine  _et al._ (2019); Bluvstein  _et al._ (2022) and $99.5\%$ Evered  _et
al._ (2023); Bluvstein  _et al._ (2023) have been demonstrated in experiments.
Specifically, the CZ gate between two atoms can be implemented in $\sim 200$
ns, which greatly suppresses the error from the decoherence of the Rydberg
state Evered  _et al._ (2023).

##### Long coherence time.

The coherence time of the qubits varies according to the species of qubits and
the trapped atoms, ranging from a few microseconds to a few seconds Norcia
_et al._ (2019); Ebadi  _et al._ (2021); Jenkins  _et al._ (2022); Ma  _et
al._ (2022). The coherence time can be further enhanced using dynamical
decoupling to $1.5$ s for Rb atoms Bluvstein  _et al._ (2022, 2023) and $3.7$
s for Yb atoms Jenkins  _et al._ (2022).

##### Qubit measurements.

The measurement of the trapped neutral atoms is similar to the method used in
trapped ion systems. Fast measurements with descent measurement fidelity on
the qubit state have been demonstrated, which can be used in quantum error
correction Xu  _et al._ (2021); Singh  _et al._ (2023); Deist  _et al._
(2022); Graham  _et al._ (2023).

##### Connectivity.

Compared to the trapped ion systems, the neutral atoms cannot be trapped using
RF electromagnetic traps. Instead, neutral atoms are optically trapped into
optical lattices. With the optical trapping techniques, higher dimensional
optical lattices have been realized Wang  _et al._ (2015, 2016); Xia  _et al._
(2015); Maller  _et al._ (2015); Graham  _et al._ (2019). In addition, due to
the large radius of the atom Rydberg state (can reach a few micrometers Urban
_et al._ (2009); Gaëtan  _et al._ (2009)), the entangling gates can be applied
to two nonadjacent atoms, which potentially increase the connectivity of the
neutral atom qubits.

Furthermore, atoms can be precisely transported with high fidelity by
adjusting the optical traps Ebadi  _et al._ (2021); Bluvstein  _et al._
(2024). Through this shuttling process, atoms trapped in distant locations can
be brought to nearby sites, implementing entangling gate operations. This
significantly enhances the connectivity of neutral atom systems.

In essence, neutral atom systems show promise as a robust platform for AQM.
Like superconducting qubits and trapped ions, a key challenge lies in
developing error correction codes and demonstrating error suppression for full
AQM support. However, the high connectivity of atoms in neutral atom systems
makes them particularly suitable for quantum simulation applications Ebadi
_et al._ (2021); Wu  _et al._ (2021); Daley  _et al._ (2022); Gross and Bloch
(2017).

## References

  * Grover (1996) L. K. Grover, in  _Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing_, STOC ’96 (ACM, New York, NY, USA, 1996) pp. 212–219.
  * Shor (1994) P. W. Shor, in  _Proceedings of the 35th Annual Symposium on Foundations of Computer Science_, SFCS ’94 (IEEE Computer Society, Washington, DC, USA, 1994) pp. 124–134.
  * Shor (1997) P. W. Shor, SIAM Journal on Computing 26, 1484 (1997), https://doi.org/10.1137/S0097539795293172 .
  * Simon (1997) D. R. Simon, SIAM Journal on Computing 26, 1474 (1997), https://doi.org/10.1137/S0097539796298637 .
  * Bennett and Brassard (1984) C. H. Bennett and G. Brassard, in  _Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing_ (India, 1984) p. 175.
  * Xu  _et al._ (2020) F. Xu, X. Ma, Q. Zhang, H.-K. Lo, and J.-W. Pan, Rev. Mod. Phys. 92, 025002 (2020).
  * Portmann and Renner (2022) C. Portmann and R. Renner, Rev. Mod. Phys. 94, 025008 (2022).
  * DiVincenzo (2000) D. P. DiVincenzo, Fortschritte der Physik 48, 771 (2000).
  * Nielsen and Chuang (2010) M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information: 10th Anniversary Edition_ (Cambridge University Press, 2010).
  * Viola  _et al._ (2001) L. Viola, E. Knill, and R. Laflamme, Journal of Physics A: Mathematical and General 34, 7067 (2001).
  * Gambetta (2022) J. Gambetta, “Expanding the IBM Quantum roadmap to anticipate the future of quantum-centric supercomputing,”  (2022).
  * Bluvstein  _et al._ (2024) D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V. Vuletić, and M. D. Lukin, Nature 626, 58 (2024).
  * Gambetta (2023) J. Gambetta, “The hardware and software for the era of quantum utility is here,”  (2023).
  * Raussendorf and Briegel (2001) R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001).
  * Raussendorf  _et al._ (2003) R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A 68, 022312 (2003).
  * Browne and Rudolph (2005) D. E. Browne and T. Rudolph, Phys. Rev. Lett. 95, 010501 (2005).
  * Briegel  _et al._ (2009) H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, and M. Van den Nest, Nature Physics 5, 19 (2009).
  * Kadian  _et al._ (2021) K. Kadian, S. Garhwal, and A. Kumar, Computer Science Review 41, 100419 (2021).
  * Venegas-Andraca (2012) S. E. Venegas-Andraca, Quantum Information Processing 11, 1015 (2012).
  * Childs (2009) A. M. Childs, Phys. Rev. Lett. 102, 180501 (2009).
  * Singh  _et al._ (2021) S. Singh, P. Chawla, A. Sarkar, and C. M. Chandrashekar, Scientific Reports 11, 11551 (2021).
  * Shi  _et al._ (2020) Y. Shi, P. Gokhale, P. Murali, J. M. Baker, C. Duckering, Y. Ding, N. C. Brown, C. Chamberland, A. Javadi-Abhari, A. W. Cross, D. I. Schuster, K. R. Brown, M. Martonosi, and F. T. Chong, Proceedings of the IEEE 108, 1353 (2020).
  * Tomesh and Martonosi (2021) T. Tomesh and M. Martonosi, IEEE Micro 41, 33 (2021).
  * Wang  _et al._ (2020) Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, Frontiers in Physics 8 (2020), 10.3389/fphy.2020.589504.
  * Chi  _et al._ (2022) Y. Chi, J. Huang, Z. Zhang, J. Mao, Z. Zhou, X. Chen, C. Zhai, J. Bao, T. Dai, H. Yuan, M. Zhang, D. Dai, B. Tang, Y. Yang, Z. Li, Y. Ding, L. K. Oxenløwe, M. G. Thompson, J. L. O’Brien, Y. Li, Q. Gong, and J. Wang, Nature Communications 13, 1166 (2022).
  * Liu  _et al._ (2023a) P. Liu, R. Wang, J.-N. Zhang, Y. Zhang, X. Cai, H. Xu, Z. Li, J. Han, X. Li, G. Xue, W. Liu, L. You, Y. Jin, and H. Yu, Phys. Rev. X 13, 021028 (2023a).
  * González-Cuadra  _et al._ (2022) D. González-Cuadra, T. V. Zache, J. Carrasco, B. Kraus, and P. Zoller, Phys. Rev. Lett. 129, 160501 (2022).
  * Duckering (2022) C. Duckering,  _New Abstractions for Quantum Computing_ , Ph.D. thesis, University of Chicago (2022).
  * Dong and Petersen (2010) D. Dong and I. R. Petersen, IET Control Theory and Applications 4, 2651 (2010).
  * D’Alessandro (2021) D. D’Alessandro,  _Introduction to Quantum Control and Dynamics_, Advances in Applied Mathematics (CRC Press, 2021).
  * Mariantoni  _et al._ (2011a) M. Mariantoni, H. Wang, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, Nature Physics 7, 287 (2011a).
  * Mariantoni  _et al._ (2011b) M. Mariantoni, H. Wang, T. Yamamoto, M. Neeley, R. C. Bialczak, Y. Chen, M. Lenander, E. Lucero, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, Y. Yin, J. Zhao, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, Science 334, 61 (2011b), https://www.science.org/doi/pdf/10.1126/science.1208517 .
  * Besse  _et al._ (2020) J.-C. Besse, K. Reuer, M. C. Collodo, A. Wulff, L. Wernli, A. Copetudo, D. Malz, P. Magnard, A. Akin, M. Gabureac, G. J. Norris, J. I. Cirac, A. Wallraff, and C. Eichler, Nature Communications 11, 4877 (2020).
  * Reuer  _et al._ (2022) K. Reuer, J.-C. Besse, L. Wernli, P. Magnard, P. Kurpiers, G. J. Norris, A. Wallraff, and C. Eichler, Phys. Rev. X 12, 011008 (2022).
  * Gao  _et al._ (2019) Y. Y. Gao, B. J. Lester, K. S. Chou, L. Frunzio, M. H. Devoret, L. Jiang, S. M. Girvin, and R. J. Schoelkopf, Nature 566, 509 (2019).
  * Raussendorf and Harrington (2007) R. Raussendorf and J. Harrington, Phys. Rev. Lett. 98, 190504 (2007).
  * Dawson and Nielsen (2005) C. M. Dawson and M. A. Nielsen, “The solovay-kitaev algorithm,” (2005), arXiv:quant-ph/0505030 [quant-ph] .
  * Aharonov (2003) D. Aharonov, “A simple proof that toffoli and hadamard are quantum universal,” (2003), arXiv:quant-ph/0301040 [quant-ph] .
  * Shi (2002) Y. Shi, “Both toffoli and controlled-not need little help to do universal quantum computation,” (2002), arXiv:quant-ph/0205115 [quant-ph] .
  * Brylinski and Brylinski (2001) J.-L. Brylinski and R. Brylinski, “Universal quantum gates,” (2001), arXiv:quant-ph/0108062 [quant-ph] .
  * Cuomo  _et al._ (2020) D. Cuomo, M. Caleffi, and A. S. Cacciapuoti, IET Quantum Communication 1, 3 (2020).
  * Cacciapuoti  _et al._ (2020) A. S. Cacciapuoti, M. Caleffi, F. Tafuri, F. S. Cataliotti, S. Gherardini, and G. Bianchi, IEEE Network 34, 137 (2020).
  * Gottesman and Chuang (1999) D. Gottesman and I. L. Chuang, Nature 402, 390–393 (1999).
  * Huang  _et al._ (2004) Y.-F. Huang, X.-F. Ren, Y.-S. Zhang, L.-M. Duan, and G.-C. Guo, Phys. Rev. Lett. 93, 240501 (2004).
  * Wan  _et al._ (2019) Y. Wan, D. Kienzler, S. D. Erickson, K. H. Mayer, T. R. Tan, J. J. Wu, H. M. Vasconcelos, S. Glancy, E. Knill, D. J. Wineland, A. C. Wilson, and D. Leibfried, Science 364, 875 (2019), https://www.science.org/doi/pdf/10.1126/science.aaw9415 .
  * Chou  _et al._ (2018) K. S. Chou, J. Z. Blumoff, C. S. Wang, P. C. Reinhold, C. J. Axline, Y. Y. Gao, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, Nature 561, 368 (2018).
  * Cerezo  _et al._ (2021) M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Nature Reviews Physics 3, 625 (2021).
  * Calderbank and Shor (1996) A. R. Calderbank and P. W. Shor, Phys. Rev. A 54, 1098 (1996).
  * Stein  _et al._ (2023a) S. Stein, S. Sussman, T. Tomesh, C. Guinn, E. Tureci, S. F. Lin, W. Tang, J. Ang, S. Chakram, A. Li, M. Martonosi, F. T. Chong, A. A. Houck, I. L. Chuang, and M. A. DeMarco, “Microarchitectures for heterogeneous superconducting quantum computers,” (2023a), arXiv:2305.03243 [quant-ph] .
  * Stein  _et al._ (2023b) S. Stein, F. Hua, C. Liu, C. Guinn, J. Ang, E. Zhang, S. Chakram, Y. Ding, and A. Li, “Multi-mode cavity centric architectures for quantum simulation,” (2023b), arXiv:2309.15994 [quant-ph] .
  * Liu  _et al._ (2023b) C. Liu, M. Wang, S. A. Stein, Y. Ding, and A. Li, “Quantum memory: A missing piece in quantum computing units,” (2023b), arXiv:2309.14432 [quant-ph] .
  * Das and Chakrabarti (2008) A. Das and B. K. Chakrabarti, Rev. Mod. Phys. 80, 1061 (2008).
  * Hauke  _et al._ (2020) P. Hauke, H. G. Katzgraber, W. Lechner, H. Nishimori, and W. D. Oliver, Reports on Progress in Physics 83, 054401 (2020).
  * Albash and Lidar (2018) T. Albash and D. A. Lidar, Rev. Mod. Phys. 90, 015002 (2018).
  * Johnson  _et al._ (2011) M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, E. M. Chapple, C. Enderud, J. P. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, C. J. S. Truncik, S. Uchaikin, J. Wang, B. Wilson, and G. Rose, Nature 473, 194 (2011).
  * Lucas (2014) A. Lucas, Frontiers in Physics 2 (2014), 10.3389/fphy.2014.00005.
  * Djidjev  _et al._ (2018) H. N. Djidjev, G. Chapuis, G. Hahn, and G. Rizk, “Efficient combinatorial optimization using quantum annealing,” (2018), arXiv:1801.08653 [quant-ph] .
  * Zaribafiyan  _et al._ (2017) A. Zaribafiyan, D. J. J. Marchand, and S. S. Changiz Rezaei, Quantum Information Processing 16, 136 (2017).
  * Lechner  _et al._ (2015) W. Lechner, P. Hauke, and P. Zoller, Science Advances 1, e1500838 (2015), https://www.science.org/doi/pdf/10.1126/sciadv.1500838 .
  * Tang  _et al._ (2018) H. Tang, C. Di Franco, Z.-Y. Shi, T.-S. He, Z. Feng, J. Gao, K. Sun, Z.-M. Li, Z.-Q. Jiao, T.-Y. Wang, M. S. Kim, and X.-M. Jin, Nature Photonics 12, 754 (2018).
  * Gong  _et al._ (2021a) M. Gong, S. Wang, C. Zha, M.-C. Chen, H.-L. Huang, Y. Wu, Q. Zhu, Y. Zhao, S. Li, S. Guo, H. Qian, Y. Ye, F. Chen, C. Ying, J. Yu, D. Fan, D. Wu, H. Su, H. Deng, H. Rong, K. Zhang, S. Cao, J. Lin, Y. Xu, L. Sun, C. Guo, N. Li, F. Liang, V. M. Bastidas, K. Nemoto, W. J. Munro, Y.-H. Huo, C.-Y. Lu, C.-Z. Peng, X. Zhu, and J.-W. Pan, Science 372, 948 (2021a), https://www.science.org/doi/pdf/10.1126/science.abg7812 .
  * Esposito  _et al._ (2022) C. Esposito, M. R. Barros, A. Durán Hernández, G. Carvacho, F. Di Colandrea, R. Barboza, F. Cardano, N. Spagnolo, L. Marrucci, and F. Sciarrino, npj Quantum Information 8, 34 (2022).
  * Van den Nest  _et al._ (2006) M. Van den Nest, A. Miyake, W. Dür, and H. J. Briegel, Phys. Rev. Lett. 97, 150504 (2006).
  * Brown and Roberts (2020) B. J. Brown and S. Roberts, Phys. Rev. Res. 2, 033305 (2020).
  * Monroe  _et al._ (2021) C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Rev. Mod. Phys. 93, 025001 (2021).
  * Lv  _et al._ (2018) D. Lv, S. An, Z. Liu, J.-N. Zhang, J. S. Pedernales, L. Lamata, E. Solano, and K. Kim, Phys. Rev. X 8, 021027 (2018).
  * Ebadi  _et al._ (2021) S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuletić, and M. D. Lukin, Nature 595, 227 (2021).
  * Daley  _et al._ (2022) A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Nature 607, 667 (2022).
  * Bloch  _et al._ (2008) I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).
  * Devoret  _et al._ (2004) M. H. Devoret, A. Wallraff, and J. M. Martinis, “Superconducting qubits: A short review,” (2004), arXiv:cond-mat/0411174 [cond-mat.mes-hall] .
  * Girvin  _et al._ (2009) S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, Physica Scripta 2009, 014012 (2009).
  * Kjaergaard  _et al._ (2020) M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Annual Review of Condensed Matter Physics 11, 369 (2020), https://doi.org/10.1146/annurev-conmatphys-031119-050605 .
  * Kockum and Nori (2019) A. F. Kockum and F. Nori, “Quantum bits with josephson junctions,” in  _Fundamentals and Frontiers of the Josephson Effect_, edited by F. Tafuri (Springer International Publishing, Cham, 2019) pp. 703–741.
  * Blais  _et al._ (2021) A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Rev. Mod. Phys. 93, 025005 (2021).
  * Häffner  _et al._ (2008) H. Häffner, C. Roos, and R. Blatt, Physics Reports 469, 155 (2008).
  * Monroe and Kim (2013) C. Monroe and J. Kim, Science 339, 1164 (2013).
  * Bruzewicz  _et al._ (2019) C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Applied Physics Reviews 6 (2019), https://doi.org/10.1063/1.5088164, 021314.
  * Brown  _et al._ (2021) K. R. Brown, J. Chiaverini, J. M. Sage, and H. Häffner, Nature Reviews Materials 6, 892 (2021).
  * Wu  _et al._ (2021) X. Wu, X. Liang, Y. Tian, F. Yang, C. Chen, Y.-C. Liu, M. K. Tey, and L. You, Chinese Physics B 30, 020305 (2021).
  * Saffman  _et al._ (2010) M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys. 82, 2313 (2010).
  * Bluvstein  _et al._ (2022) D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara, H. Pichler, M. Greiner, V. Vuletić, and M. D. Lukin, Nature 604, 451 (2022).
  * Manucharyan  _et al._ (2009) V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. Devoret, Science 326, 113 (2009).
  * Koch  _et al._ (2007) J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 76, 042319 (2007).
  * Chow  _et al._ (2010) J. M. Chow, L. DiCarlo, J. M. Gambetta, F. Motzoi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 82, 040305 (2010).
  * Bermudez  _et al._ (2017) A. Bermudez, X. Xu, R. Nigmatullin, J. O’Gorman, V. Negnevitsky, P. Schindler, T. Monz, U. G. Poschinger, C. Hempel, J. Home, F. Schmidt-Kaler, M. Biercuk, R. Blatt, S. Benjamin, and M. Müller, Phys. Rev. X 7, 041061 (2017).
  * Pogorelov  _et al._ (2021) I. Pogorelov, T. Feldker, C. D. Marciniak, L. Postler, G. Jacob, O. Krieglsteiner, V. Podlesnic, M. Meth, V. Negnevitsky, M. Stadler, B. Höfer, C. Wächter, K. Lakhmanskiy, R. Blatt, P. Schindler, and T. Monz, PRX Quantum 2, 020343 (2021).
  * Levine  _et al._ (2018) H. Levine, A. Keesling, A. Omran, H. Bernien, S. Schwartz, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Phys. Rev. Lett. 121, 123603 (2018).
  * Madjarov  _et al._ (2020) I. S. Madjarov, J. P. Covey, A. L. Shaw, J. Choi, A. Kale, A. Cooper, H. Pichler, V. Schkolnik, J. R. Williams, and M. Endres, Nature Physics 16, 857 (2020).
  * Picken  _et al._ (2018) C. J. Picken, R. Legaie, K. McDonnell, and J. D. Pritchard, Quantum Science and Technology 4, 015011 (2018).
  * Morgado and Whitlock (2021) M. Morgado and S. Whitlock, AVS Quantum Science 3 (2021), doi.org/10.1116/5.0036562, 023501.
  * Harty  _et al._ (2014) T. P. Harty, D. T. C. Allcock, C. J. Ballance, L. Guidoni, H. A. Janacek, N. M. Linke, D. N. Stacey, and D. M. Lucas, Phys. Rev. Lett. 113, 220501 (2014).
  * Gaebler  _et al._ (2016) J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, Phys. Rev. Lett. 117, 060505 (2016).
  * Ballance  _et al._ (2016) C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, Phys. Rev. Lett. 117, 060504 (2016).
  * Wang  _et al._ (2017) Y. Wang, M. Um, J. Zhang, S. An, M. Lyu, J.-N. Zhang, L. M. Duan, D. Yum, and K. Kim, Nature Photonics 11, 646 (2017).
  * Wang  _et al._ (2021) P. Wang, C.-Y. Luan, M. Qiao, M. Um, J. Zhang, Y. Wang, X. Yuan, M. Gu, J. Zhang, and K. Kim, Nature Communications 12, 233 (2021).
  * Sheng  _et al._ (2018) C. Sheng, X. He, P. Xu, R. Guo, K. Wang, Z. Xiong, M. Liu, J. Wang, and M. Zhan, Phys. Rev. Lett. 121, 240501 (2018).
  * Graham  _et al._ (2019) T. M. Graham, M. Kwon, B. Grinkemeyer, Z. Marra, X. Jiang, M. T. Lichtman, Y. Sun, M. Ebert, and M. Saffman, Phys. Rev. Lett. 123, 230501 (2019).
  * Levine  _et al._ (2019) H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuletić, H. Pichler, and M. D. Lukin, Phys. Rev. Lett. 123, 170503 (2019).
  * Houck  _et al._ (2008) A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, J. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 101, 080502 (2008).
  * DiCarlo  _et al._ (2010) L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, Nature 467, 574 (2010).
  * Barends  _et al._ (2013) R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, Phys. Rev. Lett. 111, 080502 (2013).
  * Rigetti and Devoret (2010) C. Rigetti and M. Devoret, Phys. Rev. B 81, 134507 (2010).
  * Chow  _et al._ (2011) J. M. Chow, A. D. Córcoles, J. M. Gambetta, C. Rigetti, B. R. Johnson, J. A. Smolin, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen, and M. Steffen, Phys. Rev. Lett. 107, 080502 (2011).
  * Nesterov  _et al._ (2022) K. N. Nesterov, C. Wang, V. E. Manucharyan, and M. G. Vavilov, Phys. Rev. Appl. 18, 034063 (2022).
  * Dogan  _et al._ (2023) E. Dogan, D. Rosenstock, L. Le Guevel, H. Xiong, R. A. Mencia, A. Somoroff, K. N. Nesterov, M. G. Vavilov, V. E. Manucharyan, and C. Wang, Phys. Rev. Appl. 20, 024011 (2023).
  * Stehlik  _et al._ (2021) J. Stehlik, D. M. Zajac, D. L. Underwood, T. Phung, J. Blair, S. Carnevale, D. Klaus, G. A. Keefe, A. Carniol, M. Kumph, M. Steffen, and O. E. Dial, Phys. Rev. Lett. 127, 080505 (2021).
  * Sung  _et al._ (2021) Y. Sung, L. Ding, J. Braumüller, A. Vepsäläinen, B. Kannan, M. Kjaergaard, A. Greene, G. O. Samach, C. McNally, D. Kim, A. Melville, B. M. Niedzielski, M. E. Schwartz, J. L. Yoder, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Phys. Rev. X 11, 021058 (2021).
  * Marxer  _et al._ (2023) F. Marxer, A. Vepsäläinen, S. W. Jolin, J. Tuorila, A. Landra, C. Ockeloen-Korppi, W. Liu, O. Ahonen, A. Auer, L. Belzane, V. Bergholm, C. F. Chan, K. W. Chan, T. Hiltunen, J. Hotari, E. Hyyppä, J. Ikonen, D. Janzso, M. Koistinen, J. Kotilahti, T. Li, J. Luus, M. Papic, M. Partanen, J. Räbinä, J. Rosti, M. Savytskyi, M. Seppälä, V. Sevriuk, E. Takala, B. Tarasinski, M. J. Thapa, F. Tosto, N. Vorobeva, L. Yu, K. Y. Tan, J. Hassel, M. Möttönen, and J. Heinsoo, PRX Quantum 4, 010314 (2023).
  * Bao  _et al._ (2022) F. Bao, H. Deng, D. Ding, R. Gao, X. Gao, C. Huang, X. Jiang, H.-S. Ku, Z. Li, X. Ma, X. Ni, J. Qin, Z. Song, H. Sun, C. Tang, T. Wang, F. Wu, T. Xia, W. Yu, F. Zhang, G. Zhang, X. Zhang, J. Zhou, X. Zhu, Y. Shi, J. Chen, H.-H. Zhao, and C. Deng, Phys. Rev. Lett. 129, 010502 (2022).
  * Nesterov  _et al._ (2018) K. N. Nesterov, I. V. Pechenezhskiy, C. Wang, V. E. Manucharyan, and M. G. Vavilov, Phys. Rev. A 98, 030301 (2018).
  * Nguyen  _et al._ (2022) L. B. Nguyen, G. Koolstra, Y. Kim, A. Morvan, T. Chistolini, S. Singh, K. N. Nesterov, C. Jünger, L. Chen, Z. Pedramrazi, B. K. Mitchell, J. M. Kreikebaum, S. Puri, D. I. Santiago, and I. Siddiqi, PRX Quantum 3, 037001 (2022).
  * Hilder  _et al._ (2022) J. Hilder, D. Pijn, O. Onishchenko, A. Stahl, M. Orth, B. Lekitsch, A. Rodriguez-Blanco, M. Müller, F. Schmidt-Kaler, and U. G. Poschinger, Phys. Rev. X 12, 011032 (2022).
  * Ospelkaus  _et al._ (2011) C. Ospelkaus, U. Warring, Y. Colombe, K. R. Brown, J. M. Amini, D. Leibfried, and D. J. Wineland, Nature 476, 181 (2011).
  * Yavuz  _et al._ (2006) D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, Phys. Rev. Lett. 96, 063001 (2006).
  * Xia  _et al._ (2015) T. Xia, M. Lichtman, K. Maller, A. W. Carr, M. J. Piotrowicz, L. Isenhower, and M. Saffman, Phys. Rev. Lett. 114, 100503 (2015).
  * Levine  _et al._ (2022) H. Levine, D. Bluvstein, A. Keesling, T. T. Wang, S. Ebadi, G. Semeghini, A. Omran, M. Greiner, V. Vuletić, and M. D. Lukin, Phys. Rev. A 105, 032618 (2022).
  * Saner  _et al._ (2023) S. Saner, O. Băzăvan, M. Minder, P. Drmota, D. J. Webb, G. Araneda, R. Srinivas, D. M. Lucas, and C. J. Ballance, “Breaking the entangling gate speed limit for trapped-ion qubits using a phase-stable standing wave,” (2023), arXiv:2305.03450 [quant-ph] .
  * Evered  _et al._ (2023) S. J. Evered, D. Bluvstein, M. Kalinowski, S. Ebadi, T. Manovitz, H. Zhou, S. H. Li, A. A. Geim, T. T. Wang, N. Maskara, H. Levine, G. Semeghini, M. Greiner, V. Vuletic, and M. D. Lukin, “High-fidelity parallel entangling gates on a neutral atom quantum computer,” (2023), arXiv:2304.05420 [quant-ph] .
  * Bluvstein  _et al._ (2023) D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V. Vuletić, and M. D. Lukin, Nature (2023), 10.1038/s41586-023-06927-3.
  * Krinner  _et al._ (2022) S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Herrmann, G. J. Norris, C. K. Andersen, M. Müller, A. Blais, C. Eichler, and A. Wallraff, Nature 605, 669 (2022).
  * Acharya  _et al._ (2023a) R. Acharya, I. Aleiner, R. Allen, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, R. Babbush, D. Bacon, J. C. Bardin, J. Basso, A. Bengtsson, S. Boixo, G. Bortoli, A. Bourassa, J. Bovaird, L. Brill, M. Broughton, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bushnell, Y. Chen, Z. Chen, B. Chiaro, J. Cogan, R. Collins, P. Conner, W. Courtney, A. L. Crook, B. Curtin, D. M. Debroy, A. Del Toro Barba, S. Demura, A. Dunsworth, D. Eppens, C. Erickson, L. Faoro, E. Farhi, R. Fatemi, L. Flores Burgos, E. Forati, A. G. Fowler, B. Foxen, W. Giang, C. Gidney, D. Gilboa, M. Giustina, A. Grajales Dau, J. A. Gross, S. Habegger, M. C. Hamilton, M. P. Harrigan, S. D. Harrington, O. Higgott, J. Hilton, M. Hoffmann, S. Hong, T. Huang, A. Huff, W. J. Huggins, L. B. Ioffe, S. V. Isakov, J. Iveland, E. Jeffrey, Z. Jiang, C. Jones, P. Juhas, D. Kafri, K. Kechedzhi, J. Kelly, T. Khattar, M. Khezri, M. Kieferová, S. Kim, A. Kitaev, P. V. Klimov, A. R. Klots, A. N. Korotkov, F. Kostritsa, J. M. Kreikebaum, D. Landhuis, P. Laptev, K.-M. Lau, L. Laws, J. Lee, K. Lee, B. J. Lester, A. Lill, W. Liu, A. Locharla, E. Lucero, F. D. Malone, J. Marshall, O. Martin, J. R. McClean, T. McCourt, M. McEwen, A. Megrant, B. Meurer Costa, X. Mi, K. C. Miao, M. Mohseni, S. Montazeri, A. Morvan, E. Mount, W. Mruczkiewicz, O. Naaman, M. Neeley, C. Neill, A. Nersisyan, H. Neven, M. Newman, J. H. Ng, A. Nguyen, M. Nguyen, M. Y. Niu, T. E. O’Brien, A. Opremcak, J. Platt, A. Petukhov, R. Potter, L. P. Pryadko, C. Quintana, P. Roushan, N. C. Rubin, N. Saei, D. Sank, K. Sankaragomathi, K. J. Satzinger, H. F. Schurkus, C. Schuster, M. J. Shearn, A. Shorter, V. Shvarts, J. Skruzny, V. Smelyanskiy, W. C. Smith, G. Sterling, D. Strain, M. Szalay, A. Torres, G. Vidal, B. Villalonga, C. Vollgraff Heidweiller, T. White, C. Xing, Z. J. Yao, P. Yeh, J. Yoo, G. Young, A. Zalcman, Y. Zhang, N. Zhu, and G. Q. AI, Nature 614, 676 (2023a).
  * Stricker  _et al._ (2020) R. Stricker, D. Vodola, A. Erhard, L. Postler, M. Meth, M. Ringbauer, P. Schindler, T. Monz, M. Müller, and R. Blatt, Nature 585, 207 (2020).
  * Sørensen and Mølmer (1999) A. Sørensen and K. Mølmer, Phys. Rev. Lett. 82, 1971 (1999).
  * Schreier  _et al._ (2008) J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R. Johnson, J. M. Chow, J. M. Gambetta, J. Majer, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. B 77, 180502 (2008).
  * Chang  _et al._ (2013) J. B. Chang, M. R. Vissers, A. D. Córcoles, M. Sandberg, J. Gao, D. W. Abraham, J. M. Chow, J. M. Gambetta, M. Beth Rothwell, G. A. Keefe, M. Steffen, and D. P. Pappas, Applied Physics Letters 103 (2013), 10.1063/1.4813269, 012602.
  * Jin  _et al._ (2015) X. Y. Jin, A. Kamal, A. P. Sears, T. Gudmundsen, D. Hover, J. Miloshi, R. Slattery, F. Yan, J. Yoder, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Phys. Rev. Lett. 114, 240501 (2015).
  * Place  _et al._ (2021) A. P. M. Place, L. V. H. Rodgers, P. Mundada, B. M. Smitham, M. Fitzpatrick, Z. Leng, A. Premkumar, J. Bryon, A. Vrajitoarea, S. Sussman, G. Cheng, T. Madhavan, H. K. Babla, X. H. Le, Y. Gang, B. Jäck, A. Gyenis, N. Yao, R. J. Cava, N. P. de Leon, and A. A. Houck, Nature Communications 12, 1779 (2021).
  * Wang  _et al._ (2022a) C. Wang, X. Li, H. Xu, Z. Li, J. Wang, Z. Yang, Z. Mi, X. Liang, T. Su, C. Yang, G. Wang, W. Wang, Y. Li, M. Chen, C. Li, K. Linghu, J. Han, Y. Zhang, Y. Feng, Y. Song, T. Ma, J. Zhang, R. Wang, P. Zhao, W. Liu, G. Xue, Y. Jin, and H. Yu, npj Quantum Information 8, 3 (2022a).
  * Somoroff  _et al._ (2023) A. Somoroff, Q. Ficheux, R. A. Mencia, H. Xiong, R. Kuzmin, and V. E. Manucharyan, Phys. Rev. Lett. 130, 267001 (2023).
  * Wallraff  _et al._ (2004) A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 162 (2004).
  * Walter  _et al._ (2017) T. Walter, P. Kurpiers, S. Gasparinetti, P. Magnard, A. Potočnik, Y. Salathé, M. Pechal, M. Mondal, M. Oppliger, C. Eichler, and A. Wallraff, Phys. Rev. Appl. 7, 054020 (2017).
  * Sunada  _et al._ (2022) Y. Sunada, S. Kono, J. Ilves, S. Tamate, T. Sugiyama, Y. Tabuchi, and Y. Nakamura, Phys. Rev. Appl. 17, 044016 (2022).
  * Swiadek  _et al._ (2023) F. Swiadek, R. Shillito, P. Magnard, A. Remm, C. Hellings, N. Lacroix, Q. Ficheux, D. C. Zanuz, G. J. Norris, A. Blais, S. Krinner, and A. Wallraff, “Enhancing dispersive readout of superconducting qubits through dynamic control of the dispersive shift: Experiment and theory,” (2023), arXiv:2307.07765 [quant-ph] .
  * Sunada  _et al._ (2024) Y. Sunada, K. Yuki, Z. Wang, T. Miyamura, J. Ilves, K. Matsuura, P. A. Spring, S. Tamate, S. Kono, and Y. Nakamura, PRX Quantum 5, 010307 (2024).
  * Acharya  _et al._ (2023b) R. Acharya, I. Aleiner, R. Allen, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, R. Babbush, D. Bacon, J. C. Bardin, J. Basso, A. Bengtsson, S. Boixo, G. Bortoli, A. Bourassa, J. Bovaird, L. Brill, M. Broughton, B. B. Buckley, D. A. Buell, T. Burger, B. Burkett, N. Bushnell, Y. Chen, Z. Chen, B. Chiaro, J. Cogan, R. Collins, P. Conner, W. Courtney, A. L. Crook, B. Curtin, D. M. Debroy, A. Del Toro Barba, S. Demura, A. Dunsworth, D. Eppens, C. Erickson, L. Faoro, E. Farhi, R. Fatemi, L. Flores Burgos, E. Forati, A. G. Fowler, B. Foxen, W. Giang, C. Gidney, D. Gilboa, M. Giustina, A. Grajales Dau, J. A. Gross, S. Habegger, M. C. Hamilton, M. P. Harrigan, S. D. Harrington, O. Higgott, J. Hilton, M. Hoffmann, S. Hong, T. Huang, A. Huff, W. J. Huggins, L. B. Ioffe, S. V. Isakov, J. Iveland, E. Jeffrey, Z. Jiang, C. Jones, P. Juhas, D. Kafri, K. Kechedzhi, J. Kelly, T. Khattar, M. Khezri, M. Kieferová, S. Kim, A. Kitaev, P. V. Klimov, A. R. Klots, A. N. Korotkov, F. Kostritsa, J. M. Kreikebaum, D. Landhuis, P. Laptev, K.-M. Lau, L. Laws, J. Lee, K. Lee, B. J. Lester, A. Lill, W. Liu, A. Locharla, E. Lucero, F. D. Malone, J. Marshall, O. Martin, J. R. McClean, T. McCourt, M. McEwen, A. Megrant, B. Meurer Costa, X. Mi, K. C. Miao, M. Mohseni, S. Montazeri, A. Morvan, E. Mount, W. Mruczkiewicz, O. Naaman, M. Neeley, C. Neill, A. Nersisyan, H. Neven, M. Newman, J. H. Ng, A. Nguyen, M. Nguyen, M. Y. Niu, T. E. O’Brien, A. Opremcak, J. Platt, A. Petukhov, R. Potter, L. P. Pryadko, C. Quintana, P. Roushan, N. C. Rubin, N. Saei, D. Sank, K. Sankaragomathi, K. J. Satzinger, H. F. Schurkus, C. Schuster, M. J. Shearn, A. Shorter, V. Shvarts, J. Skruzny, V. Smelyanskiy, W. C. Smith, G. Sterling, D. Strain, M. Szalay, A. Torres, G. Vidal, B. Villalonga, C. Vollgraff Heidweiller, T. White, C. Xing, Z. J. Yao, P. Yeh, J. Yoo, G. Young, A. Zalcman, Y. Zhang, N. Zhu, and G. Q. AI, Nature 614, 676 (2023b).
  * Andersen  _et al._ (2020) C. K. Andersen, A. Remm, S. Lazar, S. Krinner, N. Lacroix, G. J. Norris, M. Gabureac, C. Eichler, and A. Wallraff, Nature Physics 16, 875 (2020).
  * Gong  _et al._ (2021b) M. Gong, X. Yuan, S. Wang, Y. Wu, Y. Zhao, C. Zha, S. Li, Z. Zhang, Q. Zhao, Y. Liu, F. Liang, J. Lin, Y. Xu, H. Deng, H. Rong, H. Lu, S. C. Benjamin, C.-Z. Peng, X. Ma, Y.-A. Chen, X. Zhu, and J.-W. Pan, National Science Review 9 (2021b), 10.1093/nsr/nwab011, nwab011.
  * Wrachtrup and Jelezko (2006) J. Wrachtrup and F. Jelezko, Journal of Physics: Condensed Matter 18, S807 (2006).
  * Doherty  _et al._ (2011) M. W. Doherty, N. B. Manson, P. Delaney, and L. C. L. Hollenberg, New Journal of Physics 13, 025019 (2011).
  * Doherty  _et al._ (2013) M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. Hollenberg, Physics Reports 528, 1 (2013), the nitrogen-vacancy colour centre in diamond.
  * Fuchs  _et al._ (2009) G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, and D. D. Awschalom, Science 326, 1520 (2009).
  * Fuchs  _et al._ (2010) G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, C. D. Weis, T. Schenkel, and D. D. Awschalom, Nature Physics 6, 668 (2010).
  * Abobeih  _et al._ (2018) M. H. Abobeih, J. Cramer, M. A. Bakker, N. Kalb, M. Markham, D. J. Twitchen, and T. H. Taminiau, Nature Communications 9, 2552 (2018).
  * Maurer  _et al._ (2012) P. C. Maurer, G. Kucsko, C. Latta, L. Jiang, N. Y. Yao, S. D. Bennett, F. Pastawski, D. Hunger, N. Chisholm, M. Markham, D. J. Twitchen, J. I. Cirac, and M. D. Lukin, Science 336, 1283 (2012).
  * Bradley  _et al._ (2019) C. E. Bradley, J. Randall, M. H. Abobeih, R. C. Berrevoets, M. J. Degen, M. A. Bakker, M. Markham, D. J. Twitchen, and T. H. Taminiau, Phys. Rev. X 9, 031045 (2019).
  * Pompili  _et al._ (2021) M. Pompili, S. L. N. Hermans, S. Baier, H. K. C. Beukers, P. C. Humphreys, R. N. Schouten, R. F. L. Vermeulen, M. J. Tiggelman, L. dos Santos Martins, B. Dirkse, S. Wehner, and R. Hanson, Science 372, 259 (2021).
  * Bartling  _et al._ (2022) H. P. Bartling, M. H. Abobeih, B. Pingault, M. J. Degen, S. J. H. Loenen, C. E. Bradley, J. Randall, M. Markham, D. J. Twitchen, and T. H. Taminiau, Phys. Rev. X 12, 011048 (2022).
  * Hermans  _et al._ (2022) S. L. N. Hermans, M. Pompili, H. K. C. Beukers, S. Baier, J. Borregaard, and R. Hanson, Nature 605, 663 (2022).
  * Xie  _et al._ (2023) T. Xie, Z. Zhao, S. Xu, X. Kong, Z. Yang, M. Wang, Y. Wang, F. Shi, and J. Du, Phys. Rev. Lett. 130, 030601 (2023).
  * Jelezko  _et al._ (2004) F. Jelezko, T. Gaebel, I. Popa, M. Domhan, A. Gruber, and J. Wrachtrup, Phys. Rev. Lett. 93, 130501 (2004).
  * Abobeih  _et al._ (2022) M. H. Abobeih, Y. Wang, J. Randall, S. J. H. Loenen, C. E. Bradley, M. Markham, D. J. Twitchen, B. M. Terhal, and T. H. Taminiau, Nature 606, 884 (2022).
  * Bernien  _et al._ (2013) H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. H. Taminiau, M. Markham, D. J. Twitchen, L. Childress, and R. Hanson, Nature 497, 86 (2013).
  * Pfaff  _et al._ (2014) W. Pfaff, B. J. Hensen, H. Bernien, S. B. van Dam, M. S. Blok, T. H. Taminiau, M. J. Tiggelman, R. N. Schouten, M. Markham, D. J. Twitchen, and R. Hanson, Science 345, 532 (2014).
  * Hensen  _et al._ (2015) B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, Nature 526, 682 (2015).
  * Cramer  _et al._ (2016) J. Cramer, N. Kalb, M. A. Rol, B. Hensen, M. S. Blok, M. Markham, D. J. Twitchen, R. Hanson, and T. H. Taminiau, Nature Communications 7, 11526 (2016).
  * Taminiau  _et al._ (2014) T. H. Taminiau, J. Cramer, T. van der Sar, V. V. Dobrovitski, and R. Hanson, Nature Nanotechnology 9, 171 (2014).
  * Taminiau  _et al._ (2012) T. H. Taminiau, J. J. T. Wagenaar, T. van der Sar, F. Jelezko, V. V. Dobrovitski, and R. Hanson, Phys. Rev. Lett. 109, 137602 (2012).
  * Dong  _et al._ (2020) W. Dong, F. A. Calderon-Vargas, and S. E. Economou, New Journal of Physics 22, 073059 (2020).
  * Balasubramanian  _et al._ (2009) G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, Nature Materials 8, 383 (2009).
  * Bar-Gill  _et al._ (2013) N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, Nature Communications 4, 1743 (2013).
  * Byer (1974) R. L. Byer, Annual Review of Materials Science 4, 147 (1974), https://doi.org/10.1146/annurev.ms.04.080174.001051 .
  * Chen and Liu (1986) C.-t. Chen and G.-z. Liu, Annual Review of Materials Science 16, 203 (1986), https://doi.org/10.1146/annurev.ms.16.080186.001223 .
  * Turchette  _et al._ (1995) Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, Phys. Rev. Lett. 75, 4710 (1995).
  * Knill  _et al._ (2001) E. Knill, R. Laflamme, and G. J. Milburn, Nature 409, 46 (2001).
  * Duan and Kimble (2004) L.-M. Duan and H. J. Kimble, Phys. Rev. Lett. 92, 127902 (2004).
  * O’Brien  _et al._ (2003) J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, Nature 426, 264 (2003).
  * Basset  _et al._ (2021) F. B. Basset, M. Valeri, E. Roccia, V. Muredda, D. Poderini, J. Neuwirth, N. Spagnolo, M. B. Rota, G. Carvacho, F. Sciarrino, and R. Trotta, Science Advances 7, eabe6379 (2021).
  * Chuang and Yamamoto (1995) I. L. Chuang and Y. Yamamoto, Phys. Rev. A 52, 3489 (1995).
  * Brendel  _et al._ (1999) J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, Phys. Rev. Lett. 82, 2594 (1999).
  * Jayakumar  _et al._ (2014) H. Jayakumar, A. Predojević, T. Kauten, T. Huber, G. S. Solomon, and G. Weihs, Nature Communications 5, 4251 (2014).
  * Sun  _et al._ (2022) P.-F. Sun, Y. Yu, Z.-Y. An, J. Li, C.-W. Yang, X.-H. Bao, and J.-W. Pan, Phys. Rev. Lett. 128, 060502 (2022).
  * Lee  _et al._ (2019) J. P. Lee, B. Villa, A. J. Bennett, R. M. Stevenson, D. J. P. Ellis, I. Farrer, D. A. Ritchie, and A. J. Shields, Quantum Science and Technology 4, 025011 (2019).
  * Ortu  _et al._ (2022) A. Ortu, A. Holzäpfel, J. Etesse, and M. Afzelius, npj Quantum Information 8, 29 (2022).
  * Olislager  _et al._ (2010) L. Olislager, J. Cussey, A. T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. P. Huy, Phys. Rev. A 82, 013804 (2010).
  * Lu  _et al._ (2019) H.-H. Lu, J. M. Lukens, B. P. Williams, P. Imany, N. A. Peters, A. M. Weiner, and P. Lougovski, npj Quantum Information 5, 24 (2019).
  * Sabattoli  _et al._ (2022) F. A. Sabattoli, L. Gianini, A. Simbula, M. Clementi, A. Fincato, F. Boeuf, M. Liscidini, M. Galli, and D. Bajoni, Opt. Lett. 47, 6201 (2022).
  * Clementi  _et al._ (2023) M. Clementi, F. A. Sabattoli, M. Borghi, L. Gianini, N. Tagliavacche, H. El Dirani, L. Youssef, N. Bergamasco, C. Petit-Etienne, E. Pargon, J. E. Sipe, M. Liscidini, C. Sciancalepore, M. Galli, and D. Bajoni, Nature Communications 14, 176 (2023).
  * Kok  _et al._ (2007) P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 (2007).
  * Ralph  _et al._ (2001) T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, Phys. Rev. A 65, 012314 (2001).
  * Zou  _et al._ (2002) X. Zou, K. Pahlke, and W. Mathis, Phys. Rev. A 65, 064305 (2002).
  * Pittman  _et al._ (2002) T. B. Pittman, B. C. Jacobs, and J. D. Franson, Phys. Rev. Lett. 88, 257902 (2002).
  * Pittman  _et al._ (2003) T. B. Pittman, M. J. Fitch, B. C. Jacobs, and J. D. Franson, Phys. Rev. A 68, 032316 (2003).
  * Kimble (2008) H. J. Kimble, Nature 453, 1023 (2008).
  * Zhan and Sun (2020) Y. Zhan and S. Sun, Phys. Rev. Lett. 125, 223601 (2020).
  * Pichler  _et al._ (2017) H. Pichler, S. Choi, P. Zoller, and M. D. Lukin, Proceedings of the National Academy of Sciences 114, 11362 (2017), https://www.pnas.org/doi/pdf/10.1073/pnas.1711003114 .
  * Sun  _et al._ (2016) S. Sun, H. Kim, G. S. Solomon, and E. Waks, Nature Nanotechnology 11, 539 (2016).
  * Shi and Waks (2021) Y. Shi and E. Waks, Phys. Rev. A 104, 013703 (2021).
  * Bartolucci  _et al._ (2023) S. Bartolucci, P. Birchall, H. Bombín, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, F. Pastawski, T. Rudolph, and C. Sparrow, Nature Communications 14, 912 (2023).
  * Lindner and Rudolph (2009) N. H. Lindner and T. Rudolph, Phys. Rev. Lett. 103, 113602 (2009).
  * Economou  _et al._ (2010) S. E. Economou, N. Lindner, and T. Rudolph, Phys. Rev. Lett. 105, 093601 (2010).
  * Buterakos  _et al._ (2017) D. Buterakos, E. Barnes, and S. E. Economou, Phys. Rev. X 7, 041023 (2017).
  * Russo  _et al._ (2019) A. Russo, E. Barnes, and S. E. Economou, New Journal of Physics 21, 055002 (2019).
  * Varnava  _et al._ (2006) M. Varnava, D. E. Browne, and T. Rudolph, Phys. Rev. Lett. 97, 120501 (2006).
  * Varnava  _et al._ (2007) M. Varnava, D. E. Browne, and T. Rudolph, New Journal of Physics 9, 203 (2007).
  * Hilaire  _et al._ (2021) P. Hilaire, E. Barnes, S. E. Economou, and F. Grosshans, Phys. Rev. A 104, 052623 (2021).
  * Zhan  _et al._ (2023) Y. Zhan, P. Hilaire, E. Barnes, S. E. Economou, and S. Sun, Quantum 7, 924 (2023).
  * Li  _et al._ (2015) Y. Li, P. C. Humphreys, G. J. Mendoza, and S. C. Benjamin, Phys. Rev. X 5, 041007 (2015).
  * Scully and Zubairy (1997) M. Scully and M. Zubairy,  _Quantum Optics_, Quantum Optics (Cambridge University Press, 1997).
  * Lvovsky  _et al._ (2009) A. I. Lvovsky, B. C. Sanders, and W. Tittel, Nature Photonics 3, 706 (2009).
  * Heshami  _et al._ (2016) K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, Journal of Modern Optics 63, 2005 (2016), pMID: 27695198\.
  * Fleischhauer  _et al._ (2005) M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005).
  * Lukin (2003) M. D. Lukin, Rev. Mod. Phys. 75, 457 (2003).
  * Ledingham  _et al._ (2012) P. M. Ledingham, W. R. Naylor, and J. J. Longdell, Phys. Rev. Lett. 109, 093602 (2012).
  * Ferguson  _et al._ (2016) K. R. Ferguson, S. E. Beavan, J. J. Longdell, and M. J. Sellars, Phys. Rev. Lett. 117, 020501 (2016).
  * Alexander  _et al._ (2006) A. L. Alexander, J. J. Longdell, M. J. Sellars, and N. B. Manson, Phys. Rev. Lett. 96, 043602 (2006).
  * Gorshkov  _et al._ (2007) A. V. Gorshkov, A. André, M. Fleischhauer, A. S. Sørensen, and M. D. Lukin, Phys. Rev. Lett. 98, 123601 (2007).
  * Hosseini  _et al._ (2011) M. Hosseini, B. M. Sparkes, G. Campbell, P. K. Lam, and B. C. Buchler, Nature Communications 2, 174 (2011).
  * Dudin  _et al._ (2013) Y. O. Dudin, L. Li, and A. Kuzmich, Phys. Rev. A 87, 031801 (2013).
  * Ding  _et al._ (2013) D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, Nature Communications 4, 2527 (2013).
  * Nicolas  _et al._ (2014) A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, Nature Photonics 8, 234 (2014).
  * Ding  _et al._ (2015a) D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, Phys. Rev. Lett. 114, 050502 (2015a).
  * Ding  _et al._ (2015b) D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, B.-S. Shi, and G.-C. Guo, Nature Photonics 9, 332 (2015b).
  * Parigi  _et al._ (2015) V. Parigi, V. D’Ambrosio, C. Arnold, L. Marrucci, F. Sciarrino, and J. Laurat, Nature Communications 6, 7706 (2015).
  * Saunders  _et al._ (2016) D. J. Saunders, J. H. D. Munns, T. F. M. Champion, C. Qiu, K. T. Kaczmarek, E. Poem, P. M. Ledingham, I. A. Walmsley, and J. Nunn, Phys. Rev. Lett. 116, 090501 (2016).
  * Katz and Firstenberg (2018) O. Katz and O. Firstenberg, Nature Communications 9, 2074 (2018).
  * Hsiao  _et al._ (2018) Y.-F. Hsiao, P.-J. Tsai, H.-S. Chen, S.-X. Lin, C.-C. Hung, C.-H. Lee, Y.-H. Chen, Y.-F. Chen, I. A. Yu, and Y.-C. Chen, Phys. Rev. Lett. 120, 183602 (2018).
  * Jiang  _et al._ (2019) N. Jiang, Y. F. Pu, W. Chang, C. Li, S. Zhang, and L. M. Duan, npj Quantum Information 5, 28 (2019).
  * Wang  _et al._ (2019) Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S.-L. Zhu, Nature Photonics 13, 346 (2019).
  * Li  _et al._ (2020) C. Li, N. Jiang, Y.-K. Wu, W. Chang, Y.-F. Pu, S. Zhang, and L.-M. Duan, Phys. Rev. Lett. 124, 240504 (2020).
  * Dideriksen  _et al._ (2021) K. B. Dideriksen, R. Schmieg, M. Zugenmaier, and E. S. Polzik, Nature Communications 12, 3699 (2021).
  * Wang  _et al._ (2022b) Y. Wang, A. N. Craddock, R. Sekelsky, M. Flament, and M. Namazi, Phys. Rev. Appl. 18, 044058 (2022b).
  * Meßner  _et al._ (2023) L. Meßner, E. Robertson, L. Esguerra, K. Lüdge, and J. Wolters, Opt. Express 31, 10150 (2023).
  * Buser  _et al._ (2022) G. Buser, R. Mottola, B. Cotting, J. Wolters, and P. Treutlein, PRX Quantum 3, 020349 (2022).
  * Moiseev and Kröll (2001) S. A. Moiseev and S. Kröll, Phys. Rev. Lett. 87, 173601 (2001).
  * Nilsson and Kröll (2005) M. Nilsson and S. Kröll, Optics Communications 247, 393 (2005).
  * Kraus  _et al._ (2006) B. Kraus, W. Tittel, N. Gisin, M. Nilsson, S. Kröll, and J. I. Cirac, Phys. Rev. A 73, 020302 (2006).
  * Hétet  _et al._ (2008) G. Hétet, J. J. Longdell, A. L. Alexander, P. K. Lam, and M. J. Sellars, Phys. Rev. Lett. 100, 023601 (2008).
  * McAuslan  _et al._ (2012) D. L. McAuslan, J. G. Bartholomew, M. J. Sellars, and J. J. Longdell, Phys. Rev. A 85, 032339 (2012).
  * Saglamyurek  _et al._ (2011) E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussières, M. George, R. Ricken, W. Sohler, and W. Tittel, Nature 469, 512 (2011).
  * Clausen  _et al._ (2011) C. Clausen, I. Usmani, F. Bussières, N. Sangouard, M. Afzelius, H. de Riedmatten, and N. Gisin, Nature 469, 508 (2011).
  * Zhou  _et al._ (2012) Z.-Q. Zhou, W.-B. Lin, M. Yang, C.-F. Li, and G.-C. Guo, Phys. Rev. Lett. 108, 190505 (2012).
  * Jin  _et al._ (2022) M. Jin, Y.-Z. Ma, Z.-Q. Zhou, C.-F. Li, and G.-C. Guo, Science Bulletin 67, 676 (2022).
  * Ma  _et al._ (2021a) Y.-Z. Ma, M. Jin, D.-L. Chen, Z.-Q. Zhou, C.-F. Li, and G.-C. Guo, Nature Communications 12, 4378 (2021a).
  * Seri  _et al._ (2017) A. Seri, A. Lenhard, D. Rieländer, M. Gündoğan, P. M. Ledingham, M. Mazzera, and H. de Riedmatten, Phys. Rev. X 7, 021028 (2017).
  * Kutluer  _et al._ (2017) K. Kutluer, M. Mazzera, and H. de Riedmatten, Phys. Rev. Lett. 118, 210502 (2017).
  * Laplane  _et al._ (2017) C. Laplane, P. Jobez, J. Etesse, N. Gisin, and M. Afzelius, Phys. Rev. Lett. 118, 210501 (2017).
  * Holzäpfel  _et al._ (2020) A. Holzäpfel, J. Etesse, K. T. Kaczmarek, A. Tiranov, N. Gisin, and M. Afzelius, New Journal of Physics 22, 063009 (2020).
  * Businger  _et al._ (2020) M. Businger, A. Tiranov, K. T. Kaczmarek, S. Welinski, Z. Zhang, A. Ferrier, P. Goldner, and M. Afzelius, Phys. Rev. Lett. 124, 053606 (2020).
  * Askarani  _et al._ (2021) M. F. Askarani, A. Das, J. H. Davidson, G. C. Amaral, N. Sinclair, J. A. Slater, S. Marzban, C. W. Thiel, R. L. Cone, D. Oblak, and W. Tittel, Phys. Rev. Lett. 127, 220502 (2021).
  * Ma  _et al._ (2021b) Y. Ma, Y.-Z. Ma, Z.-Q. Zhou, C.-F. Li, and G.-C. Guo, Nature Communications 12, 2381 (2021b).
  * Ledingham  _et al._ (2010) P. M. Ledingham, W. R. Naylor, J. J. Longdell, S. E. Beavan, and M. J. Sellars, Phys. Rev. A 81, 012301 (2010).
  * Vernaz-Gris  _et al._ (2018) P. Vernaz-Gris, K. Huang, M. Cao, A. S. Sheremet, and J. Laurat, Nature Communications 9, 363 (2018).
  * Zhong  _et al._ (2020) C. Zhong, Z. Wang, C. Zou, M. Zhang, X. Han, W. Fu, M. Xu, S. Shankar, M. H. Devoret, H. X. Tang, and L. Jiang, Phys. Rev. Lett. 124, 010511 (2020).
  * Deng  _et al._ (2023) Y.-H. Deng, Y.-C. Gu, H.-L. Liu, S.-Q. Gong, H. Su, Z.-J. Zhang, H.-Y. Tang, M.-H. Jia, J.-M. Xu, M.-C. Chen, J. Qin, L.-C. Peng, J. Yan, Y. Hu, J. Huang, H. Li, Y. Li, Y. Chen, X. Jiang, L. Gan, G. Yang, L. You, L. Li, H.-S. Zhong, H. Wang, N.-L. Liu, J. J. Renema, C.-Y. Lu, and J.-W. Pan, Phys. Rev. Lett. 131, 150601 (2023).
  * Borregaard  _et al._ (2020) J. Borregaard, H. Pichler, T. Schröder, M. D. Lukin, P. Lodahl, and A. S. Sørensen, Phys. Rev. X 10, 021071 (2020).
  * Jing and Razavi (2022) Y. Jing and M. Razavi, Phys. Rev. Appl. 18, 024041 (2022).
  * Ruf  _et al._ (2021) M. Ruf, N. H. Wan, H. Choi, D. Englund, and R. Hanson, Journal of Applied Physics 130, 070901 (2021), https://pubs.aip.org/aip/jap/article-pdf/doi/10.1063/5.0056534/15266654/070901_1_online.pdf .
  * Jobez  _et al._ (2016) P. Jobez, N. Timoney, C. Laplane, J. Etesse, A. Ferrier, P. Goldner, N. Gisin, and M. Afzelius, Phys. Rev. A 93, 032327 (2016).
  * Keselman  _et al._ (2011) A. Keselman, Y. Glickman, N. Akerman, S. Kotler, and R. Ozeri, New Journal of Physics 13, 073027 (2011).
  * Ruster  _et al._ (2016) T. Ruster, C. T. Schmiegelow, H. Kaufmann, C. Warschburger, F. Schmidt-Kaler, and U. G. Poschinger, Applied Physics B 122, 254 (2016).
  * Toyoda  _et al._ (2010) K. Toyoda, S. Haze, R. Yamazaki, and S. Urabe, Phys. Rev. A 81, 032322 (2010).