Quantum |  all |  addr, QMC

In Table 5, we compare the four operation modes of a QRAM, where we
specifically show the entanglement feature of the outcome state. We noticed
that because of the coherent addressing feature, the address and the bus qubit
are entangled with the quantum memory after the reading and writing queries,
except in the case of reading the classical data using CNOT gates (or any
other methods to copy the classical data to the bus qubit). The feature of
generating entanglement between the address qubits and QMC qubits inside the
memory is unique to QRAM operations, which is caused by the coherently
addressing of the bus qubits to the quantum memory array. This feature can be
useful to generate large-scale quantum entanglement. However, in the current
stage of the quantum memory research, to our best knowledge, there is no
specific usage for these operations. Instead, reading classical data
coherently out of the quantum memory using CNOT-type classical copying
operation can realize the quantum oracle in Eq. (12), while leaving the memory
disentangled with the rest of the system, which becomes the main application
of QRAMs. In this sense, QRAMs can be used as a classical data quantum encoder
as an I/O unit or an implementation of the quantum oracle in Eq. (12), rather
than traditionally believed quantum memory devices.

When a QRAM is used in this mode, the QRAM has two stages, (1) classical data
loading, (2) coherent address. In stage (1), the classical data ($\mathbf{x}$)
is loaded into the quantum memory (or even a classical memory module, as long
as classically controlled gates on the bus qubits are available Hann  _et al._
(2019)), which can be represented as

$\displaystyle
W(\mathbf{x})\ket{0}^{(\text{QM})}\rightarrow\ket{\mathbf{x}}^{(\text{QM})}.$
(23)

In stage (2), the bus qubit is initialized at $\ket{0}$ state, and the address
qubits are prepared. The bus and the address qubits are sent into the QRAM to
coherently address the memory. This can be expressed by

$\displaystyle
R(\ket{addr},\ket{0})\ket{\mathbf{x}}^{(\text{QM})}\rightarrow\ket{addr\,\&\,\mathbf{x}}\otimes\ket{\mathbf{x}}^{(\text{QM})},$
(24)

where the output state $\ket{addr\,\&\,\mathbf{x}}$ respects a state of both
the address and the bus qubits, which respect the quantum oracle in Eq. (12).

####  III.3.6 Device requirements

As there is no experimental demonstration of a working QRAM, how to build a
QRAM, even how to layout different components in a QRAM, is still an open
question and requires a lot of effort in material fabrication, control
techniques, and device optimizations. Instead of analyzing the performance of
the available quantum memory devices, we highlight a few desired properties of
QRAMs to better fulfill the usage of QRAMs, i.e., coherently addressing the
classical data.

When the QRAM is used to encode classical data into quantum states, which are
then used in quantum algorithms to be processed, e.g., in linear algebra
operations and quantum machine learning algorithms, the classical data can be
enormous. In order to address the classical data efficiently, the
corresponding quantum routing systems also need to be massive. This requires
QRAM to have a high integration of required quantum switches and the memory
qubits inside the quantum routing module. Secondly, in order to achieve the
quantum advantage of the algorithms using the quantum oracle, it is necessary
to have a fast oracle call. Therefore, each call of the QRAM needs to be fast
such that the quantum algorithms can still outperform their classical
counterparts.

In addition, depending on the specific usage of QRAM in different quantum
algorithms, the latency requirement of the QRAM can be different. For example,
if the QRAM is to encode classical data into a quantum state, which is then
used to be processed with a deep quantum circuit, as long as the QRAM call is
short enough compared to the circuit implementation time, the latency of the
QRAM is acceptable. On the other side, if the quantum circuit is short, the
QRAM calls need to have a small latency.

##  IV Quantum Memory Functional Units in Quantum Processing Unit architecture

Figure 7: The proposed architecture of a quantum processing unit (QPU). We
envision a QPU should include a quantum function unit (QFU), quantum memory
(QM), quantum bus (Q-Bus) for communication with the QPU, quantum input-output
interface (QIO) unit for an interface with classical data, and quantum network
interconnect component (QNIC) for quantum data and quantum communication. A
classical control unit is also necessary to control different quantum modules
in the QPU. The main memory components are colored blue.

Nowadays, due to the rapid development of new fabrication technology, quantum
manipulation, and quantum error correction and mitigation techniques, the
field of quantum computing and quantum storage has been progressing rapidly
and maturing. However, current quantum computing research, especially in the
realm of designing and fabricating quantum computing devices, primarily
focuses on integrating more quantum registers to demonstrate their
performance. Simultaneously, due to limitations imposed by physical and
practical conditions, it is extremely challenging to place millions or even
billions of quantum registers within the same device and maintain high
connectivity. On the other hand, as we have seen from our previous sections,
quantum memory techniques are increasingly mature. Therefore, it becomes
possible to consider how to utilize quantum memory as an essential component
within future quantum processing units (QPU).

Drawing inspiration from the architecture of classical computers, in Fig. 7,
we show the main components that can be contained in the future design of the
QPU architecture. A detailed design of QPU architecture is beyond the scope of
our paper, and hence we show our envision of the future QPUs without
discussing the detailed designs of each functioning unit. We believe a future
QPU will include the following functional units, (1) a quantum functional unit
(QFU) for implementing quantum gate operations, (2) a quantum memory unit (QM)
that can contain a quantum cache (Q-Cache) and a larger main quantum memory,
(3) quantum bus (Q-Bus) for quantum communication within a QPU, (4) quantum
input-output interface (QIO) for classical data loading, and (5) quantum
network interface components (QNIC) for quantum data loading and
communication.

QFU is the central unit to implement quantum algorithms. It can contain a
small number of quantum registers that support universal gate sets. When the
quantum algorithm is performed, the quantum registers in QFUs implement the
quantum gates required by the quantum algorithm. Limited by the size of the
QFU, quantum memory is needed. The main QM and Q-Cache are two different
memory modules that can store the quantum information for later usage in the
QFUs, which we will discuss in Sec. IV.1 and Sec. IV.2, respectively. As RW
operations on the quantum memory devices require bus qubits, the Q-Bus is for
quantum communication between different modules inside the QPU.

The QIO and the QNIC modules are responsible for communication between the QPU
and the other quantum and classical devices. QIO module provides classical
interfaces with other devices. Specifically, when classical data is prepared
and needs to be processed by the quantum computer, the QIO unit is responsible
for encoding the classical data into quantum states. Especially, we focus on
the memory device in the QIO unit, where efficiently loading classical data
into the QPU can leverage QRAMs as we discussed in Sec. III.3. The
functionality of QRAMs inside the QPU architecture will be further discussed
in Sec. IV.4.

In the QNIC components, we consider including a quantum communication
interface with other quantum devices. For example, it can enable quantum
communication with other quantum sensors, which can generate quantum data for
the QPU to process. It can also enable coupling with other QPUs for
distributed quantum computing tasks Ang  _et al._ (2022); Wu  _et al._ (2022)
and communication with a quantum internet for long-range quantum communication
with other quantum devices. One of the key components to ensure reliable
quantum communication with other devices is quantum buffers, which we will
discuss further in Sec. IV.3.

In the rest of the section, we direct our attention to the modules consisting
of quantum memory. Particularly, we focus on the quantum memory units colored
blue in Fig. 7, which include main quantum memory, quantum cache (Q-Cache),
quantum buffer (Q-Buffer), and QRAM in the QIO. We deliberate on their
utilities at the architecture level and their design requirements.
Specifically, we evaluate their memory qubit coherence (M.Q.C), addressing
coherence (A.C), qubit integration (Q.I.), read and write parallelization (RW
Para.), and operation speed (O.S.). We will discuss each of the quantum memory
functional units in the following sub-sections. In Table 6, we give a brief
comparison between different types of quantum memory units in terms of these
metrics.

Table 6:  Comparison between the requirements of different quantum memory modules in the architecture of QPUs. Here we focus on: memory qubit coherence (M.Q.C), addressing coherence (A.C), large qubit integration (Q.I.), read and write parallelization (RW Para.), and read and write operation speed (O.S.). We rate them in a total score 3: 1 means low requirement of the property, low capability to achieve. Seq. stands for sequential operations. |  M.Q.C |  A.C. |  Q.I. |  RW Para. |  O.S.
---|---|---|---|---|---
Main QM |  3 |  Classical |  3 |  3 |  1
Q-Cache |  2 |  Classical |  1 |  2 |  3
Q Buffer |  2 to 3 |  Classical |  1 |  1 |  1 to 2
QRAM |  Classical |  3 |  3 |  Seq. |  2 to 3

###  IV.1 Quantum Memory Unit

Analog to the role of memory in the current classical computing architecture,
quantum memory is one of the central components in our architecture design. In
order to expand the capability of QFU, it is necessary to use quantum memory
to store quantum information carried by the qubits that are not immediately
involved in the quantum operations.

Similar to classical memory, a quantum memory unit should support the
operation of reading from and writing to a QMC with a given address.
Specifically, coherently addressing multiple QMCs is not necessary. As we
discussed in Sec. III.2, RAQM can be used to realize a quantum memory unit in
the QPU architecture.

The quantum memory unit should also satisfy a few design requirements. The
quantum memory unit is similar to a classical computer’s main memory. As the
quantum state may need to be stored in the quantum memory for an extended
period of time, the quantum memory should have a low error rate to ensure the
information is still authentic. Therefore, (1) the quantum memory unit should
have a much longer storage time in terms of computing operation time, i.e.,
the device’s $\beta$ metric should be large. This is the most important
requirement to fulfill. (2) The quantum memory unit should have a large number
of QMCs integrated into the device while maintaining low cross-talk errors, as
the quantum memory unit needs to store all quantum information required in a
quantum algorithm. (3) The RW operation time of the quantum memory unit should
be small, ideally. Furthermore, (4) enabling addressing QMCs in parallel would
be more beneficial for increasing the bandwidth of the RW operations, and
simultaneously ensuring its addressability ($\gamma<1$).

However, due to the long coherence time of the QMCs from the requirement (1),
if in the NISQ era where the QMCs are not error corrected, the QMC with longer
coherence time means lower coupling to the environment, which usually causes a
long reading and writing time. In the FTQC era where QEC is used, suppressing
the error rate to increase the storage time necessitates a large code distance
and more physical qubits, which can slow down the logical SWAP gate
operations. In addition, a large integration of QMCs can occupy a relatively
large physical space, which makes the quantum memory unit spatially separated
from the QFU. All these factors lead to extending the RW time of the quantum
memory. However, to ensure the first two requirements, the requirement of
latency ($\beta$) can be slightly released.

Figure 8: The interface of a quantum memory unit or a Q-Cache. The classical
address information ($addr$) and quantum registers (labeled as
$b_{\text{in}}$) interact with the quantum memory, while the quantum registers
$b$ with the QMC with address $addr$ state is outputted, which is labeled as
$b_{\text{out}}$.

With the design requirement of the quantum memory unit, we then consider how
to utilize this device in the upper stacks. The reading and writing functions
are similar to the RAQM discussed in Sec. III.2. Specifically, the quantum
memory unit should have a classical input to take in the address information.
The input can take another quantum register, which carries the quantum
information in the writing process while loading the quantum information from
the memory in the reading process. We stress that unlike classical memory,
where the reading and writing processes are unidirectional, i.e., the
information is copied from and to the quantum memory, accessing the QMCs is
bidirectional in the quantum case. Therefore, there is no hard distinction
between the reading and writing processes, and both reading and writing
processes can be represented by the model in Fig. 8. However, in order to
better organize the programming and highlight where the nontrivial quantum
state is stored, it would be useful to have both read and write functions
enabled, although the underlying physical operations are essentially the same.

###  IV.2 Quantum Cache (Q-Cache)

Quantum cache (Q-Cache) is another quantum memory functional unit inside the
architecture of QPUs. According to Sec. IV.1, the quantum memory unit can have
relatively long latency. In order to speed up quantum computation, analog to
classical computing systems, a quantum cache (Q-Cache) can be utilized.
Specifically, if the quantum information carried by a certain qubit is
relatively frequently visited, instead of swapping it from and to the main
quantum memory every time the operation is done, the information can be stored
inside a Q-Cache, which can provide faster RW operations.

Therefore, for the purpose of speeding up quantum computation, Q-Cache has a
few design requirements. (1), the latency of the RW operations on a Q-Cache
needs to be small, which is required by the operation speed of a Q-Cache. In
order to achieve the low latency of the RW operations, a Q-Cache can choose
bare qubits with shorter coherence time in the NISQ device, while choosing QEC
codes with smaller code distance in the fault-tolerant device, compared to the
ones used in the quantum memory unit. Therefore, (2) a Q-Cache may have
moderately long storage times. We claim that the Q-Cache should satisfy

$\displaystyle\alpha_{\text{ex, QC}}/\beta_{\text{QM}}=\frac{T_{\text{storage,
QC}}\eta_{\text{QC}}}{T_{\text{RW,
QM}}/\eta_{\text{QM}}}>r_{\text{threshold}}\sim 2,$ (25)

where ‘QC’ labels the properties of Q-Cache, and ‘QM’ stands for the quantum
memory unit. This means the storage time of the Q-Cache should be at least
longer than storing and retrieving the quantum information from the quantum
memory unit. If the quantum information carried by a qubit idles for a
duration longer than twice the RW time of the main quantum memory, it would be
better to be stored in the main memory, as it can experience less error. In
the actual design of a QPU, the threshold value $r_{\text{threshold}}$ can be
further optimized. In addition, to further improve the latency of the Q-Cache,
the Q-Cache is expected to be located on the same chip of QFUs or nearby. Due
to the spatial limitation, (3) the number of QMCs can be small. On the other
hand, to increase the communication bandwidth, (4) operating RW of Q-Cache in
parallel through multiple banks following the classical memory design is
desired.

As the Q-Cache can also be implemented by RAQMs, the interface of a Q-Cache is
similar to the quantum memory unit, which is discussed in Sec. IV.1. The
interface of a Q-Cache can also be represented by Fig. 8.

###  IV.3 Quantum Buffer

The quantum buffer is another quantum memory functional unit inside the QPU
architecture. In the process of implementing a quantum algorithm or quantum
operation, some resource states are probabilistically generated. Therefore, it
is necessary to include quantum buffers to store these states and retrieve
them when they are requested. The quantum buffer can be widely used in quantum
communication components, especially in the QNIC shown in Fig. 7. There are
two possible applications, interfacing with quantum sensors and with quantum
networks for quantum communication. Specifically, quantum sensors can prepare
quantum states that encode the sensing information. The quantum states can be
imported into the QPU for further processing. However, the quantum sensing
process can be slow compared to the quantum computing cycles in the QFU, and
various quantum sensors operate at different speeds, which necessitate quantum
buffers to receive the quantum state and make them ready to be processed by
the QFU.

Another application is to use a quantum buffer to buffer information from a
quantum internet. Long-range quantum communication usually relies on
entanglement generation and state teleportation Bennett  _et al._ (1993);
Briegel  _et al._ (1998). However, the remote entanglement generation,
involving state purification and photon measurements, is probabilistic in
nature. Therefore, when a qubit is successfully entangled with the remote
quantum system, it can be stored in the quantum buffer for later communication
use Wu  _et al._ (2022). Notably, the usage of quantum buffers is not limited
to QNIC units. Indeed, whenever there is a need to store the probabilistically
generated resource states, a quantum buffer can be utilized. One of the
examples would be in the magic state distillation process, where the quantum
buffer can store the generated high-fidelity magic states for later use in
implementing surface code Toffoli gates.

To fulfill these implementations, quantum buffers are required to work between
two quantum systems. Without losing the generality, one of the quantum systems
can be viewed as an information saver, which generates the quantum information
and saves it to the quantum buffer, while the other one is the information
loader, which loads the quantum information depending on its processing need.
Therefore, there are two characteristic time periods, one is the time for
generating the quantum state ($T_{\text{g}}$), which is from the information
saver, while the other one is the duration of state consumption
($T_{\text{c}}$), which is from the information loader. Therefore, it requires
(1) the external storage ratio of the quantum buffer to be long compared to
the slower process, i.e.,

$\displaystyle\alpha_{\text{ex,
QB}}=\frac{T_{\text{storage}}\eta}{\max(T_{\text{g}},T_{\text{c}})},$ (26)

is a metric for a quantum buffer design and $\alpha_{\text{ex, QB}}\gg 1$
ideally. On the other hand, (2) the latency of the quantum buffer needs to be
small compared to the faster process, i.e., ideally,

$\displaystyle\beta_{\text{QB}}=\frac{T_{\text{RW}}/\eta}{\min(T_{\text{g}},T_{\text{c}})}\ll
1.$ (27)

The integration of the quantum buffer may not be large. It is unnecessary to
buffer a huge number of quantum states, as the oldest copies can be discarded.
Ideally, wasting quantum states is not efficient, and hence (4) we require
$N\sim T_{\text{g}}/T_{\text{c}}$. The parallelization feature may not be
required and depends on the requirements on the consumption side. If multiple
states can be consumed simultaneously, parallelizing the reading process is
needed. However, simultaneously reading from and writing to the quantum buffer
should not be allowed.

Since the metrics of quantum buffers include generation and consumption times,
quantum buffers can be designed for quantum tasks differently. Notably, even
for the same task, both the generation time $T_{\text{g}}$ and the consumption
time $T_{\text{c}}$ can depend on the algorithms and protocols. For example,
in the task of quantum communication between the current QPU and a remote
quantum device, the entangled state purification protocols can have different
yield and time Ang  _et al._ (2022), which can affect $T_{\text{g}}$.
Therefore, given quantum buffer properties, optimizing the algorithms on both
generation and consumption sides to satisfy the above requirement, as well as
co-design the quantum hardware and algorithms, can be interesting directions
to proceed Wu  _et al._ (2022).

(a) Q-Buffer Interface

(b) Reading (c) Writing

Figure 9: The interfaces of the quantum buffer are shown in (a), while the
reading and write processes are shown in (b) and (c), respectively.

The quantum buffer is slightly different from the memory. In most cases, a
single quantum buffer stores a certain type of quantum state, while each state
may have different state fidelity. The state can be constantly generated with
different time intervals and can be requested from other quantum units in the
other quantum function units. The random access feature is not necessary for a
quantum buffer. Instead, a quantum buffer can be constructed by an array of
QMCs with, for example, the first-in-first-out (FIFO) policy. In Fig. 9a, we
show the interface of a quantum buffer. The quantum buffer can take in a
single bit of classical data for the instruction of reading or writing
operations. It also takes a quantum register to interact with the quantum
buffer to store or retrieve the quantum information. The quantum buffer can
return the bus register along with a single bit of classical data to show
whether the query operation is successful. The state of the returned bus
register can depend on whether the query is successful or not.

In Fig. 9b, we show the reading process, which is demonstrated by the RW bit
being set to be $0$. In the reading process, the state stored inside the
quantum buffer is requested from other QPU modules. In the reading process,
the bus register is set to be $\ket{0}$. If there are quantum states stored in
the quantum buffer, which is available to be retrieved, the reading query is
successful with a returning value $1$ in the output ‘S/F’ bit. The bus
register swaps the stored state out, labeled as $\ket{S}$. On the other hand,
if there are no available quantum states inside the quantum buffer, the
reading query fails, with S/F returning $0$. This is similar to an underflow
situation in a classical buffer. If this happens, the bus register is then
returned with state $\ket{0}$ without interacting with the buffer QMCs.

The writing process of a quantum buffer is shown in Fig. 9c. In the writing
mode, a quantum state carried by the bus register must be stored in the
quantum buffer. When the quantum buffer is not full, i.e., not all quantum
memory registers are used in the quantum buffer, the control unit of the
quantum buffer will locate the unused memory registers, and the bus register
state can be successfully stored by swapping its state to this memory
register. The S/F bit will output $0$ while the bus register is set back to
state $\ket{0}$, which corresponds to the original state of the memory
register. On the other hand, if the quantum buffer is already full, to be
consistent with the reading process, the quantum buffer will return $1$ in the
S/F output bit, while holding the quantum bus register not to interact with
the memory bit. This corresponds to a classical overflow situation.

We note that although in our design shown in Fig. 9, we consider sequentially
reading and writing quantum states of a single bus register, reading multiple
registers in the same query of quantum buffer should be supported. In this
case, the quantum buffer can take multiple bus registers as input, depending
on its specific implementation. When the underflow or overflow situation
happens, the quantum buffer should hold the reading and writing queries until
all the states or storage quantum memory registers are ready.

###  IV.4 QRAM in QIO

As mentioned in Sec. III.3, QRAMs work distinctly from the other quantum
memory units. In our architecture design, QRAMs can be used in the QIO unit,
where the classical data is loaded to the QRAM, while according to the
algorithms, a set of address qubits are prepared to be sent into the QRAM as
the address information, which coherently addressing the classical data and
preparing a bus-address entangled states as the output. As we have discussed
in Sec. III.3.6, in order to efficiently obtain this goal, a few design
requirements have been discussed.

In the architecture stack, the QRAM can be built in as a general-purpose
device for encoding classical data into quantum states and quantum compression
of classical data. In addition, QRAMs can also be built within special
function units that can perform algorithms with quantum speedups. For example,
a QRAM can be built into the special function units for the Grover search
algorithm Grover (1996), which can speed up the database search. The main
design requirement of QRAMs lies in having fast and reliable queries. In the
framework of quantum memory devices, it is equivalent to small reading
latency, where

$\displaystyle\beta_{\text{QRAM}}=\frac{T_{\text{R}}/\eta}{T_{\text{Circ}}}\ll
1,$ (28)

where $T_{\text{Circ}}$ is the time for implementing the quantum operations
between two QRAM queries. On the other hand, the QRAM is designed to interact
with classical data, which is usually in a large size. Therefore, to have a
compact integration is greatly important.

(a) QRAM Interface

(b) QRAM data loading (c) QRAM query

Figure 10: The model of a QRAM module. The interface of a QRAM is shown in
(a). The QRAM classical data loading process is shown in (b), while the QRAM
query process is in (c).

When a QRAM is used as an interface between classical data and quantum
devices, in Fig. 10, we show an abstracted model for a QRAM. The interface of
a QRAM is shown in Fig. 10a. A QRAM can have two modes: one mode is to load
classical data into the memory, while the other mode is to coherently address
the classical data. A QRAM can take a single bit of classical data as the mode
specification (labeled as ‘R/W’). In addition, a QRAM can also take classical
data as an input, or take a set of quantum address registers and a bus
register as inputs.

In the classical data loading mode, where the R/W classical register is set to
$1$, a classical data $\vec{x}$ is sent to the QRAM. The QRAM loads the
classical data $\hat{x}$ to the QRAM’s memory components. In order to use the
QRAM to quantum encode the classical data, or as quantum oracles shown in Eq.
(12) in quantum algorithms, the R/W classical register is set to $0$, which
shows the QRAM is in the quantum query mode. In this mode, the QRAM takes a
set of quantum registers, which include address and bus registers. After a
QRAM read query, the address and bus registers are returned. Their states are
now changed according to the classical data $\hat{x}$ that has been stored
inside the memory based on Eq. (12).

##  V Quantum memory programming model

With the available quantum memory units, utilize the quantum memory modules in
the future quantum programs should be available, which enables quantum-memory-
aware program design. Currently, quantum programs at the assembly level are
described as a quantum circuit using OpenQASM Cross  _et al._ (2017, 2022).
QASM focuses on representing quantum circuits, including initialization and
resetting quantum registers, applying quantum gates, performing measurements,
and classically controlled gate operations. However, present quantum programs
such as QASM are all centered around quantum registers, overlooking quantum
memory. Therefore, we investigate possible changes in the current gate-based
quantum programming languages and APIs to incorporate quantum memory.

In order to adapt quantum assembly languages, such as QASM, for quantum memory
utilities, we propose to introduce four quantum-memory-related primitives:

  * (1)

mem $size$. It declares the quantum memory requirements used in the
corresponding program, specifying the total number of quantum memory cells
($size$). It should be used at the beginning of an assembly code. When the
quantum program is executed on a quantum device, the required memory size will
be passed to the quantum hardware controller to check if the hardware can
support the required memory size.

  * (2)

ld $q$ = [$addr$]. It loads the quantum state stored in the quantum memory to
the corresponding quantum register or qubits. When the argument $q$ is a
qubit, the state of the QMC with address $addr$ will be loaded to the qubit,
while the QMC is reset. If the argument $q$ is a quantum register with $n$
qubits, the states of QMCs starting from address $addr$ are loaded to the
corresponding register, while these QMCs are reset.

  * (3)

st [$addr$] = $q$. It stores the quantum state carried by the quantum register
or qubit to the quantum memory cells with address addr. Similar to the
primitive ld, the argument $q$ can be either a qubit or a quantum register.

  * (4)

mreset $addr$. It resets the quantum memory cell with address $addr$. If the
$addr$ is missing, it will reset all the quantum memory cells declared.

With the four basic primitives to operate on quantum memory, more complex
quantum memory management strategies can be implemented. The interfaces of the
quantum buffer and quantum cache can be implemented as APIs, contained in
quantum libraries for operating the specific quantum devices.

On the other hand, QRAM is a different type of quantum function unit distinct
from quantum memory. In order to operate QRAMs in the assembly language, we
proposed to include the following three primitives.

  * (1)

qram name[$addr\\_len$,$word\\_len$]. It declares the QRAM needed in the
program, and gives it a name name. The declaration requires two arguments:
$addr\\_len$ specifies the number of address qubits needed to address all the
memory registers, and $word\\_len$ specifies the word length of each memory
register, i.e., the number of memory cells consisting of a single register.

  * (2)

qinit name [$x$]. It writes classical data into the QRAM. The classical data
needs to be coherently addressed via the QRAM. The argument $x$ should be a
classical array of registers. The size should be compatible with the QRAM
declared in the code.

  * (3)

qld name($q\\_b$)[$q\\_addr$]. It performs coherent addressing of the content
in the QRAM with the quantum address that is represented by the state of the
address qubits $q\\_addr$. The quantum register $q\\_b$ is also given as the
bus qubit.

These three primitives provide the essential functionalities of QRAM
operations, which can be utilized to construct more sophisticated quantum
programs and libraries, including realizing Grover search, quantum data
lookup, etc. In Table 7, we summarize the primitives we proposed here.

Table 7:  A summary of quantum memory primitives to be included in QASM. Device |  Code Format | Meaning
---|---|---
RAQM |  mem $size$ | declear memory usage
|  ld $q$ = [$addr$] | load quantum memory to q-register
|  st [$addr$] = $q$ | store q-register to memory
|  mreset $addr$ | reset the memory
RAQM |  qram name[$addr\\_len,word\\_len$] | declear the usage of a QRAM
|  qinit name [$x$] | load classical data to QRAM
|  qld name($q\\_b$)[$q\\_addr$] | coherently address the QRAM

In addition to the primitives to initiate and operate the QRAM, as a QRAM can
load classical data into its memory media, we feel it is necessary to include
array operations on classical data, which is recently supported by OpenQASM
3.0 Cross  _et al._ (2022). With the quantum memory enabled, array operations
on qubits and quantum registers are also necessary. Therefore, we propose to
include slice operations on quantum registers and quantum arrays. For example,

⬇

1qubit[4] q;

2mem 4;

3st [0] = q[1:];

saves the qubit register q[1], q[2], and q[3] to memory address $0$, $1$, and
$2$.

Figure 11: The amplitude encoding and QFT on the encoded state. The red arrows
show the loading quantum states from and saving to the quantum memory. Listing
1: Code example for Fig. 11

⬇

1OPENQASM 3;

2gate cr(n) c, t {

3 angle $\theta$ = 2$\pi$/power(2,n)

4 ctrl @ U(0, 0, $\theta$) c, t;

5}

6

7qubit[4] q;

8qubit[1] b;

9qubit[1] aux;

10bit[2] caux;

11

12// an example vector with binary values

13bit[16] vec = [0,1,1,0,0,1,1,1,0,0,1,1,0,1,0]

14

15mem 4; // specify the memory requirement

16qram qr[4,1]; // specify the QRAM requirement

17qinit qr [vec]; // load the data into QRAM

18

19h q;

20ldqram qr q b; // use QRAM

21// save q into memory waiting for the measurement on aux qubit

22st [0] = q;

23

24cx b aux;

25measure aux -> caux[0];

26

27if (caux[0]==1){

28 ld q = [0];

29 qld qr(b)[q]; // use QRAM

30

31 measure b -> caux[1];

32 st [0] = q[1:];

33

34 int j = 0;

35 for i in [0:2]{

36 h q[i];

37 j = i+1;

38 while(j<4){

39 if(i==0) ld q[j] = [i];

40 cr(j-i+1) q[j], q[i];

41 j = j+1;

42 }

43 st [i] = q[i];

44 }

45

46 h q[3]

47 st [3] = q[3];

48}

To describe the functionalities of the quantum memory and QRAM primitives, we
include an example of the modified QASM code for amplitude encoding the
classical data and then performing the quantum Fourier transform. The input
data is a classical binary vector vec. The quantum circuit is shown in Fig.
11. We consider using probabilistic amplitude encoding process Hann (2021).
The encoding succeeds when the measurement on the auxiliary qubit is
$\ket{1}$. The quantum Fourier transform (QFT) sequence uses control-$R_{k}$
gates, where

$R_{k}=\left[\begin{array}[]{cc}1&0\\\ 0&e^{i2\pi/2^{k}}\end{array}\right],$
(29)

where $k$ is an integer Nielsen and Chuang (2010). The outcome state is saved
in the quantum memory with addresses $0$ to $4$.

In addition to the primitives we proposed to include, we stress that including
quantum memory devices can also impact the design of quantum transpilers used
at higher programming levels. For example, in the current stage, a quantum
algorithm is written in terms of its quantum circuit, which needs to be
transpiled into a QASM code based on standardized gate operations by gate
decomposition and simplification. The OpenQASM code can then be converted into
instructions (like the control pulses to control the physical qubits) that can
directly interact with the quantum hardware. When quantum memory is available
in the architecture of quantum computing systems, a memory-aware quantum
transpiler is needed to manage the quantum memory usage and optimize the
performance of quantum algorithms.

Higher-level quantum programming languages are also expected to coordinate
with the presence of quantum memory devices. For example, quantum software
development toolkits, such as Qiskit Qiskit contributors (2023), should
include libraries that support RAQM and QRAM operations, quantum memory
management, as well as using QRAMs to speed up quantum algorithms, such as the
Grover search, quantum Fourier Transform, etc. With the memory-device-aware
transpilers available, these libraries can be transpiled to QASM codes that
can operate quantum memory units to leverage their functionalities. On top of
that, a more sophisticated quantum algebra library taking advantage of
quantum-memory-aware optimization and QRAMs can be fully built for general-
purpose quantum computers.

Furthermore, as addresses are introduced to operate on the quantum memory,
arithmetic calculations on the quantum memory addresses should also be
supported, while more sophisticated quantum data structures can be introduced
into the quantum programming languages, e.g., quantum array. In addition, it
is necessary to include pointers for quantum memory to enable dynamical memory
allocation and efficient data access.

##  VI Quantum Memory Applications

Quantum memory is a crucial component in quantum computing systems. We explore
the use of quantum memory in the architecture of a single quantum processing
unit (QPU), but its potential extends far beyond that. Quantum memory can find
various applications in quantum computing and quantum information processing
tasks.

In our discussion, quantum memory is usually used for saving quantum
information. One example is to use the QMCs in quantum memory modules to store
a multi-qubit quantum state. With a large integration of QMCs, a quantum
database can be built. A huge number of QMCs can be utilized to store a large
quantum state or numerous copies of intermediate-size quantum states. However,
a quantum database that only consists of quantum memory for quantum
information storage is not completely feasible. One limitation is from the
quantum no-cloning theorem. The quantum state stored in the quantum database
cannot be copied. When the quantum information is requested and measured, the
information is destroyed and no longer stored in the database. Conversely, in
a classical database, when the information is requested, it will not destroy
the information stored in the database. Therefore, we envision that a pure
quantum database is not suitable for permanent information storage. Instead, a
quantum data center Liu  _et al._ (2022) in the future can consist of both
classical database and quantum memory, such that the everlasting information
can be stored by a classical method in the classical memory, while the quantum
memory can be used to store quantum information, either encoded from the
classical information for further processing, or received from other quantum
parties to be processed with the classical information.

However, encoding classical information into quantum states can compress the
data for quantum processing. For example, using amplitude encoding, a
classical binary data string with length $2^{N}$ can be encoded into a quantum
state of $N$ qubits. Fortunately, this encoding can be obtained efficiently
with QRAMs Hann (2021). After encoding the classical data into quantum states,
it can then be efficiently utilized in quantum algorithms, such as quantum
machine learning Biamonte  _et al._ (2017); Cerezo  _et al._ (2022).
Furthermore, the quantum encoding of the classical information can also be
more efficient in performing remote information processing. For example, if a
client only needs to compare a large data with size $2^{N}$ on his/her end to
a chunk of data stored in the data center, classically, the data needs to be
sent for comparison. With the help of the swap test Buhrman  _et al._ (2001),
encoding the classical data into quantum states and sending a copy of the
encoded quantum states for comparison only requires sending $N$ qubits. If the
loss of the quantum network and the imperfections can be well controlled, for
a given required accuracy, only polynomially many copies of the quantum state
are needed.

Furthermore, as pointed out in Ref. Liu  _et al._ (2022), with the help of
quantum communication protocols, a quantum data center can also help improve
communication privacy. In Ref. Liu  _et al._ (2022), Liu et al. proposed a
protocol for multi-party private quantum communications, which ensures the
privacy of the communication between multiple untrusted parties.

Other than quantum data centers, quantum buffers can be used in quantum
networks and quantum internet. A quantum buffer can be combined with quantum
repeaters in the communication nodes, while the entangled states can be stored
for quantum communication and purification. The initial application of quantum
buffer in quantum networks has been discussed in Ref. Wu  _et al._ (2022).

Therefore, we believe that a quantum data center combined with quantum
internet and quantum communication can be a great application example for
quantum memory and QRAM devices. With these applications, we believe quantum
memory research will be increasingly attractive to research in related fields.

##  VII Conclusion and outlook

In the rapidly evolving landscape of quantum computing and quantum information
science, building efficient and reliable quantum memory devices stands as a
cornerstone for unlocking the full potential of quantum technologies. In this
paper, we survey different aspects of quantum memory, from the physical
systems for quantum memory materials to quantum memory devices, e.g., the RAQM
and QRAM. We envision that the rapid development of quantum computing systems
and the improvement of various physical systems enable more complicated and
high-performance memory devices, which provide the opportunity to investigate
the higher-stack design of quantum computing systems, especially focusing on
the pivotal role of quantum memory in quantum processing units. We then
discuss the quantum memory architecture design and the corresponding
programming model, as well as their possible applications. Specifically, we
point out the possible memory modules inside the QPU architecture, and define
their software-oriented functionalities. We hope this article can present not
only the status of the research in the hardware and device stacks of quantum
memory to higher-stack researchers but also a view of quantum memory devices
from the top stacks and show the hardware researchers the possible directions
to improve their devices. We believe that it is time to start considering
quantum memory and its role in quantum computing systems. We hope that our
article helps to motivate people to enter this exciting field.

##  VIII 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.

## References

  * Shor (1999) Peter W Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM review 41, 303–332 (1999).
  * Gisin  _et al._ (2002a) Nicolas Gisin, Grégoire Ribordy, Wolfgang Tittel, and Hugo Zbinden, “Quantum cryptography,” Reviews of modern physics 74, 145 (2002a).
  * Farhi  _et al._ (2014) Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, “A quantum approximate optimization algorithm,” arXiv preprint arXiv:1411.4028 (2014).
  * Han and Kim (2002) Kuk-Hyun Han and Jong-Hwan Kim, “Quantum-inspired evolutionary algorithm for a class of combinatorial optimization,” IEEE transactions on evolutionary computation 6, 580–593 (2002).
  * Georgescu  _et al._ (2014) Iulia M Georgescu, Sahel Ashhab, and Franco Nori, “Quantum simulation,” Reviews of Modern Physics 86, 153 (2014).
  * Kandala  _et al._ (2017) Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M Chow, and Jay M Gambetta, “Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets,” Nature 549, 242–246 (2017).
  * Rebentrost  _et al._ (2018) Patrick Rebentrost, Brajesh Gupt, and Thomas R Bromley, “Quantum computational finance: Monte carlo pricing of financial derivatives,” Physical Review A 98, 022321 (2018).
  * Woerner and Egger (2019) Stefan Woerner and Daniel J Egger, “Quantum risk analysis,” npj Quantum Information 5, 1–8 (2019).
  * 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, “Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682–686 (2015).
  * Ren  _et al._ (2017) Ji-Gang Ren, Ping Xu, Hai-Lin Yong, Liang Zhang, Sheng-Kai Liao, Juan Yin, Wei-Yue Liu, Wen-Qi Cai, Meng Yang, Li Li, Kui-Xing Yang, Xuan Han, Yong-Qiang Yao, Ji Li, Hai-Yan Wu, Song Wan, Lei Liu, Ding-Quan Liu, Yao-Wu Kuang, Zhi-Ping He, Peng Shang, Cheng Guo, Ru-Hua Zheng, Kai Tian, Zhen-Cai Zhu, Nai-Le Liu, Chao-Yang Lu, Rong Shu, Yu-Ao Chen, Cheng-Zhi Peng, Jian-Yu Wang, and Jian-Wei Pan, “Ground-to-satellite quantum teleportation,” Nature 549, 70–73 (2017).
  * Chen  _et al._ (2021a) Jiu-Peng Chen, Chi Zhang, Yang Liu, Cong Jiang, Wei-Jun Zhang, Zhi-Yong Han, Shi-Zhao Ma, Xiao-Long Hu, Yu-Huai Li, Hui Liu, Fei Zhou, Hai-Feng Jiang, Teng-Yun Chen, Hao Li, Li-Xing You, Zhen Wang, Xiang-Bin Wang, Qiang Zhang, and Jian-Wei Pan, “Twin-field quantum key distribution over a 511 km optical fibre linking two distant metropolitan areas,” Nature Photonics 15, 570–575 (2021a).
  * Liu  _et al._ (2023a) Yang Liu, Wei-Jun Zhang, Cong Jiang, Jiu-Peng Chen, Chi Zhang, Wen-Xin Pan, Di Ma, Hao Dong, Jia-Min Xiong, Cheng-Jun Zhang, Hao Li, Rui-Chun Wang, Jun Wu, Teng-Yun Chen, Lixing You, Xiang-Bin Wang, Qiang Zhang, and Jian-Wei Pan, “Experimental twin-field quantum key distribution over 1000 km fiber distance,” Phys. Rev. Lett. 130, 210801 (2023a).
  * Bennett and Brassard (1984) C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in  _Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing_ (India, 1984) p. 175.
  * Gisin  _et al._ (2002b) Nicolas Gisin, Grégoire Ribordy, Wolfgang Tittel, and Hugo Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002b).
  * Xu  _et al._ (2020) Feihu Xu, Xiongfeng Ma, Qiang Zhang, Hoi-Kwong Lo, and Jian-Wei Pan, “Secure quantum key distribution with realistic devices,” Rev. Mod. Phys. 92, 025002 (2020).
  * Portmann and Renner (2022) Christopher Portmann and Renato Renner, “Security in quantum cryptography,” Rev. Mod. Phys. 94, 025008 (2022).
  * Kimble (2008) H. J. Kimble, “The quantum internet,” Nature 453, 1023–1030 (2008).
  * Cacciapuoti  _et al._ (2020) Angela Sara Cacciapuoti, Marcello Caleffi, Francesco Tafuri, Francesco Saverio Cataliotti, Stefano Gherardini, and Giuseppe Bianchi, “Quantum internet: Networking challenges in distributed quantum computing,” IEEE Network 34, 137–143 (2020).
  * Wei  _et al._ (2022) Shi-Hai Wei, Bo Jing, Xue-Ying Zhang, Jin-Yu Liao, Chen-Zhi Yuan, Bo-Yu Fan, Chen Lyu, Dian-Li Zhou, You Wang, Guang-Wei Deng, Hai-Zhi Song, Daniel Oblak, Guang-Can Guo, and Qiang Zhou, “Towards real-world quantum networks: A review,” Laser & Photonics Reviews 16, 2100219 (2022).
  * Chow  _et al._ (2021) Jerry Chow, Oliver Dial, and Jay Gambetta, “IBM Quantum breaks the 100‑qubit processor barrier,”  (2021).
  * IBM (2022) “IBM Unveils 400 Qubit-Plus Quantum Processor and Next-Generation IBM Quantum System Two,”  (2022).
  * Arute  _et al._ (2019) Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G. S. L. Brandao, David A. Buell, Brian Burkett, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew P. Harrigan, Michael J. Hartmann, Alan Ho, Markus Hoffmann, Trent Huang, Travis S. Humble, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul V. Klimov, Sergey Knysh, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Mike Lindmark, Erik Lucero, Dmitry Lyakh, Salvatore Mandrà, Jarrod R. McClean, Matthew McEwen, Anthony Megrant, Xiao Mi, Kristel Michielsen, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John C. Platt, Chris Quintana, Eleanor G. Rieffel, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Kevin J. Sung, Matthew D. Trevithick, Amit Vainsencher, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, and John M. Martinis, “Quantum supremacy using a programmable superconducting processor,” Nature 574, 505–510 (2019).
  * Moses  _et al._ (2023) S. A. Moses, C. H. Baldwin, M. S. Allman, R. Ancona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, P. Blanchard, M. Bohn, J. G. Bohnet, N. C. Brown, N. Q. Burdick, W. C. Burton, S. L. Campbell, J. P. Campora III au2, C. Carron, J. Chambers, J. W. Chan, Y. H. Chen, A. Chernoguzov, E. Chertkov, J. Colina, J. P. Curtis, R. Daniel, M. DeCross, D. Deen, C. Delaney, J. M. Dreiling, C. T. Ertsgaard, J. Esposito, B. Estey, M. Fabrikant, C. Figgatt, C. Foltz, M. Foss-Feig, D. Francois, J. P. Gaebler, T. M. Gatterman, C. N. Gilbreth, J. Giles, E. Glynn, A. Hall, A. M. Hankin, A. Hansen, D. Hayes, B. Higashi, I. M. Hoffman, B. Horning, J. J. Hout, R. Jacobs, J. Johansen, L. Jones, J. Karcz, T. Klein, P. Lauria, P. Lee, D. Liefer, C. Lytle, S. T. Lu, D. Lucchetti, A. Malm, M. Matheny, B. Mathewson, K. Mayer, D. B. Miller, M. Mills, B. Neyenhuis, L. Nugent, S. Olson, J. Parks, G. N. Price, Z. Price, M. Pugh, A. Ransford, A. P. Reed, C. Roman, M. Rowe, C. Ryan-Anderson, S. Sanders, J. Sedlacek, P. Shevchuk, P. Siegfried, T. Skripka, B. Spaun, R. T. Sprenkle, R. P. Stutz, M. Swallows, R. I. Tobey, A. Tran, T. Tran, E. Vogt, C. Volin, J. Walker, A. M. Zolot, and J. M. Pino, “A race track trapped-ion quantum processor,” (2023), arXiv:2305.03828 [quant-ph] .
  * IonQ Staff (2023) IonQ Staff, “IonQ Aria: Practical Performance,” https://ionq.com/resources/ionq-aria-practical-performance (2023), Accessed: 2023-06-16.
  * Ebadi  _et al._ (2021) Sepehr Ebadi, Tout T. Wang, Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Omran, Dolev Bluvstein, Rhine Samajdar, Hannes Pichler, Wen Wei Ho, Soonwon Choi, Subir Sachdev, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin, “Quantum phases of matter on a 256-atom programmable quantum simulator,” Nature 595, 227–232 (2021).
  * Wurtz  _et al._ (2023) Jonathan Wurtz, Alexei Bylinskii, Boris Braverman, Jesse Amato-Grill, Sergio H. Cantu, Florian Huber, Alexander Lukin, Fangli Liu, Phillip Weinberg, John Long, Sheng-Tao Wang, Nathan Gemelke, and Alexander Keesling, “Aquila: Quera’s 256-qubit neutral-atom quantum computer,” (2023), arXiv:2306.11727 [quant-ph] .
  * Gambetta (2022) Jay Gambetta, “Expanding the IBM Quantum roadmap to anticipate the future of quantum-centric supercomputing,”  (2022).
  * Kaushal  _et al._ (2020) V. Kaushal, B. Lekitsch, A. Stahl, J. Hilder, D. Pijn, C. Schmiegelow, A. Bermudez, M. Müller, F. Schmidt-Kaler, and U. Poschinger, “Shuttling-based trapped-ion quantum information processing,” AVS Quantum Science 2, 014101 (2020).
  * Krinner  _et al._ (2019) S. Krinner, S. Storz, P. Kurpiers, P. Magnard, J. Heinsoo, R. Keller, J. Lütolf, C. Eichler, and A. Wallraff, “Engineering cryogenic setups for 100-qubit scale superconducting circuit systems,” EPJ Quantum Technology 6, 2 (2019).
  * Gheorghiu and Mosca (2019) Vlad Gheorghiu and Michele Mosca, “Benchmarking the quantum cryptanalysis of symmetric, public-key and hash-based cryptographic schemes,” (2019), arXiv:1902.02332 [quant-ph] .
  * Roetteler  _et al._ (2017) Martin Roetteler, Michael Naehrig, Krysta M. Svore, and Kristin Lauter, “Quantum resource estimates for computing elliptic curve discrete logarithms,” in  _Advances in Cryptology – ASIACRYPT 2017_ , edited by Tsuyoshi Takagi and Thomas Peyrin (Springer International Publishing, Cham, 2017) pp. 241–270.
  * Gidney and Ekerå (2021) Craig Gidney and Martin Ekerå, “How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits,” Quantum 5, 433 (2021).
  * Preskill (2018) John Preskill, “Quantum Computing in the NISQ era and beyond,” Quantum 2, 79 (2018).
  * Monroe and Kim (2013) C. Monroe and J. Kim, “Scaling the ion trap quantum processor,” Science 339, 1164–1169 (2013).
  * Nielsen and Chuang (2010) Michael A. Nielsen and Isaac L. Chuang,  _Quantum Computation and Quantum Information: 10th Anniversary Edition_ (Cambridge University Press, 2010).
  * Calderbank and Shor (1996) A. R. Calderbank and Peter W. Shor, “Good quantum error-correcting codes exist,” Phys. Rev. A 54, 1098–1105 (1996).
  * Fowler  _et al._ (2012) Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland, “Surface codes: Towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
  * Acharya  _et al._ (2023) Rajeev Acharya, Igor Aleiner, Richard Allen, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Juan Atalaya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Joao Basso, Andreas Bengtsson, Sergio Boixo, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Ben Chiaro, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexander L. Crook, Ben Curtin, Dripto M. Debroy, Alexander Del Toro Barba, Sean Demura, Andrew Dunsworth, Daniel Eppens, Catherine Erickson, Lara Faoro, Edward Farhi, Reza Fatemi, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, William Giang, Craig Gidney, Dar Gilboa, Marissa Giustina, Alejandro Grajales Dau, Jonathan A. Gross, Steve Habegger, Michael C. Hamilton, Matthew P. Harrigan, Sean D. Harrington, Oscar Higgott, Jeremy Hilton, Markus Hoffmann, Sabrina Hong, Trent Huang, Ashley Huff, William J. Huggins, Lev B. Ioffe, Sergei V. Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Pavol Juhas, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Tanuj Khattar, Mostafa Khezri, Mária Kieferová, Seon Kim, Alexei Kitaev, Paul V. Klimov, Andrey R. Klots, Alexander N. Korotkov, Fedor Kostritsa, John Mark Kreikebaum, David Landhuis, Pavel Laptev, Kim-Ming Lau, Lily Laws, Joonho Lee, Kenny Lee, Brian J. Lester, Alexander Lill, Wayne Liu, Aditya Locharla, Erik Lucero, Fionn D. Malone, Jeffrey Marshall, Orion Martin, Jarrod R. McClean, Trevor McCourt, Matt McEwen, Anthony Megrant, Bernardo Meurer Costa, Xiao Mi, Kevin C. Miao, Masoud Mohseni, Shirin Montazeri, Alexis Morvan, Emily Mount, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Hartmut Neven, Michael Newman, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Murphy Yuezhen Niu, Thomas E. O’Brien, Alex Opremcak, John Platt, Andre Petukhov, Rebecca Potter, Leonid P. Pryadko, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Negar Saei, Daniel Sank, Kannan Sankaragomathi, Kevin J. Satzinger, Henry F. Schurkus, Christopher Schuster, Michael J. Shearn, Aaron Shorter, Vladimir Shvarts, Jindra Skruzny, Vadim Smelyanskiy, W. Clarke Smith, George Sterling, Doug Strain, Marco Szalay, Alfredo Torres, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff Heidweiller, Theodore White, Cheng Xing, Z. Jamie Yao, Ping Yeh, Juhwan Yoo, Grayson Young, Adam Zalcman, Yaxing Zhang, Ningfeng Zhu, and Google Quantum AI, “Suppressing quantum errors by scaling a surface code logical qubit,” Nature 614, 676–681 (2023).
  * Hennessy and Patterson (2011) John L Hennessy and David A Patterson,  _Computer architecture: a quantitative approach_ (Elsevier, 2011).
  * Patterson and Hennessy (2016) David A Patterson and John L Hennessy,  _Computer organization and design ARM edition: the hardware software interface_ (Morgan kaufmann, 2016).
  * Eastin and Knill (2009) Bryan Eastin and Emanuel Knill, “Restrictions on transversal encoded quantum gate sets,” Phys. Rev. Lett. 102, 110502 (2009).
  * Bravyi and Kitaev (2005) Sergey Bravyi and Alexei Kitaev, “Universal quantum computation with ideal clifford gates and noisy ancillas,” Phys. Rev. A 71, 022316 (2005).
  * Bombin and Martin-Delgado (2009) H Bombin and M A Martin-Delgado, “Quantum measurements and gates by code deformation,” Journal of Physics A: Mathematical and Theoretical 42, 095302 (2009).
  * Horsman  _et al._ (2012) Dominic Horsman, Austin G Fowler, Simon Devitt, and Rodney Van Meter, “Surface code quantum computing by lattice surgery,” New Journal of Physics 14, 123011 (2012).
  * Brown  _et al._ (2017) Benjamin J. Brown, Katharina Laubscher, Markus S. Kesselring, and James R. Wootton, “Poking holes and cutting corners to achieve clifford gates with the surface code,” Phys. Rev. X 7, 021029 (2017).
  * Litinski (2019) Daniel Litinski, “A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery,” Quantum 3, 128 (2019).
  * Jochym-O’Connor and Laflamme (2014) Tomas Jochym-O’Connor and Raymond Laflamme, “Using concatenated quantum codes for universal fault-tolerant quantum gates,” Phys. Rev. Lett. 112, 010505 (2014).
  * Anderson  _et al._ (2014) Jonas T. Anderson, Guillaume Duclos-Cianci, and David Poulin, “Fault-tolerant conversion between the steane and reed-muller quantum codes,” Phys. Rev. Lett. 113, 080501 (2014).
  * Terhal (2015) Barbara M. Terhal, “Quantum error correction for quantum memories,” Rev. Mod. Phys. 87, 307–346 (2015).
  * Kovalev and Pryadko (2013) Alexey A. Kovalev and Leonid P. Pryadko, “Fault tolerance of quantum low-density parity check codes with sublinear distance scaling,” Phys. Rev. A 87, 020304 (2013).
  * Gidney  _et al._ (2021) Craig Gidney, Michael Newman, Austin Fowler, and Michael Broughton, “A Fault-Tolerant Honeycomb Memory,” Quantum 5, 605 (2021).
  * Bravyi  _et al._ (2023) Sergey Bravyi, Andrew W. Cross, Jay M. Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J. Yoder, “High-threshold and low-overhead fault-tolerant quantum memory,” (2023), arXiv:2308.07915 [quant-ph] .
  * Lvovsky  _et al._ (2009) Alexander I. Lvovsky, Barry C. Sanders, and Wolfgang Tittel, “Optical quantum memory,” Nature Photonics 3, 706–714 (2009).
  * Hedges  _et al._ (2010) Morgan P. Hedges, Jevon J. Longdell, Yongmin Li, and Matthew J. Sellars, “Efficient quantum memory for light,” Nature 465, 1052–1056 (2010).
  * Heshami  _et al._ (2016) Khabat Heshami, Duncan G. England, Peter C. Humphreys, Philip J. Bustard, Victor M. Acosta, Joshua Nunn, and Benjamin J. Sussman, “Quantum memories: emerging applications and recent advances,” Journal of Modern Optics 63, 2005–2028 (2016), pMID: 27695198\.
  * Briegel  _et al._ (1998) H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: The role of imperfect local operations in quantum communication,” Phys. Rev. Lett. 81, 5932–5935 (1998).
  * Sangouard  _et al._ (2011) Nicolas Sangouard, Christoph Simon, Hugues de Riedmatten, and Nicolas Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33–80 (2011).
  * Bennett  _et al._ (1993) Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters, “Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
  * Raussendorf and Briegel (2001) Robert Raussendorf and Hans J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188–5191 (2001).
  * Raussendorf  _et al._ (2003) Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
  * Benjamin  _et al._ (2009) S.C. Benjamin, B.W. Lovett, and J.M. Smith, “Prospects for measurement-based quantum computing with solid state spins,” Laser & Photonics Reviews 3, 556–574 (2009).
  * Bartolucci  _et al._ (2021) Sara Bartolucci, Patrick Birchall, Hector Bombin, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, Fernando Pastawski, Terry Rudolph, and Chris Sparrow, “Fusion-based quantum computation,” (2021), arXiv:2101.09310 [quant-ph] .
  * Zhang  _et al._ (2021) Zichang Zhang, Chenzhi Yuan, Si Shen, Hao Yu, Ruiming Zhang, Heqing Wang, Hao Li, You Wang, Guangwei Deng, Zhiming Wang, Lixing You, Zhen Wang, Haizhi Song, Guangcan Guo, and Qiang Zhou, “High-performance quantum entanglement generation via cascaded second-order nonlinear processes,” npj Quantum Information 7, 123 (2021).
  * Marxer  _et al._ (2023) Fabian Marxer, Antti Vepsäläinen, Shan W. Jolin, Jani Tuorila, Alessandro Landra, Caspar Ockeloen-Korppi, Wei Liu, Olli Ahonen, Adrian Auer, Lucien Belzane, Ville Bergholm, Chun Fai Chan, Kok Wai Chan, Tuukka Hiltunen, Juho Hotari, Eric Hyyppä, Joni Ikonen, David Janzso, Miikka Koistinen, Janne Kotilahti, Tianyi Li, Jyrgen Luus, Miha Papic, Matti Partanen, Jukka Räbinä, Jari Rosti, Mykhailo Savytskyi, Marko Seppälä, Vasilii Sevriuk, Eelis Takala, Brian Tarasinski, Manish J. Thapa, Francesca Tosto, Natalia Vorobeva, Liuqi Yu, Kuan Yen Tan, Juha Hassel, Mikko Möttönen, and Johannes Heinsoo, “Long-distance transmon coupler with cz-gate fidelity above $99.8\mathrm{\%}$,” PRX Quantum 4, 010314 (2023).
  * Blais  _et al._ (2021) Alexandre Blais, Arne L. Grimsmo, S. M. Girvin, and Andreas Wallraff, “Circuit quantum electrodynamics,” Rev. Mod. Phys. 93, 025005 (2021).
  * Kjaergaard  _et al._ (2020) Morten Kjaergaard, Mollie E. Schwartz, Jochen Braumüller, Philip Krantz, Joel I.-J. Wang, Simon Gustavsson, and William D. Oliver, “Superconducting qubits: Current state of play,” Annual Review of Condensed Matter Physics 11, 369–395 (2020), https://doi.org/10.1146/annurev-conmatphys-031119-050605 .
  * Koch  _et al._ (2007) Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck, D. I. Schuster, J. Majer, Alexandre Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Charge-insensitive qubit design derived from the cooper pair box,” Phys. Rev. A 76, 042319 (2007).
  * Schreier  _et al._ (2008) J. A. Schreier, A. A. Houck, Jens 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, “Suppressing charge noise decoherence in superconducting charge qubits,” Phys. Rev. B 77, 180502 (2008).
  * Houck  _et al._ (2008) A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, Jens Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Controlling the spontaneous emission of a superconducting transmon qubit,” Phys. Rev. Lett. 101, 080502 (2008).
  * Place  _et al._ (2021) Alexander P. M. Place, Lila V. H. Rodgers, Pranav Mundada, Basil M. Smitham, Mattias Fitzpatrick, Zhaoqi Leng, Anjali Premkumar, Jacob Bryon, Andrei Vrajitoarea, Sara Sussman, Guangming Cheng, Trisha Madhavan, Harshvardhan K. Babla, Xuan Hoang Le, Youqi Gang, Berthold Jäck, András Gyenis, Nan Yao, Robert J. Cava, Nathalie P. de Leon, and Andrew A. Houck, “New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds,” Nature Communications 12, 1779 (2021).
  * Wang  _et al._ (2022a) Chenlu Wang, Xuegang Li, Huikai Xu, Zhiyuan Li, Junhua Wang, Zhen Yang, Zhenyu Mi, Xuehui Liang, Tang Su, Chuhong Yang, Guangyue Wang, Wenyan Wang, Yongchao Li, Mo Chen, Chengyao Li, Kehuan Linghu, Jiaxiu Han, Yingshan Zhang, Yulong Feng, Yu Song, Teng Ma, Jingning Zhang, Ruixia Wang, Peng Zhao, Weiyang Liu, Guangming Xue, Yirong Jin, and Haifeng Yu, “Towards practical quantum computers: transmon qubit with a lifetime approaching 0.5 milliseconds,” npj Quantum Information 8, 3 (2022a).
  * Manucharyan  _et al._ (2009) Vladimir E. Manucharyan, Jens Koch, Leonid I. Glazman, and Michel H. Devoret, “Fluxonium: Single Cooper-Pair Circuit Free of Charge Offsets,” Science 326, 113–116 (2009).
  * Somoroff  _et al._ (2023) Aaron Somoroff, Quentin Ficheux, Raymond A. Mencia, Haonan Xiong, Roman Kuzmin, and Vladimir E. Manucharyan, “Millisecond coherence in a superconducting qubit,” Phys. Rev. Lett. 130, 267001 (2023).
  * Sheng  _et al._ (2018) Cheng Sheng, Xiaodong He, Peng Xu, Ruijun Guo, Kunpeng Wang, Zongyuan Xiong, Min Liu, Jin Wang, and Mingsheng Zhan, “High-fidelity single-qubit gates on neutral atoms in a two-dimensional magic-intensity optical dipole trap array,” Phys. Rev. Lett. 121, 240501 (2018).
  * Bluvstein  _et al._ (2022) Dolev Bluvstein, Harry Levine, Giulia Semeghini, Tout T. Wang, Sepehr Ebadi, Marcin Kalinowski, Alexander Keesling, Nishad Maskara, Hannes Pichler, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin, “A quantum processor based on coherent transport of entangled atom arrays,” Nature 604, 451–456 (2022).
  * Ma  _et al._ (2022) Shuo Ma, Alex P. Burgers, Genyue Liu, Jack Wilson, Bichen Zhang, and Jeff D. Thompson, “Universal gate operations on nuclear spin qubits in an optical tweezer array of ${}^{171}\mathrm{Yb}$ atoms,” Phys. Rev. X 12, 021028 (2022).
  * Jenkins  _et al._ (2022) Alec Jenkins, Joanna W. Lis, Aruku Senoo, William F. McGrew, and Adam M. Kaufman, “Ytterbium nuclear-spin qubits in an optical tweezer array,” Phys. Rev. X 12, 021027 (2022).
  * Wang  _et al._ (2017) Ye Wang, Mark Um, Junhua Zhang, Shuoming An, Ming Lyu, Jing-Ning Zhang, L. M. Duan, Dahyun Yum, and Kihwan Kim, “Single-qubit quantum memory exceeding ten-minute coherence time,” Nature Photonics 11, 646–650 (2017).
  * Wang  _et al._ (2021) Pengfei Wang, Chun-Yang Luan, Mu Qiao, Mark Um, Junhua Zhang, Ye Wang, Xiao Yuan, Mile Gu, Jingning Zhang, and Kihwan Kim, “Single ion qubit with estimated coherence time exceeding one hour,” Nature Communications 12, 233 (2021).
  * 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, Matthias Steffen, and O. E. Dial, “Tunable coupling architecture for fixed-frequency transmon superconducting qubits,” Phys. Rev. Lett. 127, 080505 (2021).
  * Sung  _et al._ (2021) Youngkyu Sung, Leon Ding, Jochen Braumüller, Antti Vepsäläinen, Bharath Kannan, Morten Kjaergaard, Ami Greene, Gabriel O. Samach, Chris McNally, David Kim, Alexander Melville, Bethany M. Niedzielski, Mollie E. Schwartz, Jonilyn L. Yoder, Terry P. Orlando, Simon Gustavsson, and William D. Oliver, “Realization of high-fidelity cz and $zz$-free iswap gates with a tunable coupler,” Phys. Rev. X 11, 021058 (2021).
  * Nesterov  _et al._ (2018) Konstantin N. Nesterov, Ivan V. Pechenezhskiy, Chen Wang, Vladimir E. Manucharyan, and Maxim G. Vavilov, “Microwave-activated controlled-$Z$ gate for fixed-frequency Fluxonium qubits,” Phys. Rev. A 98, 030301 (2018).
  * Nguyen  _et al._ (2022) Long B. Nguyen, Gerwin Koolstra, Yosep Kim, Alexis Morvan, Trevor Chistolini, Shraddha Singh, Konstantin N. Nesterov, Christian Jünger, Larry Chen, Zahra Pedramrazi, Bradley K. Mitchell, John Mark Kreikebaum, Shruti Puri, David I. Santiago, and Irfan Siddiqi, “Blueprint for a High-Performance Fluxonium Quantum Processor,” PRX Quantum 3, 037001 (2022).
  * Xiang  _et al._ (2013) Ze-Liang Xiang, Sahel Ashhab, J. Q. You, and Franco Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (2013).
  * Kurizki  _et al._ (2015) Gershon Kurizki, Patrice Bertet, Yuimaru Kubo, Klaus Mølmer, David Petrosyan, Peter Rabl, and Jörg Schmiedmayer, “Quantum technologies with hybrid systems,” Proceedings of the National Academy of Sciences 112, 3866–3873 (2015).
  * Clerk  _et al._ (2020) A. A. Clerk, K. W. Lehnert, P. Bertet, J. R. Petta, and Y. Nakamura, “Hybrid quantum systems with circuit quantum electrodynamics,” Nature Physics 16, 257–267 (2020).
  * Gao  _et al._ (2021) Yvonne Y. Gao, M. Adriaan Rol, Steven Touzard, and Chen Wang, “Practical guide for building superconducting quantum devices,” PRX Quantum 2, 040202 (2021).
  * Scully and Zubairy (1997) M.O. Scully and M.S. Zubairy,  _Quantum Optics_, Quantum Optics (Cambridge University Press, 1997).
  * Reagor  _et al._ (2013) Matthew Reagor, Hanhee Paik, Gianluigi Catelani, Luyan Sun, Christopher Axline, Eric Holland, Ioan M. Pop, Nicholas A. Masluk, Teresa Brecht, Luigi Frunzio, Michel H. Devoret, Leonid Glazman, and Robert J. Schoelkopf, “Reaching 10 ms single photon lifetimes for superconducting aluminum cavities,” Applied Physics Letters 102 (2013), 10.1063/1.4807015, 192604.
  * Romanenko  _et al._ (2020) A. Romanenko, R. Pilipenko, S. Zorzetti, D. Frolov, M. Awida, S. Belomestnykh, S. Posen, and A. Grassellino, “Three-dimensional superconducting resonators at $t<20$ mk with photon lifetimes up to $\tau=2$ s,” Phys. Rev. Appl. 13, 034032 (2020).
  * Yin  _et al._ (2013) Yi Yin, Yu Chen, Daniel Sank, P. J. J. O’Malley, T. C. White, R. Barends, J. Kelly, Erik Lucero, Matteo Mariantoni, A. Megrant, C. Neill, A. Vainsencher, J. Wenner, Alexander N. Korotkov, A. N. Cleland, and John M. Martinis, “Catch and release of microwave photon states,” Phys. Rev. Lett. 110, 107001 (2013).
  * Wenner  _et al._ (2014) J. Wenner, Yi Yin, Yu Chen, R. Barends, B. Chiaro, E. Jeffrey, J. Kelly, A. Megrant, J. Y. Mutus, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, T. C. White, Alexander N. Korotkov, A. N. Cleland, and John M. Martinis, “Catching time-reversed microwave coherent state photons with 99.4% absorption efficiency,” Phys. Rev. Lett. 112, 210501 (2014).
  * Campagne-Ibarcq  _et al._ (2018) P. Campagne-Ibarcq, E. Zalys-Geller, A. Narla, S. Shankar, P. Reinhold, L. Burkhart, C. Axline, W. Pfaff, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Deterministic remote entanglement of superconducting circuits through microwave two-photon transitions,” Phys. Rev. Lett. 120, 200501 (2018).
  * Chakram  _et al._ (2021) Srivatsan Chakram, Andrew E. Oriani, Ravi K. Naik, Akash V. Dixit, Kevin He, Ankur Agrawal, Hyeokshin Kwon, and David I. Schuster, “Seamless high-$q$ microwave cavities for multimode circuit quantum electrodynamics,” Phys. Rev. Lett. 127, 107701 (2021).
  * Milul  _et al._ (2023) Ofir Milul, Barkay Guttel, Uri Goldblatt, Sergey Hazanov, Lalit M. Joshi, Daniel Chausovsky, Nitzan Kahn, Engin Çiftyürek, Fabien Lafont, and Serge Rosenblum, “A superconducting quantum memory with tens of milliseconds coherence time,” (2023), arXiv:2302.06442 [quant-ph] .
  * Naik  _et al._ (2017) R. K. Naik, N. Leung, S. Chakram, Peter Groszkowski, Y. Lu, N. Earnest, D. C. McKay, Jens Koch, and D. I. Schuster, “Random access quantum information processors using multimode circuit quantum electrodynamics,” Nature Communications 8, 1904 (2017).
  * Grimm  _et al._ (2020) A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret, “Stabilization and operation of a kerr-cat qubit,” Nature 584, 205–209 (2020).
  * Darmawan  _et al._ (2021) Andrew S. Darmawan, Benjamin J. Brown, Arne L. Grimsmo, David K. Tuckett, and Shruti Puri, “Practical quantum error correction with the xzzx code and kerr-cat qubits,” PRX Quantum 2, 030345 (2021).
  * Chamberland  _et al._ (2022) Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G.S.L. Brandão, “Building a fault-tolerant quantum computer using concatenated cat codes,” PRX Quantum 3, 010329 (2022).
  * Gertler  _et al._ (2021) Jeffrey M. Gertler, Brian Baker, Juliang Li, Shruti Shirol, Jens Koch, and Chen Wang, “Protecting a bosonic qubit with autonomous quantum error correction,” Nature 590, 243–248 (2021).
  * Grimsmo and Puri (2021) Arne L. Grimsmo and Shruti Puri, “Quantum error correction with the gottesman-kitaev-preskill code,” PRX Quantum 2, 020101 (2021).
  * Campagne-Ibarcq  _et al._ (2020) P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio, M. Mirrahimi, and M. H. Devoret, “Quantum error correction of a qubit encoded in grid states of an oscillator,” Nature 584, 368–372 (2020).
  * Hu  _et al._ (2019) L. Hu, Y. Ma, W. Cai, X. Mu, Y. Xu, W. Wang, Y. Wu, H. Wang, Y. P. Song, C. L. Zou, S. M. Girvin, L-M. Duan, and L. Sun, “Quantum error correction and universal gate set operation on a binomial bosonic logical qubit,” Nature Physics 15, 503–508 (2019).
  * Stein  _et al._ (2023) Samuel Stein, Fei Hua, Chenxu Liu, Charles Guinn, James Ang, Eddy Zhang, Srivatsan Chakram, Yufei Ding, and Ang Li, “Canvas: Cavity-centric architectures for novel variational algorithm support,” (2023), in preparation.
  * Häffner  _et al._ (2008) H. Häffner, C.F. Roos, and R. Blatt, “Quantum computing with trapped ions,” Physics Reports 469, 155–203 (2008).
  * Bruzewicz  _et al._ (2019) Colin D. Bruzewicz, John Chiaverini, Robert McConnell, and Jeremy M. Sage, “Trapped-ion quantum computing: Progress and challenges,” Applied Physics Reviews 6 (2019), https://doi.org/10.1063/1.5088164, 021314.
  * Brown  _et al._ (2021) Kenneth R. Brown, John Chiaverini, Jeremy M. Sage, and Hartmut Häffner, “Materials challenges for trapped-ion quantum computers,” Nature Reviews Materials 6, 892–905 (2021).
  * Neuhauser  _et al._ (1980) W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, “Localized visible ${\mathrm{ba}}^{+}$ mono-ion oscillator,” Phys. Rev. A 22, 1137–1140 (1980).
  * Dehmelt (1990) Hans Dehmelt, “Experiments with an isolated subatomic particle at rest,” Rev. Mod. Phys. 62, 525–530 (1990).
  * Chiaverini  _et al._ (2005) J. Chiaverini, R. B. Blakestad, J. Britton, J. D. Jost, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Surface-electrode architecture for ion-trap quantum information processing,” Quantum Info. Comput. 5, 419–439 (2005).
  * Seidelin  _et al._ (2006) S. Seidelin, J. Chiaverini, R. Reichle, J. J. Bollinger, D. Leibfried, J. Britton, J. H. Wesenberg, R. B. Blakestad, R. J. Epstein, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, N. Shiga, and D. J. Wineland, “Microfabricated Surface-Electrode Ion Trap for Scalable Quantum Information Processing,” Phys. Rev. Lett. 96, 253003 (2006).
  * Labaziewicz  _et al._ (2008) Jaroslaw Labaziewicz, Yufei Ge, Paul Antohi, David Leibrandt, Kenneth R. Brown, and Isaac L. Chuang, “Suppression of Heating Rates in Cryogenic Surface-Electrode Ion Traps,” Phys. Rev. Lett. 100, 013001 (2008).
  * 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, “High-fidelity preparation, gates, memory, and readout of a trapped-ion quantum bit,” 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, “High-fidelity universal gate set for ${{}^{9}\mathrm{Be}}^{+}$ ion qubits,” 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, “High-fidelity quantum logic gates using trapped-ion hyperfine qubits,” Phys. Rev. Lett. 117, 060504 (2016).
  * Keselman  _et al._ (2011) A Keselman, Y Glickman, N Akerman, S Kotler, and R Ozeri, “High-fidelity state detection and tomography of a single-ion zeeman qubit,” 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, “A long-lived zeeman trapped-ion qubit,” Applied Physics B 122, 254 (2016).
  * 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, “Fault-tolerant parity readout on a shuttling-based trapped-ion quantum computer,” Phys. Rev. X 12, 011032 (2022).
  * 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, “Assessing the progress of trapped-ion processors towards fault-tolerant quantum computation,” Phys. Rev. X 7, 041061 (2017).
  * Toyoda  _et al._ (2010) Kenji Toyoda, Shinsuke Haze, Rekishu Yamazaki, and Shinji Urabe, “Quantum gate using qubit states separated by terahertz,” Phys. Rev. A 81, 032322 (2010).
  * Pogorelov  _et al._ (2021) I. Pogorelov, T. Feldker, Ch. 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, “Compact ion-trap quantum computing demonstrator,” PRX Quantum 2, 020343 (2021).
  * Blümel  _et al._ (2021) Reinhold Blümel, Nikodem Grzesiak, Neal Pisenti, Kenneth Wright, and Yunseong Nam, “Power-optimal, stabilized entangling gate between trapped-ion qubits,” npj Quantum Information 7, 147 (2021).
  * Jeon  _et al._ (2023) Honggi Jeon, Jiyong Kang, Jaeun Kim, Wonhyeong Choi, Kyunghye Kim, and Taehyun Kim, “Experimental realization of entangled coherent states in two-dimensional harmonic oscillators of a trapped ion,” (2023), arXiv:2305.00820 [quant-ph] .
  * 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] .
  * Stute  _et al._ (2012) A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstätter, T. E. Northup, and R. Blatt, “Tunable ion–photon entanglement in an optical cavity,” Nature 485, 482–485 (2012).
  * Stute  _et al._ (2013) A. Stute, B. Casabone, B. Brandstätter, K. Friebe, T. E. Northup, and R. Blatt, “Quantum-state transfer from an ion to a photon,” Nature Photonics 7, 219–222 (2013).
  * Begley  _et al._ (2016) Stephen Begley, Markus Vogt, Gurpreet Kaur Gulati, Hiroki Takahashi, and Matthias Keller, “Optimized multi-ion cavity coupling,” Phys. Rev. Lett. 116, 223001 (2016).
  * Takahashi  _et al._ (2020) Hiroki Takahashi, Ezra Kassa, Costas Christoforou, and Matthias Keller, “Strong coupling of a single ion to an optical cavity,” Phys. Rev. Lett. 124, 013602 (2020).
  * Christoforou  _et al._ (2020) Costas Christoforou, Corentin Pignot, Ezra Kassa, Hiroki Takahashi, and Matthias Keller, “Enhanced ion–cavity coupling through cavity cooling in the strong coupling regime,” Scientific Reports 10, 15693 (2020).
  * Blinov  _et al._ (2004) B. B. Blinov, D. L. Moehring, L. M. Duan, and C. Monroe, “Observation of entanglement between a single trapped atom and a single photon,” Nature 428, 153–157 (2004).
  * Bock  _et al._ (2018) Matthias Bock, Pascal Eich, Stephan Kucera, Matthias Kreis, Andreas Lenhard, Christoph Becher, and Jürgen Eschner, “High-fidelity entanglement between a trapped ion and a telecom photon via quantum frequency conversion,” Nature Communications 9, 1998 (2018).
  * Kobel  _et al._ (2021) Pascal Kobel, Moritz Breyer, and Michael Köhl, “Deterministic spin-photon entanglement from a trapped ion in a fiber fabry–perot cavity,” npj Quantum Information 7, 6 (2021).
  * Cabrillo  _et al._ (1999) C. Cabrillo, J. I. Cirac, P. García-Fernández, and P. Zoller, “Creation of entangled states of distant atoms by interference,” Phys. Rev. A 59, 1025–1033 (1999).
  * Kok  _et al._ (2007) Pieter Kok, W. J. Munro, Kae Nemoto, T. C. Ralph, Jonathan P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
  * Duan  _et al._ (2001) L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
  * Casabone  _et al._ (2013) B. Casabone, A. Stute, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt, and T. E. Northup, “Heralded entanglement of two ions in an optical cavity,” Phys. Rev. Lett. 111, 100505 (2013).
  * Niffenegger  _et al._ (2020) R. J. Niffenegger, J. Stuart, C. Sorace-Agaskar, D. Kharas, S. Bramhavar, C. D. Bruzewicz, W. Loh, R. T. Maxson, R. McConnell, D. Reens, G. N. West, J. M. Sage, and J. Chiaverini, “Integrated multi-wavelength control of an ion qubit,” Nature 586, 538–542 (2020).
  * Saha  _et al._ (2023) Uday Saha, James D. Siverns, John Hannegan, Mihika Prabhu, Qudsia Quraishi, Dirk Englund, and Edo Waks, “Routing single photons from a trapped ion using a photonic integrated circuit,” Phys. Rev. Appl. 19, 034001 (2023).
  * Krutyanskiy  _et al._ (2023) V. Krutyanskiy, M. Galli, V. Krcmarsky, S. Baier, D. A. Fioretto, Y. Pu, A. Mazloom, P. Sekatski, M. Canteri, M. Teller, J. Schupp, J. Bate, M. Meraner, N. Sangouard, B. P. Lanyon, and T. E. Northup, “Entanglement of trapped-ion qubits separated by 230 meters,” Phys. Rev. Lett. 130, 050803 (2023).
  * Dhara  _et al._ (2022) Prajit Dhara, Norbert M. Linke, Edo Waks, Saikat Guha, and Kaushik P. Seshadreesan, “Multiplexed quantum repeaters based on dual-species trapped-ion systems,” Phys. Rev. A 105, 022623 (2022).
  * Verdú  _et al._ (2009) J. Verdú, H. Zoubi, Ch. Koller, J. Majer, H. Ritsch, and J. Schmiedmayer, “Strong magnetic coupling of an ultracold gas to a superconducting waveguide cavity,” Phys. Rev. Lett. 103, 043603 (2009).
  * Bohnet  _et al._ (2016) Justin G. Bohnet, Brian C. Sawyer, Joseph W. Britton, Michael L. Wall, Ana Maria Rey, Michael Foss-Feig, and John J. Bollinger, “Quantum spin dynamics and entanglement generation with hundreds of trapped ions,” Science 352, 1297–1301 (2016).
  * Zhang  _et al._ (2017) J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z. X. Gong, and C. Monroe, “Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator,” Nature 551, 601–604 (2017).
  * Kiesenhofer  _et al._ (2023) Dominik Kiesenhofer, Helene Hainzer, Artem Zhdanov, Philip C. Holz, Matthias Bock, Tuomas Ollikainen, and Christian F. Roos, “Controlling two-dimensional coulomb crystals of more than 100 ions in a monolithic radio-frequency trap,” PRX Quantum 4, 020317 (2023).
  * 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, “Rydberg-mediated entanglement in a two-dimensional neutral atom qubit array,” Phys. Rev. Lett. 123, 230501 (2019).
  * Levine  _et al._ (2019) Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Omran, Tout T. Wang, Sepehr Ebadi, Hannes Bernien, Markus Greiner, Vladan Vuletić, Hannes Pichler, and Mikhail D. Lukin, “Parallel implementation of high-fidelity multiqubit gates with neutral atoms,” Phys. Rev. Lett. 123, 170503 (2019).
  * Morgado and Whitlock (2021) M. Morgado and S. Whitlock, “Quantum simulation and computing with Rydberg-interacting qubits,” AVS Quantum Science 3 (2021), doi.org/10.1116/5.0036562, 023501.
  * Levine  _et al._ (2018) Harry Levine, Alexander Keesling, Ahmed Omran, Hannes Bernien, Sylvain Schwartz, Alexander S. Zibrov, Manuel Endres, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin, “High-fidelity control and entanglement of rydberg-atom qubits,” Phys. Rev. Lett. 121, 123603 (2018).
  * Madjarov  _et al._ (2020) Ivaylo S. Madjarov, Jacob P. Covey, Adam L. Shaw, Joonhee Choi, Anant Kale, Alexandre Cooper, Hannes Pichler, Vladimir Schkolnik, Jason R. Williams, and Manuel Endres, “High-fidelity entanglement and detection of alkaline-earth rydberg atoms,” Nature Physics 16, 857–861 (2020).
  * Picken  _et al._ (2018) C J Picken, R Legaie, K McDonnell, and J D Pritchard, “Entanglement of neutral-atom qubits with long ground-rydberg coherence times,” Quantum Science and Technology 4, 015011 (2018).
  * Norcia  _et al._ (2019) Matthew A. Norcia, Aaron W. Young, William J. Eckner, Eric Oelker, Jun Ye, and Adam M. Kaufman, “Seconds-scale coherence on an optical clock transition in a tweezer array,” Science 366, 93–97 (2019).
  * Barnes  _et al._ (2022) Katrina Barnes, Peter Battaglino, Benjamin J. Bloom, Kayleigh Cassella, Robin Coxe, Nicole Crisosto, Jonathan P. King, Stanimir S. Kondov, Krish Kotru, Stuart C. Larsen, Joseph Lauigan, Brian J. Lester, Mickey McDonald, Eli Megidish, Sandeep Narayanaswami, Ciro Nishiguchi, Remy Notermans, Lucas S. Peng, Albert Ryou, Tsung-Yao Wu, and Michael Yarwood, “Assembly and coherent control of a register of nuclear spin qubits,” Nature Communications 13, 2779 (2022).
  * Lukin  _et al._ (2001) M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
  * Jaksch  _et al._ (2000) D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85, 2208–2211 (2000).
  * Urban  _et al._ (2009) E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of rydberg blockade between two atoms,” Nature Physics 5, 110–114 (2009).
  * Langenfeld  _et al._ (2020) Stefan Langenfeld, Olivier Morin, Matthias Körber, and Gerhard Rempe, “A network-ready random-access qubits memory,” npj Quantum Information 6, 86 (2020).
  * Liu  _et al._ (2023b) Yanxin Liu, Zhihui Wang, Pengfei Yang, Qinxia Wang, Qing Fan, Shijun Guan, Gang Li, Pengfei Zhang, and Tiancai Zhang, “Realization of strong coupling between deterministic single-atom arrays and a high-finesse miniature optical cavity,” Phys. Rev. Lett. 130, 173601 (2023b).
  * Hosseini  _et al._ (2011) M. Hosseini, B. M. Sparkes, G. Campbell, P. K. Lam, and B. C. Buchler, “High efficiency coherent optical memory with warm rubidium vapour,” Nature Communications 2, 174 (2011).
  * Dudin  _et al._ (2013) Y. O. Dudin, L. Li, and A. Kuzmich, “Light storage on the time scale of a minute,” Phys. Rev. A 87, 031801 (2013).
  * Ding  _et al._ (2013) Dong-Sheng Ding, Zhi-Yuan Zhou, Bao-Sen Shi, and Guang-Can Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nature Communications 4, 2527 (2013).
  * Nicolas  _et al._ (2014) A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nature Photonics 8, 234–238 (2014).
  * Ding  _et al._ (2015a) Dong-Sheng Ding, Wei Zhang, Zhi-Yuan Zhou, Shuai Shi, Guo-Yong Xiang, Xi-Shi Wang, Yun-Kun Jiang, Bao-Sen Shi, and Guang-Can Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015a).
  * Ding  _et al._ (2015b) Dong-Sheng Ding, Wei Zhang, Zhi-Yuan Zhou, Shuai Shi, Bao-Sen Shi, and Guang-Can Guo, “Raman quantum memory of photonic polarized entanglement,” Nature Photonics 9, 332–338 (2015b).
  * Parigi  _et al._ (2015) Valentina Parigi, Vincenzo D’Ambrosio, Christophe Arnold, Lorenzo Marrucci, Fabio Sciarrino, and Julien Laurat, “Storage and retrieval of vector beams of light in a multiple-degree-of-freedom quantum memory,” 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, “Cavity-enhanced room-temperature broadband raman memory,” Phys. Rev. Lett. 116, 090501 (2016).
  * Katz and Firstenberg (2018) Or Katz and Ofer Firstenberg, “Light storage for one second in room-temperature alkali vapor,” Nature Communications 9, 2074 (2018).
  * Hsiao  _et al._ (2018) Ya-Fen Hsiao, Pin-Ju Tsai, Hung-Shiue Chen, Sheng-Xiang Lin, Chih-Chiao Hung, Chih-Hsi Lee, Yi-Hsin Chen, Yong-Fan Chen, Ite A. Yu, and Ying-Cheng Chen, “Highly efficient coherent optical memory based on electromagnetically induced transparency,” 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, “Experimental realization of 105-qubit random access quantum memory,” npj Quantum Information 5, 28 (2019).
  * Wang  _et al._ (2019) Yunfei Wang, Jianfeng Li, Shanchao Zhang, Keyu Su, Yiru Zhou, Kaiyu Liao, Shengwang Du, Hui Yan, and Shi-Liang Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nature Photonics 13, 346–351 (2019).
  * Li  _et al._ (2020) C. Li, N. Jiang, Y.-K. Wu, W. Chang, Y.-F. Pu, S. Zhang, and L.-M. Duan, “Quantum communication between multiplexed atomic quantum memories,” Phys. Rev. Lett. 124, 240504 (2020).
  * Dideriksen  _et al._ (2021) Karsten B. Dideriksen, Rebecca Schmieg, Michael Zugenmaier, and Eugene S. Polzik, “Room-temperature single-photon source with near-millisecond built-in memory,” Nature Communications 12, 3699 (2021).
  * Wang  _et al._ (2022b) Yang Wang, Alexander N. Craddock, Rourke Sekelsky, Mael Flament, and Mehdi Namazi, “Field-deployable quantum memory for quantum networking,” Phys. Rev. Appl. 18, 044058 (2022b).
  * Meßner  _et al._ (2023) Leon Meßner, Elizabeth Robertson, Luisa Esguerra, Kathy Lüdge, and Janik Wolters, “Multiplexed random-access optical memory in warm cesium vapor,” Opt. Express 31, 10150–10158 (2023).
  * Buser  _et al._ (2022) Gianni Buser, Roberto Mottola, Björn Cotting, Janik Wolters, and Philipp Treutlein, “Single-photon storage in a ground-state vapor cell quantum memory,” PRX Quantum 3, 020349 (2022).
  * Hattermann  _et al._ (2017) H. Hattermann, D. Bothner, L. Y. Ley, B. Ferdinand, D. Wiedmaier, L. Sárkány, R. Kleiner, D. Koelle, and J. Fortágh, “Coupling ultracold atoms to a superconducting coplanar waveguide resonator,” Nature Communications 8, 2254 (2017).
  * Rabl  _et al._ (2006) P. Rabl, D. DeMille, J. M. Doyle, M. D. Lukin, R. J. Schoelkopf, and P. Zoller, “Hybrid quantum processors: Molecular ensembles as quantum memory for solid state circuits,” Phys. Rev. Lett. 97, 033003 (2006).
  * Petrosyan  _et al._ (2009) David Petrosyan, Guy Bensky, Gershon Kurizki, Igor Mazets, Johannes Majer, and Jörg Schmiedmayer, “Reversible state transfer between superconducting qubits and atomic ensembles,” Phys. Rev. A 79, 040304 (2009).
  * Tu  _et al._ (2022) Hai-Tao Tu, Kai-Yu Liao, Zuan-Xian Zhang, Xiao-Hong Liu, Shun-Yuan Zheng, Shu-Zhe Yang, Xin-Ding Zhang, Hui Yan, and Shi-Liang Zhu, “High-efficiency coherent microwave-to-optics conversion via off-resonant scattering,” Nature Photonics 16, 291–296 (2022).
  * Note (1) We acknowledge that the hyperfine states of Rb atom used in these two works are different. Here we aim to give an estimate of the possible implementation of the QMCs.
  * Thiele  _et al._ (2015) T. Thiele, J. Deiglmayr, M. Stammeier, J.-A. Agner, H. Schmutz, F. Merkt, and A. Wallraff, “Imaging electric fields in the vicinity of cryogenic surfaces using rydberg atoms,” Phys. Rev. A 92, 063425 (2015).
  * Garcia  _et al._ (2019) S. Garcia, M. Stammeier, J. Deiglmayr, F. Merkt, and A. Wallraff, “Single-shot nondestructive detection of rydberg-atom ensembles by transmission measurement of a microwave cavity,” Phys. Rev. Lett. 123, 193201 (2019).
  * Morgan and Hogan (2020) A. A. Morgan and S. D. Hogan, “Coupling rydberg atoms to microwave fields in a superconducting coplanar waveguide resonator,” Phys. Rev. Lett. 124, 193604 (2020).
  * Wang  _et al._ (2015) Yang Wang, Xianli Zhang, Theodore A. Corcovilos, Aishwarya Kumar, and David S. Weiss, “Coherent addressing of individual neutral atoms in a 3d optical lattice,” Phys. Rev. Lett. 115, 043003 (2015).
  * Wang  _et al._ (2016) Yang Wang, Aishwarya Kumar, Tsung-Yao Wu, and David S. Weiss, “Single-qubit gates based on targeted phase shifts in a 3D neutral atom array,” Science 352, 1562–1565 (2016).
  * Xia  _et al._ (2015) T. Xia, M. Lichtman, K. Maller, A. W. Carr, M. J. Piotrowicz, L. Isenhower, and M. Saffman, “Randomized benchmarking of single-qubit gates in a 2d array of neutral-atom qubits,” Phys. Rev. Lett. 114, 100503 (2015).
  * Maller  _et al._ (2015) K. M. Maller, M. T. Lichtman, T. Xia, Y. Sun, M. J. Piotrowicz, A. W. Carr, L. Isenhower, and M. Saffman, “Rydberg-blockade controlled-not gate and entanglement in a two-dimensional array of neutral-atom qubits,” Phys. Rev. A 92, 022336 (2015).
  * Endres  _et al._ (2016) Manuel Endres, Hannes Bernien, Alexander Keesling, Harry Levine, Eric R. Anschuetz, Alexandre Krajenbrink, Crystal Senko, Vladan Vuletic, Markus Greiner, and Mikhail D. Lukin, “Atom-by-atom assembly of defect-free one-dimensional cold atom arrays,” Science 354, 1024–1027 (2016).
  * Bernien  _et al._ (2017) Hannes Bernien, Sylvain Schwartz, Alexander Keesling, Harry Levine, Ahmed Omran, Hannes Pichler, Soonwon Choi, Alexander S. Zibrov, Manuel Endres, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature 551, 579–584 (2017).
  * McAuslan  _et al._ (2012) D. L. McAuslan, J. G. Bartholomew, M. J. Sellars, and J. J. Longdell, “Reducing decoherence in optical and spin transitions in rare-earth-metal-ion–doped materials,” Phys. Rev. A 85, 032339 (2012).
  * Saglamyurek  _et al._ (2011) Erhan Saglamyurek, Neil Sinclair, Jeongwan Jin, Joshua A. Slater, Daniel Oblak, Félix Bussières, Mathew George, Raimund Ricken, Wolfgang Sohler, and Wolfgang Tittel, “Broadband waveguide quantum memory for entangled photons,” Nature 469, 512–515 (2011).
  * Clausen  _et al._ (2011) Christoph Clausen, Imam Usmani, Félix Bussières, Nicolas Sangouard, Mikael Afzelius, Hugues de Riedmatten, and Nicolas Gisin, “Quantum storage of photonic entanglement in a crystal,” Nature 469, 508–511 (2011).
  * Ledingham  _et al._ (2012) Patrick M. Ledingham, William R. Naylor, and Jevon J. Longdell, “Experimental realization of light with time-separated correlations by rephasing amplified spontaneous emission,” Phys. Rev. Lett. 109, 093602 (2012).
  * Zhou  _et al._ (2012) Zong-Quan Zhou, Wei-Bin Lin, Ming Yang, Chuan-Feng Li, and Guang-Can Guo, “Realization of reliable solid-state quantum memory for photonic polarization qubit,” Phys. Rev. Lett. 108, 190505 (2012).
  * Ferguson  _et al._ (2016) Kate R. Ferguson, Sarah E. Beavan, Jevon J. Longdell, and Matthew J. Sellars, “Generation of light with multimode time-delayed entanglement using storage in a solid-state spin-wave quantum memory,” Phys. Rev. Lett. 117, 020501 (2016).
  * Jin  _et al._ (2022) Ming Jin, You-Zhi Ma, Zong-Quan Zhou, Chuan-Feng Li, and Guang-Can Guo, “A faithful solid-state spin-wave quantum memory for polarization qubits,” Science Bulletin 67, 676–678 (2022).
  * Ma  _et al._ (2021a) You-Zhi Ma, Ming Jin, Duo-Lun Chen, Zong-Quan Zhou, Chuan-Feng Li, and Guang-Can Guo, “Elimination of noise in optically rephased photon echoes,” Nature Communications 12, 4378 (2021a).
  * Seri  _et al._ (2017) Alessandro Seri, Andreas Lenhard, Daniel Rieländer, Mustafa Gündoğan, Patrick M. Ledingham, Margherita Mazzera, and Hugues de Riedmatten, “Quantum correlations between single telecom photons and a multimode on-demand solid-state quantum memory,” Phys. Rev. X 7, 021028 (2017).
  * Kutluer  _et al._ (2017) Kutlu Kutluer, Margherita Mazzera, and Hugues de Riedmatten, “Solid-state source of nonclassical photon pairs with embedded multimode quantum memory,” Phys. Rev. Lett. 118, 210502 (2017).
  * Laplane  _et al._ (2017) Cyril Laplane, Pierre Jobez, Jean Etesse, Nicolas Gisin, and Mikael Afzelius, “Multimode and long-lived quantum correlations between photons and spins in a crystal,” Phys. Rev. Lett. 118, 210501 (2017).
  * Holzäpfel  _et al._ (2020) Adrian Holzäpfel, Jean Etesse, Krzysztof T Kaczmarek, Alexey Tiranov, Nicolas Gisin, and Mikael Afzelius, “Optical storage for 0.53 s in a solid-state atomic frequency comb memory using dynamical decoupling,” 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, “Optical spin-wave storage in a solid-state hybridized electron-nuclear spin ensemble,” Phys. Rev. Lett. 124, 053606 (2020).
  * Askarani  _et al._ (2021) Mohsen Falamarzi Askarani, Antariksha Das, Jacob H. Davidson, Gustavo C. Amaral, Neil Sinclair, Joshua A. Slater, Sara Marzban, Charles W. Thiel, Rufus L. Cone, Daniel Oblak, and Wolfgang Tittel, “Long-lived solid-state optical memory for high-rate quantum repeaters,” Phys. Rev. Lett. 127, 220502 (2021).
  * Ma  _et al._ (2021b) Yu Ma, You-Zhi Ma, Zong-Quan Zhou, Chuan-Feng Li, and Guang-Can Guo, “One-hour coherent optical storage in an atomic frequency comb memory,” Nature Communications 12, 2381 (2021b).
  * Afzelius  _et al._ (2009) Mikael Afzelius, Christoph Simon, Hugues de Riedmatten, and Nicolas Gisin, “Multimode quantum memory based on atomic frequency combs,” Phys. Rev. A 79, 052329 (2009).
  * Zhu  _et al._ (2022) Tian-Xiang Zhu, Chao Liu, Ming Jin, Ming-Xu Su, Yu-Ping Liu, Wen-Juan Li, Yang Ye, Zong-Quan Zhou, Chuan-Feng Li, and Guang-Can Guo, “On-demand integrated quantum memory for polarization qubits,” Phys. Rev. Lett. 128, 180501 (2022).
  * Probst  _et al._ (2013) S. Probst, H. Rotzinger, S. Wünsch, P. Jung, M. Jerger, M. Siegel, A. V. Ustinov, and P. A. Bushev, “Anisotropic rare-earth spin ensemble strongly coupled to a superconducting resonator,” Phys. Rev. Lett. 110, 157001 (2013).
  * Probst  _et al._ (2015) S. Probst, H. Rotzinger, A. V. Ustinov, and P. A. Bushev, “Microwave multimode memory with an erbium spin ensemble,” Phys. Rev. B 92, 014421 (2015).
  * Wolfowicz  _et al._ (2015) Gary Wolfowicz, Hannes Maier-Flaig, Robert Marino, Alban Ferrier, Hervé Vezin, John J. L. Morton, and Philippe Goldner, “Coherent storage of microwave excitations in rare-earth nuclear spins,” Phys. Rev. Lett. 114, 170503 (2015).
  * Doherty  _et al._ (2011) M W Doherty, N B Manson, P Delaney, and L C L Hollenberg, “The negatively charged nitrogen-vacancy centre in diamond: the electronic solution,” New Journal of Physics 13, 025019 (2011).
  * Doherty  _et al._ (2013) Marcus W. Doherty, Neil B. Manson, Paul Delaney, Fedor Jelezko, Jörg Wrachtrup, and Lloyd C.L. Hollenberg, “The nitrogen-vacancy colour centre in diamond,” Physics Reports 528, 1–45 (2013), the nitrogen-vacancy colour centre in diamond.
  * Balasubramanian  _et al._ (2009) Gopalakrishnan Balasubramanian, Philipp Neumann, Daniel Twitchen, Matthew Markham, Roman Kolesov, Norikazu Mizuochi, Junichi Isoya, Jocelyn Achard, Johannes Beck, Julia Tissler, Vincent Jacques, Philip R. Hemmer, Fedor Jelezko, and Jörg Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nature Materials 8, 383–387 (2009).
  * Bar-Gill  _et al._ (2013) N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, “Solid-state electronic spin coherence time approaching one second,” Nature Communications 4, 1743 (2013).
  * Abobeih  _et al._ (2018) M. H. Abobeih, J. Cramer, M. A. Bakker, N. Kalb, M. Markham, D. J. Twitchen, and T. H. Taminiau, “One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin environment,” Nature Communications 9, 2552 (2018).
  * 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, “Fault-tolerant operation of a logical qubit in a diamond quantum processor,” Nature 606, 884–889 (2022).
  * Togan  _et al._ (2010) E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010).
  * 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, “Heralded entanglement between solid-state qubits separated by three metres,” Nature 497, 86–90 (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, “Unconditional quantum teleportation between distant solid-state quantum bits,” Science 345, 532–535 (2014).
  * 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, “Realization of a multinode quantum network of remote solid-state qubits,” Science 372, 259–264 (2021).
  * Hermans  _et al._ (2022) S. L. N. Hermans, M. Pompili, H. K. C. Beukers, S. Baier, J. Borregaard, and R. Hanson, “Qubit teleportation between non-neighbouring nodes in a quantum network,” Nature 605, 663–668 (2022).
  * Hausmann  _et al._ (2013) B. J. M. Hausmann, B. J. Shields, Q. Quan, Y. Chu, N. P. de Leon, R. Evans, M. J. Burek, A. S. Zibrov, M. Markham, D. J. Twitchen, H. Park, M. D. Lukin, and M. Loncǎr, “Coupling of nv centers to photonic crystal nanobeams in diamond,” Nano Letters 13, 5791–5796 (2013).
  * Riedel  _et al._ (2017) Daniel Riedel, Immo Söllner, Brendan J. Shields, Sebastian Starosielec, Patrick Appel, Elke Neu, Patrick Maletinsky, and Richard J. Warburton, “Deterministic enhancement of coherent photon generation from a nitrogen-vacancy center in ultrapure diamond,” Phys. Rev. X 7, 031040 (2017).
  * 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, “Room-Temperature Quantum Bit Memory Exceeding One Second,” Science 336, 1283–1286 (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, “A ten-qubit solid-state spin register with quantum memory up to one minute,” Phys. Rev. X 9, 031045 (2019).
  * 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, “Entanglement of spin-pair qubits with intrinsic dephasing times exceeding a minute,” Phys. Rev. X 12, 011048 (2022).
  * Xie  _et al._ (2023) Tianyu Xie, Zhiyuan Zhao, Shaoyi Xu, Xi Kong, Zhiping Yang, Mengqi Wang, Ya Wang, Fazhan Shi, and Jiangfeng Du, “99.92%-fidelity cnot gates in solids by noise filtering,” Phys. Rev. Lett. 130, 030601 (2023).
  * Wang  _et al._ (2023) Guoqing Wang, Ariel Rebekah Barr, Hao Tang, Mo Chen, Changhao Li, Haowei Xu, Andrew Stasiuk, Ju Li, and Paola Cappellaro, “Characterizing temperature and strain variations with qubit ensembles for their robust coherence protection,” Phys. Rev. Lett. 131, 043602 (2023).
  * Diniz  _et al._ (2011) I. Diniz, S. Portolan, R. Ferreira, J. M. Gérard, P. Bertet, and A. Auffèves, “Strongly coupling a cavity to inhomogeneous ensembles of emitters: Potential for long-lived solid-state quantum memories,” Phys. Rev. A 84, 063810 (2011).
  * Zhu  _et al._ (2011) Xiaobo Zhu, Shiro Saito, Alexander Kemp, Kosuke Kakuyanagi, Shin-ichi Karimoto, Hayato Nakano, William J. Munro, Yasuhiro Tokura, Mark S. Everitt, Kae Nemoto, Makoto Kasu, Norikazu Mizuochi, and Kouichi Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nature 478, 221–224 (2011).
  * Kubo  _et al._ (2011) Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J.-F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid quantum circuit with a superconducting qubit coupled to a spin ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
  * Kubo  _et al._ (2012) Y. Kubo, I. Diniz, A. Dewes, V. Jacques, A. Dréau, J.-F. Roch, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Storage and retrieval of a microwave field in a spin ensemble,” Phys. Rev. A 85, 012333 (2012).
  * Heshami  _et al._ (2014) Khabat Heshami, Charles Santori, Behzad Khanaliloo, Chris Healey, Victor M. Acosta, Paul E. Barclay, and Christoph Simon, “Raman quantum memory based on an ensemble of nitrogen-vacancy centers coupled to a microcavity,” Phys. Rev. A 89, 040301 (2014).
  * Kalachev  _et al._ (2019) A Kalachev, A Berezhnoi, P Hemmer, and O Kocharovskaya, “Raman quantum memory based on an ensemble of silicon-vacancy centers in diamond,” Laser Physics 29, 104001 (2019).
  * Bhaskar  _et al._ (2020) M. K. Bhaskar, R. Riedinger, B. Machielse, D. S. Levonian, C. T. Nguyen, E. N. Knall, H. Park, D. Englund, M. Lončar, D. D. Sukachev, and M. D. Lukin, “Experimental demonstration of memory-enhanced quantum communication,” Nature 580, 60–64 (2020).
  * Starling  _et al._ (2023) David J. Starling, Katia Shtyrkova, Ian Christen, Ryan Murphy, Linsen Li, Kevin C. Chen, Dave Kharas, Xingyu Zhang, John Cummings, W. John Nowak, Eric Bersin, Robert J. Niffenegger, Madison Sutula, Dirk Englund, Scott Hamilton, and P. Benjamin Dixon, “Fully packaged multichannel cryogenic quantum memory module,” Phys. Rev. Appl. 19, 064028 (2023).
  * Stas  _et al._ (2022) P.-J. Stas, Y. Q. Huan, B. Machielse, E. N. Knall, A. Suleymanzade, B. Pingault, M. Sutula, S. W. Ding, C. M. Knaut, D. R. Assumpcao, Y.-C. Wei, M. K. Bhaskar, R. Riedinger, D. D. Sukachev, H. Park, M. Lončar, D. S. Levonian, and M. D. Lukin, “Robust multi-qubit quantum network node with integrated error detection,” Science 378, 557–560 (2022).
  * Siyushev  _et al._ (2017) Petr Siyushev, Mathias H. Metsch, Aroosa Ijaz, Jan M. Binder, Mihir K. Bhaskar, Denis D. Sukachev, Alp Sipahigil, Ruffin E. Evans, Christian T. Nguyen, Mikhail D. Lukin, Philip R. Hemmer, Yuri N. Palyanov, Igor N. Kupriyanov, Yuri M. Borzdov, Lachlan J. Rogers, and Fedor Jelezko, “Optical and microwave control of germanium-vacancy center spins in diamond,” Phys. Rev. B 96, 081201 (2017).
  * Debroux  _et al._ (2021) Romain Debroux, Cathryn P. Michaels, Carola M. Purser, Noel Wan, Matthew E. Trusheim, Jesús Arjona Martínez, Ryan A. Parker, Alexander M. Stramma, Kevin C. Chen, Lorenzo de Santis, Evgeny M. Alexeev, Andrea C. Ferrari, Dirk Englund, Dorian A. Gangloff, and Mete Atatüre, “Quantum control of the tin-vacancy spin qubit in diamond,” Phys. Rev. X 11, 041041 (2021).
  * Maurand  _et al._ (2016) R. Maurand, X. Jehl, D. Kotekar-Patil, A. Corna, H. Bohuslavskyi, R. Laviéville, L. Hutin, S. Barraud, M. Vinet, M. Sanquer, and S. De Franceschi, “A cmos silicon spin qubit,” Nature Communications 7, 13575 (2016).
  * Vandersypen  _et al._ (2017) L. M. K. Vandersypen, H. Bluhm, J. S. Clarke, A. S. Dzurak, R. Ishihara, A. Morello, D. J. Reilly, L. R. Schreiber, and M. Veldhorst, “Interfacing spin qubits in quantum dots and donors—hot, dense, and coherent,” npj Quantum Information 3, 34 (2017).
  * Fedele  _et al._ (2021) Federico Fedele, Anasua Chatterjee, Saeed Fallahi, Geoffrey C. Gardner, Michael J. Manfra, and Ferdinand Kuemmeth, “Simultaneous operations in a two-dimensional array of singlet-triplet qubits,” PRX Quantum 2, 040306 (2021).
  * Zajac  _et al._ (2018) D. M. Zajac, A. J. Sigillito, M. Russ, F. Borjans, J. M. Taylor, G. Burkard, and J. R. Petta, “Resonantly driven CNOT gate for electron spins,” Science 359, 439–442 (2018).
  * Mills  _et al._ (2022) Adam R. Mills, Charles R. Guinn, Michael J. Gullans, Anthony J. Sigillito, Mayer M. Feldman, Erik Nielsen, and Jason R. Petta, “Two-qubit silicon quantum processor with operation fidelity exceeding 99%,” Science Advances 8, eabn5130 (2022).
  * Burkard  _et al._ (2023) Guido Burkard, Thaddeus D. Ladd, Andrew Pan, John M. Nichol, and Jason R. Petta, “Semiconductor spin qubits,” Rev. Mod. Phys. 95, 025003 (2023).
  * Loss and DiVincenzo (1998) Daniel Loss and David P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57, 120–126 (1998).
  * Xue  _et al._ (2022) Xiao Xue, Maximilian Russ, Nodar Samkharadze, Brennan Undseth, Amir Sammak, Giordano Scappucci, and Lieven M. K. Vandersypen, “Quantum logic with spin qubits crossing the surface code threshold,” Nature 601, 343–347 (2022).
  * Yoneda  _et al._ (2018) Jun Yoneda, Kenta Takeda, Tomohiro Otsuka, Takashi Nakajima, Matthieu R. Delbecq, Giles Allison, Takumu Honda, Tetsuo Kodera, Shunri Oda, Yusuke Hoshi, Noritaka Usami, Kohei M. Itoh, and Seigo Tarucha, “A quantum-dot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%,” Nature Nanotechnology 13, 102–106 (2018).
  * Tanttu  _et al._ (2023) Tuomo Tanttu, Wee Han Lim, Jonathan Y. Huang, Nard Dumoulin Stuyck, Will Gilbert, Rocky Y. Su, MengKe Feng, Jesus D. Cifuentes, Amanda E. Seedhouse, Stefan K. Seritan, Corey I. Ostrove, Kenneth M. Rudinger, Ross C. C. Leon, Wister Huang, Christopher C. Escott, Kohei M. Itoh, Nikolay V. Abrosimov, Hans-Joachim Pohl, Michael L. W. Thewalt, Fay E. Hudson, Robin Blume-Kohout, Stephen D. Bartlett, Andrea Morello, Arne Laucht, Chih Hwan Yang, Andre Saraiva, and Andrew S. Dzurak, “Stability of high-fidelity two-qubit operations in silicon,” (2023), arXiv:2303.04090 [quant-ph] .
  * Noiri  _et al._ (2022) Akito Noiri, Kenta Takeda, Takashi Nakajima, Takashi Kobayashi, Amir Sammak, Giordano Scappucci, and Seigo Tarucha, “Fast universal quantum gate above the fault-tolerance threshold in silicon,” Nature 601, 338–342 (2022).
  * Cerfontaine  _et al._ (2020) Pascal Cerfontaine, Tim Botzem, Julian Ritzmann, Simon Sebastian Humpohl, Arne Ludwig, Dieter Schuh, Dominique Bougeard, Andreas D. Wieck, and Hendrik Bluhm, “Closed-loop control of a gaas-based singlet-triplet spin qubit with 99.5% gate fidelity and low leakage,” Nature Communications 11, 4144 (2020).
  * Takeda  _et al._ (2020) K. Takeda, A. Noiri, J. Yoneda, T. Nakajima, and S. Tarucha, “Resonantly driven singlet-triplet spin qubit in silicon,” Phys. Rev. Lett. 124, 117701 (2020).
  * Nichol  _et al._ (2017) John M. Nichol, Lucas A. Orona, Shannon P. Harvey, Saeed Fallahi, Geoffrey C. Gardner, Michael J. Manfra, and Amir Yacoby, “High-fidelity entangling gate for double-quantum-dot spin qubits,” npj Quantum Information 3, 3 (2017).
  * Weinstein  _et al._ (2023) Aaron J. Weinstein, Matthew D. Reed, Aaron M. Jones, Reed W. Andrews, David Barnes, Jacob Z. Blumoff, Larken E. Euliss, Kevin Eng, Bryan H. Fong, Sieu D. Ha, Daniel R. Hulbert, Clayton A. C. Jackson, Michael Jura, Tyler E. Keating, Joseph Kerckhoff, Andrey A. Kiselev, Justine Matten, Golam Sabbir, Aaron Smith, Jeffrey Wright, Matthew T. Rakher, Thaddeus D. Ladd, and Matthew G. Borselli, “Universal logic with encoded spin qubits in silicon,” Nature 615, 817–822 (2023).
  * DiVincenzo  _et al._ (2000) D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard, and K. B. Whaley, “Universal quantum computation with the exchange interaction,” Nature 408, 339–342 (2000).
  * Bacon  _et al._ (2000) D. Bacon, J. Kempe, D. A. Lidar, and K. B. Whaley, “Universal fault-tolerant quantum computation on decoherence-free subspaces,” Phys. Rev. Lett. 85, 1758–1761 (2000).
  * Stockklauser  _et al._ (2017) A. Stockklauser, P. Scarlino, J. V. Koski, S. Gasparinetti, C. K. Andersen, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, “Strong coupling cavity qed with gate-defined double quantum dots enabled by a high impedance resonator,” Phys. Rev. X 7, 011030 (2017).
  * Mi  _et al._ (2017) X. Mi, J. V. Cady, D. M. Zajac, P. W. Deelman, and J. R. Petta, “Strong coupling of a single electron in silicon to a microwave photon,” Science 355, 156–158 (2017).
  * Scarlino  _et al._ (2019) P. Scarlino, D. J. van Woerkom, U. C. Mendes, J. V. Koski, A. J. Landig, C. K. Andersen, S. Gasparinetti, C. Reichl, W. Wegscheider, K. Ensslin, T. Ihn, A. Blais, and A. Wallraff, “Coherent microwave-photon-mediated coupling between a semiconductor and a superconducting qubit,” Nature Communications 10, 3011 (2019).
  * Mi  _et al._ (2018) X. Mi, M. Benito, S. Putz, D. M. Zajac, J. M. Taylor, Guido Burkard, and J. R. Petta, “A coherent spin–photon interface in silicon,” Nature 555, 599–603 (2018).
  * Landig  _et al._ (2018) A. J. Landig, J. V. Koski, P. Scarlino, U. C. Mendes, A. Blais, C. Reichl, W. Wegscheider, A. Wallraff, K. Ensslin, and T. Ihn, “Coherent spin–photon coupling using a resonant exchange qubit,” Nature 560, 179–184 (2018).
  * Landig  _et al._ (2019) A. J. Landig, J. V. Koski, P. Scarlino, C. Müller, J. C. Abadillo-Uriel, B. Kratochwil, C. Reichl, W. Wegscheider, S. N. Coppersmith, Mark Friesen, A. Wallraff, T. Ihn, and K. Ensslin, “Virtual-photon-mediated spin-qubit–transmon coupling,” Nature Communications 10, 5037 (2019).
  * Samkharadze  _et al._ (2018) N. Samkharadze, G. Zheng, N. Kalhor, D. Brousse, A. Sammak, U. C. Mendes, A. Blais, G. Scappucci, and L. M. K. Vandersypen, “Strong spin-photon coupling in silicon,” Science 359, 1123–1127 (2018).
  * Andrews  _et al._ (2014) R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nature Physics 10, 321–326 (2014).
  * Bagci  _et al._ (2014) T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves through a nanomechanical transducer,” Nature 507, 81–85 (2014).
  * Shao  _et al._ (2019) Linbo Shao, Mengjie Yu, Smarak Maity, Neil Sinclair, Lu Zheng, Cleaven Chia, Amirhassan Shams-Ansari, Cheng Wang, Mian Zhang, Keji Lai, and Marko Lončar, “Microwave-to-optical conversion using lithium niobate thin-film acoustic resonators,” Optica 6, 1498–1505 (2019).
  * Forsch  _et al._ (2020) Moritz Forsch, Robert Stockill, Andreas Wallucks, Igor Marinković, Claus Gärtner, Richard A. Norte, Frank van Otten, Andrea Fiore, Kartik Srinivasan, and Simon Gröblacher, “Microwave-to-optics conversion using a mechanical oscillator in its quantum ground state,” Nature Physics 16, 69–74 (2020).
  * Zhong  _et al._ (2020) Changchun Zhong, Zhixin Wang, Changling Zou, Mengzhen Zhang, Xu Han, Wei Fu, Mingrui Xu, S. Shankar, Michel H. Devoret, Hong X. Tang, and Liang Jiang, “Proposal for heralded generation and detection of entangled microwave–optical-photon pairs,” Phys. Rev. Lett. 124, 010511 (2020).
  * Xu  _et al._ (2021a) Yuntao Xu, Ayed Al Sayem, Linran Fan, Chang-Ling Zou, Sihao Wang, Risheng Cheng, Wei Fu, Likai Yang, Mingrui Xu, and Hong X. Tang, “Bidirectional interconversion of microwave and light with thin-film lithium niobate,” Nature Communications 12, 4453 (2021a).
  * Mirhosseini  _et al._ (2020) Mohammad Mirhosseini, Alp Sipahigil, Mahmoud Kalaee, and Oskar Painter, “Superconducting qubit to optical photon transduction,” Nature 588, 599–603 (2020).
  * Vasileiadis  _et al._ (2021) Thomas Vasileiadis, Jeena Varghese, Visnja Babacic, Jordi Gomis-Bresco, Daniel Navarro Urrios, and Bartlomiej Graczykowski, “Progress and perspectives on phononic crystals,” Journal of Applied Physics 129, 160901 (2021).
  * Bachtold  _et al._ (2022) Adrian Bachtold, Joel Moser, and M. I. Dykman, “Mesoscopic physics of nanomechanical systems,” Rev. Mod. Phys. 94, 045005 (2022).
  * Heinrich  _et al._ (2021) Andreas J. Heinrich, William D. Oliver, Lieven M. K. Vandersypen, Arzhang Ardavan, Roberta Sessoli, Daniel Loss, Ania Bleszynski Jayich, Joaquin Fernandez-Rossier, Arne Laucht, and Andrea Morello, “Quantum-coherent nanoscience,” Nature Nanotechnology 16, 1318–1329 (2021).
  * Seis  _et al._ (2022) Yannick Seis, Thibault Capelle, Eric Langman, Sampo Saarinen, Eric Planz, and Albert Schliesser, “Ground state cooling of an ultracoherent electromechanical system,” Nature Communications 13, 1507 (2022).
  * Beccari  _et al._ (2022) A. Beccari, D. A. Visani, S. A. Fedorov, M. J. Bereyhi, V. Boureau, N. J. Engelsen, and T. J. Kippenberg, “Strained crystalline nanomechanical resonators with quality factors above 10 billion,” Nature Physics 18, 436–441 (2022).
  * MacCabe  _et al._ (2020) Gregory S. MacCabe, Hengjiang Ren, Jie Luo, Justin D. Cohen, Hengyun Zhou, Alp Sipahigil, Mohammad Mirhosseini, and Oskar Painter, “Nano-acoustic resonator with ultralong phonon lifetime,” Science 370, 840–843 (2020).
  * Teufel  _et al._ (2011) J. D. Teufel, T. Donner, Dale Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
  * Wollack  _et al._ (2022) E. Alex Wollack, Agnetta Y. Cleland, Rachel G. Gruenke, Zhaoyou Wang, Patricio Arrangoiz-Arriola, and Amir H. Safavi-Naeini, “Quantum state preparation and tomography of entangled mechanical resonators,” Nature 604, 463–467 (2022).
  * von Lüpke  _et al._ (2022) Uwe von Lüpke, Yu Yang, Marius Bild, Laurent Michaud, Matteo Fadel, and Yiwen Chu, “Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics,” Nature Physics 18, 794–799 (2022).
  * Stockill  _et al._ (2022) Robert Stockill, Moritz Forsch, Frederick Hijazi, Grégoire Beaudoin, Konstantinos Pantzas, Isabelle Sagnes, Rémy Braive, and Simon Gröblacher, “Ultra-low-noise microwave to optics conversion in gallium phosphide,” Nature Communications 13, 6583 (2022).
  * Bravyi and Kitaev (1998) S. B. Bravyi and A. Yu. Kitaev, “Quantum codes on a lattice with boundary,” (1998), arXiv:quant-ph/9811052 [quant-ph] .
  * Dennis  _et al._ (2002) Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, “Topological quantum memory,” Journal of Mathematical Physics 43, 4452–4505 (2002).
  * Brown  _et al._ (2016) Benjamin J. Brown, Daniel Loss, Jiannis K. Pachos, Chris N. Self, and James R. Wootton, “Quantum memories at finite temperature,” Rev. Mod. Phys. 88, 045005 (2016).
  * Alicea  _et al._ (2011) Jason Alicea, Yuval Oreg, Gil Refael, Felix von Oppen, and Matthew P. A. Fisher, “Non-abelian statistics and topological quantum information processing in 1d wire networks,” Nature Physics 7, 412–417 (2011).
  * Sarma  _et al._ (2015) Sankar Das Sarma, Michael Freedman, and Chetan Nayak, “Majorana zero modes and topological quantum computation,” npj Quantum Information 1, 15001 (2015).
  * Chirolli  _et al._ (2022) Luca Chirolli, Norman Y. Yao, and Joel E. Moore, “Swap gate between a majorana qubit and a parity-protected superconducting qubit,” Phys. Rev. Lett. 129, 177701 (2022).
  * Roy and DiVincenzo (2017) Ananda Roy and David P. DiVincenzo, “Topological quantum computing,” (2017), arXiv:1701.05052 [quant-ph] .
  * Lahtinen and Pachos (2017) Ville Lahtinen and Jiannis K. Pachos, “A Short Introduction to Topological Quantum Computation,” SciPost Phys. 3, 021 (2017).
  * Liu  _et al._ (2022) Junyu Liu, Connor T. Hann, and Liang Jiang, “Quantum data center: Theories and applications,” (2022), arXiv:2207.14336 [quant-ph] .
  * Chang  _et al._ (2019) W. Chang, C. Li, Y.-K. Wu, N. Jiang, S. Zhang, Y.-F. Pu, X.-Y. Chang, and L.-M. Duan, “Long-distance entanglement between a multiplexed quantum memory and a telecom photon,” Phys. Rev. X 9, 041033 (2019).
  * O’Sullivan  _et al._ (2022) James O’Sullivan, Oscar W. Kennedy, Kamanasish Debnath, Joseph Alexander, Christoph W. Zollitsch, Mantas Šimėnas, Akel Hashim, Christopher N. Thomas, Stafford Withington, Irfan Siddiqi, Klaus Mølmer, and John J. L. Morton, “Random-access quantum memory using chirped pulse phase encoding,” Phys. Rev. X 12, 041014 (2022).
  * Besse  _et al._ (2020) Jean-Claude Besse, Kevin Reuer, Michele C. Collodo, Arne Wulff, Lucien Wernli, Adrian Copetudo, Daniel Malz, Paul Magnard, Abdulkadir Akin, Mihai Gabureac, Graham J. Norris, J. Ignacio Cirac, Andreas Wallraff, and Christopher Eichler, “Realizing a deterministic source of multipartite-entangled photonic qubits,” Nature Communications 11, 4877 (2020).
  * Reuer  _et al._ (2022) Kevin Reuer, Jean-Claude Besse, Lucien Wernli, Paul Magnard, Philipp Kurpiers, Graham J. Norris, Andreas Wallraff, and Christopher Eichler, “Realization of a universal quantum gate set for itinerant microwave photons,” Phys. Rev. X 12, 011008 (2022).
  * Delaney  _et al._ (2022) RD Delaney, MD Urmey, S Mittal, BM Brubaker, JM Kindem, PS Burns, CA Regal, and KW Lehnert, “Superconducting-qubit readout via low-backaction electro-optic transduction,” Nature 606, 489–493 (2022).
  * Sahu  _et al._ (2022) Rishabh Sahu, William Hease, Alfredo Rueda, Georg Arnold, Liu Qiu, and Johannes M Fink, “Quantum-enabled operation of a microwave-optical interface,” Nat. Commun. 13, 1276 (2022).
  * Xu  _et al._ (2021b) Yuntao Xu, Ayed Al Sayem, Linran Fan, Chang-Ling Zou, Sihao Wang, Risheng Cheng, Wei Fu, Likai Yang, Mingrui Xu, and Hong X Tang, “Bidirectional interconversion of microwave and light with thin-film lithium niobate,” Nat. Commun. 12, 4453 (2021b).
  * Ang  _et al._ (2022) James Ang, Gabriella Carini, Yanzhu Chen, Isaac Chuang, Michael Austin DeMarco, Sophia E. Economou, Alec Eickbusch, Andrei Faraon, Kai-Mei Fu, Steven M. Girvin, Michael Hatridge, Andrew Houck, Paul Hilaire, Kevin Krsulich, Ang Li, Chenxu Liu, Yuan Liu, Margaret Martonosi, David C. McKay, James Misewich, Mark Ritter, Robert J. Schoelkopf, Samuel A. Stein, Sara Sussman, Hong X. Tang, Wei Tang, Teague Tomesh, Norm M. Tubman, Chen Wang, Nathan Wiebe, Yong-Xin Yao, Dillon C. Yost, and Yiyu Zhou, “Architectures for multinode superconducting quantum computers,” (2022), arXiv:2212.06167 [quant-ph] .
  * Giovannetti  _et al._ (2008a) Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone, “Quantum random access memory,” Phys. Rev. Lett. 100, 160501 (2008a).
  * Giovannetti  _et al._ (2008b) Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone, “Architectures for a quantum random access memory,” Phys. Rev. A 78, 052310 (2008b).
  * Hong  _et al._ (2012) Fang-Yu Hong, Yang Xiang, Zhi-Yan Zhu, Li-zhen Jiang, and Liang-neng Wu, “Robust quantum random access memory,” Phys. Rev. A 86, 010306 (2012).
  * Harrow  _et al._ (2009) Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, “Quantum algorithm for linear systems of equations,” Phys. Rev. Lett. 103, 150502 (2009).
  * Hann  _et al._ (2019) Connor T. Hann, Chang-Ling Zou, Yaxing Zhang, Yiwen Chu, Robert J. Schoelkopf, S. M. Girvin, and Liang Jiang, “Hardware-efficient quantum random access memory with hybrid quantum acoustic systems,” Phys. Rev. Lett. 123, 250501 (2019).
  * Chen  _et al._ (2021b) K. C. Chen, W. Dai, C. Errando-Herranz, S. Lloyd, and D. Englund, “Scalable and high-fidelity quantum random access memory in spin-photon networks,” PRX Quantum 2, 030319 (2021b).
  * Asaka  _et al._ (2021) Ryo Asaka, Kazumitsu Sakai, and Ryoko Yahagi, “Quantum random access memory via quantum walk,” Quantum Science and Technology 6, 035004 (2021).
  * Asaka  _et al._ (2023a) Ryo Asaka, Kazumitsu Sakai, and Ryoko Yahagi, “Two-level quantum walkers on directed graphs. i. universal quantum computing,” Phys. Rev. A 107, 022415 (2023a).
  * Asaka  _et al._ (2023b) Ryo Asaka, Kazumitsu Sakai, and Ryoko Yahagi, “Two-level quantum walkers on directed graphs. ii. application to quantum random access memory,” Phys. Rev. A 107, 022416 (2023b).
  * Park  _et al._ (2019) Daniel K. Park, Francesco Petruccione, and June-Koo Kevin Rhee, “Circuit-based quantum random access memory for classical data,” Scientific Reports 9, 3949 (2019).
  * Matteo  _et al._ (2020) Olivia Di Matteo, Vlad Gheorghiu, and Michele Mosca, “Fault-tolerant resource estimation of quantum random-access memories,” IEEE Transactions on Quantum Engineering 1, 1–13 (2020).
  * Paler  _et al._ (2020) Alexandru Paler, Oumarou Oumarou, and Robert Basmadjian, “Parallelizing the queries in a bucket-brigade quantum random access memory,” Phys. Rev. A 102, 032608 (2020).
  * Arunachalam  _et al._ (2015) Srinivasan Arunachalam, Vlad Gheorghiu, Tomas Jochym-O’Connor, Michele Mosca, and Priyaa Varshinee Srinivasan, “On the robustness of bucket brigade quantum ram,” New Journal of Physics 17, 123010 (2015).
  * Hann  _et al._ (2021) Connor T. Hann, Gideon Lee, S.M. Girvin, and Liang Jiang, “Resilience of quantum random access memory to generic noise,” PRX Quantum 2, 020311 (2021).
  * Chen  _et al._ (2023) Zhao-Yun Chen, Cheng Xue, Yun-Jie Wang, Tai-Ping Sun, Huan-Yu Liu, Xi-Ning Zhuang, Meng-Han Dou, Tian-Rui Zou, Yuan Fang, Yu-Chun Wu, and Guo-Ping Guo, “Efficient and error-resilient data access protocols for a limited-sized quantum random access memory,” (2023), arXiv:2303.05207 [quant-ph] .
  * Xu  _et al._ (2023) Shifan Xu, Connor T. Hann, Ben Foxman, Steven M. Girvin, and Yongshan Ding, “Systems architecture for quantum random access memory,” (2023), arXiv:2306.03242 [quant-ph] .
  * Hann (2021) Connor Hann, “Practicality of quantum random access memory,”  (2021), Ph.D. Dissertation.
  * Phalak  _et al._ (2023) Koustubh Phalak, Avimita Chatterjee, and Swaroop Ghosh, “Quantum random access memory for dummies,” (2023), arXiv:2305.01178 [quant-ph] .
  * Grover (1996) Lov K. Grover, “A fast quantum mechanical algorithm for database search,” in  _Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing_, STOC ’96 (ACM, New York, NY, USA, 1996) pp. 212–219.
  * Wu  _et al._ (2022) Anbang Wu, Yufei Ding, and Ang Li, “Collcomm: Enabling efficient collective quantum communication based on epr buffering,” (2022), arXiv:2208.06724 [quant-ph] .
  * Cross  _et al._ (2017) Andrew W. Cross, Lev S. Bishop, John A. Smolin, and Jay M. Gambetta, “Open quantum assembly language,” (2017), arXiv:1707.03429 [quant-ph] .
  * Cross  _et al._ (2022) Andrew Cross, Ali Javadi-Abhari, Thomas Alexander, Niel De Beaudrap, Lev S. Bishop, Steven Heidel, Colm A. Ryan, Prasahnt Sivarajah, John Smolin, Jay M. Gambetta, and Blake R. Johnson, “Openqasm 3: A broader and deeper quantum assembly language,” ACM Transactions on Quantum Computing 3 (2022), 10.1145/3505636.
  * Qiskit contributors (2023) Qiskit contributors, “Qiskit: An open-source framework for quantum computing,”  (2023).
  * Biamonte  _et al._ (2017) Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd, “Quantum machine learning,” Nature 549, 195–202 (2017).
  * Cerezo  _et al._ (2022) M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J. Coles, “Challenges and opportunities in quantum machine learning,” Nature Computational Science 2, 567–576 (2022).
  * Buhrman  _et al._ (2001) Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf, “Quantum fingerprinting,” Phys. Rev. Lett. 87, 167902 (2001).