# Benchmarking Digital-Analog Quantum Computation

Vicente Pina Canelles<EMAIL_ADDRESS>IQM Germany GmbH,
Nymphenburgerstrasse 86, 80636 Munich, Germany  Department of Physics and
Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-
Universität München, Theresienstrasse 37, 80333 Munich, Germany  Manuel G.
Algaba  IQM Germany GmbH, Nymphenburgerstrasse 86, 80636 Munich, Germany
Hermanni Heimonen  IQM Finland Oy, Keilaranta 19, 02150 Espoo, Finland.  Miha
Papič  IQM Germany GmbH, Nymphenburgerstrasse 86, 80636 Munich, Germany
Department of Physics and Arnold Sommerfeld Center for Theoretical Physics,
Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 Munich,
Germany  Mario Ponce  IQM Germany GmbH, Nymphenburgerstrasse 86, 80636 Munich,
Germany  Department of Physics and Arnold Sommerfeld Center for Theoretical
Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333
Munich, Germany  Jami Rönkkö  IQM Finland Oy, Keilaranta 19, 02150 Espoo,
Finland.  Manish J. Thapa  IQM Germany GmbH, Nymphenburgerstrasse 86, 80636
Munich, Germany  Inés de Vega  IQM Germany GmbH, Nymphenburgerstrasse 86,
80636 Munich, Germany  Department of Physics and Arnold Sommerfeld Center for
Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse
37, 80333 Munich, Germany  Adrian Auer  IQM Germany GmbH, Nymphenburgerstrasse
86, 80636 Munich, Germany

(August 27, 2024)

###### Abstract

Digital-Analog Quantum Computation (DAQC) has recently been proposed as an
alternative to the standard paradigm of digital quantum computation. DAQC
creates entanglement through a continuous or analog evolution of the whole
device, rather than by applying two-qubit gates. This manuscript describes an
in-depth analysis of DAQC by extending its implementation to arbitrary
connectivities and by performing the first systematic study of its scaling
properties. We specify the analysis for three examples of quantum algorithms,
showing that except for a few specific cases, DAQC is in fact disadvantageous
with respect to the digital case.

††preprint: APS/123-QED

##  I Introduction

Digital-Analog Quantum Computation (DAQC) has recently emerged as a new
paradigm, posing an alternative to standard Digital Quantum Computation (DQC)
[1, 2, 3, 4, 5, 6, 7, 8, 9]. The objective of DAQC is to utilize the natural
evolution of a quantum device, generated by a given entangling Hamiltonian,
with engineered control only over single-qubit gates (SQGs) to perform quantum
computations. The reasoning is that such systems ought to be much simpler to
experimentally operate than the fine control required for digital quantum
computation. The analog evolution of the device has also been argued to be
more robust against control errors than two-qubit gates (TQGs) [10], which are
the typical entangling operations utilized in DQC. While this analog evolution
is not universal (it cannot implement an arbitrary unitary evolution),
combining it with SQGs can make it universal, by effectively engineering the
evolution under arbitrary Hamiltonians.

Although ideas similar to DAQC have been discussed in Refs. [11, 12, 13],
where multi-qubit gates combined with single qubit gates are explored, the
proposal in Refs. [1, 2] is particularly interesting because it does not
require a specific qubit connectivity, as we show in this work, and can
therefore be applied to many different promising quantum computing
architectures (such as trapped ions, superconducting circuits or Rydberg
atoms). In this regard, Refs. [1, 2] have described the method for
implementing universal quantum computation using DAQC for two types of
devices, defined by their connectivities: all-to-all (ATA), in which all
qubits are directly coupled to all other qubits [1], and an open linear chain,
in which all qubits are coupled to their nearest neighbors in a one-
dimensional array [2]. Additionally to several simulation protocols [9, 7, 8],
DAQC algorithms have been proposed to implement the Quantum Fourier Transform
(QFT) routine [3, 4], a instance of the Quantum Phase Estimation (QPE)
algorithm [4], the Harrow-Hassidim-Lloyd algorithm for solving linear systems
of equations [5], and the Quantum Approximate Optimization Algorithm (QAOA)
[6]. These works offer preliminary analyses on the scaling properties of DAQC
with respect to DQC, as well as on the impact of noise and errors in their
performance.

Thus, while DAQC appears as a promising alternative to the digital case, a
complete analysis of its implementation in arbitrary device connectivities,
its scaling properties, as well as the impact of errors in its performance is
still missing in the literature [14]. This manuscript tackles this problem by
providing the first systematic study of the limitations and potential of DAQC:
(i) we provide a generalization of this paradigm for any type of connectivity,
(ii) we provide an analysis of the error scaling of DAQC with the number of
qubits of the device, considering a detailed account of the number of
operations introduced, identifying and accounting for major sources of error,
and (iii) focusing on the QFT algorithm and the Greenberger-Horne-Zeilinger
(GHZ) state [15] preparation, we show how specific connectivities, in this
case a star layout, may have a positive impact in the scaling properties of
DAQC due to a reduction in the number of analog blocks. Throughout our study,
we consider the two versions of DAQC proposed in the literature: stepwise
DAQC, consisting of a sequential approach where all the interactions are
simultaneously switched on and off between layers of single qubit gates,
simplifying the theoretical analysis, and the experimentally attractive banged
DAQC where an always-on multi-qubit interaction is overlayed with fast single-
qubit pulses.

Our analysis allows us to conclude that, in general, DAQC scales unfavorably
with respect to the digital paradigm, except for some cases in which a
specific algorithm is implemented on a tailored device connectivity. In this
regard, the closer the computation’s Hamiltonian is to that of the device, the
better the scaling properties of DAQC.

The manuscript is organized as follows. In Section II, we develop general
stepwise and banged DAQC methods for devices with an arbitrary connectivity.
In Section III, we perform a theoretical analysis of the error scaling of
DAQC, and compare it to DQC. In Section IV, we write a specific DAQC protocol
for a star-connectivity device, similar to the one for a one-dimensional chain
of [2], that significantly improves the results of DAQC. In Section V, we
perform an error analysis of the QFT algorithm, and, in Section VI, of the GHZ
state preparation routine. Finally, in Section VII, we perform numerical
simulations for three digital-analog algorithms and compare their performance
against their digital counterparts, in terms of fidelity and time of
execution.

##  II Digital-analog quantum computation with arbitrary connectivity

Currently existing DAQC algorithms have been developed for an ATA qubit
connectivity [1, 3, 4, 6, 9] and for a one-dimensional qubit chain with
nearest neighbor couplings [2, 9]. However, promising quantum computing
architectures like those based on superconducting qubits consist of planar
devices where only local interactions with nearest neighbors can be natively
implemented. In this section we provide a generalization by developing a
protocol for implementing universal digital-analog quantum computation on a
device with an arbitrary connectivity. A more succinct method to achieve such
a general protocol is described in [9], which utilizes the ATA case as a
starting point. For the sake of completeness, in this section we describe our
method from the ground up.

###  II.1 Resource and Target Hamiltonian

Throughout this manuscript, we distinguish between resource and target
Hamiltonians:

  * •

The resource Hamiltonian is the entangling Hamiltonian according to which the
qubits of a device evolve naturally, when all interactions are turned on [1].
Its coupling coefficients are assumed to be constant and non-tunable during
the computation, though they can be turned on or off simultaneously as
desired. In the following, we denote resource Hamiltonians as $\bar{H}$.

  * •

The target Hamiltonian is the entangling Hamiltonian that generates a specific
unitary which we wish to implement. Its coupling coefficients can be chosen
arbitrarily, depending on the computation to be implemented. We denote target
Hamiltonians as $H$.

We assume that the resource Hamiltonians are of $ZZ$-Ising type and that the
target Hamiltonians that we wish to implement are also of the $ZZ$-Ising type,

$\displaystyle\bar{H}_{\mathcal{C}}$
$\displaystyle=\sum_{(j,k)\in\mathcal{C}}\bar{g}_{jk}Z^{j}Z^{k}\,,$ (1)
$\displaystyle H_{\mathcal{C}}$
$\displaystyle=\sum_{(j,k)\in\mathcal{C}}g_{jk}Z^{j}Z^{k}\,,$ (2)

where, formally, we have defined the connectivity of a device (i.e., that of
its resource Hamiltonian) as the collection of $c$ pairs of qubits that are
connected, and we write it as $\mathcal{C}=\\{(j,k)\\}$, where $j,k$ are qubit
indices and $k>j$. Additionally, $Z^{j}$ is the Pauli-$Z$ operator acting on
qubit $j$,

$Z=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$ (3)

and $\bar{g}_{jk}$ ($g_{jk}$) are the coupling coefficients of the resource
(target) Hamiltonian. A method for other types of two-body Hamiltonians is
given in Ref. [9], which, utilizing significantly more resources, is able to
engineer target Hamiltonians with arbitrary Pauli operators using resource
Hamiltonians with other arbitrary Pauli operators. In this manuscript we study
algorithms that require only $ZZ$-type target Hamiltonians, so the assumptions
of Eqs. (1) and (2) are valid for our purposes.

An analog block is the multi-qubit entangling operation consisting on the
evolution of all qubits under the resource Hamiltonian, for a finite and
tunable time $t$,

$U_{\bar{H}_{\mathcal{C}}}(t)=\exp(-i\,t\,\bar{H}_{\mathcal{C}})\,,$ (4)

where (and from now on in this manuscript) we have set $\hbar=1$ and work in
natural units.

The evolution unitary $U_{H_{\mathcal{C}}}$ under the target Hamiltonian
$H_{\mathcal{C}}$, for some time $t_{f}$, is given by

$\displaystyle U_{H_{\mathcal{C}}}(t_{f})$
$\displaystyle=\exp(i\,t_{f}\,H_{\mathcal{C}})$ (5)
$\displaystyle=\exp\left(i\,t_{f}\,\sum_{(j,k)\in\mathcal{C}}g_{jk}Z^{j}Z^{k}\right)$
(6)
$\displaystyle=\prod_{(j,k)\in\mathcal{C}}\exp\left(i\,t_{f}\,g_{jk}\,Z^{j}Z^{k}\right)\,,$
(7)

which is equivalent to implementing $c$ two-qubit gates (TQGs) of the form

$ZZ^{jk}(\phi_{jk})=e^{i\phi_{jk}Z^{j}Z^{k}}\,,$ (8)

with phases $\phi_{jk}=t_{f}g_{jk}\mod(2\pi)$ (see Fig. 1). The set of
operations comprising such unitaries $U_{H_{\mathcal{C}}}$ and arbitrary SQGs
is universal [1]. Therefore, any quantum algorithm can be written as a
combination of SQGs and the evolution under such target Hamiltonians, which
themselves can be expressed as combinations of SQGs and analog blocks as we
will explain in the following subsections. Thus, analog blocks along with SQGs
are universal. We provide an example of how a quantum algorithm can be
decomposed into such operations in Section V.

Figure 1: Digital circuit comprising $ZZ^{jk}(\phi_{jk})$ gates, equivalent to
the evolution (7) under a given target Hamiltonian (2), for some time $t_{f}$,
in a device with four qubits and ATA connectivity.

In the following subsections, we explain how one can effectively implement an
arbitrary target Hamiltonian by making use of SQGs and a given resource
Hamiltonian.

###  II.2 The stepwise digital-analog quantum circuit

The digital-analog quantum circuit we will describe in this subsection is
constructed in the so-called stepwise DAQC (sDAQC) paradigm [1], as opposed to
the banged DAQC (bDAQC) paradigm [1, 6] that will be discussed in section
II.4. The defining characteristic of sDAQC is our ability to implement analog
blocks with a defined beginning and end, by turning on and off all the
interactions simultaneously.

The resource Hamiltonian’s coupling coefficients
$\\{\bar{g}_{jk}\\}_{(j,k)\in\mathcal{C}}$ are fixed by definition, though we
assume that the qubits of the device can interact for a certain time $t$ under
the resource Hamiltonian in Eq.(1) [1]. Because of this, we are only left with
tuning the time of the evolution. The core idea of a DAQC protocol is to find
a way to effectively engineer the desired coefficients of the target
Hamiltonian, $\\{g_{jk}\\}$, by tuning the times of the analog blocks of a
digital-analog quantum circuit (see Eq. (4)), which can comprise analog blocks
and SQGs. Inspired by the methods of [1, 2], we construct a digital-analog
quantum circuit which contains $c$ analog blocks, each running for some time
$t_{mn}$ (with the indices $m,n$ running over the number of qubits, similarly
to $j,k$), that implies a transformation

$\\{t_{mn}\\}_{(m,n)\in\mathcal{C}}\longrightarrow\\{g_{jk}\\}_{(j,k)\in\mathcal{C}}\,.$
(9)

We will provide the method for calculating the appropriate runtimes
$\\{t_{mn}\\}$ which effectively implement the correct coefficients
$\\{g_{jk}\\}$ in Sec. II.3, and for now concentrate on the construction of
the DAQC circuit.

For the sake of clarity, we denote qubit indices as $(j,k)$ or $(m,n)$ as a
shorthand notation for connected qubit pairs in the set $\mathcal{C}$. We
start our considerations with a quantum circuit that consists of $c$ analog
blocks, each running for some time $t_{mn}$. In order to be able to implement
a target Hamiltonian with arbitrary coefficients, firstly we need to
effectively modify the signs of the coupling coefficients within each analog
block. This is because we use the combined evolution of these different
effective analog blocks, with modified signs, to engineer the arbitrary target
Hamiltonian. To this end, we will interleave $X$ gates in between the analog
blocks, and make use of the identity [16]

$X^{a}Z^{b}X^{a}=(-1)^{\delta_{ab}}Z^{b}\,,$ (10)

where $X$ is the Pauli-$X$ operator,

$X=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$ (11)

Then, placing an $X^{a}$ gate before and after an analog block has the effect
of flipping the signs of all the terms in $\bar{H}_{\mathcal{C}}$ involving
the qubit $a$, effectively implementing a different Hamiltonian,
$\bar{H}_{\mathcal{C}}^{\prime}$

$\displaystyle U_{\bar{H}_{\mathcal{C}}^{\prime}}(t)$
$\displaystyle=X^{a}\exp(-it\bar{H}_{\mathcal{C}})X^{a}$ (12)
$\displaystyle=X^{a}\exp\left(-it\sum_{(j,k)}\bar{g}_{jk}Z^{j}Z^{k}\right)X^{a}$
(13)
$\displaystyle=\exp\left(-it\sum_{(j,k)}\bar{g}_{jk}X^{a}Z^{j}Z^{k}X^{a}\right)$
(14)
$\displaystyle=\exp\left(-it\sum_{(j,k)}(-1)^{\delta_{aj}+\delta_{ak}}\bar{g}_{jk}Z^{j}Z^{k}\right)\,,$
(15)

where we have used the property $Re^{itH}R^{\dagger}=e^{itRHR^{\dagger}}$,
provided that $R$ is unitary [17]. Using this procedure, we can implement
effective Hamiltonians that differ from the resource Hamiltonian by one or
more sign flips, in each of the $c$ analog blocks of the circuit.

Figure 2: Digital-Analog circuit consisting of $c$ analog blocks. Each analog
block runs for time $t_{mn}$, and is preceded and followed by the gates
$X^{m}X^{n}$, for each pair of connected qubits $(m,n)\in\mathcal{C}$.

Assume our quantum circuit is similar to that of Fig. 2, where each of the
analog blocks is preceded and followed by $X$ gates placed on the same
connected qubits appearing in the connectivity $\mathcal{C}$. This specific
way of placing the $X$ gates will allow us in the next subsection to derive
the explicit relationship between the times of the analog blocks and the
coefficients of the target Hamiltonian. The evolution of a quantum state
according to this circuit is given by

$\displaystyle U_{\mathrm{DAQC}}$
$\displaystyle=\prod_{(m,n)}X^{m}X^{n}\exp(-it_{mn}\bar{H}_{\mathcal{C}})X^{m}X^{n}$
(16)
$\displaystyle=\prod_{(m,n)}\exp(-it_{mn}X^{m}X^{n}\bar{H}_{\mathcal{C}}X^{m}X^{n})$
(17)
$\displaystyle\begin{split}&=\prod_{(m,n)}\text{exp}\Bigg{(}-i\sum_{(j,k)}t_{mn}\,\bar{g}_{jk}\\\
&\hphantom{=\prod_{(m,n)}\text{exp}\Bigg{(}}\times
X^{m}X^{n}Z^{j}Z^{k}X^{m}X^{n}\Bigg{)}\,.\end{split}$ (18)

###  II.3 Runtimes of the analog blocks in stepwise DAQC

We now turn towards the calculation of the runtimes $t_{mn}$ of the analog
blocks. Utilizing Eq. (10), we can write Eq. (18) as

$\displaystyle\begin{split}U_{\mathrm{DAQC}}&=\prod_{(m,n)}\text{exp}\Bigg{(}-i\sum_{(j,k)}t_{mn}\,\bar{g}_{jk}\\\
&\hphantom{=\prod_{(m,n)}\text{exp}\Bigg{(}}\times(-1)^{\delta_{mj}+\delta_{mk}+\delta_{nj}+\delta_{nk}}Z^{j}Z^{k}\Bigg{)}\end{split}$
(19)
$\displaystyle=\prod_{(m,n)}\exp\left(-i\sum_{(j,k)}t_{mn}\,\bar{g}_{jk}\,M_{mnjk}Z^{j}Z^{k}\right)$
(20)
$\displaystyle=\exp\left(-i\sum_{(m,n)}\sum_{(j,k)}t_{mn}\,\bar{g}_{jk}\,M_{mnjk}Z^{j}Z^{k}\right)\,.$
(21)

We have defined the tensor
$M_{mnjk}\equiv(-1)^{\delta_{mj}+\delta_{mk}+\delta_{nj}+\delta_{nk}}$
containing $c$ elements taking the values $\pm 1$. We can convert these
elements $M_{mnjk}$ into a $c\times c$ matrix with entries $M_{\alpha\beta}$
by “vectorizing” the pairs of coupled qubits
$(m,n)\rightarrow\alpha;(j,k)\rightarrow\beta$ characterized by a single index
each, as explained in Appendix A. This also “vectorizes” the times
$t_{mn}\rightarrow\mathbf{t}$ and the coupling coefficients
$\bar{g}_{jk}\rightarrow\mathbf{\bar{g}}$, $g_{jk}\rightarrow\mathbf{g}$.

The interpretation of the sign of a given element $M_{\alpha\beta}$ is the
following: if $M_{\alpha\beta}=+1$ ($-1$), it means that the effective
coupling corresponding to the $\alpha$-th connection, during the $\beta$-th
analog block, is positive (negative).

Let us compare now Eq. (21), which is the evolution we implement through the
DAQC protocol, with Eq. (7), which is the evolution under the target
Hamiltonian we wish to simulate. They are equal if the following vector
equation is fulfilled,

$\mathbf{G}\,t_{f}=M\mathbf{t}\,,$ (22)

where we define each element of $\mathbf{G}$ as
$G_{\beta}\equiv\frac{g_{\beta}}{\bar{g}_{\beta}}$.

The runtimes of each analog block can therefore be calculated, such that,
effectively, the time evolution under the target Hamiltonian is implemented,
by inverting the matrix $M$,

$\mathbf{t}=M^{-1}\mathbf{G}\,t_{f}\,.$ (23)

Eq. (23) allows us to find a vector of times $\mathbf{t}$ of the analog blocks
such that the circuit described above effectively implements the evolution
under the desired target Hamiltonian, provided that the matrix $M$ is
invertible. This invertibility must be studied on a case-by-case basis, and
the SQG placement may be shifted to produce a different matrix $M$, in this
case invertible, while still making the DAQC protocol universal [1].

The case in which the resource Hamiltonian does not have only $2$-body terms,
but rather up to $M$-body terms with $M\geq 3$, is described in Appendix B.
Alternatively, an efficient way to effectively get rid of all odd-body terms
(if present) in the resource Hamiltonian is explained in Appendix C.

###  II.4 Banged DAQC

In addition to sDAQC, which was described in the subsection above, another
paradigm exists to perform an approximate digital-analog quantum computation,
called banged DAQC (bDAQC) [1, 6]. The idea is that SQGs are applied
simultaneously to an analog block, which runs throughout the whole circuit
(see Fig. 3). The motivation behind bDAQC is that it does not require us to
“turn on and off” the analog Hamiltonian throughout the quantum circuit, but
rather it stays constantly on from beginning to end. Repeatedly turning the
analog blocks on and off introduces, for example, coherent errors such as
leakage to non-computational states [18, 19]. In addition, such a procedure
also suffers from calibration errors because each time an analog block is
turned on, it needs a fine-tuned calibration of the control pulse parameters,
upon which the unitary evolution is sensitive [20].

Consequently, a slight modification in the analog times between the layers of
SQGs is required [1]. Specifically, for a quantum circuit with $l$ analog
blocks, the first and last (referred to as “boundary”) analog block times are
modified by the single qubit gate duration $\Delta t$ to

$t_{1,l}^{\prime}=t_{1,l}-\frac{3}{2}\Delta t\,,$ (24)

and the rest (referred to as “central”) of analog blocks’ times are modified
to

$t^{\prime}_{\alpha}=t_{\alpha}-\Delta
t\,,\quad\alpha\in\\{2,\ldots,l-1\\}\,.$ (25)

Figure 3: Schematic of the quantum circuit implementing a banged DAQC
algorithm, with the analog evolution running throughout the whole computation.
The analog block times $t_{\alpha}^{\prime}$ in between SQGs have been
modified according to Eqs. (24) and (25).

The evolution under the simultaneous SQGs and analog block is given by

$U_{\bar{H}+H_{s}}(\Delta t)=\exp(-i\Delta t[\bar{H}+H_{s}])\,,$ (26)

where $H_{s}$ is the Hamiltonian that generates the SQGs. In general,
$\bar{H}$ and $H_{s}$ might not commute. This introduces a reverse Trotter
error [1], due to which the bDAQC computation is not exactly equal to the
evolution generated by the target Hamiltonian anymore. This error depends,
among other things, on the duration of the SQGs $\Delta t$, and it is
different for the boundary analog blocks and for the central analog blocks,
due to different Trotterization methods. Keep in mind that, usually, the term
Trotter error is used in the case in which the ideal evolution is that of non-
commuting Hamiltonians acting simultaneously, and is introduced when
“splitting” it into sequential evolutions under each individual Hamiltonian
[21, 22]. However, we use it in the reverse case: the ideal evolution is that
of the sequential application of the Hamiltonians, and the error is introduced
when applying them simultaneously.

Consequently, there is a trade-off between the errors arising from turning on
and off the analog blocks being eliminated, and the Trotter error being
introduced.

We study this intrinsic error associated with bDAQC, and its scaling, in more
detail in Section III.4.

(a)

(b)

(c)

Figure 4: Schematic representation of error sources in a DAQC algorithm: (a)
the number of analog blocks per implementation of a target Hamiltonian
$H_{\mathcal{A}}$, for an ATA device with $4$ qubits and resource Hamiltonian
$\bar{H}_{\mathcal{A}}$, for which $c=6$; (b) errors associated to each analog
block, modeled as an uncertainty in the starting and finishing times of
application $t_{0},t_{F}$ (ramp-up and ramp-down errors), and by an error in
the coupling coefficients $\bar{g}$; (c) the number of two-qubit terms prone
to mischaracterization present in an analog block, for an ATA device with $4$
qubits and resource Hamiltonian $H_{\mathcal{A}}$.

##  III Error scaling in DAQC

The method proposed in Section II for performing DAQC introduces errors that
differ from those in DQC in several ways. In this section, we study how these
errors scale with the number of connections and qubits of the device, and how
they compare to the DQC paradigm.

We focus our analysis on the errors related to imperfect control parameters,
given that these are ubiquitous across quantum computing platforms, whereas
the nature of environmental errors can vastly change across them. However, we
make some general remarks on the latter in Section III.1.3.

###  III.1 Analog blocks

Performing a DAQC algorithm requires to construct a circuit with $c$ analog
blocks for each target Hamiltonian that needs to be simulated, where $c$ is
the total number of connections. This is due to the fact that, in Eq. (23), we
obtain a vector $\mathbf{t}$ of the analog block times which contains $c$
elements. It is important to note that, in general, $c$ analog blocks are
required to implement a target Hamiltonian, independently of how many
coefficients of said Hamiltonian are equal to $0$. As an example, we provide
the digital-analog circuit for implementing a target Hamiltonian on a
$4$-qubit device with ATA connectivity, for which $c=6$, in Fig. 4a.

Despite the fact that performing analog blocks suggests a high performance
because they implement the natural dynamics of a device, they may still be
subject to significant errors. These errors may have higher or lower relevance
depending on whether we are considering bDAQC or sDAQC, and are described in
the following subsections.

####  III.1.1 Ramp-up and ramp-down errors

Calibration errors can be produced when switching on and off the analog blocks
[20], a process during which the evolution differs from the ideal square pulse
assumed in Section II. Such errors can be modeled as an uncertainty in the
time $t$ of application of the resource Hamiltonian (see Fig. 4b).
Additionally, the ramp-up and ramp-down procedure can introduce other types of
error, such as leakage to non-computational states [18, 19].

These errors are particularly relevant within sDAQC, which requires ramping up
and down the resource Hamiltonian’s coupling coefficients repeatedly during
the algorithm, whereas bDAQC mitigates this error by requiring it only at the
beginning and end of the execution of the circuit.

####  III.1.2 Two-qubit terms in analog blocks

The resource Hamiltonian considered to implement a DAQC algorithm should be
descriptive of the natural dynamics of the device. However, there might be
several sources of characterization errors associated to their implementation:

  * •

The resource Hamiltonian might still be an approximation to the actual
dynamics for some quantum computing platforms. This is the case, for example,
in superconducting qubit devices where the native dynamics are described with
a Bose-Hubbard Hamiltonian [23], and the qubitized form of the Hamiltonian is
still an approximation to it [24, 25]. In this case, a qubitized form of the
Hamiltonian reduced to two-qubit interactions is in general valid only at
relatively short times after the activation of an analog block.

  * •

The parameters of the resource Hamiltonian might be inaccurately
characterized. All together, the resource Hamiltonian $\bar{H}_{\mathcal{C}}$
contains $c$ coupling coefficients, $\bar{g}_{jk}$. This means that the
execution of each analog block introduces $c$ terms that have a potential
mischaracterization error, even assuming that the physical Hamiltonian will
always have the exact form as in Eq. (1). Such an error can be modeled as an
uncertainty in the coupling coefficients $\bar{g}$ of the resource Hamiltonian
(see Fig. 4b).

As an example of the latter, we show the equivalence of one analog block as
two-qubit terms on a $4$-qubit device with ATA connectivity in Fig. 4c. In
general, the calibration of a large number of digital gates is simpler
compared to the calibration of an analog block of the same size, since we can
calibrate each gate individually. In this regard, the precise many-body
Hamiltonian identification needed for the successful characterization of an
analog block is still the subject of ongoing research [23, 26, 27, 28].

In addition, we know that a target Hamiltonian requires $c$ analog blocks
(Fig. 4a), and that each analog block introduces $c$ two-qubit terms (Fig.
4c), so the total number of two-qubit terms needed to implement a target
Hamiltonian is $c^{2}$. However, the error in the coefficients of the resource
Hamiltonian is multiplied by the runtime of each analog block, $t_{\alpha}$
(see Eq. (4)). Therefore, the error is not necessarily proportional to the
number of two-qubit terms, and the runtimes of the analog blocks must be
considered in the error scaling analysis.

####  III.1.3 Environmental errors

As is the digital case, the dynamics of analog blocks is subject to the impact
of its environment, which produces decoherence and information losses. While
the environment responsible for the coherence decay is the same in both the
digital and DAQC cases, the analog blocks may dissipate in a more complex and
potentially faster way specifically at longer timescales, where the presence
of non-local decaying channels involving multiple neighboring qubits may
become increasingly relevant [29, 30, 31, 32]. Depending on the physical
implementation of the qubit states, many-body effects related to collective
decay can arise in a variety of physical systems, such as e.g. atom arrays
[33, 34], quantum dots [35] and also in superconducting circuits [36, 37].

###  III.2 Depth and duration

The depth of a digital quantum circuit is defined as the number of distinct
timesteps at which gates are applied [16]. It constitutes a measure of how
long it takes to execute the quantum circuit, because each gate generally has
a fixed duration.

In the DAQC framework, the number of distinct timesteps is not directly
related to how long it takes to execute a quantum circuit, because each analog
block in general has a different duration. Therefore, we need to sum the
duration of the analog blocks and layers of single gates. Recall that the
vector of the analog block times is calculated via the matrix $M^{-1}$ (see
Eq. (23)), and we cannot make any general statements on the form of $M^{-1}$.
Thus, the duration of DAQC algorithms must be calculated and studied on a
case-by-case basis. In Section VII, we calculate numerically the total runtime
of the algorithms that we explore in this manuscript.

###  III.3 Single-qubit gates

The DAQC method also introduces extra $X$ gates to simulate one target
Hamiltonian. The exact number depends on the device’s connectivity. However,
from the method presented in Section II or from Fig. 4a, it is straightforward
to see that a constant number of $X$ gates is introduced per analog block.
Thus, the number of extra SQGs introduced per target Hamiltonian is
proportional to $c$.

###  III.4 bDAQC non-commutativity errors

As discussed above, the bDAQC paradigm does not require to switch on and off
the analog blocks repeatedly but only requires the activation of a a single
block during the whole protocol, thus reducing the corresponding calibration
errors. However, in bDAQC the non-commutativity of SQGs with the resource
Hamiltonian also introduces an error (see Section II.4), which would only
disappear for infinitely fast SQGs [1]. The non-commutativity error is
different for the boundary analog blocks (at the beginning and the end of the
DAQC circuit) and the central analog blocks. In this section, we only focus on
the central analog blocks because they appear significantly more often in DAQC
circuits and therefore have a larger total error contribution.

Specifically, when a SQG generated by a Hamiltonian $H_{s}^{a}$ applied for
some time $\Delta t$, $U^{a}=\exp(-iH_{s}^{a}\Delta t)$, is applied on qubit
$a$, the error introduced is given by [1]

$\displaystyle e_{\text{central}}$ $\displaystyle=\left\lVert
1-e^{-i\bar{H}\Delta t/2}e^{-iH_{s}^{a}\Delta t}e^{-i\bar{H}\Delta
t/2}e^{i(\bar{H}+H_{s}^{a})\Delta t}\right\rVert$ (27)
$\displaystyle=\frac{(\Delta
t)^{3}}{4}\left\lVert\big{[}[\bar{H},H_{s}^{a}],\bar{H}+2H_{s}^{a}\big{]}\right\rVert+\mathcal{O}((\Delta
t)^{4})\,.$ (28)

In the following, we work out the explicit dependence on $\Delta t$ by
carefully analyzing Eq. (28). If a SQG has a given rotation angle (for
example, if it is an $X$ gate), the amplitude of its generator Hamiltonian is
inversely proportional to the SQG’s time: $H_{s}^{a}=\frac{\pi}{2\Delta t}X$.
Since the Hamiltonian that generates the SQG, $H_{s}^{a}$, appears twice in
the nested commutators of Eq. (28), we find that the explicit dependence of
the error $e_{\text{central}}$ on the SQG gate time $\Delta t$ is linear,
$e_{\text{central}}\propto\Delta t$, in contrast to previous literature [1,
3].

Additionally, the resource Hamiltonian $\bar{H}$ also appears twice in the
nested commutators. Thus, there is an additional dependence with the degree of
qubit $a$ (i.e., the number of couplings) and with the resource Hamiltonian’s
coupling coefficients.

Specifically, the infidelity introduced by the non-commutativity of a gate
$X^{a}$ with the resource Hamiltonian $\bar{H}$ is

$\epsilon_{\text{central}}=\mathcal{O}(d_{a}\bar{g}\Delta
t+d_{a}^{2}\bar{g}^{2}\Delta t^{2})\,,$ (29)

where $d_{a}$ is the degree of qubit $a$, and $\bar{g}$ is the coupling
coefficient of the resource Hamiltonian (assumed to be homogeneous for
simplicity). An upper bound with a similar scaling is given in Ref. [6], and a
detailed derivation of the scaling given in Eq. (29) is provided in Appendix
D.

###  III.5 Compound fidelity

In this subsection, we aim to write approximate formulas for the fidelities of
DQC, sDAQC and bDAQC that account for the scaling of all the sources of error
studied in this section, and their individual infidelity contributions. In
order to do so, we make two assumptions:

  1. 1.

Each one- and two-qubit term in an evolution operator $U$ corresponding to a
SQG, TQG or analog block has a fidelity $f<1$ arising from control errors,
which is independent of all other operations.

  2. 2.

The main source of decoherence is thermal relaxation, and we consider a simple
Markovian model for it, such that the fidelity per qubit for an algorithm that
requires a time $t$ has the approximate form $F_{T_{1}}\approx e^{-t/T_{1}}$,
where $T_{1}$ is the relaxation time. Additionally, we consider this
infidelity to be independent for each qubit, and also independent from their
unitary dynamics (disregarding the complex decaying channels that can arise in
DAQC, as discussed in Section III.1.3).

Under these assumptions, the approximate total fidelity of a digital circuit
implementing one given target Hamiltonian with $c$ terms on a device with $c$
connections is

$\displaystyle\begin{split}F_{\mathrm{DQC}}\approx&(f_{\mathrm{TQG}})^{n_{\mathrm{TQT}}}\\\
&\times e^{-Nt_{\text{tot}}/T_{1}}\end{split}$ (30)
$\displaystyle\begin{split}=&(f_{\mathrm{TQG}})^{c}\\\ &\times
e^{-Nt_{\text{tot}}/T_{1}}\,,\end{split}$ (31)

where $f_{TQG}$ is the fidelity of each TQG, $n_{TQT}$ is the number of two-
qubit terms (i.e., of TQGs) and $t_{\mathrm{tot}}$ is the total execution time
of the circuit. The compound fidelity $F_{\text{DQC}}$ in Eq. (31) accounts
for that of the TQGs, and decoherence due to thermal relaxation.

On the other hand, the approximate fidelity of a stepwise digital-analog
circuit implementing the same target Hamiltonian is given by

$\displaystyle\begin{split}F_{\mathrm{sDAQC}}\approx&\left[(f_{\text{ramp}})\times(f_{\text{coupling}})\right]^{n_{\mathrm{TQT}}}\\\
&\times(f_{\mathrm{SQG}})^{n_{\mathrm{SQG}}}\\\ &\times
e^{-Nt_{\text{tot}}/T_{1}}\,,\end{split}$ (32)
$\displaystyle\begin{split}=&\left[(f_{\text{ramp}})\times(f_{\text{coupling}})\right]^{c^{2}}\\\
&\times(f_{\mathrm{SQG}})^{\mathcal{O}(c)}\\\ &\times
e^{-Nt_{\text{tot}}/T_{1}}\,,\end{split}$ (33)

where $f_{\text{ramp}}$ is the fidelity associated with the ramp-up and ramp-
down errors, $f_{\text{coupling}}$ is the fidelity associated with the
mischaracterization of $\bar{g}$, $f_{SQG}$ is the fidelity of SQGs, and
$n_{SQG}$ is the number of SQGs.

Finally, the approximate fidelity of a banged digital-analog circuit
implementing said target Hamiltonian is

$\displaystyle\begin{split}F_{\mathrm{bDAQC}}\approx&\left[(f_{\text{ramp}})^{c}\times(f_{\text{coupling}})^{n_{\mathrm{TQT}}}\right]\\\
&\times(f_{\mathrm{SQG}})^{n_{\mathrm{SQG}}}\\\ &\times
e^{-Nt_{\text{tot}}/T_{1}}\\\
&\times(1-\epsilon_{\text{central}})^{n_{\mathrm{AB}}}\,.\end{split}$ (34)
$\displaystyle\begin{split}=&\left[(f_{\text{ramp}})^{c}\times(f_{\text{coupling}})^{c^{2}}\right]\\\
&\times(f_{\mathrm{SQG}})^{\mathcal{O}(c)}\\\ &\times
e^{-Nt_{\text{tot}}/T_{1}}\\\
&\times(1-\epsilon_{\text{central}})^{c}\,,\end{split}$ (35)

where $n_{AB}$ is the number of analog blocks. In this case, the contribution
to infidelity from ramp-up and ramp-down errors gets significantly reduced,
while the infidelity from non-commutativity is introduced.

##  IV Optimized DAQC on a device with a star-connectivity

As discussed in the previous section, some of the error sources present in
DAQC are sensitive to the number of analog blocks required within the
protocol, and to the total time of the quantum circuit.

While the protocol described in Section II is general for any arbitrary
connectivity, ad hoc protocols can be developed for specific connectivities
using fewer analog blocks and, consequently, shorter algorithm runtimes. For
example, in Ref. [2], an optimized DAQC protocol is developed for a device
with a nearest-neighbors connectivity in an open, one-dimensional graph, which
reduces the number of analog blocks, and also their runtimes.

On the other hand, we focus on a device with a so-called star-connectivity,
where a central qubit is coupled to $N-1$ other external qubits (see Fig. 5a).
We can write said connectivity as
$\mathcal{S}=\\{(0,1),(0,2),\ldots,(0,N-1)\\}$, where we label the central
qubit with index $0$. Ref. [38], e.g., describes how an effective star-
connectivity device can be built out of superconducting circuits.

The main idea behind the optimized DAQC protocol [2] is to place the $X$ gates
in such a way that we obtain an $(N-1)\times(N-1)$ sign matrix $M$, that
relates the coupling coefficients of the resource and target Hamiltonians to
the analog times according to Eq. (22), of the form

$M=\begin{pmatrix}\hphantom{-}1&\hphantom{-}1&\hphantom{-}1&\cdots&\hphantom{-}1&\hphantom{-}1&\hphantom{-}1\\\
-1&\hphantom{-}1&\hphantom{-}1&\cdots&\hphantom{-}1&\hphantom{-}1&\hphantom{-}1\\\
-1&-1&\hphantom{-}1&\cdots&\hphantom{-}1&\hphantom{-}1&\hphantom{-}1\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
-1&-1&-1&\cdots&\hphantom{-}1&\hphantom{-}1&\hphantom{-}1\\\
-1&-1&-1&\cdots&-1&\hphantom{-}1&\hphantom{-}1\\\
-1&-1&-1&\cdots&-1&-1&\hphantom{-}1\end{pmatrix},$ (36)

i.e. a matrix with its elements being $1$ on and above the diagonal, and $-1$
below the diagonal.

Recall the definition of the vector $\mathbf{G}$ from Eq. (22), with elements
$G_{\beta}=g_{\beta}/\bar{g}_{\beta}$. We reorder and express, without loss of
generality, the elements of vector $\mathbf{G}$ in such a way that the
following conditions are met,

$\displaystyle G_{\beta}$ $\displaystyle\geq 0\,,$ (37) $\displaystyle
G_{\beta}$ $\displaystyle\geq G_{\beta+1}\,.$ (38)

For the first condition to be met, we may need to shift the phase of the
evolution, $\phi_{\beta}\rightarrow\phi_{\beta}^{\prime}=\phi_{\beta}-2\pi$
(see Eq. (8)), in order to change the signs of the target coefficients (recall
that $\phi_{\beta}=t_{f}g_{\beta}\mod(2\pi)$). Through this transformation, we
can change the sign of $G_{\beta}$ without affecting said unitary evolution.
For the second condition, we may need to change the order of the labels of the
coefficients in $\mathbf{G}$. Under these conditions, it is proven in Ref. [2]
that the inverse $M^{-1}$ yields runtimes for the analog blocks (see Eq. (23))
given by

$\displaystyle\frac{t_{\alpha}}{t_{f}}$
$\displaystyle=\frac{G_{\alpha}-G_{\alpha+1}}{2}\,,$ (39)
$\displaystyle\frac{t_{N-1}}{t_{f}}$
$\displaystyle=\frac{G_{1}+G_{N-1}}{2}\,.$ (40)

Also in Ref. [2], it is proven that these equations lead to the minimum number
of analog blocks, running for a minimal time, required to implement a given
target Hamiltonian. One can see in Eq. (39) how the number of analog blocks
gets reduced if $k$ elements of $\mathbf{G}$ are equal, which makes $k-1$
elements of $\mathbf{t}$ equal to zero. Also, one can see how the time of each
analog block $t_{\alpha}$ gets reduced as the difference between $G_{\alpha}$
and $G_{\alpha+1}$ gets smaller. Then, our task is to find the correct
placement of the $X$ gates in our digital-analog circuit, in order to obtain
an $M$ matrix of the form (36) for a star-connectivity.

(a)

(b)

Figure 5: (a) Scheme of a 5 qubit device with star-connectivity. Each node
corresponds to a qubit, and each edge to a coupling (i.e., to a
$\bar{g}_{j}Z^{0}Z^{j}$ term in $\bar{H}_{\mathcal{S}}$). (b) Quantum circuit
for an optimized sDAQC protocol on a star-connectivity device with $5$ qubits.

Recalling the interpretation of the $M$ matrix from Section II.3, a digital-
analog quantum circuit corresponding to the matrix $M$ can be constructed by
flipping all but one connections in the first analog block, and flipping one
less connection in the subsequent analog blocks. An example of such a circuit
for $5$ qubits is given in Fig. 5.

Additionally, recall our discussion of non-commutativity errors in bDAQC in
Section III.4 and their dependence on the qubits’ degree. This protocol also
minimizes the number of overlapping $Z^{j}Z^{a}$ terms with the $X^{a}$ gates
introduced, given that they are only acting on the external qubits, which have
degree $d=1$. Compared against other connectivities, qubits in an ATA
connectivity have $d=N-1$, and in a one-dimensional chain they have $d=2$.
Thus, this protocol also minimizes the error introduced by the non-
commutativity of the resource Hamiltonian and the single-qubit terms.

Such optimized protocols have been described only for the one-dimensional open
chain (in Ref. [2]) and for the star-connectivity (in this manuscript) so far.
This is because, in general, it is not possible to change the sign of just one
connection in an arbitrary connectivity without changing the others, which is
required to get the necessary $M$ matrix (36). Take, for example, a square
lattice: to flip a connection between two qubits, we place $X$ gates on one of
the qubits involved, but this flips three additional connections. To correct
these additional flipped signs, we can place $X$ gates on the three other
qubits involved, but this flips three additional signs each. For such a
reason, it is not possible to flip the sign of a connection in an isolated way
in an arbitrary connectivity.

##  V Digital-analog Quantum Fourier Transform

After introducing the theoretical framework of DAQC, we now present the
details of the implementation of a specific quantum algorithm utilizing DAQC:
the digital-analog Quantum Fourier Transform (QFT). This algorithm was
described and studied in Refs. [3, 4] for an ATA connectivity, along with
simulations of control errors and environmental noise. We extend the analysis
through a full study of the error scaling with the system size. We provide
such an analysis in Section V.1, and also analyze our developed implementation
of digital-analog QFT on a star-connectivity, using the optimized protocol
from Section IV, in Section V.2. In addition, for illustrative purposes, we
further present numerical simulations of the fidelity for a few qubits in
Section VII.

The QFT is a quantum routine that acts on a quantum state
$\ket{x}=\sum_{i=0}^{2^{N}-1}x_{i}\ket{i}$, where $\ket{i}$ are computational
basis states, and maps it to a Fourier-transformed quantum state
$\sum_{i=0}^{2^{N}-1}y_{i}\ket{i}$, with

$y_{k}=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}x_{j}\omega_{N}^{jk}\,,\quad
k=0,1,2,3\ldots,N-1\,,$ (41)

where $\omega_{N}=e^{2\pi i/N}$. A digital quantum circuit for this routine is
depicted in Fig. 6a using the Hadamard gate ($H$) and phase gate
$R_{z}(\theta)$,

$\displaystyle H$ $\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix},$ (42) $\displaystyle R_{z}(\theta)$
$\displaystyle=e^{-i\frac{\theta}{2}Z}=\begin{pmatrix}1&0\\\
0&e^{i\theta}\end{pmatrix},$ (43)

as well as the $ZZ(\phi)$ gate defined in Eq. (8).

(a)

(b)

Figure 6: Digital circuit for the QFT on a device with $5$ qubits, comprising
$H$, $R_{z}(\theta)$ and $ZZ(\phi)$ gates, (a) on an ATA connectivity, and (b)
on a star-connectivity, where SWAP gates have been introduced as necessary.

###  V.1 ATA-QFT error analysis

In order to translate the digital QFT into the digital-analog paradigm, we
must first identify the target Hamiltonians we need to implement. We do that
by taking the biggest blocks of consecutive $ZZ(\phi)$ gates. As an example,
we show the first and last such target Hamiltonians for the ATA-QFT in Fig.
6a, and from that it becomes clear that $N-1$ such blocks are needed.

The sources of possible errors were specified in Section III and their effects
on the compound fidelity summarized in Eqs. (33) and (35). For an ATA device,
the number of connections $c$ is given by $c=N(N-1)/2$. For the digital-analog
ATA-QFT, we therefore find the following scaling behaviour with the number of
qubits $N$:

  1. 1.

Number of analog blocks: The ATA-QFT circuit is constructed as
$\mathcal{O}(N)$ target Hamiltonians. Each target Hamiltonian requires
$c=\mathcal{O}(N^{2})$ analog blocks (see Section III.1). Thus, the total
number of analog blocks is $\mathcal{O}(N^{3})$.

  2. 2.

Number of two-qubit terms: Each analog block contains $c=\mathcal{O}(N^{2})$
two-qubit terms (see Section III.1.2). Thus, the total number of two qubit
terms in all the analog blocks is $\mathcal{O}(N^{5})$.

  3. 3.

Duration: The digital ATA-QFT can be implemented in depth $\mathcal{O}(N)$
[39, 40]. On the other hand, the total duration of the DAQC algorithm depends
on the resulting matrix $M$ for each case, and thus we cannot say anything
about it a priori (see Section III.2). We numerically compute the duration of
this algorithm in Section VII, and, by fitting a curve to the simulated data,
we extract a scaling of the duration $\mathcal{O}(N^{2.05})$.

  4. 4.

Number of single-qubit gates: Each target Hamiltonian requires
$c=\mathcal{O}(N^{2})$ $X$ gates (see Section III.3). Thus, the total number
of SQGs is $\mathcal{O}(N^{3})$.

  5. 5.

bDAQC non-commutativity: Each qubit has a degree $d=\mathcal{O}(N)$, so from
Eq. (29), and setting $\Delta t,\bar{g}$ to be constant, each analog block
introduces an error that scales as
$\epsilon_{\text{central}}=\mathcal{O}(N^{2})$ (see Section III.4). There are
$\mathcal{O}(N^{3})$ analog blocks, so the contribution from bDAQC to the
compound fidelity in Eq. (35) scales as
$(1-\mathcal{O}(N^{2}))^{\mathcal{O}(N^{3})}$.

We summarize these scalings, and compare them to those of a purely digital
implementation, in Table 1(a).

The worse error scaling for DAQC in the case of ATA-QFT is, in part, a result
of the sparsity of two-qubit terms in QFT. DQC requires only one TQG for each
non-zero term of the target Hamiltonians; however, DAQC is introducing
superfluous analog blocks needed to effectively cancel the non-zero couplings
of the resource Hamiltonian. On top of that, each of these analog blocks
introduces a large number of two-qubit terms, as compared to just one two-
qubit term per TQG.

Table 1: Scaling of the error sources in DQC and DAQC with the number of
qubits $N$, from which the total fidelity of the quantum circuit can be
calculated using Eqs. (30), (32) and (34) for DQC, sDAQC and bDAQC
respectively, for the following algorithms: (a) QFT on an ATA connectivity,
for which all sources of error scale better in DQC than in DAQC; (b) QFT on a
star-connectivity, for which the sources of error in DAQC scale better than in
the ATA case, with an advantage with respect to DQC in the time of execution;
and (c) GHZ state preparation on a star-connectivity, for which we do not
consider bDAQC, and all the sources of error in DAQC scale similarly or better
than those in DQC.

(a) ATA-QFT

| $\mathbf{n_{AB}}$ | $\mathbf{n_{TQT}}$ | $\mathbf{n_{SQG}}$ | $\mathbf{t}$ | $\mathbf{\epsilon_{\text{central}}}$
---|---|---|---|---|---
DQC | - | $\mathcal{O}(N^{2})$ | $\mathcal{O}(N)$ | $\mathcal{O}(N)$ | -
sDAQC | $\mathcal{O}(N^{3})$ | $\mathcal{O}(N^{5})$ | $\mathcal{O}(N^{3})$ | $\mathcal{O}(N^{2.05})$ | -
bDAQC | $\mathcal{O}(N^{3})$ | $\mathcal{O}(N^{5})$ | $\mathcal{O}(N^{3})$ | $\mathcal{O}(N^{2.05})$ | $\mathcal{O}(N^{2})$

(b) Star-QFT

| $\mathbf{n_{AB}}$ | $\mathbf{n_{TQT}}$ | $\mathbf{n_{SQG}}$ | $\mathbf{t}$ | $\mathbf{\epsilon_{\text{central}}}$
---|---|---|---|---|---
DQC | - | $\mathcal{O}(N^{2})$ | $\mathcal{O}(N)$ | $\mathcal{O}(N^{2})$ | -
sDAQC | $\mathcal{O}(N^{2})$ | $\mathcal{O}(N^{3})$ | $\mathcal{O}(N^{2})$ | $\mathcal{O}(N)$ | -
bDAQC | $\mathcal{O}(N^{2})$ | $\mathcal{O}(N^{3})$ | $\mathcal{O}(N^{2})$ | $\mathcal{O}(N)$ | $\mathcal{O}(1)$

(c) Star-GHZ

| $\mathbf{n_{AB}}$ | $\mathbf{n_{TQT}}$ | $\mathbf{n_{SQG}}$ | $\mathbf{t}$
---|---|---|---|---
DQC | - | $\mathcal{O}(N)$ | $\mathcal{O}(N)$ | $\mathcal{O}(N)$
sDAQC | $1$ | $\mathcal{O}(N)$ | $\mathcal{O}(N)$ | $\mathcal{O}(1)$

###  V.2 Star-QFT error analysis

In this subsection, we analyze the scaling of the error sources for the
digital-analog QFT implemented on a star-connectivity, using the optimized
protocol of Section IV, and work out the improvement compared to the ATA-QFT.

Implementing the QFT on a star-connectivity introduces the need for
$\rm{SWAP}$ gates placed between each target Hamiltonian, as can be seen in
Fig. 6b, and each $\rm{SWAP}$ gate requires six additional analog blocks when
translated to a digital-analog implementation (see Appendix E).

For a star-connectivity device, the number of connections $c$ is again given
by $c=N-1$. For the digital-analog Star-QFT, we therefore find the following
scaling behaviour with the number of qubits $N$:

  1. 1.

Number of analog blocks: Similarly to the ATA-QFT, the Star-QFT circuit is
constructed as $\mathcal{O}(N)$ target Hamiltonians. However, in this case,
each target Hamiltonian requires $c=\mathcal{O}(N)$ analog blocks, because the
other analog blocks get cancelled. On the other hand, in total, the need for
$\mathcal{O}(N)$ SWAP gates introduces $\mathcal{O}(N)$ analog blocks. Thus,
the total number of analog blocks is $\mathcal{O}(N^{2})$.

  2. 2.

Number of two-qubit terms: Each analog block contains $c=\mathcal{O}(N)$ two-
qubit terms. Therefore, the total number of two-qubit terms in all the analog
blocks is $\mathcal{O}(N^{3})$.

  3. 3.

Duration: The digital Star-QFT can be implemented in depth
$\mathcal{O}(N^{2})$. In the digital-analog circuit, the $n$-th target
Hamiltonian has $n$ null coupling coefficients (meaning that $g_{jk}=0$),
which eliminates $n-1$ analog blocks (recalling the discussion of Eq. (39)).
In addition, the difference between one coupling coefficient and the next
decreases exponentially (see the exponentially decreasing phases in Fig. 6b).
Recall from Eq. (39) that the analog times are proportional to the difference
between the coefficient of each term in the Hamiltonian and the following.
Thus, the contribution to the analog times of each target Hamiltonian is
asymptotically constant, and the duration of the whole algorithm is decreased
to $\mathcal{O}(N)$.

  4. 4.

Number of single-qubit gates: Each target Hamiltonian requires
$\mathcal{O}(N)$ $X$ gates. Thus, the total number of SQGs is
$\mathcal{O}(N^{2})$.

  5. 5.

bDAQC non-commutativity: Each external qubit, on which $X$ gates are applied,
has one coupling, so from Eq. (29), each analog block introduces an infidelity
that scales as $\mathcal{O}(1)$, when $\Delta t,\bar{g}$ are set to be
constant. There are $\mathcal{O}(N^{2})$ analog blocks, so the contribution to
the compound fidelity (see Eq. (35)) introduced by bDAQC scales as
$(1-\mathcal{O}(1))^{\mathcal{O}(N^{2})}$.

A full comparison of these scalings to the purely digital implementation of
the Star-QFT can be found in Table 1(b).

The errors for Star-QFT scale slower when compared to the ATA-QFT, but the
scaling is generally still worse than in DQC. This is because, even though the
number of analog blocks scales similarly to the number of TQGs required for
the DQC algorithm, each analog block introduces more two-qubit terms that are
prone to mischaracterization.

In this case, the duration of the algorithm scales linearly in DAQC while it
scales quadratically in DQC, so it presents an advantage in that regard.
Additionally, the intrinsic error introduced by bDAQC is smaller than that
introduced by the two-qubit terms, so bDAQC has the potential of a bigger
improvement than in the ATA case.

##  VI Digital-analog GHZ state preparation

In this section, we introduce another example of a digital-analog quantum
algorithm and describe the DAQC protocol for generating the maximally
entangled Greenberger-Horne-Zeilinger (GHZ) state [15] in a star-connectivity
device with $N$ qubits,

$\ket{\mathrm{GHZ}_{N}}=\frac{\ket{0}^{\otimes N}+\ket{1}^{\otimes
N}}{\sqrt{2}}\,.$ (44)

(a)

(b)

Figure 7: Quantum circuits implementing the GHZ state preparation protocol on
a star-connectivity with $5$ qubits: (a) in the DQC paradigm, (b) in the sDAQC
paradigm.

In Fig. 7a, we show the digital circuit for generating the GHZ state on $N=5$
qubits by utilizing $ZZ(-\frac{\pi}{4})$ gates. We can write the $N-1$
consecutive $ZZ$ gates appearing in the digital circuit as the evolution
operator

$U=e^{-i\sum_{j=1}^{N}\frac{\pi}{4}Z^{0}Z^{j}}\,,$ (45)

where we have set the time of the evolution $t_{f}=\frac{\pi}{4g}$. Writing a
digital-analog circuit for this algorithm is now possible following the method
described in Section IV.

In this section we assume the resource Hamiltonian is homogeneous, i.e., all
its coupling coefficients $\bar{g}\equiv\bar{g}_{0j}$ are equal, and also that
they are independent of the number of qubits $N$. Then, all elements of the
vector $\bf{G}$ are equal (see Sec. IV, Eq. (39)), given that both the
resource and the target Hamiltonians are homogeneous. This means that there is
only one distinct $G_{\beta}$ and the digital-analog quantum circuit,
therefore, only requires a single analog block with appropriate runtime (see
Fig. 7b).

The runtime of the analog block is given by the relation between the
coefficients of the target and the resource Hamiltonians, $t=g/\bar{g}$. This
runtime $t$ is independent of the number of qubits, whereas the number of TQGs
needed in the digital paradigm, and thus also the runtime of the algorithm,
scales linearly with the number of qubits. A comparison of the scaling of the
digital-analog and the purely digital implementations of this algorithm can be
found in Table 1(c). In this case, we do not consider bDAQC, because only one
analog block is present and thus bDAQC presents no advantage.

For this algorithm, we can see that DAQC scales favorably when compared to the
DQC paradigm. This is because, in this case, the target and resource
Hamiltonians are related just by a multiplicative factor, and thus just one
analog block is required to simulate the target Hamiltonian. This way, DAQC is
not introducing superfluous resources as it was for the two previous examples,
while it is actually reducing the duration of the circuit execution.

##  VII Numerical simulations

After deriving the scaling of the sources of error of the three algorithms in
Section V and Section VI, we validate our derivations through numerical
simulations under a certain error model for different numbers of qubits, and
extract average execution fidelities as well as durations of circuit
execution.

###  VII.1 Error model and methods

DQC employs single- and two-qubit gates, whereas DAQC employs single-qubit
gates and analog blocks. As discussed in Section III, we model the errors
caused by all these operations by introducing errors in their control
parameters. We introduce a coherent and an incoherent contribution of these
control errors to the total infidelity by implementing two different
modifications to the control parameters: (1) systematic errors that are
constant throughout each noisy simulation of the quantum circuit and (2)
stochastic errors that are randomly chosen every time an operation gets
applied, for every run of circuit simulation.

For each SQG generated by a Hamiltonian $H_{s}^{a}$,
$U_{H_{s}^{a}}(\theta)=\exp(-i\theta H_{s}^{a})$, we modify the angle of the
rotation as

$\displaystyle\begin{split}\theta\rightarrow\theta^{\prime}=\theta\times(1+\Delta\theta+\delta\theta)\,,\end{split}$
(46)

where $\Delta\theta$ is the systematic error, and $\delta\theta$ is the
stochastic error.

For each TQG, $ZZ(\phi)=\exp(-i\phi Z^{j}Z^{k})$, we modify the phase of the
rotation as

$\displaystyle\begin{split}\phi\rightarrow\phi^{\prime}=\phi\times(1+\Delta\phi+\delta\phi)\,,\end{split}$
(47)

where $\Delta\phi$ is the systematic error, and $\delta\phi$ is the stochastic
error.

Finally, for each analog block,
$U_{\bar{H}_{\mathcal{C}}}(t)=\exp(-it\sum\bar{g}_{jk}Z^{j}Z^{k})$, we modify
the runtime and coupling coefficients of the resource Hamiltonian as

$\displaystyle t$ $\displaystyle\rightarrow t^{\prime}=t\times(1+\Delta
t+\delta t)\,,$ (48) $\displaystyle\bar{g}$
$\displaystyle\rightarrow\bar{g}^{\prime}=\bar{g}\times(1+\Delta\bar{g}+\delta\bar{g})\,,$
(49)

where, again, $\Delta t,\Delta\bar{g}$ (systematic), and $\delta
t,\delta\bar{g}$ (stochastic) are unitless.

For the case of QFT, because we are interested in the fidelity of the process
regardless of the initial state, we compute the ideal unitary implemented by
the quantum circuit, and average over the erroneous unitaries’ fidelities. We
define the fidelity of one erroneous unitary $U$ with respect to its ideal
$\tilde{U}$ as its average fidelity over all possible initial states [41],

$F_{U}=\frac{n+\left|\text{Tr}(\tilde{U}^{\dagger}U)\right|^{2}}{n(n+1)}\,,$
(50)

where $n=2^{N}$ is the dimensionality of the Hilbert space.

On the other hand, for the case of the GHZ state preparation, because we are
interested in the final state only, we compute the ideal state and average
over the erroneous states’ fidelities. We define the fidelity of one erroneous
final state $\ket{\psi}$ with respect to its ideal state (44) as:

$F_{\psi}=\left|\braket{\psi}{\mathrm{GHZ}_{N}}\right|^{2}.$ (51)

For the sake of specificity, we choose the error parameters of the simulations
to match those of a state-of-the-art superconducting QPU. We sample all errors
from a Gaussian distribution $\mathcal{N}(\mu=0,\sigma)$ centered around $0$,
where $\sigma$ is chosen so that each type of operation has a given average
fidelity: $99.99\%$ for SQGs, $99.9\%$ for TQGs, and $99.95\%$ for each two
qubit term in analog blocks. These figures are calculated executing the
erroneous gates $10000$ times, and averaging the resulting erroneous
unitaries’ fidelities accorcing to Eq. (50).

This choice for the fidelities of each operation entails considerably better
SQGs than TQGs, and analog blocks that introduce less error per two-qubit term
than each TQG. Additionally, in the case of bDAQC, the error associated with
the runtimes is applied only in the first and last analog blocks of the
quantum circuit (see Section III.1.1). Finally, all the values of $\sigma$ are
also chosen so that the coherent errors account for $25\%$ of the infidelity
per operation, and incoherent errors account for $75\%$ of it.

We simulate the circuits for digital and digital-analog ATA-QFT, Star-QFT and
Star-GHZ, and compute the noisy fidelities for each case, after applying the
errors described above. We do this by running $1000$ iterations of noisy
circuits, computing the resulting erroneous unitaries and final states
according to Eqs. (50) and (51), respectively, and averaging to obtain
$\langle F_{U}\rangle$ in the case of QFT or $\langle F_{\psi}\rangle$ in the
case of the GHZ state preparation.

The quantum circuits must be compiled to a specific basis gate set, which may
be different for each quantum computing platform. For specificity, we focus
our simulations on one consisting of superconducting qubits. Therefore, the
native SQGs that we assume can be implemented in the devices are the $R_{xy}$
and $R_{z}$ gates,

$\displaystyle R_{xy}^{a}(\theta,\phi)$
$\displaystyle=e^{-i\frac{\theta}{2}\left(\cos\phi X^{a}+\sin\phi
Y^{a}\right)}\,,$ (52) $\displaystyle R_{z}^{a}(\theta)$
$\displaystyle=e^{-i\frac{\theta}{2}Z^{a}}\,,$ (53)

where $X$ and $Z$ are the Pauli-$X$ and Pauli-$Z$ matrices, respectively (see
Eqs. (11) and (3)), and $Y$ is the Pauli-$Y$ matrix,

$Y=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix}\,.$ (54)

For superconducting quantum computers, the $R_{xy}^{a}(\theta,\phi)$ gate can
be physically implemented via a microwave drive [42]. The gate
$R_{z}^{a}(\theta)$ does not need to be physically implemented because it can
be accounted for virtually by readjusting the phase of the subsequent gates
applied on qubit $a$ [43]. On the other hand, the native TQG we assume is the
$ZZ^{jk}(\phi)$ gate previously defined in Eq. (8).

We set the resource Hamiltonians to be homogeneous, with coupling coefficient
$\bar{g}=10$ MHz and the SQG times $\Delta t=5$ ns. On the other hand, we
assume that the time it takes to implement a TQG does not depend on the phase
of its rotation, and it is $t_{\text{TQG}}=50$ ns, as is realistic for
superconducting transmon qubits coupled via tuneable couplers.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8: Top row: Fidelity of DQC, sDAQC and bDAQC algorithms as a function
of the number of qubits for the following algorithms: (a) ATA-QFT, for which
both versions of DAQC perform worse than DQC; (b) Star-QFT, for which both
versions of DAQC perform worse than DQC, but bDAQC performs better than sDAQC
for $N>6$; and (c) Star-GHZ, for which sDAQC performs better than DQC. Bottom
row: Duration of the DQC and sDAQC circuits of algorithms as a function of the
number of qubits: (d) ATA-QFT, for which we have fit the data to the closest
signomial expression, $\mathcal{O}(N^{2.05})$, and the DAQC algorithm’s
execution is longer than DQC for the whole range; (e) Star-QFT, for which the
DQC algorithm’s execution is longer than DAQC for $N>25$; and (f) Star-GHZ,
for which the DQC algorithm’s execution always is longer than DAQC.

###  VII.2 Results

Considering the error model and methods described in the previous subsection,
we have performed our simulations using the open-source quantum computing
library Qiskit [44] and the results are presented in Fig. 8.

In the top row of Fig. 8, we plot the fidelity of the digital and digital-
analog computations for each algorithm we have studied (ATA-QFT, Star-QFT and
Star-GHZ), as a function of the number of qubits.

For ATA-QFT, we skip the case of $N=4$. This is because, as explained in Ref.
[1], the matrix $M$ that results for the ATA connectivity with $N=4$ is not
invertible, and thus the times of the analog blocks cannot be calculated via
Eq. (23). Furthermore, also for ATA-QFT, we simulate the bDAQC circuit only
for $3$, $5$ and $6$ qubits as those are the only cases in which the
compilation described in Ref. [1] can be applied. For any $N>6$, Eq. (23) may
return negative runtimes for the analog blocks. Indeed, the ATA-QFT needs the
implementation of analog blocks with negative runtimes, which is not physical.
A protocol for obtaining a digital-analog circuit with only non-negative
runtimes is given in Ref. [9], though it requires the construction of an $M$
matrix whose size grows exponentially with the number of qubits, thus
rendering it impractical.

Likewise, in the bottom row of Fig. 8, we plot the time of execution of the
quantum circuits as a function of the number of qubits. Because we do not need
to actually simulate the quantum circuits, we can extend these figures to a
higher number of qubits. The times of bDAQC circuits are not represented
because they are very similar to those of sDAQC, undergoing just the minor
modifications of Eqs. (24) and (25).

We now discuss these results in detail for each algorithm.

####  VII.2.1 All-to-all Quantum Fourier Transform

As can be seen from Fig. 8a, the fidelities of QFT in both DAQC paradigms are
below the fidelity in DQC over the entire range of the number of qubits $N$
studied. One reason is that, even though each two-qubit term in analog blocks
is more error-robust, the number of two-qubit terms is much smaller in DQC.
For example, in the case of $N=3$, DQC has 3 two-qubit terms whereas DAQC has
18 of them. Additionally, in DAQC, the two-qubit terms are repeatedly applied
on the same pairs of qubits, leading to a higher accumulation of coherent
errors throughout the computation [45]. The much worse scaling of DAQC
compared to DQC only exacerbates the difference in fidelity for a bigger
number of qubits. Recall from Section V.1 that part of the difference in
scaling comes from the fact that each target Hamiltonian has many null
coupling coefficients, (i.e., $g_{jk}=0$, see Fig. 6a), which means that a
small number of TQGs is needed to implement them in the DQC paradigm, while no
such reduction of resources occurs necessarily in DAQC.

Regarding the time of the computation, plotted in Fig. 8d, we have fit the
data to the best signomial expression, which gives a scaling of
$\mathcal{O}(N^{2.05})$, whereas the computation time for DQC scales linearly
as expected. The detrimental impact of decoherence is therefore much larger
for DAQC than for DQC. Additionally, a spike is present in the range
$N\in(5,8)$. This non-monotonic behavior is due to a property of the matrix
$M$ for the ATA connectivity: usually, its inverse $M^{-1}$ has a balance of
positive and negative elements that makes it so the contributions to analog
times are partially cancelled in Eq. (23); however, for $N=6,7$ in the ATA
connectivity, its elements are all negative and positive, respectively.

####  VII.2.2 Star Quantum Fourier Transform

From Fig. 8b, one can see that the fidelities for DAQC are also below the DQC
fidelity for the whole range of $N$ studied in this case. However, the
fidelities of DAQC are higher with respect to the ATA case, as was expected
from the analysis of the error scaling in Section V.2. Thus, we see explicitly
the dependence of the performance of DAQC on the connectivity and the
compilation used for the digital-analog circuit. Additionally, in this case,
the intrinsic error associated with bDAQC scales slower than the error
associated with the analog blocks (see Table 1), so the trade-off is favorable
to bDAQC and it outperforms sDAQC for $N>6$.

As for the time of computation, in Fig. 8e, we see that for DAQC, it grows
linearly, while for DQC, it grows quadratically, in such a way that for
$N>25$, the duration of the digital algorithm surpasses that of the digital-
analog. Recall from Section V.2 that this is because the coupling coefficients
of each target Hamiltonian in QFT decrease exponentially, and so does the
difference between one and the next. This means, according to Eq. (39), that
the times of the analog blocks for each target Hamiltonian also decrease
exponentially. This leads to a total contribution of the times that is
asymptotically constant for each target Hamiltonian, and therefore linear for
the whole algorithm. On the other hand, we assume that TQGs require a constant
time no matter how small their phase is, and the quadratic contribution to
time arises.

In this case, while the infidelity coming from control errors is greater for
DAQC, there may be a trade-off with the infidelity arising from decoherence
and other environmental noise related to the time of execution of the quantum
circuits, which is greater for DQC than for DAQC for a big enough number of
qubits. The total fidelity under both sources of noise is calculated
approximately in Appendix F, where we conclude that the trade-off can be
favorable for DAQC for certain ranges of parameters, for example, if the
execution of TQGs is very slow, and/or if the relaxation time $T_{1}$ of the
qubits is very short.

It is important to keep in mind the difficulty of implementing analog blocks
with exponentially decreasing runtimes. On the same note, the digital QFT
algorithm can be approximated to a very good degree by the Approximate Quantum
Fourier Transform (AQFT) [46], which ignores some of the TQGs, whose phases
decrease exponentially.

####  VII.2.3 Star GHZ state preparation

In Fig. 8c, we can see that the fidelities for sDAQC are better than those of
DQC, because the number of two-qubit terms is the same and the analog
evolution is more resilient to the control errors.

Additionally, and as expected, the time of the digital-analog algorithm is
constant whereas that of the digital algorithm scales linearly with $N$ (see
Fig. 8f). This is because, in a way, the digital-analog algorithm is
parallelizing all the two-qubit terms while the digital algorithm requires
that we apply them sequentially, one after the other.

Therefore, DAQC presents an advantage with respect to DQC when we can express
the evolution of many consecutive two-qubit gates in a digital algorithm as a
very reduced number of analog blocks, and combining them with SQGs.

##  VIII Conclusions

In the past few years the DAQC paradigm has been presented as an alternative
path to perform universal quantum computation that combines the robustness of
analog quantum computing with the flexibility of the digital approach.

In this manuscript we have systematically analyzed its performance by studying
and simulating the scaling of errors with respect to the digital case.
Furthermore, we have considered the most general situation, i.e. regardless of
the connectivity or the algorithm to be implemented. Our analysis shows a
clearly disadvantageous error scaling of DAQC with respect to the digital
case, coming mainly from the number of analog blocks needed to engineer one
target Hamiltonian, and the number of two-qubit terms introduced per each
analog block. While for DAQC the implementation of one target Hamiltonian
entails the introduction of $c^{2}$ two-qubit terms (where $c$ is the number
of connections of the device), for DQC it only entails, at most, the
introduction of $c$ two-qubit terms.

To illustrate our scaling analysis, we have analyzed the performance of DAQC
with respect to the digital case for two different algorithms, the QFT and the
GHZ state preparation algorithm, on two different connectivities: ATA and a
star configuration. We have consistently found DAQC to be less efficient in
terms of fidelities, except for the case in which the device’s resource
Hamiltonian closely matches the algorithm’s target Hamiltonian. In this
situation, it can be argued that the resulting quantum circuit corresponds to
a purely analog implementation, with SQGs applied before and after the analog
evolution (see, e.g., Fig. 7b). While this implies the need for tailoring the
device’s connectivity to match that of the algorithm, this case shows a
promising advantage as it parallelizes the two-qubit interactions that would
otherwise be applied sequentially in the digital paradigm, and takes full
advantage of the potentially more error-resilient analog evolution. Thus, we
foresee potential areas of application of DAQC in quantum simulation [47, 10,
48, 49, 7, 50, 8, 51, 9] and variational algorithms in which fast generation
of entanglement across the whole device is desirable [47, 52, 53, 54].

###### Acknowledgements.

We would like to acknowledge the support of our colleagues in IQM, and
specially thank T. Liu and B. G. Taketani. We also thank M. Sanz for fruitful
discussions at the early stages of our work, as well as P. García-Molina.
Finally, we acknowledge the support from the German Federal Ministry of
Education and Research (BMBF) under DAQC (grant No. 13N15686) and Q-Exa (grant
No. 13N16062).

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##  Appendix A The sign matrix M

The set of elements $M_{mnjk}$ has four indices. Let us “reorder” these
elements in such a way that they can be arranged into a matrix, so that we
will be able to invert it. In order to do that, we vectorize the pairs of
indices $(m,n)\rightarrow\alpha$ and $(j,k)\rightarrow\beta$, assigning to
each pair a single number, ordered from smallest to biggest $m$ ($j$), then
from smallest to biggest $n$ ($k$). For example, for an ATA $3$-qubit device:

$\displaystyle(m,n)=(0,1)$ $\displaystyle\rightarrow\alpha=1\,,$ (55)
$\displaystyle(m,n)=(0,2)$ $\displaystyle\rightarrow\alpha=2\,,$ (56)
$\displaystyle(m,n)=(1,2)$ $\displaystyle\rightarrow\alpha=3\,.$ (57)

This way, each pair of indices $(m,n)$ is uniquely mapped to a single index
$\alpha$. The same is done with each pair of indices $(j,k)$, which is
uniquely mapped into a single index $\beta$. This way, we are also able to map
$M_{mnjk}$ to $M_{\alpha\beta}$.

The general formula for this mapping in the ATA case for $N$ qubits is [1]

$\displaystyle(m,n)$ $\displaystyle\rightarrow\alpha=N(m-1)-m(m+1)/2+n\,,$
(58) $\displaystyle(j,k)$ $\displaystyle\rightarrow\beta=N(j-1)-j(j+1)/2+k\,.$
(59)

On the other hand, the inverse transformation $\alpha\rightarrow(m,n)$ is
given by:

$\displaystyle\begin{split}n&=1+H_{1}\left[\frac{\alpha}{N}\right]+H_{1}\left[\frac{\alpha}{2N-2}\right]\\\
&+H_{1}\left[\frac{\alpha}{3N-5}\right]+\cdots+H_{1}\left[\frac{\alpha}{N(N-1)/2}\right],\end{split}$
(60) $\displaystyle m$ $\displaystyle=\alpha-N(n-1)+\frac{n(n+1)}{2},$ (61)

where $H_{1}$ is the Heaviside step function at $1$,

$H_{1}[x]=\left\\{\begin{array}[]{ll}0&x<1\\\ 1&x\geq 1\\\ \end{array}\right.$
(62)

##  Appendix B DAQC for Hamiltonians with up to M-body terms

In Section II, we have laid out a method for performing DAQC using a
Hamiltonian with $2$-body terms. In this section, we generalize this method
for the case of resource Hamiltonians that have additional, up to $M$-body
terms. We characterize such a Hamiltonian by the collection of pairs of
connected qubits $(j,k)\in\mathcal{C}_{2}$, the collection of triplets of
connected qubits $(j,k,l)\in\mathcal{C}_{3}$… in general, $\mathcal{C}_{b}$,
and their corresponding coupling strengths $\bar{g}^{b}$

$\bar{H}_{M}=\sum_{(j,k)}\bar{g}^{2}_{jk}Z^{j}Z^{k}+\sum_{(j,k,l)}\bar{g}^{3}_{jkl}Z^{j}Z^{k}Z^{l}+\ldots$
(63)

Each $\mathcal{C}_{b}$ has $c_{b}$ elements. This way, the total connectivity
of the $M$-body resource Hamiltonian is

$\mathcal{C}=\bigcup_{b=2}^{M}\mathcal{C}_{b}\,,$ (64)

which has a total of $c=\sum_{b}^{M}c_{b}$ elements. As a specific example,
the total number of terms, $c$, appearing in an ATA Hamiltonian with up to
$M$-body terms is given by:

$c=\binom{N}{2}+\binom{N}{3}+\ldots+\binom{N}{M}\,.$ (65)

If we are able to use the resource Hamiltonian (63) to implement an arbitrary
target Hamiltonian with the same structure,

${H}_{M}=\sum_{(j,k)}{g}^{2}_{jk}Z^{j}Z^{k}+\sum_{(j,k,l)}{g}^{3}_{jkl}Z^{j}Z^{k}Z^{l}+\ldots\,,$
(66)

then we can get rid of the higher body terms by setting $g^{b}=0$ for all
$b>2$. Alternatively, if our problem at hand has such higher body terms, we
can use them to our advantage. Such interaction terms may appear, e.g., in
fermionic [55, 56] and lattice gauge theory quantum simulations [57], and
quantum optimization [58, 59, 60].

Let us construct a digital-analog quantum circuit consisting of $c$ analog
blocks. The first $c_{2}$ analog blocks are preceded and followed by
$X^{m}X^{n}$ gates, in exactly the same way as described in Section II (see
Fig. 2). The following $c_{3}$ analog blocks are preceded and followed by
$X^{m}X^{n}X^{p}$ gates, with $(m,n,p)\in\mathcal{C}_{3}$. This pattern is
repeated until the analog blocks are exhausted. This quantum circuit is a
generalization of the one described in Section II, for which we were
restricting ourselves to $\mathcal{C}=\mathcal{C}_{2}$.

This way, the unitary evolution of such a circuit is

$\displaystyle\begin{split}U_{DAQC}=&\prod_{(m,n)}X^{m}X^{n}\exp\Bigg{(}i\,t_{mn}^{2}\Big{[}\sum_{(j,k)}\bar{g}_{jk}^{2}Z^{j}Z^{k}+\sum_{(j,k,l)}\bar{g}_{jkl}^{3}Z^{j}Z^{k}Z^{l}+\ldots\Big{]}\Bigg{)}X^{m}X^{n}\\\
&\times\prod_{(m,n,p)}X^{m}X^{n}X^{p}\exp\Bigg{(}i\,t_{mnp}^{3}\Big{[}\sum_{(j,k)}\bar{g}_{jk}^{2}Z^{j}Z^{k}+\sum_{(j,k,l)}\bar{g}_{jkl}^{3}Z^{j}Z^{k}Z^{l}+\ldots\Big{]}\Bigg{)}X^{m}X^{n}X^{p}\\\
&\times\cdots\end{split}$ (67)
$\displaystyle\begin{split}=&\prod_{(m,n)}\exp\Bigg{(}i\,t_{mn}^{2}\Big{[}\sum_{(j,k)}M_{jkmn}^{(2,2)}\bar{g}_{jk}^{2}Z^{j}Z^{k}+\sum_{(j,k,l)}M_{jklmn}^{(3,2)}\bar{g}_{jkl}^{3}Z^{j}Z^{k}Z^{l}+\ldots\Big{]}\Bigg{)}\\\
&\times\prod_{(m,n,p)}\exp\Bigg{(}i\,t_{mnp}^{3}\Big{[}\sum_{(j,k)}M_{jkmnp}^{(2,3)}\bar{g}_{jk}^{2}Z^{j}Z^{k}+\sum_{(j,k,l)}M_{jklmnp}^{(3,3)}\bar{g}_{jkl}^{3}Z^{j}Z^{k}Z^{l}+\ldots\Big{]}\Bigg{)}\\\
&\times\cdots\,,\end{split}$ (68)

where we have introduced the collections of elements $M^{(a,b)}_{jk\ldots
mn\ldots}$, which can take on the values $\pm 1$. These elements are
calculated as

$M^{(a,b)}_{jk\ldots mn\ldots}=(-1)^{\alpha},\text{where
}\alpha={\sum_{\begin{subarray}{c}\nu=\\{j,k\ldots\\}\\\
\mu=\\{m,n\ldots\\}\end{subarray}}\delta_{\nu\mu}}\,,$ (69)

for which the ones in Eq. (21) are a special case with $\nu=\\{i,j\\}$ and
$\mu=\\{m,n\\}$. Now, each of these collections of elements can be rearranged
into a matrix $M^{(a,b)}$ of dimensions $c_{a}\times c_{b}$, following a
process similar to the one in Appendix A. When comparing our DAQC evolution
(68) and the evolution under the target Hamiltonian (66) for some time
$t_{f}$, we can see that they are equal when the following set of vector
equations holds:

$\displaystyle\begin{split}t_{f}\,\mathbf{G}^{2}&=M^{(2,2)}\hphantom{\vdots}\mathbf{t}^{2}+M^{(2,3)}\hphantom{\vdots}\mathbf{t}^{3}+\cdots\\\
t_{f}\,\mathbf{G}^{3}&=M^{(3,2)}\hphantom{\vdots}\mathbf{t}^{2}+M^{(3,3)}\hphantom{\vdots}\mathbf{t}^{3}+\cdots\\\
\vdots\hphantom{G^{2}}&=\hphantom{M^{(3}}\vdots\hphantom{\mathbf{t}^{2,2)}}+\hphantom{M^{(3}}\vdots\hphantom{\mathbf{t}^{3,3)}}+\ddots\end{split}$
(70)

Again, Eq. (22) is a special case of this set of equations, in which we
restrict ourselves only to the first term of the RHS of the first equation.
This set of equations can be written as just one vector equation, where a
joint matrix of dimensions $c\times c$ appears, comprising all the $M^{(a,b)}$
matrices:

$t_{f}\,\begin{pmatrix}\mathbf{G}^{2}\\\ \mathbf{G}^{3}\\\
\vdots\end{pmatrix}=\begin{pmatrix}M^{(2,2)}&M^{(2,3)}&\cdots\\\
M^{(3,2)}&M^{(3,3)}&\cdots\\\
\vdots&\vdots&\ddots\end{pmatrix}\begin{pmatrix}\mathbf{t}^{2}\\\
\mathbf{t}^{3}\\\ \vdots\end{pmatrix},$ (71)

which we can write as
$t_{f}\mathbf{G}_{\text{joint}}=M_{\text{joint}}\mathbf{t}_{\text{joint}}$ for
compactness. By solving this equation through the inversion of the joint
matrix $M_{\text{joint}}$, we can calculate the time each analog block must
run for in our digital-analog circuit:

$\mathbf{t}_{\text{joint}}=M_{\text{joint}}^{-1}\mathbf{G}_{\text{joint}}t_{f}\,.$
(72)

Expressing the relationship between the times and the couplings of the target
Hamiltonian in this single equation is very useful, because then the only
condition we need to impose for this relation to hold is the invertibility of
the joint matrix, as opposed to the invertibility of each individual
$M^{(a,b)}$ matrix.

##  Appendix C Cancelling undesired odd-body terms in the resource Hamiltonian

By substituting an analog block of time $t$ by two analog blocks of time $t/2$
each, and placing $X$ gates on all qubits before and after one of the two
analog blocks (see Fig. 9), we can effectively cancel all odd-body terms. This
is because flipping the sign of the connections of all qubits leaves the even-
body terms untouched, but flips the sign of all odd-body terms, as exemplified
here for two- and three-body terms:

$\displaystyle\left(\prod_{m=0}^{N}X^{m}\right)Z^{j}Z^{k}\left(\prod_{m=0}^{N}X^{m}\right)$
$\displaystyle=(-1)^{2}Z^{j}Z^{k}=Z^{j}Z^{k}\,,$ (73)
$\displaystyle\left(\prod_{m=0}^{N}X^{m}\right)Z^{j}Z^{k}Z^{l}\left(\prod_{m=0}^{N}X^{m}\right)$
$\displaystyle=(-1)^{3}Z^{j}Z^{k}Z^{l}=-Z^{j}Z^{k}Z^{l}\,.$ (74)

Thus, evolving by times $t/2$ with the original and flipped signs cancels all
odd-body terms, while evolving according to all even-body terms for a total
time $t$. This procedure introduces, at most, $2N$ single-qubit gates per
analog block, while leaving the total analog times intact. The single-qubit
gate depth is increased, at most, by two per analog block.

Figure 9: Digital-analog circuit implementing an analog block with all odd-
body terms cancelled, for a $5$-qubit device.

##  Appendix D Derivation of bDAQC non-commutativity error

in Ref. [1], it is shown that the first and last analog blocks (i.e., a
constant number of analog blocks) of a digital-analog circuit introduce errors
$e_{\text{boundary}}=\mathcal{O}(\Delta t^{2})$.

Each central analog block, however, is shown to introduce an error

$\displaystyle e_{\text{central}}$ $\displaystyle=\left\lVert
1-e^{-i\bar{H}\Delta t/2}e^{-iH_{s}\Delta t}e^{-i\bar{H}\Delta
t/2}e^{i(\bar{H}+H_{s})\Delta t}\right\rVert$ (75)
$\displaystyle=\frac{(\Delta
t)^{3}}{4}\left\lVert\big{[}[\bar{H},H_{s}],\bar{H}+2H_{s}\big{]}\right\rVert+\mathcal{O}(\Delta
t^{4})\,,$ (76)

where $H_{s}$ is the Hamiltonian generating the SQG that overlaps the resource
Hamiltonian. Let’s assume that this SQG is an $X^{a}$ gate, and thus
$H_{s}=\frac{\pi}{2\Delta t}X^{a}$. By explicitly plugging this, and the
resource Hamiltonian $\bar{H}=\sum_{j,k}\bar{g}_{jk}Z^{j}Z^{k}$ into (76), we
can get the explicit expression of $e_{\text{central}}$. The innermost
commutator in Eq. (76) is

$\displaystyle\left[\bar{H},H_{s}\right]$
$\displaystyle=\left[\sum_{k=1}^{d}\bar{g}Z^{a}Z^{k},\frac{\pi}{2\Delta
t}X^{a}\right]$ (77) $\displaystyle=\frac{d\bar{g}\pi}{2\Delta
t}\left[Z^{a}Z^{k},X^{a}\right]$ (78)
$\displaystyle=\frac{d\bar{gi}\pi}{\Delta t}Y^{a}Z^{k}\,.$ (79)

Plugging this into the outermost commutator yields

$\displaystyle[[\bar{H},H_{s}],\bar{H}+2H_{s}]$
$\displaystyle=\frac{d\bar{g}\pi i}{\Delta
t}\big{[}Y^{a}Z^{k},\bar{H}+2H_{s}\big{]}$ (80)
$\displaystyle=\frac{d\bar{g}\pi i}{\Delta
t}\left([Y^{a}Z^{k},\bar{H}]+[Y^{a}Z^{k},2H_{s}]\right)\,.$ (81)

Each of the two commutators above yields:

$\displaystyle\left[Y^{a}Z^{a},\bar{H}\right]$
$\displaystyle=\left[Y^{a}Z^{a},\sum_{k=1}^{d}\bar{g}Z^{a}Z^{k}\right]$ (82)
$\displaystyle=2d\bar{g}iX^{a}I^{k}$ (83)
$\displaystyle\left[Y^{a}Z^{a},2H_{s}\right]$
$\displaystyle=\left[Y^{a}Z^{a},\frac{\pi}{\Delta t}X^{a}\right]$ (84)
$\displaystyle=\frac{-2\pi i}{\Delta t}Z^{a}Z^{k}\,.$ (85)

Computing the total infidelity, we get

$e_{\text{central}}=\frac{(\Delta
t)^{3}}{4}\sqrt{(d\bar{g})^{4}\left(\frac{2\pi}{\Delta
t}\right)^{2}+(2d\bar{g})^{2}\left(\frac{\pi}{\Delta t}\right)^{4}}\,.$ (86)

##  Appendix E Digital-analog SWAP gates on a star-connectivity

The $\rm{SWAP}$ gate acts on two qubits by exchanging their states,
$\ket{j}\ket{k}\rightarrow\ket{k}\ket{j}$

$\rm{SWAP}=\begin{pmatrix}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\end{pmatrix}.$ (87)

A $\rm{SWAP}$ gate applied on the central qubit and one of the external qubits
can be expressed in terms of two-qubit Pauli rotations as:

$\displaystyle\rm{SWAP}$
$\displaystyle=\exp\big{(}i\frac{\pi}{4}(X^{0}X^{k}+Y^{0}Y^{k}+Z^{0}Z^{k})\big{)}$
(88)
$\displaystyle=\exp\big{(}i\frac{\pi}{4}X^{0}X^{k}\big{)}\exp\big{(}i\frac{\pi}{4}Y^{0}Y^{k}\big{)}\exp\big{(}i\frac{\pi}{4}Z^{0}Z^{k}\big{)}$
(89)
$\displaystyle\begin{split}&=\Big{[}H^{0}H^{k}\exp\big{(}i\frac{\pi}{4}Z^{0}Z^{k}\big{)}H^{0}H^{k}\Big{]}\\\
&\times\Big{[}S^{0}S^{k}H^{0}H^{k}\exp\big{(}i\frac{\pi}{4}Z^{0}Z^{k}\big{)}H^{k}H^{0}S^{\dagger
k}S^{\dagger 0}\Big{]}\\\
&\times\exp\big{(}i\frac{\pi}{4}Z^{0}Z^{k}\big{)}\,.\end{split}$ (90)

In the last equality, we have decomposed the $\rm{SWAP}$ gate into three
different evolutions under a $Z^{0}Z^{k}$ Hamiltonian, each of which we can
interpret as a target Hamiltonian with all couplings $g_{jk}=0$ except for
$g_{0k}=\frac{\pi}{4t_{f}}$. Following the optimized DAQC protocol described
in Section IV, each of these target Hamiltonians requires two analog blocks,
accounting for a total of $6$ analog blocks needed to implement a $\rm{SWAP}$
gate.

Figure 10: Average fidelity of the Star-QFT algorithm for DQC, sDAQC and bDAQC, considering control errors, obtained in Section VII. The dashed lines represent the fitted curves of the form (92). Table 2: Parameters of the fitting of the curve (92) to the data in Fig. 10. The parameter $f$ is smaller for DAQC in part because, even though the fidelity per two-qubit term was chosen to be higher for DAQC in the simulations, their successive action on the same pairs of qubits throughout the algorithm accumulates coherent errors, thus effectively decreasing their overall fidelity [45]. Similarly, the parameter $b$ is worse than the scaling of the number of two-qubit terms predicted in Section V.2 for the three paradigms because of this accumulation of coherent errors, and from the absortion of the secondary sources of error into a single parameter. | $f$ | $a$ | $b$ | $c$
---|---|---|---|---
DQC | $0.99986$ | $0.92985$ | $2.3882$ | $-1.58\times 10^{-4}$
sDAQC | $0.99831$ | $0.06445$ | $3.8571$ | $-2.94\times 10^{-4}$
bDAQC | $0.99858$ | $0.42443$ | $2.8559$ | $-0.02373$

##  Appendix F Trade-off between control errors and environmental noise in
Star-QFT

(a)

(b)

Figure 11: Approximate total fidelity of the Star-QFT algorithm, calculated
according to Eq. (91), in the DQC, sDAQC and bDAQC paradigms, for different
execution times of the TQGs, and (a) a relatively large $T_{1}=500\mu s$, and
(b) a relatively small $T_{1}=50\mu s$. The trade-off between control errors
and decoherence can be favorable to DAQC in a parameter regime with very
limited $T_{1}$ and very long TQGs, as can be seen in (b), where DAQC
outperforms DQC for $N>11$ when $t_{TQG}=0.15\mu s$, and for $N>5$ when
$t_{TQG}=0.3\mu s$.

While the DQC algorithm for the Star-QFT has a better performance than DAQC
regarding the infidelity coming from control errors (see Fig. 8b), it has a
longer execution time (see Fig. 8e). In turn, long execution times imply that
the algorithm becomes more affected by environmental decoherence, so a trade-
off may arise for a large enough number of qubits, in which decoherence
accounts for a bigger effect on the infidelity.

In order to analyze such a trade-off, we study the relationship between the
scaling of both sources of infidelity. We assume, like we did in Section
III.5, that the main source of decoherence is thermal relaxation, and consider
a simple Markovian model for it. Additionally, we consider this infidelity to
be independent for each qubit, and also independent from their unitary
dynamics. Therefore, the total fidelity of the computation is given by

$F_{\text{total}}\approx\langle F_{U}\rangle\times e^{-Nt/T_{1}}\,,$ (91)

where $F_{U}$ is the unitary evolution’s fidelity, as defined in Eq. (50) and
represented in Fig. 8b, $N$ is the number of qubits, $t$ is the execution time
of the quantum circuit and $T_{1}$ is the thermal relaxation time. While
approximate, this expression can give us insight into the interplay between
the scaling of the two sources of infidelity.

In order to extend $\langle F_{U}\rangle$ to a higher number of qubits, for
which the effects of decoherence become more relevant, we fit the fidelity
data of Fig. 8e for DQC, sDAQC and bDAQC to a function of the form

$\langle F_{U}\rangle\approx f^{a(N^{b})}+c\,,$ (92)

where we have assumed that each operation incurs in an independent infidelity,
and where $f,a,b,c$ are the parameters resulting from the function fitting,
which are given in Table 2. We plot the fitted curves on top of the simulated
data in Fig. 10.

Finally, in Fig. 11, we plot the resulting total fidelity calculated as in Eq.
(91) for different scenarios, in which TQGs have execution times of $50$,
$150$ and $300ns$, and in which $T_{1}$ is either $50$ or $500\mu s$. For the
regime in which $T_{1}$ is very short, and the time of the TQGs is very long,
the trade-off is favorable to DAQC, as can be seen in Fig. 11b, for which
$T_{1}=50\mu s$, and DAQC outperforms DQC for $t_{\text{TQG}}=150ns$ and
$t_{\text{TQG}}=300ns$.