# Comparing planar quantum computing platforms at the quantum speed limit

Daniel Basilewitsch<EMAIL_ADDRESS>Institute for Theoretical
Physics, University of Innsbruck, A-6020 Innsbruck, Austria  Clemens Dlaska
Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck,
Austria  Wolfgang Lechner  Institute for Theoretical Physics, University of
Innsbruck, A-6020 Innsbruck, Austria  Parity Quantum Computing GmbH, A-6020
Innsbruck, Austria

(August 28, 2024)

###### Abstract

An important aspect that strongly impacts the experimental feasibility of
quantum circuits is the ratio of gate times and typical error time scales.
Algorithms with circuit depths that significantly exceed the error time scales
will result in faulty quantum states and error correction is inevitable. We
present a comparison of the theoretical minimal gate time, i.e., the quantum
speed limit (QSL), for realistic two- and multi-qubit gate implementations in
neutral atoms and superconducting qubits. Subsequent to finding the QSLs for
individual gates by means of optimal control theory we use them to quantify
the circuit QSL of the quantum Fourier transform and the quantum approximate
optimization algorithm. In particular, we analyze these quantum algorithms in
terms of circuit run times and gate counts both in the standard gate model and
the parity mapping. We find that neutral atom and superconducting qubit
platforms show comparable weighted circuit QSLs with respect to the system
size.

##  I Introduction

Quantum computers promise to solve computational problems that are deemed hard
or even intractable for classical computers. Their potential applications
include prime-factoring of large integers [1], quantum simulation [2], quantum
chemistry [3], combinatorial optimization [4], and even problems in finance
[5]. Currently, quantum computing is in the so-called noisy intermediate-scale
quantum (NISQ) era [6], characterized by imperfect qubit control, and qubit
numbers that prohibit quantum error correction [7] for relevant problem sizes.
Nevertheless, recent proof-of-principle experiments [8, 9, 10, 11]
demonstrated that a computational quantum advantage over classical computers
can be reached already with NISQ hardware. However, it remains a crucial
challenge to go beyond the proof-of-principle stage, i.e., to demonstrate a
quantum advantage for practically relevant computational tasks on resource-
limited present-day devices.

To reach a practical quantum advantage regime [2] in NISQ-era digital quantum
computing it is of crucial importance to execute quantum algorithms as
efficiently as possible in order to minimize the time for noise mechanisms to
impair the quantum information processing. This effectively makes a
minimization of the quantum algorithm run times and gate counts desirable — a
task that can be addressed in various ways. One option is to find an
algorithm’s optimal circuit representation, i.e., a circuit requiring a
minimal circuit depth together with a minimal gate count for a given set of
available gates. Quantum circuit optimization has, e.g., been done
heuristically [12, 13] or by machine learning techniques [14, 15] and several
open-source packages are readily available [16, 17, 18].

Another option for minimizing the algorithm run time is to minimize the time
for each elemental quantum gate of a given quantum circuit. While protocols
for fast, high-fidelity quantum gates are nowadays routinely available and
implemented on all major quantum computing platforms like neutral atoms [19,
20], superconducting circuits [21, 8] or trapped ions [22, 23], the ideal
would be to execute every quantum gate at its quantum speed limit (QSL). In
general, the QSL denotes the shortest time needed to accomplish a given task
[24]. It constitutes a fundamental limit in time and depends on the system
under consideration, i.e., its Hamiltonian and the control knobs available to
steer the dynamics.

Here, we determine the QSLs of quantum gates for two major quantum computing
platforms that allow for two-dimensional (2D) qubit arrangements — neutral
atoms and superconducting circuits 111Due to the restriction to 2D
architectures, we do not consider trapped ion platforms in this study. Our
study reveals how close the current experimental gate protocols are in
comparison to their QSLs and thus exemplifies what can theoretically still be
gained from further speeding up gate protocols. Moreover, provided that every
gate could be experimentally realized at the QSL, our analysis gives an
estimate of how many gates can be executed realistically before decoherence
takes over and renders longer quantum circuits practically infeasible. To this
end, we consider two prototypical quantum computing algorithms: (i) the
quantum Fourier transform (QFT), required for Shor’s algorithm for integer
factorization [1], and (ii) the quantum approximate optimization algorithm
(QAOA), used to solve combinatorial optimization problems [4]. Considering
standard NISQ devices for both neutral atoms and superconducting circuits with
qubits arranged in a 2D grid architecture with only nearest-neighbor
connectivity, we calculate the circuit run times with gates at the QSL for
both algorithms. This allows for a direct comparison of both platforms in
terms of the maximal problem sizes that should currently be feasible on their
NISQ representatives.

A common challenge arising in 2D platforms with nearest-neighbor connectivity
is the requirement to perform gates between non-neighboring qubits. In the
standard gate model (SGM), such gates can be replaced by sequences of
universal single- and two-qubit gates using the available local connectivity.
However, this comes at the price of increasing the circuit depths and gate
counts. As an alternative to the SGM, we also examine circuit representations
using the so-called parity mapping (PM). In brevity, the PM for quantum
computing [26] and quantum optimization [27, 28] is a problem-independent
hardware blueprint that only requires nearest-neighbor connectivity at the
cost of increased qubit numbers. Since for QAOA circuits in the PM it is
beneficial to use local three- and four-qubit gates [29, 30], we also
determine their QSLs on both platforms.

It should be noted that this work focuses entirely on the determination of the
QSLs for various quantum gates and how to turn these into a fair comparison of
circuit run times across platforms. A more detailed discussion of gate
protocols for specific quantum gates or platforms can, e.g., be found in Refs.
[31, 32, 33] for neutral atoms or in Refs. [34, 35, 36] for superconducting
circuits. Moreover, it should be noted that we only consider the gate times,
respectively circuit run times, as well as the number of gates as indicators
for the feasibility of quantum circuits. While both quantities are doubtlessly
important, they are by no means the only quantities impacting a circuit’s
feasibility. A holistic figure of merit assessing a circuit’s feasibility
would also need to account for state preparation and measurement errors and
various other error sources.

The paper is organized as follows. In Sec. II we present our main result,
i.e., an overview of circuit run times when using gate times from the
literature and gate times at the QSL — evaluated both for neutral atoms and
superconducting circuits as well as for circuits in the SGM and the PM. In
Sec. III we then present the details of our numerical model. Section IV
introduces the basic notion of QSLs and how quantum optimal control theory
(OCT) can be used for determining QSLs. A detailed discussion of QSLs as well
as which combinations of available control fields allow to reach the QSLs, is
given in Sec. V. Section VI concludes.

##  II Main result: quantum algorithms at the quantum speed limit

In this section, we compare the run times of QFT and QAOA quantum circuits
using gate set implementations available on neutral atom and superconducting
circuit hardware. To do so we consider two scenarios. In the first scenario,
we use literature values for gate times of state-of-the-art gate
implementations of the minimal universal gate set (referred to as “standard
gate set” (SGS) from now on) native to each platform, which thus represents
the canonical way of converting quantum algorithms into executable quantum
circuits. In the second scenario, we use an extended set of gates available at
each platform with gate times at the QSL (referred to as “QSL gate set” (QGS)
from now on), which thus yields the current fundamental limits in circuit run
times. We describe both gate sets in Sec. II.1 and use them to analyze circuit
run times of QFT and QAOA quantum circuits both in the SGM in Sec. II.2 and
the PM in Sec. II.3. A brief review of the basic concepts and quantum circuits
of a QFT and a single QAOA step within the SGM and the PM is given in Appendix
A.

|  neutral atoms |  superconducting circuits
---|---|---
standard gate set (SGS) | gate | time (ns) | gate | time (ns)
local |  $\sim 1000$ [20] | local |  $25$ [37]
CZ |  $350$ [20] | Sycamore |  $12$ [37]
QSL gate set (QGS) | gate | time (ns) | gate | time (ns)
local |  $\sim 1000$ [20] | local |  $25$ [37]
CNOTa | $300$ | CNOT | $14$
CZ | $350$ | CZ | $10$
SWAP | $400$ | SWAP | $12$
ZZZ | $600$ | ZZZ | $24$
ZZZZ | $600$ | ZZZZ | $80$

  * a

The CNOT gate is listed for completeness but not used in any circuits for
neutral atoms, see Sec. V for details.

Table 1: Overview of different gate sets and corresponding gate times on
neutral atoms and superconducting circuits. While the row “standard gate set”
(SGS) represents typical gates and times used on the respective platform, the
multi-qubit gates in the row named “QSL gate set” (QGS) correspond to an
extended gate set with gate times at the QSL.  Figure 1: Comparison of the
circuit run times and gate counts for the quantum Fourier transform (QFT) and
quantum approximate optimization algorithm (QAOA) between neutral atoms and
superconducting circuits. The left (right) column corresponds to the QFT
(QAOA) and the orange (purple) marker correspond to neutral atoms
(superconducting circuits). The circuit run times for both algorithms and
various problem instances of different size, $N=9,16,\dots,81$ qubits for the
QFT and $N=9,16,\dots,121$ qubits for the QAOA, are given in panels (a) and
(c), respectively. The numbers of two- and multi-qubit gates in the
corresponding circuits are given in panels (b) and (d), respectively. The data
for the squares [circles] is obtained using the standard gate model (SGM)
using the gates and times from the standard gate set (SGS) [QSL gate set
(QGS)], cf. Table 1. In contrast, the plus signs [crosses] represent the
results when the same algorithms are realized in the parity mapping (PM) using
the SGS [QGS]. The pentagons correspond to an altered 2D architectures
allowing for next-to-nearest neighbor (NNN) coupling between qubits. The run
time for each circuit is given in units of typical platform-specific error
times. The gray area in panels (a) and (c) indicates where the circuit run
times exceeds the error times.  Figure 2: Reduction in weighted circuit run
times and (two- and multi-qubit) gate counts for the scenarios and data shown
in Fig. 1. In panels (a) and (b) the circuits in the standard gate model (SGM)
and parity mapping (PM) are compared when using the “QSL gate set” (QGS)
instead of the “standard gate set” (SGS). In panels (c) and (d) the circuits
in the SGS and QGS are compared when using the PM instead of the SGM. All
presented numbers are the average over problem instances of different sizes,
i.e., different numbers of qubits.

###  II.1 Standard and QSL gate sets

In the SGM, each quantum algorithm is converted into quantum circuits using
gates from a universal set of quantum gates. Such a universal set typically
contains all the single-qubit gates and at least one entangling two-qubit
gate, e.g., the CNOT gate [38]. The row named “standard gate set” in Table 1
summarizes the native universal gate sets and their typical gate times for
both platforms at comparable gate fidelities. For neutral atoms, we consider
the controlled-Z gate, $\mathrm{CZ}=\mathrm{diag}\\{1,1,1,-1\\}$, as the
typical entangling two-qubit gate. This is a common choice for neutral atoms
[20] as it has been successfully used for implementing quantum algorithms [39,
40]. For superconducting circuits, we have chosen the iSWAP-like Sycamore gate
as an entangling two-qubit gate, motivated by its short gate time and
successful usage in recent quantum advantage experiments with quantum
processors based on tunable couplers [8, 9].

In addition, we consider an extended, platform-independent set of quantum
gates operated at the QSL (see “QSL gate set” in Table 1). This set consists
of all single-qubit gates as well as several multi-qubit gates implementable
on both platforms: CZ, CNOT, SWAP, ZZZ, and ZZZZ. The availability of a wider
range of gates allows for more flexibility in finding circuit representations
with fewer gates and shorter circuit run times. In addition, we also assume
that every gate in this set is executed at the QSL, which allows for another
speed-up of circuit run times. The run times obtained for the QGS should
therefore be viewed as an estimate for the QSL of the circuit itself, i.e.,
the circuit QSL — provided that no other, potentially faster and/or better
suited gates are available 222Note that in practice, gate sets are typically
tailored to match the quantum algorithm in question. However, this goes beyond
the scope of this work. Note that the details regarding the method to
determine the QSLs and their results are presented in Secs. IV and V. The row
“QSL gate set” in Table 1 summarizes the gates of the extended set and lists
the QSL times for both platforms.

Table 1 indicates that absolute circuit run times for neutral atoms will be
longer compared to those for superconducting circuits since their elemental
gate times differ by more than an order of magnitude. However, in order to
ensure a fair comparison of circuit run times the absolute gate times need to
be weighted by the finite coherence time and lifetime of qubits and other
levels involved in the gate mechanisms.

For neutral atoms, we take both the coherence time of the qubit states, given
by the dephasing time $T_{2}^{*}=$4\text{\,}\mathrm{ms}$$ [39], and the
lifetime of the Rydberg state,
$T_{\mathrm{Ryd}}=$150\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ [39], as
typical error time scales against which we compare circuit run times. We take
the former as reference for single-qubit gates, since they don’t occupy the
Rydberg level 333Assuming that the qubit is encoded in the atomic ground state
manifold, such as hyperfine states.,and the latter for two-qubit gates, where
controlled transitions via Rydberg levels constitute the primary gate
mechanism 444Note that we do not consider the lifetime of the intermediate
levels between qubit and Rydberg states. For superconducting circuits, we take
their intrinsic $T_{1}$ time, $T_{1}=$15\text{\,}\mathrm{\SIUnitSymbolMicro
s}$$ [8], as typical error time scale since it applies for both single- and
two-qubit gates. However, note that these choices are rather conservative. For
neutral atoms, any two-qubit gate dynamics will naturally also involve the
qubit levels, which have much longer coherence times. Weighting two-qubit
gates exclusively by the Rydberg lifetime thus overestimates the lifetime-
induced error probability. For superconducting circuits, longer $T_{1}$ times
up to $500\text{\,}\mathrm{\SIUnitSymbolMicro s}$ have already been reported
[44].

###  II.2 Circuit times in the standard gate model

In the following, we calculate the circuit run times for QFTs and single QAOA
steps in the SGM using gates from both the SGS and the QGS. As already
outlined above, we assume both hardware platforms to consist of qubits
arranged in 2D arrays with only nearest-neighbor connectivity (see Sec. III
for details regarding the model). This requires replacing all gates between
non-connected qubits with gate sequences between physically connected qubits.

####  II.2.1 Standard gate set circuits

Using gates from the SGS, the circuit’s gate sequences consist of single-qubit
gates plus CZ gates in case of neutral atoms and single-qubit gates plus
Sycamore gates in case of superconducting circuits (cf. Table 1). Since the
circuit for the QFT does by construction not require any gate between non-
neighboring qubits [45], we only replace the controlled-phase gate and SWAP
gate by gate sequences from the SGS. The situation changes for the QAOA
circuits, as it assumes an all-to-all connected architecture and finding a
circuit representation with minimal depth while only requiring nearest-
neighbor connectivity is likely NP-hard [46]. As a remedy, we use the open-
source _pytket_ compiler [16] to translate the QAOA’s bare quantum circuits to
executable quantum circuits that are in agreement with the hardware’s nearest-
neighbor connectivity and SGSs. We furthermore use the compiler’s quantum
circuit optimization features to optimize the circuits in order to minimize
gate counts and circuit depths.

The squares in Fig. 1 (a) and (c) show the resulting run times for quantum
circuits corresponding to QFTs and QAOA steps, respectively, when executed on
neutral atoms (orange) and superconducting circuits (purple). Within each
line, the size of the problem instance increases from the lower left to the
upper right. Note that for a fair comparison between platforms, all gate times
have been weighted by each platform’s intrinsic error time scales as described
previously. Circuits with run times significantly exceeding the platform’s
intrinsic, level-dependent error time scales, highlighted by the gray area in
Fig. 1 (a) and (c), will most likely yield unreliable results. We observe that
despite the different time scales for gates, the weighted circuit run times
are almost identical for both platforms and for both the QFT and QAOA, see
squares in Fig. 1 (a) and (c), respectively. However, only problem instances
with relatively small qubit numbers seem to be currently doable on both
platforms.

While the circuit run time is one deciding factor in whether it is feasible on
current NISQ hardware, this measure neglects the fact that every gate comes
with an intrinsic error probability. Assuming gate errors on the order of
$0.1-1\%$, which are realistic both for neutral atoms [20, 40] and
superconducting circuits [37], it is clear that also the gate count limits the
feasibility of quantum circuits and lower gate counts are thus preferable. To
this end, the squares in Fig. 1 (b) and (d) show the number of two-qubit gates
required for QFT and single-step QAOA circuits, respectively. Note that
circuit representations in terms of the SGS require the same number of two-
qubit gates both on neutral atom and superconducting qubit hardware. Hence,
both cases are represented by a single line in Figs. 1 (b) and (d).

####  II.2.2 QSL gate set circuits

The circles in Fig. 1 (a) and (c) show the circuit run times for the same
quantum algorithms and complexity levels as used for the squares but when the
QGS is employed to generate circuit representations. The corresponding two-
qubit gate counts are illustrated by the circles in Fig. 1 (b) and (d). We
observe an overall reduction in circuit run times and gate counts for both
platforms, both algorithms and any considered problem instance. This
improvement is a combined effect of using maximally fast quantum gates at the
QSL and taking advantage of the increased flexibility provided by the extended
set of gates. Especially the availability of SWAP gates needs to be stressed
as it directly reduces the gate count and thus leads to a reduction in circuit
run time even without an additional speedup in gate times. In order to
quantify the improvements achievable through the QGS, Fig. 2 (a) lists the
average reduction in circuit run times and gate counts for each algorithm and
platform.

Our analysis demonstrates to which extent circuit run times and gate counts
can — from a theoretical perspective — still be improved if standard gates are
replaced by an extended gate set with gate times at the QSL. However, even
with the improved gate set and its reduction in run time and gate count, most
of the quantum circuits, i.e., circles in Fig. 1, remain likely infeasible
using current NISQ hardware. Quantum error correction codes could in principle
address the issue of circuit run times and gate counts exceeding the limits
set by finite lifetimes and gate errors. Nevertheless, since both lifetimes
[47, 44] and qubit numbers [48] are constantly increasing, more complex
quantum circuits will likely reach the feasible regime in the near future.

###  II.3 Circuits times in the parity mapping

Besides the representation of quantum algorithms using the SGM, we now
consider the representation of the same algorithms within the PM. While the PM
was originally designed to tackle combinatorial optimization problems via
quantum annealing [27], it can also be utilized for digital quantum
optimization algorithms such as QAOA [30] as well as to achieve universal
quantum computing [26]. At its core, the PM circumvents the need for long-
range interactions between qubits, which in turn renders gates between non-
adjacent qubits obsolete. However, this comes at the expense of requiring more
physical qubits and many-body constraints on $2\times 2$ plaquettes of qubits
as specified in detail in Appendix A.2. Similar to our analysis for the SGM
(see Sec. II.2) we use either gates from the SGS or the QGS in the following.

####  II.3.1 Standard gate set circuits

The plus signs in Fig. 1 (a) [(b)] show the circuit run times [two-qubit gate
counts] of QFTs in the PM for the same problem instances as used for the SGM
(squares). In both cases and for both platforms, the gates and times of the
SGS have been used. We observe a reduction in circuit run times and gate
counts for both platforms with the comparison being made with respect to the
values for the SGM [see also Fig. 2 (c)]. The same comparison can be done for
the QAOA steps with their circuit run times and gate counts given by the plus
signs in Fig. 1 (c) and (d), respectively. Regarding single-step QAOA resource
requirements, we observe a significant reduction in circuit run times [see
also Fig. 2 (c)] due to the constant circuit depth in the PM [29, 30].
However, note that the circuit depth differs for neutral atoms and
superconducting circuits due to the different coupling mechanisms between
qubits with neutral atoms having the deeper circuits, cf. Appendix A.2. The
reduction in circuit run time is accompanied by an increase in two-qubit gate
counts compared to the SGM, which we attribute to the decomposition of the
constraint gates, cf. Eq. (17), into the natively available gates within each
platform.

####  II.3.2 QSL gate set circuits

The crosses in Fig. 1 show the results for circuit representations in the PM
when employing gates and times from the QGS. For the QFT on neutral atoms, we
do not observe any further reduction in circuit time or gate counts compared
to its representation utilizing the SGS. This is because its circuit
representations [49] for neutral atoms contain only single-qubit and CZ gates
— gates for which the gate times are identical within the SGS and the QGS. In
contrast, for superconducting circuits, we still observe an improvement in
circuit time since the CZ gate becomes directly available in the QGS and must
no longer be replaced by Sycamore and single-qubit gates. However, the
representations in the SGS and QGS only differ by single-qubit gates and hence
their two-qubit gate count is identical and also identical to that of the
neutral atoms. This is reflected in Fig. 1 (c) by a single line of
superimposed plus signs and crosses.

For the single-step QAOA circuits in the PM, we observe a significant
reduction both in circuit run time and gate counts on both platforms when the
QGS is used instead of the SGS. This is due to the availability of the multi-
qubit gates $\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$, cf. Eq. (17),
where the gate-count reduction originates from avoiding single- and two-body
gate decompositions. In addition, it turns out to be much faster to use
control pulses that directly implement these multi-qubit gates as opposed to
serially applying control pulses to implement the required single- and two-
qubit gates. Figure 2 (b) summarizes the average run time and gate count
reductions (in terms of two- and multi-qubit gates) when replacing the SGS
with the QGS within the PM.

In contrast to panels (a) and (b) of Fig. 2, where the gate set changes and
the circuits stay in the SGM or PM, panels (c) and (d) examine the opposite
scenario, i.e., the gate sets are kept constant but the circuit models change.
In detail, Fig. 2 (c) and (d) show the average circuit run time and gate count
reduction when the SGM is replaced by the PM while the gate set is given by
the SGS and QGS, respectively. Especially Fig. 2 (d) needs to be emphasized as
it reveals to which extent the PM allows to reduce circuit run times and gate
counts compared to representations of the same circuits in the SGM. As
mentioned above and detailed in Appendix A.2, the PM-specific improvements
come at the only expense of requiring more qubits. For the current state of
NISQ hardware and independent from the platform, we believe the SGM to be
better suited for QFTs, as more problem instances seem to be feasible, judging
by circuit run time and required qubits. In contrast, for QAOA using the QGS,
we find the described parity representation of circuits (despite its overhead-
induced inferior success probabilities compared to direct SGM-QAOA 555Note
that this statement does not include additional PM-QAOA-performance-increasing
strategies such as described in Ref. [101]) to be an advantageous option, as
the run time and gate count is drastically reduced for each QAOA step compared
to the SGM. In particular, the need for more PM-QAOA steps compared to SGM-
QAOA steps might be compensatable given the resource reduction per PM-QAOA
step. Given the ongoing upscaling of quantum computers in terms of qubit
numbers, see, e.g., IBM’s quantum roadmap [48], the PM seems to be a viable
option for QAOA.

As a final remark, it should be noted that we only discussed optimization
problems with all-to-all connectivity, cf. Eq. (12). However, many realistic
optimization problems have rather sparse connectivity, which would lead to a
significant reduction in required qubits [28] while maintaining its strength
of constant circuit depth.

##  III Modeling neutral atom and superconducting circuit platforms

In Sec. II we have presented the main result of our work — namely a
calculation and comparison of circuit run times and gate counts using neutral
atoms and superconducting circuits as quantum computing platform. In this
section, we now introduce the detailed physical models used for both
platforms. Since our focus is the description of dynamics, we focus on the
respective Hamiltonians including the various control knobs typically
available to steer the systems and implement quantum gates.

###  III.1 Neutral atoms

Arrays of trapped neutral atoms laser coupled to highly excited Rydberg states
are a promising platform for quantum computing [51, 52] and quantum simulation
[53] as qubits can for example be encoded in long-lived hyperfine ground
states. High-fidelity single-qubit gates can be achieved using microwave
fields [54], two-photon Raman transitions [55], or a combination of microwaves
and gradient fields for individual-qubit addressing [56, 57, 58]. In contrast,
entangling operations between atoms, i.e., many-body gates, are typically
realized via strongly interacting Rydberg levels [59, 60] and various control
schemes for two- and multi-qubit gates have been experimentally demonstrated
[61, 62, 63, 20, 64].

Such gates have been used in recent experiments [39, 40] with up to several
hundreds of atoms arranged in a planar geometry. If we consider the smallest
building block of such a 2D array, it consists of $N=4$ atoms with each atom
described by three relevant levels. In the rotating frame, its Hamiltonian
reads ($\hbar=1$)

$\begin{split}\mathsf{H}(t)&=-\sum_{n=1}^{N}\Delta_{n}(t)\ket{r_{n}}\bra{r_{n}}+\sum_{\begin{subarray}{c}n,m=1\\\
n<m\end{subarray}}^{N}V_{nm}\ket{r_{n}r_{m}}\bra{r_{n}r_{m}}\\\
&\quad+\frac{1}{2}\sum_{n=1}^{N}\sum_{l=\uparrow,\downarrow}\Big{[}\Omega_{l,n}(t)e^{\textup{i}\varphi_{l,n}}\ket{r_{n}}\bra{l_{n}}+\mathrm{H.c.}\Big{]},\end{split}$
(1)

where $\ket{\downarrow_{n}}$ and $\ket{\uparrow_{n}}$ denote the qubit levels
of the $n$th atom and $\ket{r_{n}}$ its Rydberg level. The Rabi frequencies of
the two laser fields coupling the $\ket{\downarrow_{n}}$ and
$\ket{\uparrow_{n}}$ states of the $n$th atom to their respective Rydberg
level $\ket{r_{n}}$ are denoted by $\Omega_{\downarrow,n}(t)$ and
$\Omega_{\uparrow,n}(t)$ 666In practice, the coupling between qubit states and
the Rydberg level is typically achieved via a two-photon process and an
intermediate level [93], which we neglect for simplicity. The phases of the
respective laser fields are given by $\varphi_{\downarrow,n}(t)$ and
$\varphi_{\uparrow,n}(t)$ while $\Delta_{n}(t)$ denotes the detuning of the
laser field from the exact transition frequency. We assume $\Delta_{n}(t)$ to
be identical for both fields. The van der Waals (vdW) interaction between atom
$n$ and $m$ is denoted by $V_{nm}$. Note that in Hamiltonian (1) we have
omitted fields responsible for single-qubit gates, i.e., fields coupling the
two qubit levels, since our focus is on multi-qubit gates for which the
Rydberg levels and their interaction are the primarily gate mechanism.

If not stated otherwise, we assume the laser fields to act identically on all
atoms, i.e., $\Omega_{\downarrow,n}(t)=\Omega_{\downarrow}(t)$,
$\Omega_{\uparrow,n}(t)=\Omega_{\uparrow}(t)$ and
$\varphi_{\downarrow,n}(t)=\varphi_{\downarrow}(t)$,
$\varphi_{\uparrow,n}(t)=\varphi_{\uparrow}(t)$ and $\Delta_{n}(t)=\Delta(t)$
for all $n$. In the following, we moreover assume by default a pseudo-2D
architecture with $V_{nm}=V$ as described in more detail in Ref. [66].
Effectively, this corresponds to an architecture with equally strong nearest
neighbor (NN) and next-to-nearest neighbor (NNN) coupling. However, it should
be noted that we do not exploit the NNN couplings when constructing the
quantum circuits in Sec. II. It thus does not affect any of the single- or
two-qubit gate times presented in Table 1 as these times would be identical
for an actual, planar 2D architecture where $V$ is simply the coupling
strength between nearest neighbors. The assumption of of the pseudo-2D
architecture does only affect the multi-qubit constraint gates
$\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$, cf. Eq. (17), which are —
within our study — only relevant for the PM using the QGS, cf. the crosses in
Fig. 1 (c).

###  III.2 Superconducting Circuits

Qubits encoded in the lowest energy levels of superconducting circuits are
another promising platform for quantum computing [67]. Since their parameters
can be to some extent chosen during their fabrication process, superconducting
circuits come in various variants and parameter regimes with transmon qubits
[68] being currently the most prominent ones for quantum computing. Their
qubit levels can be manipulated via microwave fields to either implement
single-qubit gates [69, 70] or two-qubit gates [71, 72] with high fidelity. In
contrast to neutral atom qubits, which interact directly via their Rydberg
levels, qubits encoded in superconducting circuits often interact indirectly
via intermediate coupling elements [73]. In an architecture, where such
couplers are made tunable [74], the effective interaction strength between the
qubits becomes tunable as well. Such qubit architectures have been
successfully used in recent quantum advantage experiments [8, 9] and thus are
a prototypical NISQ quantum computing platform.

The qubits in such a tunable coupler architecture are arranged on a 2D lattice
with nearest-neighbor couplings. In the following, we take the architecture
from Ref. [8] as a reference. The smallest building block within such a system
consists of $N=4$ qubits. The Hamiltonian for this subsystem reads [8]

$\begin{split}\mathsf{H}(t)&=\sum_{n=1}^{N}\left[\omega_{n}(t)\mathsf{b}_{n}^{\dagger}\mathsf{b}_{n}-\frac{\alpha_{n}}{2}\mathsf{b}_{n}^{\dagger}\mathsf{b}_{n}^{\dagger}\mathsf{b}_{n}\mathsf{b}_{n}\right]\\\
&\quad+\sum_{n=1}^{N}\left(\mathsf{b}_{n}+\mathsf{b}_{n}^{\dagger}\right)\Omega_{n}(t)\cos(\bar{\omega}_{n}(t)t)+\mathsf{H}_{\mathrm{int}}(t)\end{split}$
(2)

with

$\begin{split}\mathsf{H}_{\mathrm{int}}(t)&=g_{12}(t)\left(\mathsf{b}_{1}^{\dagger}\mathsf{b}_{2}+\mathsf{b}_{1}\mathsf{b}_{2}^{\dagger}\right)+\frac{g_{12}^{2}(t)}{|\eta|}\mathsf{b}_{1}^{\dagger}\mathsf{b}_{1}\mathsf{b}_{2}^{\dagger}\mathsf{b}_{2}\\\
&\quad+g_{23}(t)\left(\mathsf{b}_{2}^{\dagger}\mathsf{b}_{3}+\mathsf{b}_{2}\mathsf{b}_{3}^{\dagger}\right)+\frac{g_{23}^{2}(t)}{|\eta|}\mathsf{b}_{2}^{\dagger}\mathsf{b}_{2}\mathsf{b}_{3}^{\dagger}\mathsf{b}_{3}\\\
&\quad+g_{34}(t)\left(\mathsf{b}_{3}^{\dagger}\mathsf{b}_{4}+\mathsf{b}_{3}\mathsf{b}_{4}^{\dagger}\right)+\frac{g_{34}^{2}(t)}{|\eta|}\mathsf{b}_{3}^{\dagger}\mathsf{b}_{3}\mathsf{b}_{4}^{\dagger}\mathsf{b}_{4}\\\
&\quad+g_{41}(t)\left(\mathsf{b}_{4}^{\dagger}\mathsf{b}_{1}+\mathsf{b}_{4}\mathsf{b}_{1}^{\dagger}\right)+\frac{g_{41}^{2}(t)}{|\eta|}\mathsf{b}_{4}^{\dagger}\mathsf{b}_{4}\mathsf{b}_{1}^{\dagger}\mathsf{b}_{1},\end{split}$
(3)

where $\omega_{n}(t)$ is the frequency-tunable level splitting of transmon
$n$, $\alpha_{n}$ its anharmonicity and $\mathsf{b}_{n}$ its annihilation
operator. The tunable coupling between transmons $n$ and $m$ is denoted by
$g_{nm}(t)$ and $\eta$ is the non-linearity of the involved transmons, which
is roughly constant $\alpha_{n}\approx\eta$ for all $n$. The time-dependent
amplitude and frequency of a local $X$-type control field on transmon $n$,
e.g., some microwave field, is denoted by $\Omega_{n}(t)$ and
$\bar{\omega}_{n}(t)$, respectively. For numerical reasons, it is advantageous
to change into a rotating frame. We change into a rotating frame with
frequency $\omega_{\mathrm{rot}}$ via the transformation

$\displaystyle\mathsf{H}^{\prime}(t)$
$\displaystyle=\mathsf{O}^{\dagger}(t)\mathsf{H}(t)\mathsf{O}(t)-\textup{i}\mathsf{O}^{\dagger}(t)\frac{\mathrm{d}\mathsf{O}(t)}{\mathrm{d}t},$
(4a) $\displaystyle\mathsf{O}(t)$
$\displaystyle=\exp\left\\{-\textup{i}\omega_{\mathrm{rot}}\left(\sum_{n=1}^{N}\mathsf{b}_{n}^{\dagger}\mathsf{b}_{n}\right)t\right\\}$
(4b)

and find

$\begin{split}\mathsf{H}^{\prime}(t)&=\sum_{n=1}^{N}\left[\left(\omega_{n}(t)-\omega_{\mathrm{rot}}\right)\mathsf{b}_{n}^{\dagger}\mathsf{b}_{n}-\frac{\alpha_{n}}{2}\mathsf{b}_{n}^{\dagger}\mathsf{b}_{n}^{\dagger}\mathsf{b}_{n}\mathsf{b}_{n}\right]\\\
&\quad+\sum_{n=1}^{N}\frac{1}{2}\left[\mathsf{b}_{n}\left(\bar{\Omega}_{n,\mathrm{re}}(t)+\textup{i}\bar{\Omega}_{n,\mathrm{im}}(t)\right)\right.\\\
&\qquad\qquad\left.+\mathsf{b}_{n}^{\dagger}\left(\bar{\Omega}_{n,\mathrm{re}}(t)-\textup{i}\bar{\Omega}_{n,\mathrm{im}}(t)\right)\right]+\mathsf{H}_{\mathrm{int}}(t),\end{split}$
(5)

where we have introduced the auxiliary control fields

$\displaystyle\bar{\Omega}_{n,\mathrm{re}}(t)$
$\displaystyle=\mathfrak{Re}\left\\{\Omega_{n}(t)e^{-\textup{i}(\omega_{\mathrm{rot}}-\bar{\omega}_{n}(t))t}\right\\},$
(6a) $\displaystyle\bar{\Omega}_{n,\mathrm{im}}(t)$
$\displaystyle=\mathfrak{Im}\left\\{\Omega_{n}(t)e^{-\textup{i}(\omega_{\mathrm{rot}}-\bar{\omega}_{n}(t))t}\right\\}.$
(6b)

While $\Omega_{n}(t)$ and $\bar{\omega}_{n}(t)$ are the actual physical
control fields, we may take $\bar{\Omega}_{n,\mathrm{re}}(t)$ and
$\bar{\Omega}_{n,\mathrm{im}}(t)$ as auxiliary control fields which capture
the time-dependent nature of $\Omega_{n}(t)$ and $\bar{\omega}_{n}(t)$ in the
rotating frame and the latter can be reobtained from
$\bar{\Omega}_{n,\mathrm{re}}(t)$ and $\bar{\Omega}_{n,\mathrm{im}}(t)$.

##  IV Determining quantum speed limits via quantum optimal control

After having introduced the physical model for neutral atoms and
superconducting circuits in Sec. III, we now review the basic notion of QSLs
in Sec. IV.1 and of quantum optimal control theory in Sec. IV.2 since both
build the theoretical, respectively methodical, foundation for the results
presented in this work. Our method is described in Sec. IV.3.

###  IV.1 Quantum speed limits

The notion of quantum speed limits (QSLs) naturally arises in the context of
quantum control problems. To this end, let us consider a quantum system
described by the Hamiltonian
$\mathsf{H}(t)=\mathsf{H}(\\{\mathcal{E}_{k}(t)\\})$, which depends on a set
of control fields, $\\{\mathcal{E}_{k}(t)\\}$, that can be externally tuned,
e.g., by the time-dependent amplitudes, phases or detunings in Eqs. (1) or
(5). A quantum control problem is then defined by a set of initial states,
$\\{\ket{\psi_{l}^{\mathrm{in}}}\\}$, that should be transferred into a set of
target states, $\\{\ket{\psi_{l}^{\mathrm{trgt}}}\\}$,

$\displaystyle\Ket{\psi_{l}^{\mathrm{trgt}}}=\mathsf{U}(T,0;\\{\mathcal{E}_{k}(t)\\})\Ket{\psi_{l}^{\mathrm{in}}},\quad\forall
l,$ (7)

where $\mathsf{U}(T,0;\\{\mathcal{E}_{k}(t)\\})$ is the system’s time-
evolution operator and $T$ the total time. Any choice of
$\\{\mathcal{E}_{k}(t)\\}$ which fulfills Eq. (7) is considered a solution to
the control problem. It is important to note that solutions to quantum control
problems are usually not unique. Even for a fixed protocol duration $T$ there
typically exist many and often infinite many solutions.

The QSL for a given control problem is defined by the shortest protocol
duration $T_{\mathrm{QSL}}$ for which at least one solution exists, i.e., for
which at least one set of control fields $\\{\mathcal{E}_{k}(t)\\}$ exists
that fulfills Eq. (7). In the context of quantum computing and the NISQ era,
where time is a limited resource due to decoherence, it is desirable to
implement quantum gates at the QSL. In order to calculate $T_{\mathrm{QSL}}$
analytically, $\mathsf{U}(T,0;\\{\mathcal{E}_{k}(t)\\})$ must be analytically
calculable for _any_ set $\\{\mathcal{E}_{k}(t)\\}$ of conceivable control
fields — a requirement that typically limits an analytical calculation of
$T_{\mathrm{QSL}}$ to simple systems [75].

Besides an analytical determination, there are various methods to approximate
$T_{\mathrm{QSL}}$. One prominent method is to calculate a lower bound
$T_{\mathrm{bound}}\leq T_{\mathrm{QSL}}$ in order to get an estimate for
$T_{\mathrm{QSL}}$ itself. For the simplest case of a state-to-state control
problem, lower bounds can be calculated analytically [24]. In contrast, in
case of multiple pairs of initial and final states, which describe the
implementation of quantum gates, such lower bounds only exist for very simple
systems [76]. In most cases, one needs to resort to numerical tools for
estimating $T_{\mathrm{QSL}}$. In that context, quantum optimal control theory
has proven to be very useful [77] as it not only estimates $T_{\mathrm{QSL}}$
quite accurately but additionally yields the control fields that implement the
desired dynamics, i.e., realizing the transition from initial to target
states, cf. Eq. (7). Since this is our method of choice, we introduce it in
more detail in the following.

###  IV.2 Quantum optimal control theory

Quantum optimal control theory (OCT) [78] is a toolbox providing analytical
and numerical tools that allow to derive optimized control fields which solve
a given control problem, e.g., in shortest time or with minimal error.
Mathematically, an optimal control problem is formulated by introducing the
cost functional

$\begin{split}J\left[\\{\psi_{l}\\},\\{\mathcal{E}_{k}\\},T\right]&=\varepsilon_{T}\left[\\{\psi_{l}(T)\\}\right]\\\
&\quad+\int_{0}^{T}J_{t}\left[\\{\psi_{l}(t)\\},\\{\mathcal{E}_{k}(t)\\},t\right]\mathrm{d}t,\end{split}$
(8)

where $\\{\psi_{l}(t)\\}$ is a set of time-evolved states and
$\\{\mathcal{E}_{k}(t)\\}$ a set of control fields to be optimized. The error-
measure $\varepsilon_{T}$ quantifies the distance between the time-evolved
states
$\ket{\psi_{l}(T)}=\mathsf{U}(T,0;\\{\mathcal{E}_{k}(t)\\})\ket{\psi_{l}^{\mathrm{in}}}$
and the desired target states $\ket{\psi_{l}^{\mathrm{trgt}}}$ at the
protocol’s final time $T$, cf. Eq. (7). The term $J_{t}$ in Eq. (8) captures
time-dependent running costs. In most cases, the error-measure
$\varepsilon_{T}$ is the crucial figure of merit. In order to optimize for
quantum gates, we use the error-measure [79]

$\displaystyle\varepsilon_{T}\left[\\{\psi_{l}(T)\\}\right]=1-\frac{1}{N_{\mathrm{trgt}}}\sum_{l=1}^{N_{\mathrm{trgt}}}\mathfrak{Re}\left\\{\Braket{\psi_{l}^{\mathrm{trgt}}}{\psi_{l}(T)}\right\\}$
(9)

and take
$\ket{\psi_{l}^{\mathrm{trgt}}}=\mathsf{O}\ket{\psi_{l}^{\mathrm{in}}}$ as the
desired target states for the target gate $\mathsf{O}$. The set
$\\{\psi_{l}^{\mathrm{in}}\\}$ runs over the $N_{\mathrm{trgt}}$ logical basis
states affected by $\mathsf{O}$ with $N_{\mathrm{trgt}}=4,8,16$ for two-,
three- or four-body gates, respectively.

Since the cost functional $J$ is formulated such that smaller values
correspond to better solutions of the control problem, solving an optimal
control problem becomes essentially a minimization task, i.e., to find a set
of control fields $\\{\mathcal{E}_{k}^{\mathrm{opt}}(t)\\}$ that minimizes
$J$, respectively $\varepsilon_{T}$. This is an optimization problem for which
several numerical algorithms have been developed [80, 81, 82, 83, 84]. Many of
them are readily available in open-source software packages [85, 86, 87, 88,
89].

###  IV.3 Optimization procedure and method

In the context of determining $T_{\mathrm{QSL}}$ numerically via OCT, we
search for the shortest protocol duration $T$ for which the error-measure
$\varepsilon_{T}$, cf. Eq. (9), is still sufficiently small. In mathematical
terms, we therefore define an error threshold $\varepsilon_{\mathrm{max}}$ and
search for the shortest time $T$ for which

$\displaystyle\underset{\\{\mathcal{E}_{k}\\}}{\mathrm{min}}\left[\varepsilon_{T}\left[\\{\psi_{l}(T)\\}\right]\right]\leq\varepsilon_{\mathrm{max}}$
(10)

has a solution. However, the minimization over all conceivable control fields
can not be done numerically — as there are infinitely many fields to check —
and thus must be replaced by a sampling over finitely many fields in practice.
In order to explore the function space efficiently by finite sampling,
optimization algorithm as described in Sec. IV.2 can be used. Since we are
interested in the fundamental QSL, we put only minimal limitations — apart
from physically motivated limitations on amplitudes — on the form of each
control field $\mathcal{E}_{k}(t)$. Hence, we need an optimization algorithm
that is capable of exploring a function space of almost arbitrary field
shapes.

As our method of choice we use Krotov’s method [90], a gradient-based
optimization algorithms for time-continuous control fields. While a more
detailed description of Krotov’s method is given in Appendix B, its basic
working principle is outlined in the following. It consists of an iterative
update of the control fields $\\{\mathcal{E}_{k}(t)\\}$. Starting from a set
of guess control fields $\\{\mathcal{E}_{k}^{0}(t)\\}$, Krotov’s method
updates them until either $\varepsilon_{T}\leq\varepsilon_{\mathrm{max}}$ or a
maximum number of iterations is reached. This procedure can be viewed as a
local but structured search within the space of all conceivable sets of
control fields — the so-called control landscape. The locally searched area is
thereby determined by the choice of the guess fields
$\\{\mathcal{E}_{k}^{0}(t)\\}$, which set the initial starting point of the
search. While the local nature of this search might appear contradicting to
the global search required for evaluating Eq. (10), it can be turned into an
approximate global search by using various sets of randomized guess fields.
The combined effect of all these local searches “cover” a larger fraction of
the control landscape. The total procedure for approximating
$T_{\mathrm{QSL}}$ using this method is thus to start with a protocol duration
$T$ for which the optimization algorithm finds solutions, i.e., optimized
fields giving rise to $\varepsilon_{T}\leq\varepsilon_{\mathrm{max}}$, and
then to consecutively lower $T$ until none of the various sets of randomized
guess fields finds a solution anymore. Appendix C summarizes the details
regarding the generation of random guess fields as well as each field’s
parametrization within Krotov’s method.

Similar application of numerical optimal control techniques have previously
shown excellent agreement with analytically provable QSLs [91, 92]. At worst,
this method could overestimate the actual QSLs, in which case the actual QSLs
would be even smaller and the circuit and gate times in Sec. II that use the
QGS would be even better.

Figure 3: Overview of the QSLs for the various gates of the “QSL gate set”
(QGS) in Table 1. The left (right) column shows the results for neutral atoms
(superconducting circuits) for three different configurations of control
fields (specified in the main text), respectively. Each marker represents the
gate error $\varepsilon_{T}$, cf. Eq. (9), after either reaching
$\varepsilon_{T}\leq\varepsilon_{\mathrm{max}}=10^{-3}$ or $1500$ iterations
of Krotov’s method, cf. Appendix B, after starting from a set of random guess
fields generated via Eq. (22). While the lines connect the lowest errors
reached for each gate time $T$ within a given field configuration, the shaded
background color indicates the range between the lowest and highest error. The
marker density is shown at the right side of each panel as a histogram. The
parameters for neutral atoms are $V/2\pi=$40\text{\,}\mathrm{MHz}$$,
$0\leq\Omega_{\downarrow}(t),\Omega_{\uparrow}(t)\leq\Omega_{\mathrm{max}}=0.1V$
and $|\Delta(t)|\leq\Delta_{\mathrm{max}}=0.3V$. The parameters for
superconducting circuits are $-$40\text{\,}\mathrm{MHz}$\leq
g_{nm}(t)/2\pi\leq$5\text{\,}\mathrm{MHz}$$,
$$6700\text{\,}\mathrm{MHz}$\lesssim\omega_{n}(t)\lesssim$7100\text{\,}\mathrm{MHz}$$
(exact values depending on $n$ and taken from Ref. [8]) and
$-$50\text{\,}\mathrm{MHz}$\leq\bar{\Omega}_{n,\mathrm{re}}(t),\bar{\Omega}_{n,\mathrm{im}}(t)\leq$50\text{\,}\mathrm{MHz}$$.
The anharmonic ladder for each transmon has been truncated after five levels
with population in the highest level suppressed during optimization. The
optimization results for $\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$,
cf. Eq. (17), have been obtained for the maximally entangling gate at
$\gamma=\pi/4$. The results for the CNOT gate on neutral atoms have been
obtained using site-dependent control fields for each atom.

##  V Benchmarking quantum gate times on 2D architectures

In this final section, we now present the detailed results regarding the QSLs
obtained via methods described in Sec. IV for the various gates listed in
Table 1 and have been used to calculate the circuit run times in Sec. II.

###  V.1 Neutral atoms

In this section, we determine the QSLs for different gates on neutral atoms by
utilizing the control knobs available in Hamiltonian (1). To this end, we set
the vdW interaction strength between Rydberg levels to
$V/2\pi=$40\text{\,}\mathrm{MHz}$$ and assume a maximally achievable Rabi
frequency of $\Omega_{\mathrm{max}}/2\pi=0.1V=$4\text{\,}\mathrm{MHz}$$ for
both $\Omega_{\downarrow}(t)$ and $\Omega_{\uparrow}(t)$ as well as
$\Delta_{\mathrm{max}}/2\pi=0.3V=$12\text{\,}\mathrm{MHz}$$ for $|\Delta(t)|$.
These parameters are in the same regime than those reported in recent
experiments [93, 20, 39].

The markers in Fig. 3 (left column) show the achievable gate error
$\varepsilon_{T}$, cf. Eq. (9), for various gates and various gate times $T$
on neutral atoms. Each individual marker thereby indicates the result of a
single optimization with Krotov’s method [cf. Appendix (B)] i.e., the final
error $\varepsilon_{T}$ after $1500$ iterations when starting from random
guess fields generated via Eq. (22). In the following we set
$\varepsilon_{\mathrm{max}}=10^{-3}$ for all gates and stop any optimization
as soon as this threshold is reached.

####  V.1.1 Two-qubit gates

Figure 3 (a) shows the results for a CZ gate for three different, paradigmatic
configurations of control fields. The circles correspond to the “parallel”
configuration where all five possible control fields
$\Omega_{\downarrow}(t),\Omega_{\uparrow}(t),\varphi_{\downarrow}(t),\varphi_{\uparrow}(t)$
and $\Delta(t)$ have been optimized. In contrast, the crosses correspond to
the “phase” configuration, where only $\varphi_{\uparrow}(t)$ has been
optimized while $\Omega_{\uparrow}(t)=\Omega_{\mathrm{max}}$,
$\Delta(t)=\Delta_{\mathrm{max}}$ and
$\Omega_{\downarrow}(t)=\varphi_{\downarrow}(t)=0$ have been kept fixed. In
both cases, we obtain a QSL of
$T_{\mathrm{QSL}}^{\mathrm{CZ}}=$350\text{\,}\mathrm{ns}$$. The third field
configuration, which we called “sequential” configuration (diamonds), consists
of a sequential use of $\Omega_{\downarrow}(t),\varphi_{\downarrow}(t)$ and
$\Omega_{\uparrow}(t),\varphi_{\uparrow}(t)$, i.e., the first half of the
protocol we have $\Omega_{\uparrow}(t)=\varphi_{\uparrow}(t)=0$ and in the
second half $\Omega_{\downarrow}(t)=\varphi_{\downarrow}(t)=0$. This
configuration is inspired by an adiabatic protocol for implementing
$\mathrm{ZZZZ}(\gamma)$ gates, cf. Eq. (17) and Ref. [66]. Its QSL
$T_{\mathrm{QSL}}^{\mathrm{CZ}}=$700\text{\,}\mathrm{ns}$$ is twice as long
compared to the other two configurations.

In order to compare the results from the three configurations in terms of how
successful the optimization has been in finding solutions, the histogram on
the right side of Fig. 3 (a) provides the probability density for obtaining
final errors $\varepsilon_{T}$ within certain ranges. For obtaining a solution
with errors $\varepsilon_{T}\leq\varepsilon_{\mathrm{max}}$, we find the
lowest probability for the “sequential” configuration (diamonds) — coinciding
with the highest QSL — and the highest and almost identical probability for
the other two configurations. Among those two, the “phase” configuration
(crosses) needs to be emphasized in particular. From a physical perspective,
setting $\Omega_{\downarrow}(t)$ and $\varphi_{\downarrow}(t)$ to zero
automatically ensures that $\ket{\downarrow\downarrow}$ is mapped onto itself
— as required by the CZ gate. This is not automatically guaranteed by the
other two configurations and the optimization needs to explicitly ensure it
and therefore needs to solve a slightly more complex optimization problem.
However, the advantage of having one basis state automatically mapped
correctly does not translate into an advantage regarding the QSL of
$T_{\mathrm{QSL}}^{\mathrm{CZ}}=$350\text{\,}\mathrm{ns}$$ or the reachable
error in general. Interestingly, both the “parallel” and “phase”
configurations yield the same achievable lowest errors for each gate time $T$.
This is visually highlighted by the lines connecting the lowest errors per $T$
in Fig. 3 (a). Moreover, it should be stressed that these errors
$\varepsilon_{T}$ are reached for almost every set of initial guess fields,
i.e., independent of the initial starting point within the problem’s control
landscape, and obtained independently for both configurations. This supports
the conjecture that $T_{\mathrm{QSL}}^{\mathrm{CZ}}=$350\text{\,}\mathrm{ns}$$
is the actual QSL for a CZ gate and generally validates our method in
determining the QSL. From an optimal control perspective, it is interesting to
see that the flexibility originating from the extended set of available
control fields in the “parallel” configuration can not be turned into an
advantage in error or time compared to the “phase” configuration. From a
practical perspective, the latter is advantageous for experimental
realizations as it requires fewer physical resources.

While the three configurations discussed so far should only be viewed as
examples, we did not find any configuration giving rise to faster CZ gates.
Hence, we assume $T_{\mathrm{QSL}}^{\mathrm{CZ}}=$350\text{\,}\mathrm{ns}$$ to
be the fundamental QSL across all configurations. A natural comparison for
$T_{\mathrm{QSL}}^{\mathrm{CZ}}$ with a value from the literature would be the
gate time from the analytical protocol introduced in Ref. [20], especially
because it uses the same control fields as the “phase” configuration to
implement the gate. For our parameters, we find
$T_{\mathrm{lit}}^{\mathrm{CZ}}\approx$340\text{\,}\mathrm{ns}$$ as analytical
gate time, which we assume identical with our QSL
$T_{\mathrm{QSL}}^{\mathrm{CZ}}=$350\text{\,}\mathrm{ns}$$ given the rather
coarse sampling of gate times $T$ in Fig. 3 (a). However, it should be noted
that the analytical protocol of Ref. [20] implements a CZ gate only up to
local operations — operations that are already contained in our optimized gate
protocols at the QSL. Nevertheless, for a fair comparison of analytical and
QSL gate times, as needed in Sec. II, as well as for simplicity, we set both
times to $350\text{\,}\mathrm{ns}$ in Table 1.

The remaining panels (b)-(e) in the left column of Fig. 3 show the results for
other gates from the QGS of Table 1. The results for a CNOT gate are shown in
panel (b). It is the only gate for neutral atoms (among those we considered)
that requires individual instead of global fields, i.e., it is the only gate
for which we did not assume
$\Omega_{\downarrow,n}(t)=\Omega_{\downarrow}(t),\Omega_{\uparrow,n}(t)=\Omega_{\uparrow}(t),\varphi_{\downarrow,n}(t)=\varphi_{\downarrow}(t),\varphi_{\uparrow,n}(t)=\varphi_{\uparrow}(t)$
and $\Delta_{n}(t)=\Delta(t)$ for all $n$ but actually assume individual
fields with unique field shapes directed at each atom. We nevertheless
consider the same three field configurations as for the CZ gate in panel (a)
but now applied to the individual fields
$\Omega_{\downarrow,n}(t),\Omega_{\uparrow,n}(t),\varphi_{\downarrow,n}(t),\varphi_{\uparrow,n}(t)$
and $\Delta_{n}(t)$ instead of their global versions. Even with this more
general setting of control fields, we find only the “parallel” configuration
(circles) to allow for the realization of a CNOT gate with a QSL of
$T_{\mathrm{QSL}}^{\mathrm{CNOT}}=$300\text{\,}\mathrm{ns}$$. In contrast, the
“phase” and “sequential” configurations do not allow to realize a CNOT gate at
all. These results demonstrate that CNOT gates can be implemented using
exclusively the site-dependent laser couplings of the qubit and Rydberg levels
and no control knobs for single-qubit gates. However, an experimentally more
convenient option, requiring no full site-dependent control of coupling qubit
to Rydberg states, is to realize CZ gates with global laser pulses and convert
them into CNOTs via local operations. We nevertheless include the CNOT gate
and its QSL for completeness in our analysis as well as in the QGS in Table 1
but exclude it from any quantum circuit for neutral atoms in Sec. II for the
reasons just mentioned.

Figure 3 (c) shows the results for a SWAP gate. Like for the CNOT gate, we
find the “parallel” configuration — assuming again global fields that are
identical for each atom — to be the only one capable of realizing a SWAP gate.
We obtain $T_{\mathrm{QSL}}^{\mathrm{SWAP}}=$400\text{\,}\mathrm{ns}$$ as its
QSL. The other two configurations are not capable of realizing SWAP gates.
However, since the SWAP gate, in contrast to the CNOT gate, can be realized
with global control fields, we believe it to be experimentally feasible and
thus include it as a viable gate in the QGS in Table 1.

Figure 4: Panel (a) shows the entanglement power [94] of the three- and four-
qubit constraint gates $\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$, cf.
Eq. (17), as a function of $\gamma$. In contrast, panels (b) and (c) examine
the QSLs for these gates under various conditions on neutral atoms. In panel
(b), the dependence of the QSL on the parameters $\gamma$ is shown. In panel
(c), the impact of $\Delta_{\mathrm{max}}/\Omega_{\mathrm{max}}$ is
visualized. In the latter case, we have $\gamma=\pi/4$ and a fixed
$\Omega_{\mathrm{max}}$ while $\Delta_{\mathrm{max}}$ is modified. The QSLs in
panels (b) and (c) have been determined using the parameters and “phase”
configuration described in Fig. 3.

####  V.1.2 Three- and four-qubit constraint gates

So far, we have discussed the results for two-qubit gates in Fig. 3 (a)-(c).
These gates and their respective QSLs have been used in determining the
quantum circuits and calculating the corresponding circuit run times in Sec.
II — especially for those circuits in the SGM discussed in Sec. II.2. In
contrast, for QAOA circuits in the PM [29], circuit representations without
two-qubit gates exist, e.g., when the required three- and four-qubit
constraint gates $\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$, cf. Eq.
(17), are available natively and thus must not be decomposed into single- and
two-qubit gates. In the following, we determine and discuss their QSLs.

It should first be noted that, from an algorithmic point of view, it is
irrelevant whether, e.g., $\mathrm{ZZZ}(\gamma)$ or
$e^{\textup{i}\alpha}\mathrm{ZZZ}(\gamma)$, with $\alpha$ some arbitrary
phase, is realized in experiments. The latter just changes the global phase of
the quantum state during circuit execution. The same holds for the
$\mathrm{ZZZZ}(\gamma)$ gate. In both cases, we may choose $\alpha$
arbitrarily but, for practical reasons, choose it such that the states
$\ket{\downarrow\downarrow\downarrow}$ and
$\ket{\downarrow\downarrow\downarrow\downarrow}$ do not acquire a phase from
the respective constraint gates. In the following, we therefore consider the
phase-shifted constraint gates

$\displaystyle e^{\textup{i}\gamma}\mathrm{ZZZ}(\gamma),\qquad
e^{-\textup{i}\gamma}\mathrm{ZZZZ}(\gamma),$ (11)

instead of the ones from Eq. (17).

In the context of QAOA circuits in the PM, these constraint gates need to be
realized for various $\gamma$, cf. Eq. (14). However, since we can not
determine their QSL for each value of $\gamma$, we first analyze the gate’s
entangling power [94] as a function of $\gamma$ in Fig 4 (a). We observe
maximal entangling power for $\gamma=\pi/4$ for both $\mathrm{ZZZ}(\gamma)$
and $\mathrm{ZZZZ}(\gamma)$ and thus decide to first benchmark their QSLs for
that particular value as we expect the control problem in that case to be the
hardest to solve and consequently the QSLs to be the largest.

Figure 3 (d) and (e) show the optimization results for the phase-shifted
constraint gates $\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$, cf. Eq.
(11), for $\gamma=\pi/4$, respectively. We use the same three configurations
of control fields as for the two-qubit gates in panels (a)-(c) and find all
configurations to be capable of implementing the constraint gates but observe
the best performance for the “parallel” and “phase” configuration. While both
configurations yield the same QSLs of
$T_{\mathrm{QSL}}^{\mathrm{ZZZ}}=$400\text{\,}\mathrm{ns}$$ and
$T_{\mathrm{QSL}}^{\mathrm{ZZZZ}}=$500\text{\,}\mathrm{ns}$$ for
$\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$, respectively, the “phase”
configuration exhibits the better convergence behavior. Like for the CZ gate
in Fig. 3 (a), we observe every set of guess fields for this configuration to
reliably converge towards the same final error $\varepsilon_{T}$ for every
$T$. Since the “parallel” configuration yields the same achievable errors as a
function of gate time $T$ and uses by definition a different control strategy
than the “phase” configuration, we believe to have reliably identified the
QSLs for the constraint gates. In terms of experimental feasibility, the
“phase” configuration is advantageous as it requires fewer hardware and
control resources.

The fact that using $\varphi_{\uparrow}(t)$ as the only time-dependent control
field suffices for implementing $\mathrm{ZZZ}(\gamma)$ or
$\mathrm{ZZZZ}(\gamma)$ originates from considering the gates’ phase-shifted
versions of Eq. (11). In detail, since the states
$\ket{\downarrow\downarrow\downarrow}$ and
$\ket{\downarrow\downarrow\downarrow\downarrow}$ are the only states among the
$8$ or $16$ basis states of $\mathrm{ZZZ}(\gamma)$ or $\mathrm{ZZZZ}(\gamma)$
that technically require non-zero $\Omega_{\downarrow}(t)$ and
$\varphi_{\downarrow}(t)$ in order to be phase-configurable, we simply avoid
this requirement by considering the gates’ phase-shifted version. For the
remaining $7$ or $15$ basis states, which all have at least one atom initially
in the $\ket{\uparrow}$ state, the phase $\varphi_{\uparrow}(t)$ together with
a constant $\Omega_{\uparrow}(t)=\Omega_{\mathrm{max}}$ is sufficient to
correctly adjust all phases. The “phase” configuration thus represents a
hardware-efficient control scheme to realize fast $\mathrm{ZZZ}(\gamma)$ and
$\mathrm{ZZZZ}(\gamma)$ gates.

Our results furthermore reveal that the “sequential” configuration, which was
recently introduced in Ref. [66] and designed to implement high-fidelity
$\mathrm{ZZZZ}(\gamma)$ gates, seems not to be ideal when it comes to gate
time as its configuration-specific QSL is roughly twice as long as the QSLs
for the other configurations. However, it should be noted that a comparison of
the protocol from Ref. [66] with protocols at the QSL is not a fair
comparison. On the one hand, the control scheme of Ref. [66] is based on
adiabaticity — a regime that we are far away from in our numerical
calculations. On the other hand, while the parameter $\gamma$ is a tunable
variable in the adiabatic control scheme, the results in Fig. 3 (d) and (e)
are only valid for $\gamma=\pi/4$. If gates with a different $\gamma$ are
required, one explicitly needs to optimize control fields for that purpose.
While it is beyond the scope of this work to examine whether there exists an
analytical control scheme with configurable $\gamma$ at the QSL, in the
following, we nevertheless provide an analysis of the QSLs for
$\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$ beyond $\gamma=\pi/4$. In
detail, after having identified the “phase” configuration as the most reliable
configuration to determine a gate’s QSL, we provide the QSLs for other
$\gamma$ values in Fig. 4 (b). We find the QSLs to be almost constant for
$\gamma>0$ and only zero for $\gamma=0$, in which case $\mathrm{ZZZ}(\gamma)$
and $\mathrm{ZZZZ}(\gamma)$ coincide with the identity operation.
Interestingly, we do not observe a decrease in the QSLs for $\gamma=\pi/2$ in
which case the constraint gates are no longer entangling, cf. Fig. 4 (a), and
should therefore theoretically be implementable with local operations only. We
suspect that we do not see a decrease of the QSLs since we do not consider any
control fields for local operations in Hamiltonian (1) and thus need to
implement the local gates by means of the Rydberg levels.

We moreover analyze the dependence of the QSLs on the ratio
$\Delta_{\mathrm{max}}/\Omega_{\mathrm{max}}$ in Fig. 4 (c). Surprisingly, we
observe the QSLs for both $\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$
to be independent on this ratio and find $\Delta_{\mathrm{max}}=0$ to be a
viable option.

In general, we observe that the QSLs for $\mathrm{ZZZ}(\gamma)$ and
$\mathrm{ZZZZ}(\gamma)$ are only slightly larger than those of the two-qubit
gates. In view of quantum circuits for PM-QAOA, where such constraint gates
are required, it is thus advantageous to have these gates natively available,
since their representation via single- and two-qubit gates [29] consumes
significantly more time. This effect can be seen in Fig. 1 (c), where the plus
signs illustrate the data for constraint gates expanded in single- and two-
qubit gates and the crosses for the usage of native constraint gates. One
possible explanation for the short QSLs for $\mathrm{ZZZ}(\gamma)$ and
$\mathrm{ZZZZ}(\gamma)$ compared to those of the two-qubit gates might be that
the permutation symmetry of atoms within the pseudo-2D architecture [66],
i.e., $V_{nm}=V$, matches the permutation symmetry of the gate operation
itself.

Figure 5: Overview of different QSLs similar to Fig. 3 but exclusively for the
three- and four-qubit constraint gates $\mathrm{ZZZ}(\gamma)$ and
$\mathrm{ZZZZ}(\gamma)$, cf. Eq. (17). The left column shows results for
neutral atoms, obtained using the “phase” configuration of Fig. 3, for the
pseudo-2D architecture (triangles) and an actual, planar 2D architecture
(stars). The right column compares the results for superconducting circuits
using the “no-X” configuration from Fig. 3. The data correspond to the
physical architecture of Ref. [8] where only nearest neighbor (NN) couplings
between qubits are present (triangles) and where diagonal couplings, i.e.,
next-to-nearest neighbor (NNN) couplings, are added (stars).

At last, we therefore examine the impact of the pseudo-2D architecture onto
the constraint gates $\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$. For
the three- and four-qubit constraint gates, the change to an actual, planar 2D
architecture implies that while the couplings between nearest neighbors remain
$V$, diagonal couplings, i.e., couplings between next-to-nearest neighbors or,
in other words, qubits on opposite edges of a $2\times 2$ square plaquette,
are replaced by $V/8$. The stars in Fig. 5 (a) and (b) show the corresponding
results for $\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$ gates and
$\gamma=\pi/4$, respectively. While their QSLs within the pseudo-2D
architecture are $T_{\mathrm{QSL}}^{\mathrm{ZZZ}}=$400\text{\,}\mathrm{ns}$$
and $T_{\mathrm{QSL}}^{\mathrm{ZZZZ}}=$500\text{\,}\mathrm{ns}$$, they become
$T_{\mathrm{QSL},\mathrm{2D}}^{\mathrm{ZZZ}}=T_{\mathrm{QSL},\mathrm{2D}}^{\mathrm{ZZZZ}}=$600\text{\,}\mathrm{ns}$$
in the actual, planar 2D architecture. Interestingly, this corresponds only to
a relatively small increase in the QSLs for both gates. A possible explanation
might be that the gate speed for neutral atoms is primarily determined by the
maximal Rabi frequency $\Omega_{\mathrm{max}}$, which is identical for both
examples, and not so much by the interatomic interaction strength, which is
different for both architectures.

Although we believe the pseudo-2D architecture to be viable in experiments due
to the great flexibility to arrange neutral atoms [95], we nevertheless take
the QSLs for the actual, planar 2D architecture to be the reference gate times
within the QGS in Table 1 and Sec. II. However, recall that the constraint
gates are — within our study — only relevant for the QAOA circuits in the PM,
cf. Fig. 1 (c). In order to nevertheless allow for a comparison of the run
times in the actual, planar 2D architecture (orange crosses) with those using
the pseudo-2D architecture, we add the latter as orange pentagons to Fig. 1
(c).

###  V.2 Superconducting circuits

Similar to neutral atoms in Sec. V.1, we now determine and analyze the QSLs
for the same quantum gates but for superconducting circuits. The available
control knobs to implement these gates are the tunable qubit frequencies
$\omega_{n}(t)$, the tunable coupling strength $g_{nm}(t)$ between qubits and
the (auxiliary) X-type local control fields $\bar{\Omega}_{n,\mathrm{re}}(t)$
and $\bar{\Omega}_{n,\mathrm{im}}(t)$ in Hamiltonian (5). In order to remain
experimentally realistic, we take parameters from Ref. [8]. To this end, we
single out a $2\times 2$ plaquette consisting of four qubits from the
generally larger 2D architecture. We take the qubit frequencies and their
tunable range to be given by
$$6700\text{\,}\mathrm{MHz}$\lesssim\omega_{n}(t)\lesssim$7100\text{\,}\mathrm{MHz}$$
and their anharmonicities given by
$\alpha_{n}\approx$200\text{\,}\mathrm{MHz}$$, with exact values depending on
$n$. The tunable coupling strength is given by $-$40\text{\,}\mathrm{MHz}$\leq
g_{nm}(t)\leq$5\text{\,}\mathrm{MHz}$$, as reported in Ref. [8]. Moreover, we
assume the (auxiliary) X-type control fields $\bar{\Omega}_{n,\mathrm{re}}(t)$
and $\bar{\Omega}_{n,\mathrm{im}}(t)$, which encode the physical X-type
control fields $\Omega_{n}(t)$ and their tunable driving frequencies
$\bar{\omega}_{n}(t)$, to satisfy
$-$50\text{\,}\mathrm{MHz}$\leq\bar{\Omega}_{n,\mathrm{re}}(t),\bar{\Omega}_{n,\mathrm{im}}(t)\leq$50\text{\,}\mathrm{MHz}$$.

####  V.2.1 Two-qubit gates

Figure 3 (f) shows results for a CZ gate on superconducting circuits using
three different configurations of control fields — different, of course, from
those used for neutral atoms. In the “full” configuration (circles), all
available control fields,
$\omega_{n}(t),g_{nm}(t),\bar{\Omega}_{n,\mathrm{re}}(t)$ and
$\bar{\Omega}_{n,\mathrm{im}}(t)$, are time-dependent and being optimized. In
the “no-X” configuration (crosses), only $\omega_{n}(t)$ and $g_{nm}(t)$ are
optimized while the X-type control fields are set to zero,
$\bar{\Omega}_{n,\mathrm{re}}(t)=\bar{\Omega}_{n,\mathrm{im}}(t)=0$. At last,
in the “interaction” configuration only $\omega_{n}(t)$ is time-dependent and
optimized while $g_{nm}(t)=-$40\text{\,}\mathrm{MHz}$=g_{\mathrm{max}}$ is set
to its maximum magnitude and
$\bar{\Omega}_{n,\mathrm{re}}(t)=\bar{\Omega}_{n,\mathrm{im}}(t)=0$. For the
CZ gate in Fig. 3 (f), we observe all three configurations to indicate the
same QSL of $T_{\mathrm{QSL}}^{\mathrm{CZ}}=$10\text{\,}\mathrm{ns}$$. In
terms of convergence behavior, the “interaction” configuration shows the best
performance as indicated by the probability density on the right side of Fig.
3 (f). In general, the three configurations show slightly worse convergence
behavior than the three configurations for the neutral atoms, cf. Fig. 3 (a).
Nevertheless, since all three configurations indicate the same QSL and, in
general, yield the same achievable error $\varepsilon_{T}$ depending on $T$,
we believe our method to determine the QSL to yield reliable results also for
superconducting circuits and conjecture
$T_{\mathrm{QSL}}^{\mathrm{CZ}}=$10\text{\,}\mathrm{ns}$$ to be the
fundamental QSL for CZ gates. While there are reference implementations for CZ
gates on similar architectures with tunable couplers [74, 72, 96, 97], none of
these architectures matches our architecture and parameter regime. Hence, we
compare our QSL to the gate time of the fastest two-qubit gate, the Sycamore
gate, reported in Ref. [8]. We find
$T_{\mathrm{QSL}}^{\mathrm{CZ}}=$10\text{\,}\mathrm{ns}$$ to be slightly
faster than $T_{\mathrm{lit}}^{\mathrm{Syc.}}=$12\text{\,}\mathrm{ns}$$ [37].

In Fig. 3 (g), the results for a CNOT gate are shown using the same three
configuration as for the CZ gate. We find only the “full” configuration to be
capable of realizing a CNOT gate, yielding the QSL
$T_{\mathrm{QSL}}^{\mathrm{CNOT}}=$14\text{\,}\mathrm{ns}$$, while the other
two configuration are not capable of it. Among the available control knobs,
the X-type control fields $\bar{\Omega}_{n,\mathrm{re}}(t)$ and
$\bar{\Omega}_{n,\mathrm{im}}(t)$ are crucial for a CNOT gate to be feasible.
For the SWAP gate, we again find all three configuration to converge, cf. Fig.
3 (h), with the “no-X” and “interaction” configuration showing the best
convergence behavior. We find a QSL of
$T_{\mathrm{QSL}}^{\mathrm{SWAP}}=$12\text{\,}\mathrm{ns}$$.

####  V.2.2 Three- and four-qubit constraint gates

We now turn towards the three- and four-qubit constraint gates
$\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$. However, note that in the
following and in contrast to neutral atoms, we do not consider their phase-
shifted versions, cf. Eq. (11), but their original versions, cf. Eq. (17).

Figure 3 (i) and (j) show the results for the constraint gates
$\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$, respectively. We use the
same three configurations of control fields as for the two-qubit gates of
panels (f)-(h). We observe the “interaction” configuration to have the best
convergence behavior while the “no-X” configuration gives in both cases rise
to the shortest QSLs of
$T_{\mathrm{QSL}}^{\mathrm{ZZZ}}=$24\text{\,}\mathrm{ns}$$ and
$T_{\mathrm{QSL}}^{\mathrm{ZZZZ}}=$80\text{\,}\mathrm{ns}$$ for
$\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$, respectively. Both QSLs
have been determined for $\gamma=\pi/4$. The QSL for $\mathrm{ZZZ}(\gamma)$
has thereby been confirmed independently by both the “full” and “no-X”
configurations as both yield almost identical achievable errors
$\varepsilon_{T}$ as a function of gate time $T$. We thus assume the QSL
$T_{\mathrm{QSL}}^{\mathrm{ZZZ}}=$24\text{\,}\mathrm{ns}$$ to be well backed
up. The situation is different for $\mathrm{ZZZZ}(\gamma)$, for which we
observe very different convergence behaviors for the three configurations, cf.
Fig. 3 (j). While the “no-X” configuration yields the shortest QSL of
$T_{\mathrm{QSL}}^{\mathrm{ZZZZ}}=$80\text{\,}\mathrm{ns}$$, the “full”
configuration shows slightly better performance for
$T<T_{\mathrm{QSL}}^{\mathrm{ZZZZ}}$, which might suggest that even shorter
gate protocols for $\mathrm{ZZZZ}(\gamma)$ may exist but our method did not
find them due, e.g., limited numbers of guess fields for exploring the control
landscape. In the following, as well as for the calculation of circuit run
times in Sec. II, we nevertheless assume
$T_{\mathrm{QSL}}^{\mathrm{ZZZZ}}=$80\text{\,}\mathrm{ns}$$ to be the QSL for
the $\mathrm{ZZZZ}(\gamma)$ gate as it is the fastest gate time $T$ among the
three configurations of control fields for which Krotov’s method was able to
find a solution with $\varepsilon_{T}\leq\varepsilon_{\mathrm{max}}$.

Interestingly, while we observe the QSLs for $\mathrm{ZZZ}(\gamma)$ and
$\mathrm{ZZZZ}(\gamma)$ to be almost identical for neutral atoms and only
slightly longer than the QSLs for the two-qubit gates, we observe the same
only for the $\mathrm{ZZZ}(\gamma)$ gate for superconducting circuits. The
$\mathrm{ZZZZ}(\gamma)$ gate has a much longer QSL compared to the other QSLs
on that platform. To rigorously decide whether this is due to the non-ideal
convergence behavior observed in Fig. 3 (j) or has some deeper physical origin
is beyond the scope of this study. In an attempt to tackle this question
nevertheless, we consider the scenario of having additional diagonal
couplings, i.e., next-to-nearest neighbor couplings, among the transmons in
the superconducting circuit architecture. For Hamiltonian (3), this implies
adding two additional rows with couplings $g_{13}(t)$ and $g_{24}(t)$ in the
same form as the already present couplings $g_{12}(t),g_{23}(t),g_{34}(t)$ and
$g_{41}(t)$. This scenario is inspired by the pseudo-2D architecture for
neutral atoms, which also exhibits identical nearest neighbor (NN) and next-
to-nearest neighbor (NNN) couplings and where having these couplings is
advantageous. Figure 5 (c) and (d) show the results for the two cases with
only NN couplings (triangles) and with NN plus NNN couplings (stars). Besides
observing much better convergence properties for the latter case, we also
obtain improved QSLs of
$T_{\mathrm{QSL},\mathrm{NNN}}^{\mathrm{ZZZ}}=$20\text{\,}\mathrm{ns}$$ and
$T_{\mathrm{QSL},\mathrm{NNN}}^{\mathrm{ZZZZ}}=$60\text{\,}\mathrm{ns}$$ for
the $\mathrm{ZZZ}(\gamma)$ and $\mathrm{ZZZZ}(\gamma)$ gate, respectively.
However, despite improvements in this scenario, the QSL of the
$\mathrm{ZZZZ}(\gamma)$ gate does not get close to the QSL of the
$\mathrm{ZZZ}(\gamma)$ gate as it does for neutral atoms. The presence of NNN
couplings might therefore be just a partial explanation of the QSL differences
for the constraint gates between neutral atoms and superconducting circuits.
Despite the scenario with NNN couplings not reflecting the actual architecture
for superconducting circuits, we nevertheless add the circuit run times for
the QAOA circuits in the PM using these faster constraint gates to Fig. 1 (c)
for reference purposes as purple pentagons.

##  VI Conclusions

In this study, we have determined the QSLs for several common two-qubit and
two specific multi-qubit quantum gates for two promising quantum computing
platforms that allow for a 2D arrangement of qubits — neutral atoms and
superconducting circuits. We have used OCT to determine the QSLs, as it
provides a generally applicable tool that warrants a fair comparison of both
platforms.

On the level of individual quantum gates, our study allows assessing how close
gate protocols from the literature are compared to their fundamental QSLs or,
in other words, how much can (theoretically) still be gained in time if known
gate protocols are replaced by numerically optimized ones. We find the QSLs
for all investigated two-qubit gates, encompassing CNOT, CZ and SWAP, to be
very similar within each platform and close to the reference gate times for a
CZ gate in case of neutral atoms [20] and a Sycamore gate in case of
superconducting circuits [8].

On the level of quantum algorithms, our study has moreover allowed us to
determine the “QSLs” for entire quantum circuits. To this end, we have assumed
a 2D grid architecture for both platforms and qubit connectivity that allows
physical quantum gates only between neighboring qubits. However, we have
assumed these gates to be executable at the QSL. This has allowed us to
calculate the circuit run times at the “QSL” for two paradigmatic quantum
algorithms — the QFT and a single step of the QAOA. We find that the
corresponding weighted circuit run times scale comparably with respect to the
system size. Furthermore, we observe this to be independent of the chosen gate
set used to translate the quantum algorithms into executable quantum circuits,
i.e., independent of whether the SGS or QGS is used. We observe platform-
independently that the QGS yields circuit run times and gate counts that are
roughly half compared to those in the SGS. On the one hand, this demonstrates
that further speedup of circuit run times is theoretically possible on both
platforms. On the other hand, it also shows that both platforms perform
equally well when running prototypical quantum circuits using typical,
present-day NISQ hardware.

Besides a representation of the quantum circuits in the SGM, we have also
explored the representation of the same quantum algorithms in the PM [27, 29,
26]. We observe a reduction in circuit run times as well as in gate counts in
most cases. This reduction comes at the expense of requiring more physical
qubits but without the need to change the geometrical layout or the control
hardware. In this context, we want to specifically emphasize the circuit run
times of a single QAOA step in the PM. Compared to its representation in the
SGM, it offers a constant circuit depth independent on the problem complexity
but requires the implementation of local three- and four-qubit constraint
gates [29]. For superconducting circuits we find their gate times at the QSL
to be roughly similar to the run times of their decompositions into single-
and two-qubit gates. In contrast, for neutral atoms we find the direct
implementation of the constraint gates to be only slightly slower than any
single two-qubit gate and especially much faster than their decompositions
into single- and two-qubit gates. We nevertheless observe for both platforms
run times for a single QAOA step on the order of $2-4\%$ of the platform’s
intrinsic coherence time. While this corresponds to an improvement of one
order of magnitude for a problem size with $N=9$ logical qubits, this grows to
an improvement of three orders of magnitude for $N=121$. Since the gate
counts, when the native implementations of the constraint gates are used, are
also smaller in the PM compared to the SGM, we believe the PM to be
advantageous in terms of the described resources for a single QAOA step. It
minimizes errors due to finite coherence time and allows for large-depth QAOA.
However, we want to emphasize that deeper PM-QAOA circuits are in general
necessary to reach similar success probabilities compared to lower-depth SGM-
QAOA implementations.

While we want to emphasize that the feasibility of a quantum circuit depends
on more than just its run time and gate count, our benchmark study
demonstrates that at least these two factors can be theoretically further
improved using optimized gate protocols. This holds for both neutral atoms and
superconducting circuits.

###### Acknowledgements.

We would like to thank Hannes Pichler and Kilian Ender for helpful discussions
and Michael Fellner for help with the parity QFT circuits. This work was
supported by the Austrian Science Fund (FWF) through a START grant under
Project No. Y1067-N27 and I 6011. This research was funded in whole, or in
part, by the Austrian Science Fund (FWF) SFB BeyondC Project No. F7108-N38.
For the purpose of open access, the author has applied a CC BY public
copyright licence to any Author Accepted Manuscript version arising from this
submission. This project was funded within the QuantERA II Programme that has
received funding from the European Union’s Horizon 2020 research and
innovation programme under Grant Agreement No. 101017733.

##  Appendix A Quantum circuits for QFT and QAOA

###  A.1 Standard gate model

The quantum Fourier transform (QFT) is a key ingredient in Shor’s algorithm
for integer factorization [1] and thus a prototypical application for quantum
computers. While a typical circuit representation of a QFT contains
exclusively Hadamard gates, $H$, and controlled-phase gates,
$R_{n}=\mathrm{diag}\\{1,1,1,\exp\\{2\pi\textup{i}/2^{n}\\}\\}$, a more
efficient representation with fewer gates and lower circuit depths can be
constructed using SWAP gates [45].

The quantum approximate optimization algorithm (QAOA) aims at finding
approximate solution to combinatorial optimization problems [4]. For instance,
let us consider the task to find the ground state of the $N$ qubit spin glass
Hamiltonian

$\displaystyle\mathsf{H}_{\mathrm{z}}=\sum_{\begin{subarray}{c}n,m=1\\\
n<m\end{subarray}}^{N}J_{nm}\mathsf{\sigma}_{\mathrm{z}}^{(n)}\mathsf{\sigma}_{\mathrm{z}}^{(m)},$
(12)

where $J_{nm}$ denotes the interaction strength between qubits $n$ and $m$ and
$\mathsf{\sigma}_{\mathrm{z}}^{(i)}$ the Pauli-z operator on qubit $i$. The
QAOA allows to find an approximate solution, i.e., an approximate ground state
for Hamiltonian (12), by applying the procedure

$\displaystyle\ket{\psi_{\mathrm{out}}}=\prod_{k=1}^{p}e^{-\textup{i}\alpha_{k}\mathsf{H}_{\mathrm{x}}}e^{-\textup{i}\beta_{k}\mathsf{H}_{\mathrm{z}}}\ket{\psi_{\mathrm{in}}},\qquad\mathsf{H}_{\mathrm{x}}=\sum_{n=1}^{N}\mathsf{\sigma}_{\mathrm{x}}^{(n)},$
(13)

where $\ket{\psi_{\mathrm{in}}}$ is the ground state of the so-called mixing
Hamiltonian $\mathsf{H}_{\mathrm{x}}$ and $\alpha_{k},\beta_{k}\in[0,2\pi)$
are angles — physically corresponding to evolution times — that are
iteratively optimized via a classical, closed-loop feedback optimization and
the energy expectation value of $\ket{\psi_{\mathrm{out}}}$ being the
objective to minimize. The number of steps is denoted by $p$. A single step in
the QAOA is thus given by the application of the spin glass or problem
Hamiltonian $\mathsf{H}_{\mathrm{z}}$ followed by the mixing Hamiltonian
$\mathsf{H}_{\mathrm{x}}$. While the latter corresponds to a parallel
application of $\alpha_{k}$-dependent single-qubit $X$ rotations,
$R_{\mathrm{x}}^{k}$, in the associated quantum circuit — and is thus
negligible time-wise — the circuit implementation of the former requires
multiple CNOT gates as well as single-qubit phase gates,
$R_{\mathrm{z}}^{nm}=\exp\\{-\textup{i}\beta_{k}J_{nm}\mathsf{\sigma}_{\mathrm{z}}\\}$,
containing information about $J_{nm}$ and $\beta_{k}$.

###  A.2 Parity mapping

In the SGM, as described in Appendix A.1, every logical qubit is given by
exactly one physical qubit and gates on logical qubits are equivalent to gates
on physical qubits. This is different for the PM, where $K>N$ physical qubits
are required for a problem of $N$ logical qubits and gates on logical qubits
become different gates or even gate sequences on the physical qubits. However,
due to the arrangement of physical qubits according to the PM, all gates
between physical qubits are strictly local and thus require only nearest-
neighbor connectivity.

For the QFT, we need $K=N(N+1)/2$ physical qubits in the PM and find that
Hadamard gates, $H$, on logical qubits become equivalent to several single-
and two-qubit gates on neighboring physical qubits. In contrast, the logical
two-qubit controlled-phase gates, $R_{n}$, between any pair for logical qubits
is given by exactly three parallel single-qubit gates on physical qubits [26,
49]. Hence, while the logical Hadamard gates require more resources in the PM,
the logical controlled-phase gates require significantly less resources —
especially for those gates where the logical qubits are far away from each
other.

For the QAOA, we need $K=N(N-1)/2$ physical qubits in the PM and Eq. (13)
becomes [29]

$\displaystyle\ket{\psi_{\mathrm{out}}}=\prod_{k=1}^{p}e^{-\textup{i}\alpha_{k}\mathsf{H}_{\mathrm{x}}^{\mathrm{phys}}}e^{-\textup{i}\beta_{k}\mathsf{H}_{\mathrm{z}}^{\mathrm{phys}}}e^{-\textup{i}\gamma_{k}\mathsf{H}_{\mathrm{c}}}\ket{\psi_{\mathrm{in}}},$
(14)

where

$\displaystyle\mathsf{H}_{\mathrm{x}}^{\mathrm{phys}}=\sum_{k=1}^{K}\tilde{\mathsf{\sigma}}_{\mathrm{x}}^{(k)},\qquad\mathsf{H}_{\mathrm{z}}^{\mathrm{phys}}=\sum_{k=1}^{K}\tilde{J}_{k}\tilde{\mathsf{\sigma}}_{\mathrm{z}}^{(k)}$
(15)

are the modified mixing and spin glass Hamiltonians in the PM, respectively.
The local field strengths $\tilde{J}_{k}$ run over the $N(N-1)/2$ interactions
$J_{nm}$, cf. Eq. (12), and $\tilde{\mathsf{\sigma}}_{\mathrm{z}}^{(k)}$
encodes the parity of the corresponding two-qubit interaction
$\mathsf{\sigma}_{\mathrm{z}}^{(n)}\mathsf{\sigma}_{\mathrm{z}}^{(m)}$ [27].
In order to solve optimization problems using the PM, we additionally need to
constraint the dynamics to the $2^{N-1}$ dimensional subspace within the
$2^{K}$ dimensional physical Hilbert space that corresponds to the $2^{N-1}$
eigenstates of Eq. (12) that have unique eigenvalues 777Note that every
eigenstate of Eq. (12) appears at least twice due to a global spin flip
symmetry. Hence, since not every eigenstates of
$\mathsf{H}_{\mathrm{z}}^{\mathrm{phys}}$ has a logical counterpart in
$\mathsf{H}_{\mathrm{z}}$, it needs to be energetically penalized as it would
not be a valid solution to the optimization problem. This is achieved by
realizing $C=K-N+1$ local three- and four-qubit constraints via [27]

$\displaystyle\mathsf{H}_{\mathrm{c}}=\sum_{c=1}^{C}\tilde{\mathsf{\sigma}}_{\mathrm{z}}^{(k_{1})}\tilde{\mathsf{\sigma}}_{\mathrm{z}}^{(k_{2})}\tilde{\mathsf{\sigma}}_{\mathrm{z}}^{(k_{3})}\left(\tilde{\mathsf{\sigma}}_{\mathrm{z}}^{(k_{4})}\right),$
(16)

where “local” refers to $k_{1},\dots,k_{4}$ being nearest-neighbor physical
qubits. For every QAOA step in Eq. (14), this requires $C$ constraint gates of
the form (neglecting tildes and indices)

$\displaystyle\mathrm{ZZZ}(\gamma)$
$\displaystyle=\exp\left\\{-\textup{i}\gamma\mathsf{\sigma}_{\mathrm{z}}\mathsf{\sigma}_{\mathrm{z}}\mathsf{\sigma}_{\mathrm{z}}\right\\},$
(17a) $\displaystyle\mathrm{ZZZZ}(\gamma)$
$\displaystyle=\exp\left\\{-\textup{i}\gamma\mathsf{\sigma}_{\mathrm{z}}\mathsf{\sigma}_{\mathrm{z}}\mathsf{\sigma}_{\mathrm{z}}\mathsf{\sigma}_{\mathrm{z}}\right\\}$
(17b)

with $\gamma$ the effective constraint strengths, which — like in the original
QAOA scheme of Eq. (13) — are optimized in a classical, closed-loop feedback
optimization. The gates in Eq. (17) can be either realized directly [66] or by
decomposing them into single- and two-qubit gates, e.g., by using four or six
CNOT gates plus one $\gamma$-dependent single-qubit phase-gate for
$\mathrm{ZZZ}(\gamma)$ or $\mathrm{ZZZZ}(\gamma)$, respectively [29].

It is important to note that the constraint gates are the only multi-qubit
gates in Eq. (14). Independent of $N$, their implementation can be
parallelized with at most nine [66] or four [29] consecutive layers of
constraint gates for neutral atoms or superconducting circuits, respectively.
However, note that shallower circuits may be feasible by now [99]. The
difference between the two platforms originates from their different qubit-
qubit coupling mechanism. The tunable coupler architecture of superconducting
circuits [8] allows to switch off the coupling between any pair of qubits. As
a consequence, all constraint plaquettes that do not share a common qubit,
i.e., next-to-nearest neighbor plaquettes, can be implemented in parallel. A
single QAOA step thus requires at most four layers of constraint gates to
realize all of them. Neutral atoms, in contrast, interact via their Rydberg
levels and therefore require additional spatial separation between atoms that
are in their Rydberg levels but are not supposed to interact, i.e., to
suppress unwanted interactions. Assuming that a single line of atoms in non-
Rydberg levels suffices as a buffer between plaquettes that are to be
implemented in parallel, this corresponds to two lines of plaquettes as a
buffer in each spatial direction. This yields a maximal number of nine layers
of constraint gates. Despite the platform-dependent differences, all quantum
circuits corresponding to Eq. (14) have constant circuit depth.

##  Appendix B Krotov’s method for quantum optimal control

Krotov’s method [90] is an iterative, gradient-based optimization algorithm
for time-continuous control fields featuring a build-in monotonic convergence
[82]. To achieve the latter, Krotov’s method requires a specific choice of the
total optimization functional $J$, cf. Eq. (8). In detail, while the error
measure $\varepsilon_{T}$ at final time $T$ remains the relevant figure of
merit that we want to minimized, Krotov’s method achieves its minimization
only indirectly by minimizing the total functional $J$ where the time-
dependent running costs $J_{t}$ are give by [79]

$\displaystyle
J_{t}\left[\\{\psi_{l}(t)\\},\\{\mathcal{E}_{k}(t)\\},t\right]=\sum_{k}\frac{\lambda_{k}}{S_{k}(t)}\left(\mathcal{E}_{k}(t)-\mathcal{E}_{k}^{\mathrm{ref}}(t)\right)^{2},$
(18)

where $\mathcal{E}_{k}^{\mathrm{ref}}(t)$ is a reference field for the control
field $\mathcal{E}_{k}(t)$ that is to be optimized, $S_{k}(t)\in(0,1]$ is a
shape function and $\lambda_{k}>0$ a numerical parameter. With the choice of
Eq. (18), the update equation for field $\mathcal{E}_{k}(t)$ becomes [82]

$\displaystyle\mathcal{E}_{k}^{(i+1)}(t)=\mathcal{E}_{k}^{\mathrm{ref}}(t)+\frac{S_{k}(t)}{\lambda_{k}}\mathfrak{Im}\left\\{\sum_{l}\Braket{\chi^{(i)}_{l}(t)}{\frac{\partial\mathsf{H}\left[\left\\{\mathcal{E}_{k^{\prime}}\right\\}\right]}{\partial\mathcal{E}_{k}}\Big{missing}}{{}_{\\{\mathcal{E}^{(i+1)}_{k^{\prime}}(t)\\}}|\psi^{(i+1)}_{l}(t)}\right\\},$
(19)

where $\ket{\psi_{l}^{(i+1)}(t)}$ are forward-propagated states and solutions
to the Schrödinger equations

$\frac{\mathrm{d}}{\mathrm{d}t}\Ket{\psi^{(i+1)}_{l}}(t)=-\textup{i}\mathsf{H}^{(i+1)}(t)\Ket{\psi^{(i+1)}_{l}(t)}$
(20a) with boundary conditions given by the initial states
$\Ket{\psi^{(i+1)}_{l}(0)}=\Ket{\psi_{l}(0)}.$ (20b)

In contrast, $\ket{\chi^{(i)}_{l}(t)}$ are backward-propagated co-states and
solutions to the equations

$\frac{\mathrm{d}}{\mathrm{d}t}\Ket{\chi^{(i)}_{l}(t)}=\textup{i}\mathsf{H}^{(i)}(t)\Ket{\chi^{(i)}_{l}(t)}$
(21a) with boundary conditions
$\Ket{\chi^{(i)}_{l}(T)}=-\frac{\partial\varepsilon_{T}}{\partial\Bra{\psi_{l}}}\Bigg{|}_{\left\\{\Psi_{l}^{(i)}(T)\right\\}}.$
(21b)

The superscripts $i$ and $i+1$ in Eqs. (19)-(21) indicate whether the
corresponding quantity is calculated using the “old” fields from iteration $i$
or the updated fields from iteration $i+1$, respectively.

In order to turn Eq. (19) into a proper update equation, the reference field
$\mathcal{E}_{k}^{\mathrm{ref}}(t)$ is taken to be the field
$\mathcal{E}_{k}^{(i)}(t)$ from the previous iteration in which case the
second term of the left-hand side of Eq. (19) becomes its update. This choice
causes the time-dependent costs $J_{t}$, cf. Eq. (18), to gradually vanish as
the iterative procedure converges. Hence, the error-measure $\varepsilon_{T}$
at final time $T$ becomes the dominant term within the total optimization
functional $J$, cf. Eq. (8), and is thus predominantly minimized. Equation
(19) also reveals that while $\lambda_{k}$ can be used to control the general
size of the update, $S_{k}(t)$ can be used to suppress updates at certain
times.

We refer to Ref. [82] for a more detailed introduction of Krotov’s method and
to Ref. [85] for a detailed discussion about its numerical implementation.

##  Appendix C Field generation and parametrization

In this Appendix, we specify some of the technicalities for the method
described in Sec. IV.3.

We generate randomized, albeit smooth, guess fields via the formula [100]

$\displaystyle
f(t)=a_{0}+\sqrt{2}\sum_{j=1}^{m}\left[a_{j}\cos\left(\frac{2\pi
jt}{t_{1}-t_{0}}\right)+b_{j}\sin\left(\frac{2\pi
jt}{t_{1}-t_{0}}\right)\right]$ (22)

where $a_{0},a_{j},b_{j}$ are chosen randomly from a normal distribution
$N(\mu,\sigma)$ with center $\mu=0$ and variance $\sigma=1/(2m+1)$. The
integer $m$ thereby not only determines the number of frequency components in
$f(t)$ but also the frequencies of each component. Since we expect frequencies
on the same time scale than those contained in the Hamiltonian to be of
greater relevance for finding solutions, we chose $m$ randomly from an
interval matching each Hamiltonian’s frequencies.

Besides the generation of randomized guess fields, we also specify the
internal parametrization of the fields. To this end, note that some of the
physical or auxiliary control fields in Eqs. (1) or (5) are experimentally
limited in range, i.e., have a lower and upper bound that should not be
violated by the optimization algorithm. Let
$\mathcal{E}_{\mathrm{min}}\leq\mathcal{E}(t)\leq\mathcal{E}_{\mathrm{max}}$
be bounded. In order to restrict $\mathcal{E}(t)$ to its bounds and to avoid
manual truncation in case of violation of the bounds, we internally
parametrize $\mathcal{E}(t)$ via

$\displaystyle u(t)$
$\displaystyle=\mathrm{arctanh}\left(\frac{2\mathcal{E}(t)-\mathcal{E}_{\mathrm{max}}-\mathcal{E}_{\mathrm{min}}}{\mathcal{E}_{\mathrm{max}}-\mathcal{E}_{\mathrm{min}}}\right),$
(23a) $\displaystyle\mathcal{E}(t)$
$\displaystyle=\frac{\mathcal{E}_{\mathrm{max}}-\mathcal{E}_{\mathrm{min}}}{2}\tanh\left(u(t)\right)+\frac{\mathcal{E}_{\mathrm{max}}+\mathcal{E}_{\mathrm{min}}}{2}.$
(23b)

The auxiliary field $-\infty<u(t)<\infty$ is the one optimized in practice and
can be optimized boundless since, by construction, it can never violate the
boundaries of the actual field $\mathcal{E}(t)$ which it encodes.

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