#  Minimal evolution times for fast, pulse-based state preparation in silicon
spin qubits

Christopher K. Long<EMAIL_ADDRESS>Hitachi Cambridge Laboratory, J. J.
Thomson Ave., Cambridge, CB3 0HE, United Kingdom  Cavendish Laboratory,
Department of Physics, University of Cambridge, Cambridge, CB3 0HE, United
Kingdom  Nicholas J. Mayhall  Department of Chemistry, Virginia Tech,
Blacksburg, VA 24061, USA  Sophia E. Economou  Department of Physics, Virginia
Tech, Blacksburg, VA 24061, USA  Edwin Barnes  Department of Physics, Virginia
Tech, Blacksburg, VA 24061, USA  Crispin H. W. Barnes  Cavendish Laboratory,
Department of Physics, University of Cambridge, Cambridge, CB3 0HE, United
Kingdom  Frederico Martins  Hitachi Cambridge Laboratory, J. J. Thomson Ave.,
Cambridge, CB3 0HE, United Kingdom  David R. M. Arvidsson-Shukur  Hitachi
Cambridge Laboratory, J. J. Thomson Ave., Cambridge, CB3 0HE, United Kingdom
Normann Mertig  Hitachi Cambridge Laboratory, J. J. Thomson Ave., Cambridge,
CB3 0HE, United Kingdom

###### Abstract

Standing as one of the most significant barriers to reaching quantum
advantage, state-preparation fidelities on noisy intermediate-scale quantum
processors suffer from quantum-gate errors, which accumulate over time. A
potential remedy is pulse-based state preparation. We numerically investigate
the minimal evolution times (METs) attainable by optimizing (microwave and
exchange) pulses on silicon hardware. We investigate two state preparation
tasks. First, we consider the preparation of molecular ground states and find
the METs for H2, HeH+, and LiH to be $2.4\text{\,}\mathrm{ns}$,
$4.4\text{\,}\mathrm{n}\mathrm{s}$, and $27.2\text{\,}\mathrm{n}\mathrm{s}$,
respectively. Second, we consider transitions between arbitrary states and
find the METs for transitions between arbitrary four-qubit states to be below
$50\text{\,}\mathrm{ns}$. For comparison, connecting arbitrary two-qubit
states via one- and two-qubit gates on the same silicon processor requires
approximately $200\text{\,}\mathrm{ns}$. This comparison indicates that pulse-
based state preparation is likely to utilize the coherence times of silicon
hardware more efficiently than gate-based state preparation. Finally, we
quantify the effect of silicon device parameters on the MET. We show that
increasing the maximal exchange amplitude from $10\text{\,}\mathrm{MHz}$ to
$1\text{\,}\mathrm{GHz}$ accelerates the METs, e.g., for H2 from
$84.3\text{\,}\mathrm{ns}$ to $2.4\text{\,}\mathrm{ns}$. This demonstrates the
importance of fast exchange. We also show that increasing the maximal
amplitude of the microwave drive from $884\text{\,}\mathrm{kHz}$ to
$56.6\text{\,}\mathrm{MHz}$ shortens state transitions, e.g., for two-qubit
states from $1000\text{\,}\mathrm{ns}$ to $25\text{\,}\mathrm{ns}$. Our
results bound both the state-preparation times for general quantum algorithms
and the execution times of variational quantum algorithms with silicon spin
qubits.

##  I Introduction

Quantum simulation is one of the most important applications of quantum
computing. However, running quantum-simulation algorithms on fault-tolerant,
error-corrected quantum computers requires significant hardware advances and
is many years away. Instead, the near-term realization of quantum simulation
on quantum processors in the pre-fault-tolerant era is an intriguing
alternative. To that end, the dominant approach is based on variational
quantum algorithms (VQAs).

In the standard VQA approach [1, 2], state preparation is carried out by a
parameterized unitary. This unitary is transpiled into fixed two-qubit and
parameterized single-qubit gates not necessarily native or natural to the
hardware. These gates are then compiled into pulses: time-dependent terms of
the native Hamiltonian. The parameters of the quantum gates are then optimized
iteratively, such that a cost function of the prepared state is lowered
according to the variational principle of quantum mechanics. We refer to this
standard VQA model as the gate-based VQA model. Despite the enormous amount of
research in this field, quantum advantage with gate-based VQAs has not been
demonstrated. This is largely due to the short coherence times of current
quantum hardware; gate-based VQAs are too noise-sensitive [3, 4, 5].

One strategy to improve the noise sensitivity of VQAs is transforming them
into pulse-based algorithms [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Pulse-based
quantum algorithms allow for more fine-grained control of the quantum system
and forgo the overhead induced by transpilation and compilation, thus,
reducing execution times. This is demonstrated numerically by the recent
pulse-based VQA called “ctrl-VQE” on superconducting qubits [11]. In pulse-
based VQAs, the pulses for state preparation are parameterized, and the
objective function (same as in gate-based VQAs) is optimized by varying the
parameters. Given that all gate-based circuits are ultimately executed as
pulses, pulse-based VQAs generalize all their gate-based counterparts. As a
result, one can identify fundamental limits and draw conclusions about general
VQAs by studying pulse-based state preparation. A prominent example of this is
the minimum evolution time (MET), which is the fundamental limit on how long
it takes for a system to evolve from one state to another [9]. In our case,
this will be especially useful when the initial state is some classical
approximation to the solution (e.g., the Hartree-Fock state) and the final
state is the solution to the simulation problem (e.g., a Hamiltonian’s ground
state). The concept of MET is also known in the optimal-control literature and
related to ‘quantum speed limits’ [16].

Name |  | Symbol |  | Value or Constraints |  | SiMOS |  | Si/SiGe
---|---|---|---|---|---|---|---|---
Zeeman splitting |  | $B$ |  |  $28\text{\,}\mathrm{GHz}$11footnotemark: 1 |  | [17] |  | –
Zeeman-splitting Inhomogeneity |  | $\Delta B_{i}$ |  | Equally spaced in range -303011footnotemark: 1 |  | [17, 18] |  | [19]
Drive strengths |  | $g\left(t\right)$ |  |  $\left|I_{i}\left(t\right)\right|,\left|Q_{i}\left(t\right)\right|\leq$20\text{\,}\mathrm{MHz}$\times\frac{1}{2\sqrt{2}}$22footnotemark: 2 |  | [20] |  | [19]
Exchange coupling |  | $J_{i}\left(t\right)$ |  |  $$0\text{\,}\mathrm{GHz}$\leq J_{i}\left(t\right)\leq$1\text{\,}\mathrm{GHz}$$33footnotemark: 3 |  | [21] |  | [22, 23]
Microwave frequencies |  | $\omega_{i}$ |  |  $\frac{\omega_{i}}{2\pi}$ in the range 2729 |  | – |  | –
Table 1: Model parameters, Eq. 1. Values are in units where $h=1$. For an
explanation of the footnotes, see Appendix A.

The performance of pulse-based state preparation can depend dramatically on
the type of quantum processor. This is because each hardware platform has
distinct characteristics, connectivities, and control capabilities. As a
result, pulse-based state preparation constitutes a tool to gauge the ultimate
performance of a given processor. Compared to the quantification of coherence
times, gate times, and other low-level metrics, the study of METs in pulse-
based computation provides a more meaningful way to juxtapose and benchmark
different hardwares’ computational capabilities: The MET can be used to lower
bound the execution time of any quantum algorithm on a given processor.

The early work on pulse-based VQAs [9, 10, 11, 12, 13, 14] focused on
superconducting hardware. One of the main competitors to superconducting
qubits is a system of quantum-dot spin qubits in silicon [24, 25, 26, 27]. The
small size of silicon qubits and the potential for industrial scaling through
advanced semiconductor-manufacturing technology [28, 29, 30, 31, 32] make them
an attractive hardware option for quantum processors. Other beneficial
features include long coherence times [33, 34], competitive gate fidelities
[35, 36, 37, 38, 39, 18], and operability above 1K [18]. This has led to
demonstrations of a six-qubit processor [19]. Silicon hardware has also hosted
a quantum simulator of Hubbard models [40, 41]. While the realization of such
simulators constitutes important milestones, the approach of analog quantum
simulation is inherently limited to the emulation of specific Hamiltonians
that are consistent with the architecture and native interactions of the
device, such as lattice spin models. Many other problems of interest, such as
simulating molecules, are not feasible with analog-simulator approaches.
Pulse-based VQAs, on the other hand, enable universal quantum simulation.
However, their performance on silicon hardware has, until now, not been
explored.

In this paper, we investigate the performance of pulse-based state preparation
on silicon quantum hardware. A schematic of the types of devices we consider
is shown in Fig. 1(a). We use parameters extracted from recent experiments
[17, 18, 19, 21, 20, 23, 22] to estimate the METs in two important settings.
The first setting is the preparation of molecular ground states, a key
component in quantum chemistry algorithms. The second setting is the unitarily
driven transition between pairs of random states. We find that the METs for
preparing the molecular ground state of H2, HeH+, and LiH are
$2.4\text{\,}\mathrm{ns}$, $4.4\text{\,}\mathrm{ns}$, and
$27.2\text{\,}\mathrm{ns}$, respectively. This is significantly faster than
the $\sim$15ns MET computed for H2 on a transmon architecture in Ref. [9]. For
transitions between arbitrary pairs of four-qubit states, we show that the MET
is always shorter than $50\text{\,}\mathrm{ns}$. Interestingly, both values
are significantly smaller than METs of gate-based approaches, (which require
at least $200\text{\,}\mathrm{ns}$), indicating that significant noise
improvements are available via pulse-based quantum computation. Finally, we
explore how pulse parameters affect the MET. For example, we show that
increasing the maximal exchange amplitude from $10\text{\,}\mathrm{MHz}$ to
$1\text{\,}\mathrm{GHz}$ accelerates the METs of H2 from
$84.3\text{\,}\mathrm{ns}$ to $2.4\text{\,}\mathrm{ns}$.

Figure 1: Device Diagram, State preparation, and MET. (a) Diagram of a SiMOS
processor: Plunger- (semi-transparent blue) and $J$-gates (blue) on a SiO_x
layer (gray) confine electron spins in potential wells (blue curve) within
silicon (light gray). The $J$-gates can lower the potential barriers and
induce exchange interactions (qubits 3 and 4). A static magnetic field along
the $z$-direction induces Zeeman splittings. An electron-spin-resonance
antenna (orange) induces a time-dependent magnetic field. (b) Gate-based state
preparation: $\ket{\psi}$ is prepared using pre-calibrated one- and two-qubit
gates (black boxes in the circuit diagram) driven by microwave bursts (orange
lines) and voltage pulses (blue shaded regions), respectively. (c) Pulse-based
state preparation: $\ket{\psi}$ is prepared by changing microwave bursts
(orange lines) and voltage pulses (blue-shaded regions) simultaneously. (d)
Numerically calculated energy errors $\Delta(T)$’s for H2. A vertical dashed
line marks the MET. Insets show the optimal pulse shapes as a function of time
$t$ for various total evolution times $T$.

The remainder of this article is structured as follows: In Sec. II, we
introduce our silicon-processor model. In Sec. III, we review pulse-based
state preparation. In Sec. IV and Sec. V, we present METs for pulse-based
preparation of molecular ground states and Haar random state transitions,
respectively. In Sec. VI, we explore the dependence of METs on silicon control
parameters. We conclude in Sec. VII.

##  II Silicon quantum-processor model

We use a Heisenberg spin-chain model [42] to describe a typical 1D chain of
spin qubits in SiMOS heterostructures [17, 29, 31, 18] [see Fig. 1(a)]:

$\\!\hat{H}\\!\left(t\right)\\!=\\!-\\!\\!\sum_{i=1}^{N}\\!\frac{B_{i}}{2}\hat{\sigma}^{\left(i\right)}_{z}\\!+\\!\frac{g\\!\left(t\right)}{2}\\!\\!\sum_{i=1}^{N}\\!\hat{\sigma}^{\left(i\right)}_{x}-\\!\\!\\!\sum_{i=1}^{N-1}\\!\frac{J_{i}\\!\left(t\right)}{4}\vec{\hat{\sigma}}^{\left(i\right)}\cdot\vec{\hat{\sigma}}^{\left(i+1\right)}.\\!$
(1)

Here,
$\vec{\hat{\sigma}}^{\left(i\right)}=\left(\hat{\sigma}_{x}^{(i)},\hat{\sigma}_{y}^{(i)},\hat{\sigma}_{z}^{(i)}\right)$
is a vector of Pauli operators of the spin of a single electron, trapped in
the $i$th quantum dot under a plunger gate.

An external magnetic field ($\approx$1\text{\,}\mathrm{T}$$) induces a Zeeman
splitting of $B_{i}=B+\Delta B_{i}$ on the electron spin in the $i$th dot with
an average value of $B=\frac{1}{N}\sum_{i=1}^{N}B_{i}$
($\approx$28\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$) [17]. We assume
detunings of $B_{i+1}-B_{i}=\Delta B$, similar to Refs. [18, 17], where
$\Delta B$ is approximately $\qtyrange{10}{30}{}$.

One-qubit operations are achieved via electron-spin resonance using a
microwave antenna [43] [see Fig. 1(a)], which induces a time-dependent global
magnetic field in the $x$-direction. The field induced by the microwave drive
is proportional to

$\displaystyle
g\left(t\right)=\sum_{i=1}^{N}\left[I_{i}\left(t\right)\cos\left(\omega_{i}t\right)+Q_{i}\left(t\right)\sin\left(\omega_{i}t\right)\right].$
(2)

The field can be produced with an arbitrary waveform generator, where
$I_{i}\left(t\right)$ and $Q_{i}\left(t\right)$ are the in-phase and
quadrature components of $N$ microwave pulses with carrier frequency
$\omega_{i}$, respectively. Typically,
$|I_{i}\left(t\right)|,|Q_{i}\left(t\right)|\leq$$10\text{\,}\mathrm{MHz}$
[20] and $\omega_{i}$ usually varies by up to $1\text{\,}\mathrm{GHz}$ from
the Zeeman splitting (depending on the bandwidth of the arbitrary waveform
generator).

Finally, two-qubit operations are induced via nearest-neighbor exchange
couplings between spin $i$ and $i+1$. The entangling operation is controlled
by voltage pulses on the barrier gates, which alters the exchange strength
$J_{i}(t)$. Typically, $J_{i}(t)\leq$10\text{\,}\mathrm{MHz}$$ [34, 18, 39],
but exchange couplings in SiMOS as large as
$J_{i}(t)\approx$1\text{\,}\mathrm{GHz}$$ have already been measured [21].
Moreover, the control of fast exchange pulses has previously been demonstrated
in GaAs [44, 45]. As we will show below, fast exchange is crucial to achieve
short METs.

The parameters of our spin-chain model are summarized in Table 1. As discussed
above, these parameters describe realistic SiMOS devices [17, 18, 21, 20].
Similar parameters also describe electron spin-qubits in
Silicon/SiliconGermanium (Si/SiGe) heterostructures [22, 23, 36, 37, 38, 19,
46]. In Si/SiGe, the Zeeman-splitting inhomogeneity and $g\left(t\right)$
originates from micromagnets [47] and electron dipole spin resonance type
driving via the confinement gate [36, 37, 38, 19, 46], respectively. As with
SiMOS devices, experiments in Si/SiGe tend to use slow exchange
($J_{i}(t)\leq$10\text{\,}\mathrm{MHz}$$) [36, 37, 38, 19, 46]. However, fast
exchange ($J_{i}(t)\approx$1\text{\,}\mathrm{GHz}$$) has already been
demonstrated in Refs. [22, 23].

##  III Pulse-based state preparation

In conventional gate-based approaches, a target quantum state is prepared by
evolving an initial state through a sequence of quantum gates. These gates are
implemented by executing pre-optimized control pulses to realize the specific
parameterized one- and fixed two-qubit gates. In Appendix C, we also allow
two-qubit gates to be parameterized and calculate gate-based state-preparation
times. These times serve as lower bounds for state preparation with fixed two-
qubit gates. In contrast, pulse-based state preparation is realized by
directly optimizing the control pulses that drive the device state through
native interactions. Since gate-based state preparation is less flexible, in a
variational sense, pulse-based state preparation is able to execute in shorter
times.

A device-agnostic implementation of pulse-based state preparation utilizing
variational pulse-shaping [48, 11] consists of three steps: (i) The input is a
cost Hamiltonian $\hat{C}$ whose (potentially unknown) ground state
$\ket{\phi}$ we wish to prepare. For example, $\hat{C}$ can encode a molecular
Hamiltonian [49]. It can always be expressed as a linear combination of Pauli
strings [49, 2] whose expectation value, in the case of physically relevant
problems, can be sampled efficiently [2]. (ii) We fix the total evolution time
$T$ and prepare the qubit register in the state $\ket{\psi(\bm{x};T)}$. Here,
$\bm{x}$ denotes the control parameters of the silicon processor. (iii) These
parameters are iteratively tuned using a classical optimizer to obtain the
minimal expectation value

$\displaystyle
C(T)\coloneqq\min_{\bm{x}}\left\\{\matrixelement{\psi(\bm{x};T)}{\hat{C}}{\psi(\bm{x};T)}\right\\}.$
(3)

Provided that $T$ is sufficiently large and $\bm{x}$ sufficiently expressive,
variational pulse-shaping yields $\ket{\psi(\bm{x}_{\star};T)}=\ket{\phi}$ for
the optimal parameters $\bm{x}_{\star}$.

Specifically, for our device (see Sec. II), a cost Hamiltonian $\hat{C}$ and
fixed evolution time $T$, we implement variational pulse-shaping as follows:
Working in the frame rotating with the drift Hamiltonian
$\hat{H}_{D}\coloneqq-\frac{1}{2}\sum_{i=1}^{N}\\!B_{i}\hat{\sigma}^{\left(i\right)}_{z}$,
we numerically integrate the Schrödinger equation of $\hat{H}$. This yields
the state $\ket{\psi(\bm{x};T)}$ in the rotating frame. We optimize its
control parameters
$\bm{x}\coloneqq\left\\{I_{i}\left(t\right),Q_{i}\left(t\right),J_{j}\left(t\right),\omega_{i}\right\\}$
within the constraints of Table 1. We use the quadrature parameterization of
$g\left(t\right)$ as opposed to the polar parameterization to make
optimization easier [10]. Further, for convenience, we parametrize each
control using a piecewise linear function. That is, we use $M$ segments of
duration $\Delta t=T/M$. For each $t\in\left[\left(m-1\right)\Delta t,m\Delta
t\right)$ with $m=1,...,M$ we set $I_{i}\left(t\right)=I_{i,m}$,
$Q_{i}\left(t\right)=Q_{i,m}$ and $J_{j}\left(t\right)=J_{j,m}$. We then use
the gradient ascent pulse-engineered (GRAPE) algorithm [48] to find the
parameters that achieve the minimal cost $C(T)$.

Fig. 1(d) illustrates numerically calculated $C(T)$s where $\hat{C}$ encods
the Hamiltonian of H2. For large $T$, GRAPE finds control parameters, such
that $\ket{\psi(\bm{x};T)}$ approches the ground state $\ket{\phi}$ of H2 and
$C(T)$ approaches the H2 ground-state energy $C_{0}$ of $\hat{C}$. For
sufficiently small $T$, GRAPE no longer finds the ground state of H2, and the
energy error $\Delta(T)\coloneqq C(T)-C_{0}$ increases monotonically as a
function of $T$. The time below which all $\Delta(T)>10^{-7}$ Hartrees is a
numerical estimate of the minimal evolution time (MET). Below the MET, a
silicon quantum processor can no longer accurately prepare the ground state of
$\hat{C}$. Thus, hardware-specific METs provide lower bounds on the time of
desired quantum evolutions. A protocol for implementing variational pulse-
shaping and MET searches in experiments is given in Appendix B.

Figure 2: Dissociation curves and METs for molecular-ground-state preparation
of H2, HeH+ and LiH (left to right). (top) Molecular energy $C(T)$ as a
function of bond distance for a range of evolution times. At zero evolution
time, $C(0)$ corresponds to the (yellow) Hartree-Fock disassociation curve of
the initial state. At the MET, the energy $C(T=\textrm{MET})$ (blue
dissociation curve) converges to the full configuration interaction (FCI)
energy $C_{0}$ (dashed black curve). (bottom) Energy error $\Delta(T)\coloneqq
C(T)-C_{0}$ on a logarithmic color scale as a function of bond distance and
evolution time. A white line shows the silicon METs. A dashed (grey) curve
marks transmon METs [11].

##  IV Minimal evolution times for molecular ground states

In this section, we investigate the silicon processor’s METs for variational
pulse-based state preparation of molecular ground states. To facilitate
comparison with previous works on superconducting hardware, we consider the
molecules H2, HeH+, and LiH with the same molecular Hamiltonians as in Ref.
[11]. We work in the STO-3G basis and use parity encoding that block
diagonalizes the Hamiltonians. We then identify $\hat{C}$ with the block
containing the ground state. We permute our basis so that the Hartree Fock
state is given by $\ket{01}$ for H2 and HeH+, and $\ket{0011}$ for LiH—we then
take these states as our initial states. The time required to prepare these
initial states is not counted towards the MET.

Using the pulse-shaping methods of Sec. III, we numerically calculate the
molecular energies $C(T)$ of H2, HeH+, and LiH as functions of the bond
distances for a range of evolution times $T$. See Fig. 2 (top row) for
results. At zero evolution time $T=0$, the prepared state is the Hartree-Fock
state, and the molecular energy $C(0)$ is the Hartree-Fock energy. A yellow
line shows the corresponding dissociation curve. As $T$ increases, the
(increasingly blue) dissociation curves $C(T)$ converge to the full
configuration interaction (FCI) energies $C_{0}$, marked by a black dashed
curve. For $T=\mathrm{MET}$, the (purple) dissociation curve $C(\mathrm{MET})$
is identical to the (FCI) energies $C_{0}$. The color bar is truncated at the
largest MET along the bond distance.

To identify the METs for molecular ground-state preparation of H2, HeH+, and
LiH, we plot the energy error $\Delta(T)\coloneqq C(T)-C_{0}$ as a function of
bond distance and evolution time $T$. The results are shown in Fig. 2 (bottom
row). For each bond distance, $\Delta(T)$ shows a sharp increase as $T$
decreases. For numerical purposes, we estimate the MET as the point where
$\Delta(T)>10^{-7}$ Hartrees. A white line highlights the silicon METs. For
HeH+, the MET drops to zero at a large bond distance because the Hartree-Fock
state converges to the true FCI ground state. For H2, HeH+, and LiH we obtain
METs as small as $2.4\text{\,}\mathrm{ns}$,
$4.4\text{\,}\mathrm{n}\mathrm{s}$, and $27.2\text{\,}\mathrm{n}\mathrm{s}$,
respectively. For comparison, a gate-based circuit for preparing arbitrary
states on two qubits requires at least $200\text{\,}\mathrm{n}\mathrm{s}$ (see
Sec. C.2). Further, these METs are one order of magnitude faster than the
fastest single qubit $\pi$ rotations in silicon [33]. These silicon METs are
also one order of magnitude faster than the numerically-estimated METs for
superconducting transmon qubits from Ref. [11], which are shown as gray dashed
lines in Fig. 2 (bottom). Overall, these results fuel the hope that pulse-
based variational state preparation methods could significantly accelerate the
state preparation in quantum chemistry and, thus, make near-term quantum
algorithms on silicon quantum processors more resilient to noise.

Figure 3: Infidelity distributions vs. evolution time $T$ for $1,2,3$, and $4$
qubits (left to right). (Top) Infidelity of 1024 Haar-random state pairs as
histograms. Histogram colors indicate the evolution time $T$ as depicted by
the insets. (Bottom) Cumulative distribution of infidelities as a color map,
with infidelity on a log scale on the horizontal axis and evolution time $T$
on the vertical axis. (Right two columns bottom) Numerically calculated data
is below the dashed lines, while extrapolated data is above. We average the
ten distributions with $T$ just below the dashed curves for extrapolation.

We conclude this section with three technical remarks concerning our numerical
simulations: (i) We discretized the control parameters into $M=10$ segments.
(ii) To obtain the numerical data in Fig. 2, we first initialize GRAPE with a
pulse of zero amplitude and three random pulse shapes for a bond distance
where optimization is easy and select the final pulse shapes with minimal
$C(T)$. Next, we move to the neighboring bond distance, where optimization is
harder, and initialize GRAPE with three random pulse shapes and the optimal
pulse shape of the previous bond distance. Then, we select the pulse shape
with the minimal $C(T)$. For H2 and LiH, the Hartree Fock ground state is a
good approximation to the true ground state at short bond lengths. Thus, we
scan from smaller to larger bond distances. For HeH+, the Hartree-Fock state
is identical to the FCI ground state in the limit of large bond distances.
Hence, we scan from larger to smaller bond distances. (iii) We emphasize that
scanning along bond distance, coupled with GRAPE, is not expected to be
scalable to large qubit numbers. However, this work aims to investigate the
ultimate performance limits of VQAs in silicon on small qubit numbers.
Presenting a scalable method to achieve this optimal performance at large
qubit numbers is a task left for future investigation.

It is now natural to ask: How does the MET depend on the device parameters? We
answer this question in Sec. VI. But first, we consider the task of
transitioning between random states in Sec. V. Readers who are primarily
interested in quantum computational chemistry simulations may skip ahead to
Sec. VI.

##  V Haar-random state transitions

In this section, we determine the METs for transitions between arbitrary
states. As the initial state for this task is no longer close to the target
state, this will require longer METs compared to chemistry or physics
problems, where the mean-field solution is often a good starting point. We
find that for systems with two, three, or four qubits, the slowest MET is
still faster than the fastest possible one-qubit $\pi$ gate. Further, we
theoretically characterize the distribution of METs between Haar random
states.

To numerically investigate METs between arbitrary state pairs on $N$ qubits,
we first sample 1024 pairs of random input states $\ket{\psi_{0}}$ and target
states $\ket{\phi}$ from the Haar measure. If arbitrary quantum control was
available one could pick a fixed initial state and just sample random target
states. However, real hardware cannot move uniformly through Hilbert space.
Consequently, some initial states are “better” than others. For each state
pair, we use the silicon device Hamiltonian $\hat{H}$ to numerically evolve
$\ket{\psi_{0}}$ into the state $\ket{\psi(\bm{x};T)}$, maximizing its
fidelity (overlap) with $\ket{\phi}$ as

$\displaystyle
F(T)\coloneqq\max_{\bm{x}}\left\\{\left|\innerproduct{\psi(\bm{x};T)}{\phi}\right|^{2}\right\\}.$
(4)

To achieve this goal for a given state pair and a given evolution time $T$, we
use variational pulse-shaping with the cost Hamiltonian $\hat{C}$ defined as
$\hat{C}\coloneqq\mathds{1}-\ket{\phi}\bra{\phi}$. The expectation value of
$\hat{C}$ is the infidelity $C\left(T\right)\coloneqq 1-F(T)$ to the state
$\ket{\phi}$. The results of our numerical investigation are depicted in Figs.
3 and 4 for one to four qubits (left to right, respectively).

In Fig. 3, we analyze the infidelity of each state pair as a function of
evolution time $T$. The top panels show the infidelities of the 1024 state
pairs as histograms, where each histogram corresponds to a distinct evolution
time $T$. For $T=0$ (yellow histogram), the distribution of infidelities is
broad, indicating the broad range of overlaps
$\left|\innerproduct{\psi_{0}}{\phi}\right|^{2}$ between Haar-random state
pairs prior to time evolution (an analytical expression for this initial
distribution is given in Eq. 23 in Appendix D). As $T$ increases (increasingly
blue histograms), the evolved input states $\ket{\psi(\bm{x};T)}$ increase
their overlap with the corresponding target states $\ket{\phi}$. This leads to
higher fidelities $F(T)$, shifting the histograms towards lower infidelity.
Moreover, as $T$ increases, an increasing fraction of states
$\ket{\psi(\bm{x};T)}$ reaches the target state $\ket{\phi}$. For this
fraction of state pairs $F(T)=1$, leading to an increasing peak of the
histograms around infidelity equal to zero. Finally, as the MET for the
hardest-to-connect state pair is reached, the (purple) infidelity histogram
entirely localizes around infidelity equal to zero. The bottom row shows the
estimated cumulative probability distribution of the infidelity,
$\mathbb{P}\left(C\left(T\right)\leq\Delta\middle|T\right)$, of state pairs on
a log scale along the horizontal axis and as a function of the evolution time
$T$ along the vertical axis.

In Fig. 4, we approximate the cumulative distribution of the MET,
$\mathbb{P}\left(\textrm{MET}\leq T\right)$, by the fraction of the 1024 state
pairs, which reach an infidelity $C(T)<10^{-7}$ in an evolution time less than
$T$. A blue-shaded region marks the corresponding confidence interval of
99.99%. For comparison, we overlay the time required by the same processor to
implement $\frac{1}{\sqrt{2}}\left(X\pm Y\right)$ (the fastest single-qubit
$\pi$-gate), $X$, $Y$, and $\sqrt{\textrm{SWAP}}$ gates as blue vertical lines
in Fig. 4. In a one-qubit system, the time for operating the fastest
$\pi$-gate is slightly shorter than the MET of the hardest-to-connect state
pairs. However, for systems with more than two qubits, we observe that even
the slowest random state transitions can be accomplished in less than the time
set by the fastest one-qubit $\pi$-gate. This may imply that embedding hard-
to-connect single-qubit states into a system with two or more qubits could
accelerate the transition between one qubit states by using, e.g., fast
exchange interactions in the expanded space of neighboring qubits. This effect
is reminiscent of accelerated state preparation through excited states in
superconducting devices [9] and will be investigated in more detail in future
works.

Figure 4: Cumulative probability distribution of the MET for haar random state
transitions. The black solid curves are the numerical data for 1024 state
pairs. The shaded region corresponds to a 99.99% confidence interval (CI)
estimated via $10^{5}$ bootstrap resamples (see Appendix F). Vertical blue
lines mark typical gate times. The dashed purple and orange curves are the
homogeneous and isotropic (HI) fits and higher-order fits, respectively. For
details of the fit see Appendix G.

Next, to underpin our numerical studies, we derive two expressions to describe
the cumulative distribution of METs depicted in Fig. 4. The first expression
assumes that a pair of states is connected in the shortest time by traveling
with a constant speed $v$ along the geodesics of the manifold of $N$-qubit
states—i.e., the speed limit is homogeneous and isotropic. For a single qubit,
this manifold corresponds to the Bloch sphere, and we assume the fastest
transition connecting any two quantum states on the Bloch sphere corresponds
to traveling along an arc of a great circle at speed $v$. In Appendix D, we
show that this assumption gives rise to the following distribution:

$\mathbb{P}\left(\textrm{MET}\leq
T\right)=\frac{2}{B\left(d-1,\frac{1}{2}\right)}\int\limits_{0}^{\frac{vT}{2}}\differential{x}\sin^{2d-3}(x),$
(5)

for $0\leq T\leq\frac{\pi}{v}$ where $d$ is the Hilbert space dimension, and
$B\left(\bullet,\bullet\right)$ is the beta function. We fit the numerical
data by adjusting the speed parameter $v$—this gives the dashed purple curves
in Fig. 4. In general, the assumption (that a pair of states is connected in
the shortest time by traveling along a geodesic with a constant speed) will
not hold—as in our particular hardware. Our second expression,

$\displaystyle\mathbb{P}\left(\textrm{MET}\leq
T\right)=\int\limits_{0}^{\frac{\tilde{v}T}{2}}\differential{x}\sum_{n=2d-3}^{\infty}c_{n}\sin^{n}(x),$
(6) where
$\displaystyle\tilde{v}\coloneqq\frac{\pi}{\displaystyle\max_{\ket{\psi_{0}},\ket{\phi}}\textrm{MET}\left(\ket{\psi_{0}},\ket{\phi}\right)},$
(7)

derived in Appendix E, generalizes the first expression by adding correction
terms to account for the fact that the speed at which the state can move on
the manifold may depend on the location within the manifold and the direction
of motion—i.e., the speed limit is inhomogeneous and anisotropic. For example,
in our simulations of silicon hardware, we fix the maximum values for
$I_{i}\left(t\right)$ and $Q_{i}\left(t\right)$, which induce $X$ and $Y$
rotations, respectively. We can rotate a state about the
$\frac{1}{\sqrt{2}}\left(X\pm Y\right)$ axes by simultaneously driving
$I_{i}\left(t\right)$ and $Q_{i}\left(t\right)$. Thus, rotations about the
$\frac{1}{\sqrt{2}}\left(X\pm Y\right)$ axes will be a factor of $\sqrt{2}$
faster than about the $X$ or $Y$ axes. We fit a truncation of the generalized
expression [Eq. 6]; see the dashed orange curves in Fig. 4 and Appendix G.

Finally, we conclude this section with two brief technical remarks about our
numerical simulations: (i) We discretized the control parameters into $M=40$
segments. (ii) Like in Sec. IV, we implement variational pulse-shaping as
follows: First, we initialize GRAPE with three random pulse shapes at the
maximum evolution time $T$ over which we will scan. Next, we move to the next
shortest evolution time $T$ and initialize GRAPE with two random pulse shapes
and the optimal pulse shape of the previous value for $T$. Then, we select the
pulse shape with the minimal $C(T)$.

##  VI Dependence of minimal evolution times on silicon device parameters

In this section, we investigate how silicon METs depend on the device
parameters and their experimental bounds in a silicon processor. In
particular, we investigate the dependence of METs on the maximal drive
strength $\textrm{IQ}_{\max}$, the maximal exchange coupling $J_{\max}$, and
the Zeeman-splitting inhomogeneity $\Delta B$. The investigation is motivated
by gate-based qubit operations in silicon. On the one hand,
$\textrm{IQ}_{\max}$ is known to limit the speed of one-qubit operations,
which are known to be relatively slow [33]. On the other hand, $J_{\max}$ is
known to enable relatively fast two-qubit power-of-SWAP gates [33]. We
investigate whether similar parameter limitations apply to pulse-based state
preparation.

Figure 5: MET for molecular ground-state preparation vs. bond distance for
(left) H2 and (right) HeH+, respectively. Colored curves depict the MET for
independently varying (top) the maximal drive strength $\textrm{IQ}_{\max}$,
(center) the maximal exchange coupling $J_{\max}$, and (bottom) the Zeeman-
splitting inhomogeneity $\Delta B$ relative to their specified values in Table
1 as decoded by the color bars. The black dashed line in the central row is
for $J_{\max}=$10\text{\,}\mathrm{MHz}$$. Figure 6: Cumulative distribution of
METs for Haar random state transitions (colored curves) for independently
varying (top) the maximal drive strength $\textrm{IQ}_{\max}$ and (bottom) the
maximal exchange coupling $J_{\max}$ relative to the values in Table 1 as
decoded by the color bars. Shaded regions correspond to 99% confidence
intervals. Dashed lines mark the time to perform a single qubit $\left(X\pm
Y\right)/\sqrt{2}$ gate. The black curve in the bottom panel is for
$J_{\max}=$10\text{\,}\mathrm{MHz}$$.

To begin with, we investigate the influence of microwave drives, exchange
couplings, and Zeeman-splitting inhomogeneities on METs for molecular ground-
state preparation. To this end, we extend the numerical investigation of Sec.
IV and determine the METs for H2 and HeH+. The data is shown in Fig. 5, where
colored curves depict the MET. The top, center, and bottom panels, show METs
where $\textrm{IQ}_{\max}$, $J_{\max}$, and $\Delta B$ are varied
independently, relative to their values in Table 1. Roughly speaking, this
data confirms that larger $\textrm{IQ}_{\max}$, larger $J_{\max}$, and larger
$\Delta B$ lead to faster state-preparation times and thus shorter METs. In
addition, we calculate the METs for molecular ground-state preparation (dashed
black line in center panels) for $J_{\max}=$10\text{\,}\mathrm{MHz}$$—a common
value for realizing CZ [19, 36, 38, 34] or CNOT [46] gates. The corresponding
METs for H2 and HeH+ become as slow as $100\text{\,}\mathrm{ns}$ and
$10\text{\,}\mathrm{ns}$, respectively. This observation highlights the
importance of fast exchange for accelerated state preparation. A deeper
discussion of the METs for molecular ground-state preparation of H2 and HeH+
in relation to the one-qubit drives is given below.

Next, we investigate the influence of $\textrm{IQ}_{\max}$ and $J_{\max}$ on
transitions between Haar random states. To this end, we extend the numerical
investigation of Sec. V and determine the cumulative distribution of METs from
1024 pairs of Haar random states. The data is shown in Fig. 6. In the top
panel, we apply the device parameters of Table 1 and vary
$\textrm{IQ}_{\max}$. As the drive strength varies, the METs are scaled by the
same factor. In the bottom panel, we apply the device parameters of Table 1
and vary $J_{\max}$. As the exchange coupling is varied, the cumulative METs
remain almost identical. These data confirm our intuition that silicon pulse-
based METs—similar to gate-based state-preparation times—are, on average,
limited by $\textrm{IQ}_{\max}$. On the other hand, lowering $J_{\max}$ to
$125\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ would not lower METs
significantly. However, once $J_{\max}$ is reduced to
$10\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ (black curve) we find that the MET
starts to increase significantly. Again, this illustrates the importance of
fast exchange for achieving small METs.

###  VI A Molecular ground states and qubit drives

We now discuss, in more detail, how specific microwave and exchange pulses
achieve the METs in Fig. 5. In Appendix H, we show that the exchange operation
in the rotating frame of the drift Hamiltonian is power-of-SWAP like: In
particular in the $\ket{01}$, $\ket{10}$ subspace, it drives population
transfer with a phase proportional to $\Delta Bt_{0}/\hbar$. Here, $t_{0}$ is
the time at which exchange is turned on (as compared to the start of the pulse
at $t=0$). Thus, waiting until time $t_{0}$ before switching on the exchange
pulse allows us to adjust the phase between the $\ket{01}$ and $\ket{10}$
states.

First, consider the MET data for H2 in Fig. 5. For H2, the initial Hartree-
Fock state is $\ket{01}$. The true ground state (over the range scanned) is
well approximated by the state proportional to $\ket{01}+\epsilon
e^{i\phi}\ket{10}$. Notably, the absence of $\ket{00}$ and $\ket{11}$
components in the molecular ground state of H2, implies that single-qubit
gates are not required to prepare the molecular ground state of H2 from
$\ket{01}$. For that reason, changing $\textrm{IQ}_{\max}$ (top left figure of
Fig. 5) has no influence on the MET. Next, in order to prepare the ground
state of H2, we require several applications of the exchange pulse to transfer
some population from the initial Hartree-Fock state $\ket{01}$ into $\ket{10}$
with the correct phase. In Fig. 5 (center left), we can see that this transfer
tends to happen faster as $J_{\mathrm{max}}$ is increased. However, after
$J_{\mathrm{max}}$ reaches $0.5\text{\,}\mathrm{GHz}$ the acceleration of the
MET slows down. We conjecture that this slowdown is related to a wait time
required to build up the correct phase $\Delta Bt_{0}/\hbar$. Finally, in Fig.
5 (bottom left), we see that increasing the detuning $\Delta B$ tends to
accelerate the MET. We conjecture that this observation is related to the fact
that a larger $\Delta B$ allows for faster accumulation of the phase $\Delta
Bt_{0}/\hbar$.

Finally, for HeH+, we observe three different scenarios in Fig. 5, depending
on the bond length. At large bond lengths beyond
($\geq$2.8\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$), the initial Hartree-Fock
state $\ket{01}$ describes the true ground state with sufficient
accuracy—thus, the MET is zero. At intermediate bond distances (2.22.8) the
true ground state is approximately proportional to
$\ket{01}+\epsilon^{\prime}e^{i\phi^{\prime}}\ket{10}$. Neither single qubit
gates nor long wait times $t_{0}$ are required to adjust the transfer phase
$\Delta Bt_{0}/\hbar$ of the exchange pulse. Hence, in that region, the MET is
neither dependent on $\textrm{IQ}_{\mathrm{max}}$ nor $\Delta B$. However, it
is inversely proportional to the exchange $J_{\mathrm{max}}$. Finally, for
smaller bond distances (0.52.2), the true ground state is proportional to
$\ket{01}+\epsilon^{\prime\prime}e^{i\phi^{\prime\prime}}\ket{10}+\delta_{00}\ket{00}+\delta_{11}\ket{11}$,
with some amplitudes on $\ket{00}$ and $\ket{11}$. Preparing the amplitudes on
$\ket{00}$ and $\ket{11}$ requires single qubit drives and increasing
$\textrm{IQ}_{\mathrm{max}}$ decreases the MET proportionally, see Fig. 5(top
right). Finally, the exchange pulse is needed to prepare probability in
$\ket{10}$ with the right amplitude and phase. We conjecture that the trends
in $J_{\max}$ and $\Delta B$ arise from the same mechanisms for producing a
superposition of $\ket{01}$ and $\ket{10}$ with the correct phase as described
for H2.

##  VII Discussion and Conclusion

In this article, we have numerically characterized the METs for state
preparation in silicon quantum processors. These METs lower bound the ultimate
performance for both gate- and pulse-based state preparation and unitary
evolution.

We investigated METs for pulse-based preparation of molecular ground states.
We found that the ground states of H2, HeH+, and LiH can be prepared in
$2.4\text{\,}\mathrm{ns}$, $4.4\text{\,}\mathrm{n}\mathrm{s}$, and 27.2ns,
respectively. These METs are less than the $200\text{\,}\mathrm{n}\mathrm{s}$
required to prepare arbitrary states on two qubits. Additionally, they are an
order of magnitude faster than the fastest single qubit $\pi$ rotations in
silicon [33] and the numerically-estimated METs for superconducting transmon
qubits from Ref. [11].

The acceleration observed for molecular ground-state preparation is promising
for pulse-based simulations for chemistry purposes. The speed-up stems from
the weak entanglement in molecules. Conventional two-qubit gates (e.g., the
CNOT or CZ gates) can create maximal entanglement between two qubits.
Consequently, gate-based computations utilize significantly longer two-qubit
interactions than needed—this is ill-suited for molecular ground-state
preparations. Moreover, gate-based computation relies on slow single-qubit
gates to limit the amount of entanglement induced during the computation. On
the other hand, a pulse-based approach enables the generation of weak
entanglement via fast purpose-tailored pulses.

We investigated METs for transitions between Haar-random states and
demonstrated that a four-qubit silicon quantum processor can transition
between any pair of states within $50\text{\,}\mathrm{ns}$. Our data suggests
that single-qubit state transitions can be achieved faster by partially
entangling the qubits during the operation and disentangling them at the end.
This peculiar effect should be studied in more detail in the future.

Finally, we investigated the METs’ dependence on silicon device parameters
such as the maximally allowed microwave drive, the maximally allowed exchange
coupling, or the Zeeman-splitting inhomogeneity. We demonstrated that fast
exchange is vital for short state-preparation METs: Decreasing the maximal
exchange coupling from $8\text{\,}\mathrm{GHz}$ to $250\text{\,}\mathrm{MHz}$
makes little difference to METs. However, decreasing the maximal exchange
coupling any further to $10\text{\,}\mathrm{MHz}$ can, for some problem
instances, increase the MET by a factor of $\sim 100$. On the other hand, we
showed that most METs scale directly proportional to the maximal microwave
drive amplitude, which we identify as the limiting factor for most silicon
METs.

Our results give hope that pulse-based state preparation on silicon hardware
could lead to a new generation of intermediate-scale quantum algorithms that
are significantly more resilient to noise.

###### Acknowledgements.

The authors would like to thank Johnathan Thio, Kyle Sherbert, Ayush Asthana,
Guo Xuan Chan, Stefano Bosco, Leon C. Camenzind, Andrew J. Ramsay, and Charles
G. Smith for useful discussions.

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##  Appendix A Footnotes

aFrom Ref. [17]

bThe fastest single qubit gates recorded on silicon electron Loss-DiVincenzo
qubits take approximately $50\text{\,}\mathrm{ns}$ [33]. This corresponds to a
drive strength of $2\times$20\text{\,}\mathrm{MHz}$$. However, as there are
$S$ signals on the line, each with an in-phase and quadrature component, we
divide by $\sqrt{2}S$. As we work with up to four qubits in this paper, we
take $S=4$ in this expression for all devices considered to enable fair
comparisons.

cThe order of magnitude is found by using $\left|J\right|=4\frac{t^{2}}{U}$
where $t$ is the interdot hopping strength and $U$ is the double occupation
energy. Ref. [23] measures $t\leq$12\text{\,}\mathrm{GHz}$$ and Ref. [50] fits
for $U\approx$240\text{\,}\mathrm{GHz}$$ using the data from Ref. [23].
Additionally, Ref. [21] measures $t\leq$100\text{\,}\mathrm{GHz}$$ and we
estimate $U\approx$30\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$$ as half the
width of the (1,1) diamond along the detuning axis in Fig. 7 using the quoted
detuning axis lever arm of
$0.3\text{\,}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{V}$.

##  Appendix B Cost Function

Throughout this article, we consider the cost function

$\displaystyle
f(\bm{x})\coloneqq\matrixelement{\psi(\bm{x};T)}{\hat{C}}{\psi(\bm{x};T)}.$
(8)

If $\hat{C}$ acts on $n$ qubits we can estimate $f(\bm{x})$ by decomposing
$\hat{C}$ as a sum of weighted Pauli strings:

$\displaystyle\hat{C}\equiv\sum_{\hat{P}\in\mathcal{P}_{n}}c_{\hat{P}}\hat{P},$
(9)

where $\mathcal{P}_{n}$ is the set of Pauli strings on $n$ qubits and
$c_{\hat{P}}$ is the coefficient of the Pauli string
$\hat{P}\in\mathcal{P}_{n}$. Thus,

$\displaystyle
f(\bm{x})\equiv\sum_{\hat{P}\in\mathcal{P}_{n}}c_{\hat{P}}\matrixelement{\psi(\bm{x};T)}{\hat{P}}{\psi(\bm{x};T)}.$
(10)

By preparing many copies of $\ket{\psi\left(\bm{x};T\right)}$ the expectation
value
$\mathbb{E}_{\hat{P}}\coloneqq\matrixelement{\psi\left(\bm{x};T\right)}{\hat{P}}{\psi\left(\bm{x};T\right)}$
of each Pauli string $\hat{P}$ with non-zero coefficient $c_{\hat{P}}$ can be
estimated as follows: To rotate into the eigenbasis of $\hat{P}$ apply a
Haramard ($R_{x}\left(-\pi/2\right)$) gate to every qubit upon which $\hat{P}$
acts as an $X$ ($Y$) gate. If the Hadamard and $R_{x}\left(-\pi/2\right)$
gates are only calibrated to start at discrete clock times $t=m\Delta\tau$ for
$m\in\mathbb{N}$ then one simply needs to set $g(t)$ and $J(t)$ to zero for
$t\in\left[T,\left\lceil T/\Delta\tau\right\rceil\Delta\tau\right]$ and then
apply the Hadamard and $R_{x}\left(-\pi/2\right)$ gates at $t=\left\lceil
T/\Delta\tau\right\rceil\Delta\tau$. Next, we can measure the $l$ qubits in
the computational basis upon which $\hat{P}$ acts non-trivially. If the
Hamming weight of the resulting $l$-bitstring is even (odd) then
$\ket{\psi\left(\bm{x};T\right)}$ was measured as being in the $+1$ ($-1$)
eigenspace of $\hat{P}$. Let $N_{\hat{P}}^{\pm}$ be the number of measurements
in the $\pm 1$ eigenspaces of $\hat{P}$. We can estimate $\mathbb{E}_{P}$ as
$\tilde{\mathbb{E}}_{\hat{P}}\coloneqq\left(N_{\hat{P}}^{+}-N_{\hat{P}}^{-}\right)/N_{\hat{P}}$
where $N_{\hat{P}}=N_{\hat{P}}^{+}+N_{\hat{P}}^{-}$ is the total number of
measurements of $\hat{P}$. Ref. [51] proves that given
$\bar{N}\coloneqq\sum_{\hat{Q}\in\mathcal{P}_{n}}N_{\hat{Q}}$ total
measurements, the optimal distribution of these across the Pauli strings is:

$N_{\hat{P}}=\frac{\left|c_{\hat{P}}\right|}{\sum_{\hat{Q}\in\mathcal{P}_{n}}\left|c_{\hat{Q}}\right|}\bar{N}.$
(11)

Thus, using Hoeffding’s inequality [52], one can show that with

$\displaystyle
N_{\hat{P}}=\left|c_{\hat{P}}\right|\sum_{\hat{Q}\in\mathcal{P}_{n}}\left|c_{\hat{Q}}\right|\frac{2}{\varepsilon^{2}}\log\left(\frac{2}{\delta}\right)$
(12)

measurements for each $\hat{P}\in\mathcal{P}_{n}$ we can calculate the
estimator $\tilde{f}\left(\bm{x}\right)$ of $f\left(\bm{x}\right)$ within an
accuracy $\varepsilon$ with at least a probability $1-\delta$ as

$\displaystyle\tilde{f}\left(\bm{x}\right)=\sum_{\hat{P}\in\mathcal{P}_{n}}c_{\hat{P}}\tilde{\mathbb{E}}_{\hat{P}}=\sum_{\hat{P}\in\mathcal{P}_{n}}c_{\hat{P}}\left(N_{\hat{P}}^{+}-N_{\hat{P}}^{-}\right)/N_{\hat{P}}.$
(13)

##  Appendix C Bounds on the Maximal METs for one and two-qubit systems

In this appendix, we bound the maximal MET for state preparation of one- and
two-qubit systems. We utilize these bounds in the body of the paper as a
comparison and to ensure that we have scanned to a large enough total
evolution time $T$ to obtain the entire cumulative probability distribution
$\mathbb{P}\left(\textrm{MET}\leq T\right)$.

In deriving these bounds, we assume that within the gate-based model, the
quantum computer operates on a clock: parameterized gates have an associated
execution time that is independent of the parameter value. If the execution
time for a parameterized gate naturally depends on the parameter, we can just
fix the execution time as the maximal execution time over the parameter space.
For all naturally slower gates, we simply apply the identity operation for the
remainder of the execution time.

###  C.1 One-Qubit-State Preparation Maximal MET

Any unitary on one-qubit can be comprised of three Euler rotations, for
example:

$\displaystyle U\left(\alpha,\beta,\gamma\right)\equiv
R_{X}\left(\alpha\right)R_{Y}\left(\beta\right)R_{X}\left(\gamma\right).$ (14)

The rotation is characterized by three proper Euler angles:
$\alpha,\gamma\in\left(-\pi,\pi\right]$ and
$\beta\in\left[-\pi/2,\pi/2\right]$. The fastest single-qubit gate recorded in
silicon electron Loss-DiVincenzo qubits took approximately
$50\text{\,}\mathrm{ns}$ and corresponds to a $\pi$ rotation about a single
axis [53, 33]. In this article, we multiplied this single qubit gate time by
four so that a four-qubit system would not implement a faster single qubit
gate by aligning the four drive frequencies to the Rabi frequency of a single
qubit. That is we take the fastest single-qubit gate to take
$200\text{\,}\mathrm{ns}$. Thus, at most, a single-qubit gate implemented
using the three Euler rotations within our system will take
$2.5\times$200\text{\,}\mathrm{ns}$=$500\text{\,}\mathrm{ns}$$. Further, an
arbitrary single-qubit gate must take at least as long as the time for the
fastest $\pi$ rotation: $200\text{\,}\mathrm{ns}$.

###  C.2 Two-Qubit-State Preparation Maximal MET

Žnidarič  _et al._ prove in Ref. [54] that a single CNOT and four single-qubit
gates are sufficient to transition between arbitrary two-qubit states. Here,
we follow a similar proof but replace the CNOT with a power of SWAP. The
exchange operator in Eq. 41 is the native two-qubit operation in the rotating
frame considered in this article. However, we can use the single-qubit gates
in the procedure below to move back into the lab frame so that the native two-
qubit operation is once again the power of SWAP.

###### Lemma 1

Four single-qubit gates and $\textrm{SWAP}^{\alpha}$ with
$\alpha\in\left[0,1\right]$ are sufficient to transition between arbitrary
two-qubit states.

  * Proof.

We can express our initial state $\ket{\psi_{0}}$ as a Schmidt decomposition:

$\displaystyle\ket{\psi_{0}}=\cos\left(\theta\right)\ket{a_{0}}\ket{b_{0}}+\sin\left(\theta\right)\ket{a_{1}}\ket{b_{1}},$
(15)

where $\left\\{\ket{a_{0}},\ket{a_{1}}\right\\}$ and
$\left\\{\ket{b_{0}},\ket{b_{1}}\right\\}$ are orthonormal bases of each
qubit. By applying a single-qubit gate to each qubit, we can induce the
following map

$\displaystyle\ket{\psi_{0}}\mapsto\ket{\psi^{\prime}}\coloneqq\cos\left(\theta\right)\ket{0}\ket{1}+\sin\left(\theta\right)\ket{1}\ket{0}.$
(16)

Now, applying a power of SWAP gate, we will obtain

$\displaystyle\ket{\psi^{\prime}}\mapsto\ket{\phi^{\prime}}\coloneqq\cos\left(\theta^{\prime}\right)\ket{0}\ket{1}+\sin\left(\theta^{\prime}\right)\ket{1}\ket{0}.$
(17)

Finally, applying a single-qubit gate to each qubit again, we can obtain an
arbitrary state

$\displaystyle\ket{\phi^{\prime}}\mapsto\cos\left(\theta^{\prime}\right)\ket{a^{\prime}_{0}}\ket{b^{\prime}_{0}}+\sin\left(\theta^{\prime}\right)\ket{a^{\prime}_{1}}\ket{b^{\prime}_{1}}.$
(18)

In this article we limit $J$ to $1\text{\,}\mathrm{GHz}$, thus,
$\textrm{SWAP}^{\alpha}$ for $\alpha\in\left[0,1\right]$ takes less than
$0.5\text{\,}\mathrm{n}\mathrm{s}$ to impliment. Thus, the transition between
arbitrary two-qubit states can always be performed in under
$1000.5\text{\,}\mathrm{n}\mathrm{s}$ by applying the pairs of single-qubit
gates in parallel.

When preparing the ground states of H2 and HeH+ in Sec. IV we start in the
$\ket{01}$ state. Thus, for these preparations, we can start from the state
$\ket{\psi^{\prime}}$ in Eq. 16 and forgo the first layer of arbitrary single-
qubit rotations. This reduces the upper bound on the gate-based two-qubit
molecular ground-state preparation times to $500.5\text{\,}\mathrm{ns}$.

To get the lower bound, we note we must use at least two single qubit gates in
parallel and a two-qubit gate to prepare an arbitrary two-qubit state. Thus,
the arbitrary two-qubit state preparation time is lower bounded by
$200.5\text{\,}\mathrm{ns}$.

##  Appendix D Homogeneous and Isotropic Speed Limit

In this appendix, we derive the cumulative probability distribution for the
MET between Haar random input and output states in a $d$-dimensional Hilbert
space with a homogeneous and isotropic quantum speed limit. More precisely, a
Haar distributed pure state $\ket{\varphi}\sim\mu$ is a pure state generated
by multiplying a Haar random unitary by a fixed pure state.

The geodesic distance between two pure states $\ket{\psi_{0}}$ and
$\ket{\phi}$ under the Fubini-Study metric is

$\displaystyle
d\left(\ket{\psi_{0}},\ket{\phi}\right)=2\cos^{-1}\left|\innerproduct{\psi_{0}}{\phi}\right|.$
(19)

If we take the homogeneous and isotropic quantum speed limit to be $v$, then
the MET is

$\displaystyle\textrm{MET}\left(\ket{\psi_{0}},\ket{\phi}\right)=\frac{2}{v}\cos^{-1}\left|\innerproduct{\psi_{0}}{\phi}\right|.$
(20)

For notational ease, let

$\displaystyle f_{X\sim
D}\left[g\left(X\right)\right]\left(x\right)\differential{y}=\mathbb{P}_{X\sim
D}\left(g\left(x\right)\leq g(X)<g\left(x\right)+\differential{y}\right)$ (21)

be the probability density function of $g\left(X\right)$ where $X\sim D$. Note
that the invariance of the Haar distribution under unitary transformations
allows us to fix one of the states

$\displaystyle
f_{\ket{\psi_{0}},\ket{\phi}\sim\mu}\left[\left|\innerproduct{\psi_{0}}{\phi}\right|\right]=f_{\ket{\phi}\sim\mu}\left[\left|\innerproduct{\psi_{0}}{\phi}\right|\right].$
(22)

Without loss of generality, let
$\bra{\psi_{0}}=\begin{bmatrix}1&0&\cdots&0\end{bmatrix}$ and fix the global
phase of $\ket{\phi}$ such that the first entry is real. Thus,
$f_{\ket{\phi}\sim\mu}\left[\left|\innerproduct{\psi_{0}}{\phi}\right|\right]$
is equal to the probability density function of the magnitude of the first
component of a $2d-1$ dimensional real vector that is distributed uniformly
over the $2d-1$ real unit sphere. Therefore, we find [55]:

$\displaystyle
f_{\ket{\psi_{0}},\ket{\phi}\sim\mu}\left[\left|\innerproduct{\psi_{0}}{\phi}\right|\right]$
$\displaystyle=\frac{2\left(1-\left|\innerproduct{\psi_{0}}{\phi}\right|^{2}\right)^{d-2}}{B\left(d-1,\frac{1}{2}\right)}$
(23)
$\displaystyle=\frac{2\left(1-\cos^{2}\left[\frac{v}{2}\textrm{MET}\left(\ket{\psi_{0}},\ket{\phi}\right)\right]\right)^{d-2}}{B\left(d-1,\frac{1}{2}\right)}$
(24)
$\displaystyle=\frac{2\sin^{2d-4}\frac{v}{2}\textrm{MET}\left(\ket{\psi_{0}},\ket{\phi}\right)}{B\left(d-1,\frac{1}{2}\right)},$
(25)

where $B\left(\bullet,\bullet\right)$ is the beta function.

Now we can use the identity $f_{X\sim
D}\left[g\left(X\right)\right]\left(y\right)\equiv f_{X\sim
D}\left[X\right]\left(g\left(y\right)\right)\left|\derivative{g}{y}\right|^{-1}$
to show

$f_{\ket{\psi_{0}},\ket{\phi}\sim\mu}\left[\textrm{MET}\left(\ket{\psi_{0}},\ket{\phi}\right)\right]\left(T\right)\\\
=\frac{v}{B\left(d-1,\frac{1}{2}\right)}\sin^{2d-3}\left(\frac{vT}{2}\right).$
(26)

Integrating Eq. 26 and performing a change of variables gives our final result

$\displaystyle\mathbb{P}\left(\textrm{MET}\leq
T\right)=\frac{2}{B\left(d-1,\frac{1}{2}\right)}\int\limits_{0}^{\frac{vT}{2}}\sin^{2d-3}\left(x\right)\differential{x}.$
(27)

While this integral can be performed analytically for integer $d$, it is
generally cumbersome. We utilize the analytic form of the integral to
calculate the curves in Fig. 4.

##  Appendix E Inhomogeneous and Anisotropic Speed Limit

In this appendix, we will first use the homogeneous and isotropic result to
constrain the functional form of the inhomogeneous and anisotropic
distribution in Sec. E.1. Second, we will provide a possible explanation for
the vanishing derivatives of $\mathbb{P}\left(\textrm{MET}\leq T\right)$ at
the maximal MET in Sec. E.2. Finally, we will combine the two results in Sec.
E.3.

###  E.1 Series Expansion

First, note that the probability density function only has support on the
finite interval

$\displaystyle\left[0,\max_{\ket{\psi_{0}},\ket{\phi}}\textrm{MET}\left(\ket{\psi_{0}},\ket{\phi}\right)\right].$
(28)

Thus, define the quantity

$\tilde{v}\coloneqq\frac{\pi}{\displaystyle\max_{\ket{\psi_{0}},\ket{\phi}}\textrm{MET}\left(\ket{\psi_{0}},\ket{\phi}\right)},$
(29)

which will play an analogous role to the speed limit $v$. Using this
definition, we Maclaurin-expand the function

$\displaystyle g$ $\displaystyle\coloneqq
f_{\ket{\psi_{0}},\ket{\phi}\sim\mu}\left[\textrm{MET}\left(\ket{\psi_{0}},\ket{\phi}\right)\right]\circ\frac{2}{\tilde{v}}\sin^{-1}$
(30) $\displaystyle g\left(x\right)$
$\displaystyle\equiv\sum_{n=0}^{\infty}c_{n}x^{n}\quad
x\in\left[0,\varepsilon\right),$ (31)

for some radius of convergence $\varepsilon$. As $\sin^{-1}$ is an analytic
function then the power series

$f_{\ket{\psi_{0}},\ket{\phi}\sim\mu}\left[\textrm{MET}\left(\ket{\psi_{0}},\ket{\phi}\right)\right]\left(T\right)\equiv\sum_{n=0}^{\infty}c_{n}\sin^{n}\left(\frac{\tilde{v}T}{2}\right),$
(32)

converges on the interval
$\left[0,\frac{2}{\tilde{v}}\sin^{-1}\varepsilon\right)$. Further, as the
Maclaurin series converges uniformly, so too does the series in Eq. 32.
Therefore, we can integrate
$f_{\ket{\psi_{0}},\ket{\phi}\sim\mu}\left[\textrm{MET}\left(\ket{\psi_{0}},\ket{\phi}\right)\right]\left(T\right)$
by integrating each term in Eq. 32:

$\displaystyle\mathbb{P}\left(\textrm{MET}\leq T\right)$
$\displaystyle\equiv\int\limits_{0}^{\frac{\tilde{v}T}{2}}\differential{x}\sum_{n=0}^{\infty}c_{n}\sin^{n}\left(x\right),$
(33)
$\displaystyle=\sum_{n=0}^{\infty}c_{n}\int\limits_{0}^{\frac{\tilde{v}T}{2}}\differential{x}\sin^{n}\left(x\right).$
(34)

We will use this expression to find coefficients that must vanish.

First, note that

$\int\limits_{0}^{\delta}\differential{x}\sin^{n}\left(kx\right)=\mathcal{O}\left(\delta^{n+1}\right),$
(35)

as $\delta\to 0$. Thus, the first non-zero $c_{n}$ must be positive as
$\mathbb{P}\left(\textrm{MET}\leq T\right)$ is positive. Additionally,
$\mathbb{P}\left(\textrm{MET}\leq T\right)$ must be upper bounded by the
corresponding cumulative probability distribution for a $d$-dimensional
Hilbert space with a homogeneous and isotropic speed limit given by the
maximal speed limit of the inhomogeneous and anisotropic case. Thus, the first
non-zero $c_{n}$ must be for $n\geq 2d-3$. Therefore we find

$\mathbb{P}\left(\textrm{MET}\leq
T\right)\equiv\int\limits_{0}^{\frac{\tilde{v}T}{2}}\differential{x}\sum_{\mathclap{n=2d-3}}^{\infty}c_{n}\sin^{n}\left(x\right),$
(36)

where the first non-zero $c_{n}$ is positive. Only the first term in the
expansion remains for the homogeneous and isotropic case. Generally, the terms
for $n>2d-3$ are corrections due to the inhomogeneities and anisotropies.

###  E.2 Vanishing Derivatives at Maximal MET

In this section, we provide a possible explanation for the vanishing
derivatives of $\mathbb{P}\left(\textrm{MET}\leq T\right)$ observed in Fig. 4.
First, we define $T_{\star}$ as the minimum $T$ that satisfies
$\mathbb{P}\left(\textrm{MET}\leq T\right)=1$. Assuming
$\mathbb{P}\left(\textrm{MET}\leq T\right)$ is Taylor expandable about
$T=T_{\star}$ with some finite radius of convergence, then for the first $k$
derivatives of $\mathbb{P}\left(\textrm{MET}\leq T\right)$ to vanish at
$T=T_{\star}$ we require $1-\mathbb{P}\left(\textrm{MET}\leq
T\right)=\mathbb{P}\left(\textrm{MET}\geq
T\right)=\mathcal{O}\left([T-T_{\star}]^{k+1}\right)$ at $T=T_{\star}$. Thus,
for any $\tau>0$ we find $\mathbb{P}\left(\textrm{MET}\geq
T_{\star}-\tau\right)>0$.

Next, we note that $\mathbb{P}\left(\textrm{MET}\geq T\right)$ is a normalized
volume in the manifold of state pairs $\mathcal{M}$:

$\displaystyle\mathbb{P}\left(\textrm{MET}\geq T\right)$
$\displaystyle=\int\limits_{\mathclap{\textrm{MET}\left(\ket{\psi_{0}},\ket{\phi}\right)\geq
T}}\differential{\ket{\psi_{0}}}\differential{\ket{\phi}}.$ (37)

Note that for the homogeneous and isotropic case, the volume
$\mathbb{P}\left(\textrm{MET}\geq T\right)$ is linear in $(T-T_{\star})$ at
$T=T_{\star}$. We now give one possible mechanism in the inhomogeneous and
anisotropic case for the volume to scale as $\mathbb{P}\left(\textrm{MET}\geq
T\right)=\mathcal{O}\left([T-T_{\star}]^{k+1}\right)$.

Consider the subset of states pairs $\mathcal{S}\subset\mathcal{M}$ such that
$\textrm{MET}\left(\mathcal{S}\right)=T_{\star}$ and let us assume that
$\mathcal{S}$ is closed and the normal space well defined at all points in
$\mathcal{S}$. If the velocity one can move at through the manifold of quantum
states is limited to be finite at all points and in all directions, then
$\mathcal{S}$ must expand from all points a distance
$\mathcal{O}\left([T-T_{\star}]^{\alpha}\right)$ with $\alpha\geq 1$ in each
direction in the normal space to that point. Thus, the volume
$\mathbb{P}\left(\textrm{MET}\geq
T\right)=\mathcal{O}\left([T-T_{\star}]^{\alpha D}\right)$ where $D$ is the
maximal dimension of the normal spaces to $\mathcal{S}$. Therefore, if at
least the first $D-1$ derivatives of $\mathbb{P}\left(\textrm{MET}\leq
T\right)$ will vanish.

The maximal dimension of the normal space $D$ can be increased by breaking
symmetries. To gain some intuition, consider the following broken symmetry:
Suppose not all input states $\ket{\psi_{0}}$ have a corresponding output
state $\ket{\phi}$ that maximizes the MET. If the set of $\ket{\psi_{0}}$ that
do have at least one $\ket{\phi}$ that maximizes the MET are measured zero in
the manifold of pure states, then $D>1$. Therefore, at least the first
derivative of $\mathbb{P}\left(\textrm{MET}\leq T\right)$ vanishes.

###  E.3 Numerical Fitting

In numerical fitting, we have to truncate our series expansion

$\mathbb{P}\left(\textrm{MET}\leq
T\right)\equiv\int\limits_{0}^{\frac{\tilde{v}T}{2}}\differential{x}\sum_{\mathclap{n=2d-3}}^{L}c_{n}\sin^{n}\left(x\right).$
(38)

If we want to fit Eq. 38 to the whole curve of
$\mathbb{P}\left(\textrm{MET}\leq T\right)$ we are implicitly assuming the
convergence radius $\varepsilon\geq\frac{\pi}{2}$. Thus, if we know that the
first $k$ derivatives at $T_{\star}$ must vanish by symmetry arguments, we can
reduce the number of free parameters in the fit. In our work, we will assume
the first derivative vanishes, which gives the following constraint:

$c_{L}=-\sum_{\mathclap{n=2d-3}}^{L-1}c_{n}.$ (39)

##  Appendix F Bootstrapping

We apply bootstrapping to quantify the error in the cumulative distributions,
which we estimate by sampling pairs of states from the Haar distribution. Our
estimated cumulative distribution consists of 1024 input-output state pairs.
We resample, with replacement, from this sample, 1024 input-output state pairs
and use this resample to compute a new cumulative distribution. We repeat this
$10^{5}$ times to get $10^{5}$ resampled cumulative distributions. For each
value of $T$, we calculate the confidence interval and variance of the
resampled cumulative probabilities. We use the variances during the chi-
squared fits in Fig. 4 and Appendix G. When the variance of the resampled
cumulative probabilities is less than $1/1024^{2}$, we take $1/1024^{2}$ as
the variance, as this is the square of the precision of our estimate.

##  Appendix G Chi-Squared Values

In this appendix, we perform chi-squared fits using the variances from
Appendix F to the data in Fig. 4. First, we fit using the homogeneous and
isotropic approximation Eq. 27 and a normalization constraint. Second, we
perform a higher order fit using the truncation in Eq. 38, under the
constraints of normalization and Eq. 39. In the higher order fit, we include
the minimal number of terms such that chi-squared $\chi^{2}$ divided by the
number of degrees of freedom $N_{\textrm{DoF}}$ is less than unity
($\chi^{2}/N_{\textrm{DoF}}<1$) or $\chi^{2}/N_{\textrm{DoF}}$ stops
decreasing monotonically with the addition of further terms. We present the
values for $\chi^{2}/N_{\textrm{DoF}}$ for the higher order fit in Table 2.

|  |  |  | Number of |  | Terms in
---|---|---|---|---|---|---
Qubits |  | $\chi^{2}/N_{\textrm{DoF}}$ |  | Parameters |  | Expansion
1 |  | 0.62 |  | 6 |  | 7
2 |  | 0.77 |  | 4 |  | 5
3 |  | 1.16 |  | 5 |  | 6
4 |  | 7.53 |  | 7 |  | 8
Table 2: $\bm{\chi^{2}/N_{\textrm{DoF}}}$ values for higher order fits in Fig.
4. We also present the number of parameters and terms in Eq. 38 used for the
fit under the constraints of normalization and Eq. 39.

##  Appendix H Exchange in the Rotating Frame

In this appendix, we analyze the unitary evolution induced by the exchange
Hamiltonian in the rotating frame of the drift Hamiltonian:

$\hat{H}_{J}=-\sum_{i=1}^{N-1}\frac{J_{i}(t)}{4}\hat{\mathcal{E}}_{i,i+1}\left(t\right).$
(40)

On the subspace of two qubits, encoded by the spins of the $i$th and $j$th
electron, its operator in matrix form is

$\\!\\!\\!\hat{\mathcal{E}}_{i,j}\\!\left(t\right)\\!\coloneqq\\!\\!\bNiceMatrix[last-
col]1&\ket{\uparrow_{i},\uparrow_{j}}\\\
-1\\!2e^{i\left(B_{j}-\\!B_{i}\right)t\\!/\\!\hbar}\ket{\uparrow_{i},\downarrow_{j}}\\\
2e^{i\left(B_{i}-\\!B_{j}\right)t\\!/\\!\hbar}-1\ket{\downarrow_{i},\uparrow_{j}}\\!\\\
1\ket{\downarrow_{i},\downarrow_{j}}.\\!\\!\\!$ (41)

For a constant $J_{i}\left(t\right)=J$ the unitary evolution from a time
$t_{0}$ to $t_{0}+\Delta t$ in the rotating frame induced by the exchange
Hamiltonian is

$\displaystyle\hat{U}_{J}(t_{0}\to t_{0}+\Delta
t)=\begin{bmatrix}U_{0,0}&&&\\\ &U_{1,1}&U_{1,2}&\\\ &U_{2,1}&U_{2,2}&\\\
&&&U_{3,3}\\\ \end{bmatrix}.$ (42)

Using $\tilde{J}\coloneqq\sqrt{J^{2}+\Delta B^{2}}$ its non-zero entries are

$\displaystyle U_{0,0}$ $\displaystyle\\!=\\!U_{3,3}\\!=\\!\exp(i\frac{J\Delta
t}{4\hbar}),$ (43) $\displaystyle U_{1,1}$
$\displaystyle\\!=\\!\\!\left[\\!\cos(\\!\frac{\tilde{J}\Delta
t}{2\hbar}\\!)\\!\\!-\\!i\frac{\Delta
B}{\tilde{J}}\\!\sin(\\!\frac{\tilde{J}\Delta
t}{2\hbar}\\!)\\!\right]\\!\\!e^{-i(\frac{J}{4}-\frac{\Delta
B}{2})\frac{\Delta t}{\hbar}},$ (44) $\displaystyle U_{2,2}$
$\displaystyle\\!=\\!\\!\left[\\!\cos(\\!\frac{\tilde{J}\Delta
t}{2\hbar}\\!)\\!\\!+\\!i\frac{\Delta
B}{\tilde{J}}\\!\sin(\\!\frac{\tilde{J}\Delta
t}{2\hbar}\\!)\\!\right]\\!\\!e^{-i(\frac{J}{4}+\frac{\Delta
B}{2})\frac{\Delta t}{\hbar}},$ (45) $\displaystyle U_{1,2}$
$\displaystyle\\!=\\!i\frac{J}{\tilde{J}}\sin(\frac{\tilde{J}\Delta
t}{2\hbar})e^{i\Delta Bt_{0}/\hbar}e^{-i(\frac{J}{4}-\frac{\Delta
B}{2})\frac{\Delta t}{\hbar}},$ (46) $\displaystyle U_{2,1}$
$\displaystyle\\!=\\!i\frac{J}{\tilde{J}}\sin(\frac{\tilde{J}\Delta
t}{2\hbar})e^{-i\Delta Bt_{0}/\hbar}e^{-i(\frac{J}{4}+\frac{\Delta
B}{2})\frac{\Delta t}{\hbar}}.$ (47)

In the limit $\Delta B\Delta t/\hbar\to 0$ (the relevant limit for the
molecular ground-state preparation METs) we find

$\left|\Delta B\sin(\frac{\tilde{J}\Delta t}{2\hbar})\right|\leq\left|\Delta
B\frac{\tilde{J}\Delta t}{2\hbar}\right|\to 0,$ (48)

and

$\displaystyle\left|\left(1-\frac{J}{\tilde{J}}\right)\sin(\frac{\tilde{J}\Delta
t}{2\hbar})\right|$ (49)
$\displaystyle\leq\left|\left(\tilde{J}-J\right)\frac{\Delta
t}{2\hbar}\right|$ (50) $\displaystyle=\left|\left(\sqrt{J^{2}\Delta
t^{2}+\Delta B^{2}\Delta t^{2}}-J\Delta t\right)\frac{1}{2\hbar}\right|$ (51)
$\displaystyle\to 0.$ (52)

Thus,

$\displaystyle U_{1,1}$ $\displaystyle\approx\cos(\frac{\tilde{J}\Delta
t}{2\hbar})e^{-i\frac{J}{4}\frac{\Delta t}{\hbar}},$ (53) $\displaystyle
U_{2,2}$ $\displaystyle\approx\cos(\frac{\tilde{J}\Delta
t}{2\hbar})e^{-i\frac{J}{4}\frac{\Delta t}{\hbar}},$ (54) $\displaystyle
U_{1,2}$ $\displaystyle\approx i\sin(\frac{\tilde{J}\Delta
t}{2\hbar})\exp(i\frac{\Delta Bt_{0}}{\hbar})e^{-i\frac{J}{4}\frac{\Delta
t}{\hbar}},$ (55) $\displaystyle U_{2,1}$ $\displaystyle\approx
i\sin(\frac{\tilde{J}\Delta t}{2\hbar})\exp(-i\frac{\Delta
Bt_{0}}{\hbar})e^{-i\frac{J}{4}\frac{\Delta t}{\hbar}}.$ (56)

Finally, using $J\Delta t/\hbar\approx\tilde{J}\Delta t/\hbar$ in the limit
$\Delta B\Delta t/\hbar\to 0$, we find $\hat{U}_{J}(t_{0}\to t_{0}+\Delta t)$
is a power-of-SWAP like gate:

$\hat{U}_{J}(t_{0}\to t_{0}+\Delta t)\approx\\\ \exp(i\frac{\tilde{J}\Delta
t}{4\hbar})\begin{bmatrix}1\\\ &0&e^{i\Delta Bt_{0}/\hbar}\\\ &e^{-i\Delta
Bt_{0}/\hbar}&0\\\ &&&1\end{bmatrix}^{\dfrac{\tilde{J}\Delta t}{\pi\hbar}},$
(57)

which is (up to a global phase) a rotation about the $\cos(\Delta
Bt_{0}/\hbar)X+\sin(\Delta Bt_{0}/\hbar)Y$ axis of the Bloch sphere of the
$\ket{\tilde{0}}\equiv\ket{01}$, $\ket{\tilde{1}}\equiv\ket{10}$ subspace
where $\left\\{\ket{\tilde{0}},\ket{\tilde{1}}\right\\}$ is the eigenbasis of
the $Z$ operator on the subspace.