# State Preparation in the Heisenberg Model through Adiabatic Spiraling

Anthony N. Ciavarella  0000-0003-3918-4110<EMAIL_ADDRESS>InQubator for
Quantum Simulation (IQuS), Department of Physics, University of Washington,
Seattle, Washington 98195-1550, USA  Stephan Caspar  0000-0002-3658-9158
<EMAIL_ADDRESS>InQubator for Quantum Simulation (IQuS), Department of Physics,
University of Washington, Seattle, Washington 98195-1550, USA  Marc Illa
0000-0003-3570-2849<EMAIL_ADDRESS>InQubator for Quantum Simulation (IQuS),
Department of Physics, University of Washington, Seattle, Washington
98195-1550, USA  Martin J. Savage  0000-0001-6502-7106<EMAIL_ADDRESS>InQubator
for Quantum Simulation (IQuS), Department of Physics, University of
Washington, Seattle, Washington 98195-1550, USA

###### Abstract

An adiabatic state preparation technique, called the adiabatic spiral, is
proposed for the Heisenberg model. This technique is suitable for
implementation on a number of quantum simulation platforms such as Rydberg
atoms, trapped ions, or superconducting qubits. Classical simulations of small
systems suggest that it can be successfully implemented in the near future. A
comparison to Trotterized time evolution is performed and it is shown that the
adiabatic spiral is able to outperform Trotterized adiabatics.

††preprint: IQuS@UW-21-033

##  I Introduction

Quantum simulations of lattice gauge theories, such as quantum chromodynamics
(QCD) and quantum electrodynamics (QED), are anticipated, in the future, to
enable reliable predictions for non-equilibrium dynamical processes, ranging
from fragmentation in high-energy hadronic collisions, through to transport in
extreme astrophysical environments. While a quantum advantage has yet to be
established for a scientific application, including quantum field theories,
there are substantial efforts underway to perform quantum simulations that can
be compared with experiment, or impact future experiments, and impressive
progress has been made toward these objectives in the last decade [1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44,
45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63,
64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82,
83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101,
102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116,
117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128]. Currently
available hardware and our present understanding of quantum algorithms has so
far limited quantum simulations of lattice gauge theories to one- and two-
dimensions with only a small number of lattice sites [40, 69, 65, 98, 102,
115, 118, 119]. Qualitative insights can be gained from simulations of spin
models that share one or more features of QCD, QED or low-energy effective
field theories (EFTs) relevant to nuclear and particle physics. These include
models that are in the same universality class as these theories, that can be
fruitfully digitized onto qubit registers or mapped to analog quantum
simulators, such as arrays of Rydberg atoms.

The Heisenberg model with arbitrary couplings is computationally universal in
the sense that all other lattice models can be simulated in arbitrary
dimensions, particle content and interactions by simulations of Heisenberg
models [129]. Therefore, detailed understandings of quantum simulations of the
Heisenberg model inform the simulations of quantum field theories describing
the forces of nature. Translating results obtained from lattice field theories
to predictions that can be compared with experiment requires that all relevant
physical length scales are much larger than the scale of discretization of
spacetime, and universality guarantees that low-energy continuum physics can
be reproduced from simulations that are tuned near a 2nd order critical point
[130, 131, 132]. As an example, it has been proposed that universality assures
that the continuum physics of the $1+1$d O(3) NL$\sigma$M, which has been
studied as a toy model of quantum chromodynamics (QCD) due to sharing a number
of qualitative features such as asymptotic freedom, dynamical transmutation,
the generation of a non-perturbative mass gap and non-trivial $\theta$ vacua,
can be recovered from simulations of an anti-ferromagnetic Heisenberg model
[133, 134, 135, 136, 137, 138, 139, 140]. Thus, quantum simulations of the
low-energy dynamics of the anti-ferromagnetic Heisenberg model, which requires
preparing a low-energy state and evolving it forward in time, are expected to
provide key insights into strategies for simulating QCD, including state
preparation.

To enable practical quantum simulations of physical systems, preparation of
states that have energies much less than the inverse lattice spacing is
required. Schemes for simulating the Heisenberg model’s time evolution have
been proposed using digital quantum simulation [141, 142], hybrid digital-
analog simulation [143], periodically driven trapped ions [144], global
microwave pulses on Rydberg atoms [145], nuclear spins [146, 147], and Floquet
engineering Ising interactions [148]. These proposals can be used to simulate
adiabatic switching to prepare low energy states, provided a path can be found
through parameter space where the energy gap does not close. In this work, we
show how adiabatic switching can be performed using Floquet engineering and
evaluate its potential for implementation on existing quantum hardware.

##  II Adiabatic Spirals: Spiraling Toward Ground States

Figure 1: Path on the unit sphere taken by $\vec{h}(t)$ (blue) and
$\vec{e}(t)$ (green) during the course of the adiabatic spiral. On the left is
the Schrödinger picture, where $\vec{e}$ is static and $\vec{h}$ is changed
adiabatically. On the right is the interaction picture, where $\vec{h}$ is
static up to diabatic effects. In this picture $\vec{e}$ is precessing around
$\vec{h}$ while the opening angle $\theta$ changes adiabatically, resulting in
the adiabatic spiral. Straight vectors indicate the direction of the fields at
the end of the spiral at time $T$.

We recently showed that the Ising model with large external fields in the
transverse and longitudinal directions can approximate the time evolution of
the Heisenberg model at discrete time intervals [148]. This generalizes
previous work showing the Ising model with a large transverse field
approximates the dynamics of the XY model [149, 150, 151, 152, 153], and is
related to techniques used to study pre-thermalization [154, 155, 156, 157].
Explicitly, if a quantum simulator evolves under the Hamiltonian

$\hat{H}^{\text{Ising}}=\sum_{i,j}J_{i,j}\hat{Z}_{i}\hat{Z}_{j}+\sum_{i}\frac{\Omega}{2}\left(\cos{\theta}\hat{Z}_{i}+\sin{\theta}\hat{X}_{i}\right)\
\ \ ,$ (1)

then the time evolution of

$\hat{H}^{\text{Heis.}}=\sum_{i,j}J_{i,j}\left[\cos^{2}{\theta}\hat{Z}_{i}\hat{Z}_{j}+\frac{\sin^{2}{\theta}}{2}\left(\hat{X}_{i}\hat{X}_{j}+\hat{Y}_{i}\hat{Y}_{j}\right)\right]$
(2)

will be approximated at times that are integer multiples of
$t=\frac{2\pi}{\Omega}$, up to a change of basis and corrections that are
$O\left(\frac{1}{\Omega^{2}}\right)$. If the time evolution was reproduced
exactly at these time intervals, it would guarantee that
$\hat{H}^{\text{Heis.}}$ and $\hat{H}^{\text{Ising}}$ share the same
eigenstates and that their energy levels agree up to integer multiples of
$\Omega$. This would suggest that by adiabatically varying $\theta$, it should
be possible to adiabatically prepare an eigenstate of the Heisenberg model
from an eigenstate of the Ising model with no external field. When viewed in
the interaction picture where the free part of the Hamiltonian is taken to be
the single spin driving terms
$H_{0}(t)=\Omega\sum_{i}\vec{h}(t)\cdot\vec{S}_{i}$, the local $\hat{Z}$
operator becomes

$\hat{Z}^{I}_{j}(t)=U_{0}^{\dagger}(t)\hat{Z}_{j}U_{0}(t)=2~{}\vec{e}(t)\cdot\vec{S}_{j}\
\ ,$ (3)

where $\vec{S}_{j}$ is a vector of Pauli matrices and $\vec{e}(t)$ is a unit
vector. For the Hamiltonian in Eq. (1), $\vec{e}(t)$ rotates in a circle about
a vector pointing in the $\cos\theta\hat{z}+\sin\theta\hat{x}$ direction. As
$\theta$ is adiabatically varied to prepare the ground state of a Heisenberg
model, $\vec{e}(t)$ will move in a spiral motion along the surface of the
sphere as shown in Fig. 1. For this reason we call this method of state
preparation adiabatic spiraling.

In practice, the initial eigenstate of the Ising model will often be
degenerate, and this degeneracy will need to be split for the adiabatic
approximation to be valid. This can be done by modifying the single-spin terms
in the Hamiltonian. Explicitly, at time intervals of $\frac{2\pi}{\Omega}$,
the time evolution generated by

$\hat{H}^{\text{Ising}}=\sum_{i,j}J_{i,j}\hat{Z}_{i}\hat{Z}_{j}+\sum_{j}\frac{\Omega}{2}\left(\cos{\theta}\hat{Z}_{j}+\sin{\theta}\hat{X}_{j}\right)+\frac{h_{P}(j)}{2}\hat{Z_{j}}\
\ ,$ (4)

can be approximated by the Floquet operator,

$\begin{split}\hat{U}_{F}=\hat{U}^{\dagger}_{B}\exp\\{&-i\frac{2\pi}{\Omega}\sum_{ij}J_{ij}\left(\cos^{2}{\theta}\hat{Z}_{i}\hat{Z}_{j}+\frac{\sin^{2}{\theta}}{2}\left(\hat{X}_{i}\hat{X}_{j}+\hat{Y}_{i}\hat{Y}_{j}\right)\right)\\\
&-i\frac{2\pi}{\Omega}\sum_{j}\frac{1}{2}\cos\theta\ h_{P}(j)\
\hat{Z}_{j}+O\left(\frac{1}{\Omega^{2}}\right)\Bigg{\\}}\hat{U}_{B}\ \ \
,\end{split}$ (5)

where $\hat{U}_{B}$ is a local change of basis given by
$\hat{U}_{B}=\prod_{j}e^{i\frac{\theta}{2}\hat{Y}_{j}}$. The additional
single-qubit term can be tuned to create an energy penalty that breaks the
degeneracy of the initial Hamiltonian which enables the application of the
adiabatic approximation. As an explicit demonstration, we consider the anti-
ferromagnetic Heisenberg model on a 1D chain, with a Hamiltonian of the form

$\hat{H}^{\text{Heis.}}=J\sum_{j}\left[\hat{X}_{j}\hat{X}_{j+1}+\hat{Y}_{j}\hat{Y}_{j+1}+\hat{Z}_{j}\hat{Z}_{j+1}\right]\
\ \ .$ (6)

The ground state of this system can be prepared adiabatically by starting in a
state with spins alternating up and down in the $\hat{z}$ direction (a Néel
state), e.g.,
$\ket{\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow...}$, and
applying a time-dependent Hamiltonian of the form

$\hat{H}(t)=\sum_{j}\left[J\hat{Z}_{j}\hat{Z}_{j+1}+\frac{\Omega}{2}\left(\frac{1}{\sqrt{3}}\hat{Z}_{j}+f(t)\hat{X}_{j}\right)+\frac{h_{P}(t)}{2}(-1)^{j}\hat{Z}_{j}\right]\
\ \ ,$ (7)

for a time $T$, where $h_{P}(0)>0$, $h_{P}(T)=0$, $f(0)=0$, and
$f(T)=\sqrt{\frac{2}{3}}$. Explicitly, if $\ket{Vac}$ is the ground state of
the anti-ferromagnetic Heisenberg model given in Eq. (6) and
$\ket{N\acute{e}el}$ is a Néel state, then

$\ket{Vac}=\mathcal{T}e^{-i\int_{0}^{T}dt\hat{H}(t)}\ket{N\acute{e}el}\ \ ,$
(8)

where $\hat{H}(t)$ is the time-dependent Hamiltonian defined in Eq. (7).

Typically, analog quantum simulators are initialized with all qubits in their
ground state, e.g., $|\downarrow\rangle^{\otimes n}$. However, to apply the
adiabatic spiral to the anti-ferromagnetic Heisenberg model, relevant for
simulations of the O(3) NL$\sigma$M, the initial state should be a Néel state,
which can be accomplished by applying a rotation on every other qubit.
Alternately, computations could be performed in a different basis. If
$\mathbf{X}$ is defined to be a product of Pauli X’s on every other site such
that $\ket{N\acute{e}el}=\mathbf{X}\ket{\downarrow}^{\otimes n}$ where
$\ket{\downarrow}^{\otimes n}$ is the state with all spins down, then Eq. (8)
can be written as

$\ket{Vac}=\mathcal{T}e^{-i\int_{0}^{T}dt\hat{H}(t)}\mathbf{X}\ket{\downarrow}^{\otimes
n}\ \ \ .$ (9)

Multiplying both sides of this equation by $\mathbf{X}$ yields

$\mathbf{X}\ket{Vac}=\mathbf{X}\mathcal{T}e^{-i\int_{0}^{T}dt\hat{H}(t)}\mathbf{X}\ket{\downarrow}^{\otimes
n}=\mathcal{T}e^{-i\int_{0}^{T}dt\tilde{H}(t)}\ket{\downarrow}^{\otimes n}\ \
\ ,$ (10)

where

$\tilde{H}(t)=\sum_{j}\left[-J\hat{Z}_{j}\hat{Z}_{j+1}+\frac{\Omega}{2}\left(\frac{(-1)^{j}}{\sqrt{3}}\hat{Z}_{j}+f(t)\hat{X}_{j}\right)+\frac{h_{P}(t)}{2}\hat{Z}_{j}\right]\
\ \ .$ (11)

If the quantum simulator can directly implement $\tilde{H}(t)$, then an
adiabatic spiral can be used to adiabatically prepare the Heisenberg ground
state from the $\ket{\downarrow}^{\otimes n}$ state up to a basis
transformation. However, on some analog quantum simulators such as Rydberg
atom systems, the sign of the two-spin interaction is fixed to be positive.
This does not present an issue in using an adiabatic spiral, as it can be
implemented with $-\tilde{H}(t)$. This is because a change of overall sign
does not change the eigenstates or presence of an energy gap between
eigenstates, indicating that the adiabatic approximation remains valid.

##  III A Numerical Example

To implement the adiabatic spiral in Eqs. (7) and (11), specific choices have
to be made for $h_{P}(t)$ and $f(t)$. As an example, we consider the
Heisenberg comb that has recently been shown to reproduce the O(3) NL$\sigma$M
in the continuum and infinite-volume limits. It has modest qubit requirements,
making it a good candidate for near term quantum simulations [136, 137]. The
Hamiltonian is given by

$\hat{H}=\sum_{x}\left[J\vec{S}_{x,1}\cdot\vec{S}_{x+1,1}+J_{p}\vec{S}_{x,1}\cdot\vec{S}_{x,2}\right]\
\ ,$ (12)

where $\vec{S}_{x,y}$ is a vector of Pauli matrices at position $(x,y)$ on the
lattice. An adiabatic spiral can be used to prepare the ground state of this
model from a Néel state with the Hamiltonian

$\hat{H}(t)=\frac{1}{4}\sum_{x}\left[J\hat{Z}_{x,1}\hat{Z}_{x+1,1}+J_{p}\hat{Z}_{x,1}\hat{Z}_{x,2}\right]+\sum_{x,y}\left[\frac{\Omega}{2}\left(\frac{1}{\sqrt{3}}\hat{Z}_{x,y}+f(t)\hat{X}_{x,y}\right)+\frac{h_{P}(t)}{2}(-1)^{x+y}\hat{Z}_{x,y}\right]\
\ \ .$ (13)

The simplest choices for $f(t)$ and $h_{P}(t)$ are linear functions of $t$.
Once the functional forms of $f(t)$ and $h_{P}(t)$ has been chosen,
implementing the adiabatic spiral further requires choices of $\Omega$, $T$,
and $h_{P}(0)$. To ensure the eigenstates of the Ising model are as close to
eigenstates of the Heisenberg model as possible, and the conditions of the
adiabatic theorem are satisfied, $\Omega$ and $T$ should be taken to be as
large as possible. As an example, Fig. 2 shows the energy of a state obtained
using the adiabatic spiral as a function of $\Omega$ for a Heisenberg comb of
length $4$ with $J=J_{P}=1$ and a switching time of $T=25$. In this
calculation, $h_{P}(t)=0$ for all times and $f(t)$ was taken to be a linear
function. At small values of $\Omega$, the eigenstates of the Ising model and
Heisenberg model are not close and the adiabatic spiral fails. At large
$\Omega$, the adiabatic spiral is able to prepare a state with an energy below
the energy of the first excited state. The reason the energy of the state in
Fig. 2 saturates above the ground-state energy is due to a combination of the
amount of time used in the switching and the initial degeneracy in the ground
state. With the ground state comprised of degenerate energy eigenstates, the
adiabatic spiral prepares a superposition of the ground state and excited
states.

Figure 2:  The energy of the final-state obtained after implementing an
adiabatic spiral as a function $\Omega$ for a comb of length 4 with
$J=J_{P}=1$ and a switching time of $T=25$, starting from a Néel state. The
spiral utilized the Hamiltonian given in Eq. (13), with $h_{P}(t)=0$ and
$f(t)$ a linear function. The black dashed line is the energy of the ground
state and the red dashed line is the energy of the first excited state.

The energy of the state prepared by the adiabatic spiral can be lowered by
taking a non-zero value of $h_{P}(0)$. Unlike with $\Omega$ and $T$, the
energy of the state prepared by an adiabatic spiral is not monotonic in
$h_{P}(0)$. If $h_{P}(0)$ is taken to be too large, the process of switching
off the initial energy penalty in a finite amount of time can break
adiabaticity, leading to a state with larger energy being prepared. An optimal
value of $h_{P}(0)$ can be found variationally. Fig. 3 shows the energy
obtained from the adiabatic spiral with $\Omega=8$ and $h_{P}(0)=0.18$. This
value for $h_{P}(0)=0.18$ was selected by minimizing the energy obtained by
performing the adiabatic spiral with $\Omega=8$ and $T=25$.

The adiabatic spiral’s performance can be improved further by optimizing the
path taken through parameter space. If $h_{p}(t)$ is taken to be a linear
function of time, and $f(t)$ taken to be

$f(t)=\sqrt{\frac{2}{3}}\left(\frac{t}{T}+\sum_{n<N}\beta_{n}\sin\left(n\pi\frac{t}{T}\right)\right)\
\ \ ,$ (14)

an optimal path through parameter space can be found by minimizing the energy
obtained as a function of the $\beta_{n}$s [158]. Note that if the maximum
driving field strength is limited on the analog simulator, this will constrain
the values that the $\beta_{n}$s can take. For long switching times, assuming
that the maximum value of the driving field is obtained at the end of the
spiral and truncating at $\beta_{1}$, it was found that the energy obtained by
the adiabatic spiral was monotonic as a function of $\beta_{1}$. This means
that in practice, the largest value of $\beta_{1}$ allowed by the constraint
should be used so that the maximum value of the driving field is obtained at
the end of the spiral, which is $\beta_{1}=\frac{1}{\pi}$. The energy of the
state obtained by performing the adiabatic spiral with this path is also shown
in Fig. 3. At long switching times, this path can offer an improved
performance over linear switching. It was also found that including more terms
in the expansion in Eq. (14) offered minimal improvements in the performance
of the adiabatic spiral.

Figure 3:  The dependence of the final state energy, after implementing an
adiabatic spiral, on the total switching time used for a comb of length 4 with
$J=J_{P}=1$. The dark green curve shows the energy obtained when $f(t)$ and
$h_{P}(t)$ are taken to be linear functions. The light green curve shows the
energy obtained when
$f(t)=\sqrt{\frac{2}{3}}\left(\frac{t}{T}+\frac{1}{\pi}\sin(\pi\frac{t}{T})\right)$.
The black dashed line is the energy of the ground state and the red dashed
line is the energy of the first excited state.

##  IV Potential Hardware Implementations

###  IV.1 Rydberg Atoms

Figure 4: An arrangement of Rydberg atoms that can be used to perform an
analog quantum simulation of the Heisenberg comb.

Arrays of Rydberg atoms are an experimental platform that could be used to
implement the adiabatic spiral. If the atoms in the Rydberg state interact
through the Van der Waals interaction, the Hamiltonian describing their
dynamics is

$\hat{H}^{\text{Ryd.}}(t)=\frac{\Omega(t)}{2}\sum_{i}\hat{X}_{i}-\Delta(t)\sum_{i}\hat{n}_{i}+\sum_{i<j}\frac{V_{0}}{\absolutevalue{\vec{x}_{i}-\vec{x}_{j}}^{6}}\hat{n}_{i}\hat{n}_{j}\
\ ,$ (15)

where $\hat{n}_{i}$ is the Rydberg state occupation at site $i$, $\vec{x}_{i}$
is the position of site $i$ and $\hat{X}_{i}$ couples the ground state of the
atom at site $i$ to its Rydberg state. This native implementation of Ising
interactions has enabled the use of Rydberg atoms to perform analog
simulations of the Ising model in 1D and 2D [159, 160, 161, 162, 163, 164,
165, 166, 167, 168]. Their native Ising interactions allows for the
implementation of adiabatic spirals.

Floquet engineering is not the only technique that can be used to simulate the
Heisenberg interaction on Rydberg atoms. In recent work, a method of
simulating Heisenberg interactions on Rydberg atoms making use of dipole-
dipole interactions and an external microwave field was proposed [145]. This
approach uses global microwave pulses to rotate the dipole-dipole interaction
to generate time evolution approximating an XXZ Heisenberg Hamiltonian,
analogous to how one could Trotterize the Heisenberg interaction on a digital
quantum computer. While this proposal makes use of dipole-dipole interactions,
the same approach could be used for Rydberg atom simulations that use the Van
der Waals interaction to couple atoms. In that setup, an effective
$\hat{Z}_{i}\hat{Z}_{j}$ is available. In principle, the ground state of a
Heisenberg model could be prepared on this hardware using the adiabatic spiral
or by using global pulses to perform a Trotterized adiabatic switching. The
choice of which method to use will depend on the specific analog simulator
being used. As a case study, we can consider hardware parameters used in a
recent experiment to study phases of the 2D Ising model [168]. This
experimental setup used a driving field with a max amplitude of
$\Omega_{max}=2\pi\times 4.3\text{MHz}$ and had a max coherence time of
$3\mu\text{s}$. With these hardware parameters, the preparation of the ground
state of a Heisenberg comb with the adiabatic spiral parameters from the
previous section can be performed by placing atoms in an array as shown in
Fig. 4 with a lattice spacing of $a\approx 10.5\mu\text{m}$. Note that the
Hamiltonian simulated will not quite be the same as the Hamiltonian from the
previous section due to the $\frac{1}{r^{6}}$ coupling between the atoms, but
the alternating vertical positions of atoms will ensure the next-to-nearest
neighbor interactions will be suppressed. This same arrangement of atoms could
be used to implemented Trotterized adiabatics. In this approach,
$\frac{\pi}{2}$ pulses are alternated with periods of the external field
turned off to construct a Trotterized approximation of time evolution
according to

$\hat{H}(t)=\sum_{i,j}J_{i,j}\left(\hat{Z}_{i}\hat{Z}_{j}+\frac{t}{T}\left(\hat{X}_{i}\hat{X}_{j}+\hat{Y}_{i}\hat{Y}_{j}\right)\right)+h_{P}\left(1-\frac{t}{T}\right)\sum_{i}(-1)^{i}\hat{Z}_{i}\
\ \ ,$ (16)

where $(-1)^{i}$ represents an alternating pattern of signs that breaks the
initial degeneracy of the Hamiltonian. The errors in this state preparation
method come from a finite switching time, standard Trotterization errors and
the amount of time required to implement the $\frac{\pi}{2}$ pulses. Explicit
pulse sequences can be found in the Appendices A, B and C.

Figure 5 shows the energy that can be obtained using the path optimized
adiabatic spiral and Trotterized adiabatics with different numbers of Trotter
steps as a function of total runtime on the analog simulator. At short times,
a small number of Trotter steps are able to outperform the spiral. However, if
the number of Trotter steps is fixed and the switching time is increased, the
Trotterization error will grow and the energy of the state obtained eventually
will increase. This can be mitigated by performing more Trotter steps, but as
the number of Trotter steps increases the contribution to the error from the
finite pulse time will accumulate. As shown in Fig. 5, this effectively puts a
limit on the total number of Trotter steps that can be performed before
adiabatics is no longer being effectively simulated. The adiabatic spiral does
not suffer from either of these issues. Increasing the amount of time used in
the adiabatic spiral only increases the overlap with the ground state. Given
these limitations, it appears that the adiabatic spiral is a better choice for
preparing the ground state of a Heisenberg model than Trotterized adiabatics
on a Rydberg system.

Figure 5:  The dependence of the final-state energy from an adiabatic spiral
and Trotterized adiabatics on the total switching time used for a comb of
length 4 with $J=J_{P}=1$. The black dashed line is the energy of the ground
state and the red dashed line is the energy of the first excited state. The
vertical blue line indicates the coherence time of $3\mu\text{s}$ in units of
$J^{-1}$.

###  IV.2 D-Wave

Another experimental platform that could potentially implement the adiabatic
spiral is D-Wave’s quantum annealer. The D-Wave quantum annealer implements
time evolution according to the Hamiltonian,

$\hat{H}^{\text{DW}}(t)=-\frac{\text{A}(s(t))}{2}\sum_{i}\hat{X}_{i}+\frac{\text{B}(s(t))}{2}\left(\sum_{i}h_{i}\hat{Z}_{i}+\sum_{i<j}J_{ij}\hat{Z}_{i}\hat{Z}_{j}\right)\
\ ,$ (17)

where $\text{A}(s)$ and $\text{B}(s)$ are fixed functions, but $s(t)$, $h_{i}$
and $J_{ij}$ can be programmed by the user, up to some constraints. Although
it is designed to solve optimization problems, D-Wave’s hardware has been used
to perform analog simulations of the Ising model [169, 170, 171, 172, 173,
174, 175, 176, 177, 178, 179]. In previous work, D-Wave’s quantum annealer has
also been used to simulate time evolution and perform state preparation for
lattice gauge theories, however these mappings to quantum simulation have more
in common with digital quantum simulation than the analog protocol we are
proposing [68, 102, 118]. To implement the adiabatic spiral on the D-Wave
annealer, a set of qubits with a comb pattern coupling must be selected. Given
the connectivity of the hardware, this is straightforward to do. $J_{ij}$ and
$h_{i}$ should be chosen so that the implemented Hamiltonian is

$\hat{H}^{\text{DW}}(t)=-\frac{\text{A}(s(t))}{2}\sum_{x,y}\hat{X}_{x,y}+\frac{\text{B}(s(t))}{2}\left(\sum_{x,y}h(-1)^{x+y}\hat{Z}_{x,y}+\sum_{x}J\hat{Z}_{x,1}\hat{Z}_{x+1,1}+J_{p}\hat{Z}_{x,1}\hat{Z}_{x,2}\right)\
\ \ .$ (18)

Figure 6: Annealing schedule on the D-Wave’s Advantage 6.1 to implement the
adiabatic spiral with $h=2$. $\text{A}(t)$ and $\text{B}(t)$ are given in
units of MHz and time is in units of $\mu$s.  Figure 7:  Energy of the state
produced on a simulation of D-Wave’s Advantage 6.1 hardware for different
values of $J$ and $h$. The gray line is the energy of the initial state. The
black line is the vacuum energy, and the red line is the energy of the first
excited state. Note that the energy shown here is measured in units of $J$.
Figure 8:  Energy of the state produced on a simulation of D-Wave’s Advantage
6.1 as a function of time for a comb of length 4 with $J=J_{P}=0.1$ and $h=2$.
The black dashed line is the energy of the ground state and the red dashed
line is the energy of the first excited state. Note that the energy is given
in units of $J$.

The spiral can be implemented by using D-Wave’s reverse annealing function to
begin with $\text{A}(s(0))=0$, and adiabatically evolve to $s^{*}$ such that
$\text{A}(s^{*})=\sqrt{2}\text{B}(s^{*})h$. As long as $J,J_{p}\ll h$, the
adiabatic spiral will be able to prepare an equal superposition of the ground
state and the first excited state of the Heisenberg comb. The pure ground
state will not be prepared because an energy penalty cannot be implemented to
break the degeneracy within the initial state. Note that to perform a
measurement on the D-Wave QPU, the pulse schedule must end with
$\text{A}(s=1)=0$. The process of turning off $\text{A}(s)$ will create errors
in the state preparation whose size are controlled by the slew rate of
$\text{A}(s)$ and the choice of $s^{*}$. One choice of time dependence of
$\text{A}(t)$ and $\text{B}(t)$ to implement the adiabatic spiral is shown in
Fig. 6. As an explicit example, we once again consider the Heisenberg comb of
length $4$ with $J=J_{p}$. The energy of states prepared with classical
simulations (Mathematica and python) of the D-Wave Advantage 6.1 hardware for
different choices of $J$ and $h$ are shown in Fig. 7. For small values of $J$
and $h$, simulations of the D-Wave hardware are able to produce states with a
lower energy than the initial state. However, the quantum device has
calibration errors on the order of $\delta J\approx 4*10^{-3}$, making it
unlikely that these parameters can be used in practice. For larger values of
$J$, the simulations cannot reach states with energy lower than the initial
state. As an example, the energy of states as a function of time for $J=0.1$
and $h=2$ found in simulations of Advantage are shown in Fig. 8. As seen,
Advantage is expected to be able to prepare a low energy state with respect to
the Heisenberg Hamiltonian part-way through a simulation. However, in the
process of turning off $\text{A}(s)$ as required to perform a measurement, the
time evolution is still approximately adiabatic, and the system ends up in a
final state with large overlap with the initial state. If the slew rate of
Advantage could be increased, or if measurements could be performed without
fully turning off $\text{A}(t)$, it would be possible to perform measurements
of the low-energy state prepared by the adiabatic spiral and potentially use
Advantage in a scalable manner for quantum simulation of the Heisenberg model.

##  V Discussion

In this work, a method of Floquet engineering the Ising model into the
Heisenberg model was extended to perform adiabatic state preparation of the
Heisenberg ground state. Due to the number of quantum simulation platforms
able to natively implement the Ising model, this technique is likely to have
experimental applicability. It was shown that it should be feasible to
implement this technique on Rydberg atoms, and that it can out-perform a
Trotterized approach to adiabatic state preparation. The feasibility of
implementing the adiabatic spiral on D-Wave quantum annealers was also
investigated, and it was found that the requirement of turning off the
$\hat{X}$ driving field at the end of the calculation limits its efficacy. The
ability of the adiabatic spiral to prepare low-energy states of the Heisenberg
model on these hardware platforms suggests that near-term quantum simulators
can be used to perform analog quantum simulations of the $O(3)$ NL$\sigma$M,
and other such theories of importance to nuclear and particle physics. While
the case of the fully symmetric XXX Heisenberg model was focused on in this
work, the adiabatic spiral can also be applied to the XXZ Heisenberg model as
well. As the approximation of the dynamics of the Heisenberg model by an Ising
model with strong drive fields is a special case of the more generic phenomena
of prethermalization [154, 155, 156, 157], similar ideas may find
applicability for analog quantum simulators with native couplings other than
Ising.

###### Acknowledgements.

The authors would like to acknowledge Pavel Lougovski, Peter Komar and Cedric
Lin from the Amazon Braket team for contributions to developing this
technique. The views expressed are those of the authors and do not reflect the
official policy or position of AWS. The authors would also like to acknowledge
many useful conversations with Hersh Singh. The material presented here was
funded in part by the DOE QuantISED program through the theory consortium
“Intersections of QIS and Theoretical Particle Physics” at Fermilab with
Fermilab Subcontract No. 666484, and in part by U.S. Department of Energy,
Office of Science, Office of Nuclear Physics, Inqubator for Quantum Simulation
(IQuS) under Award Number DOE (NP) Award DE-SC0020970.

##  Appendix A Trotterization with Ising Model Quantum Hardware

Quantum hardware with constant background Ising model interactions $J_{i,j}$
can be used to engineer other interactions, given the freedom to manipulate a
time-dependent global magnetic field $|\vec{\omega}(t)|\leq\Omega$,

$\hat{H}^{\text{Ising}}(t)=\sum_{i,j}J_{i,j}\hat{Z}_{i}\hat{Z}_{j}+\sum_{i}\vec{\omega}(t)\cdot\vec{S}_{i}+\sum_{i}\frac{h(t)}{2}(-1)^{i}\hat{Z}_{i}\
\ \ .$ (19)

Access to a weak staggered $\hat{Z}$-field $h(t)$, which breaks degeneracies
in the Ising model to facilitate adiabatic evolution, is assumed. Other
interactions, such as $\hat{X}\hat{X}$ and $\hat{Y}\hat{Y}$, can be generated
by using the global magnetic field to perform $\frac{\pi}{2}$ pulses to change
between bases. The desired interactions can be assembled from the following
set of global analog gates,

$\displaystyle R^{\pm}_{X}(\epsilon)$
$\displaystyle=\exp{-i\epsilon\sum_{i,j}J_{i,j}\hat{Z}_{i}\hat{Z}_{j}\mp\frac{i\pi}{4}\sum_{j}\hat{X}_{j}}\
\ \ ,$ $\displaystyle R^{\pm}_{Y}(\epsilon)$
$\displaystyle=\exp{-i\epsilon\sum_{i,j}J_{i,j}\hat{Z}_{i}\hat{Z}_{j}\mp\frac{i\pi}{4}\sum_{j}\hat{Y}_{j}}\
\ \ ,$ $\displaystyle R_{Z}(t,\kappa)$
$\displaystyle=\exp{-it\sum_{i,j}J_{i,j}\hat{Z}_{i}\hat{Z}_{j}-i\frac{\kappa}{2}\sum_{j}(-1)^{j}\hat{Z}_{j}}\
\ \ .$ (20)

Due to the maximum magnetic field strength, $\Omega$, the
$\frac{\pi}{2}$-pulses, $R^{\pm}_{X,Y}$, need a minimum device time
$\epsilon=\frac{\pi}{2\Omega}$ during which the Ising interaction cannot be
”turned off”. To determine the effect $J_{i,j}$ has on pulses, the deviation
from the exact pulse to leading order in $\epsilon$ is,

$\displaystyle\Upsilon(\epsilon)$
$\displaystyle=\exp{\frac{i\pi}{4}\sum_{j}\hat{Y}_{j}}R^{+}_{Y}(\epsilon)=\exp\left\\{-i\sum_{k=1}^{\infty}\upsilon_{k}\epsilon^{k}\right\\}=\exp\left\\{-i~{}\upsilon_{1}~{}\epsilon+\mathcal{O}(\epsilon^{2})\right\\}\
\ \ .$ (21)

The exponent is given as a Magnus expansion,

$\displaystyle i\partial_{\eta}\Upsilon(\epsilon,\eta)$
$\displaystyle=\epsilon~{}\hat{H}(\eta)~{}\Upsilon(\epsilon,\eta)=\exp{i\eta\frac{\pi}{4}\sum_{j}\hat{Y}_{j}}\left[\epsilon\sum_{i,j}J_{i,j}\hat{Z}_{i}\hat{Z}_{j}\right]\exp{-i\eta\frac{\pi}{4}\sum_{j}\hat{Y}_{j}}\Upsilon(\epsilon,\eta)\
\ \ ,$ (22)

where $\Upsilon(\epsilon,0)=1$ and $\Upsilon(\epsilon,1)=\Upsilon(\epsilon)$.
The leading-order coefficient in the expansion is given by,

$\displaystyle\upsilon_{1}(\eta)$
$\displaystyle=\int_{0}^{\eta}\hat{H}(\eta^{\prime})\,d\eta^{\prime}=\sum_{i,j}J_{i,j}\left[\eta\frac{\hat{Z}_{i}\hat{Z}_{j}+\hat{X}_{i}\hat{X}_{j}}{2}-\sin^{2}\left(\frac{\pi\eta}{2}\right)\frac{\hat{X}_{i}\hat{Z}_{j}+\hat{Z}_{i}\hat{X}_{j}}{\pi}+\frac{\sin(\pi\eta)}{\pi}\frac{\hat{Z}_{i}\hat{Z}_{j}-\hat{X}_{i}\hat{X}_{j}}{2}\right]\
\ \ ,$ (23)

such that the leading-order bias of the (${\pi\over 2}$-) pulse can be
determined,

$\displaystyle R^{+}_{Y}(\epsilon)$
$\displaystyle=\exp{-\frac{i\pi}{4}\sum_{j}\hat{Y}_{j}}\exp{-i\epsilon\sum_{i,j}J_{i,j}\left(\frac{\hat{Z}_{i}\hat{Z}_{j}+\hat{X}_{i}\hat{X}_{j}}{2}-\frac{\hat{X}_{i}\hat{Z}_{j}+\hat{Z}_{i}\hat{X}_{j}}{\pi}\right)+\mathcal{O}(\epsilon^{2})}\
\ \ .$ (24)

An important extension of the result in Eq. (24) is that a $\pi$-pulse can be
obtained by setting $\eta=2$ in Eq. (23),

$\displaystyle\left(R^{+}_{Y}\right)^{2}$
$\displaystyle=\exp{-\frac{i\pi}{2}\sum_{j}\hat{Y}_{j}}\exp{-i\epsilon\sum_{i,j}J_{i,j}\left(\hat{Z}_{i}\hat{Z}_{j}+\hat{X}_{i}\hat{X}_{j}\right)+\mathcal{O}(\epsilon^{2})}\
\ ,$ (25)

such that all the unwanted mixed terms, of the form
$\hat{X}_{i}\hat{Z}_{j}+\hat{Z}_{i}\hat{X}_{j}$, cancel at leading-order in
$\epsilon$. The remaining terms can be used in forming a target Hamiltonian.
For example, an anisotropic $XXZ$-Heisenberg unitary operator can be formed
from four $\frac{\pi}{2}$-pulses,

$\displaystyle(R^{+}_{X})^{2}(R^{+}_{Y})^{2}$
$\displaystyle=\exp{-\frac{i\pi}{2}\sum_{j}\hat{Z}_{j}}\exp{-i\epsilon\sum_{i,j}J_{i,j}\left(\hat{X}_{i}\hat{X}_{j}+\hat{Y}_{i}\hat{Y}_{j}+2\hat{Z}_{i}\hat{Z}_{j}\right)+\mathcal{O}(\epsilon^{2})}\
.$ (26)

The strength of the $\hat{Z}_{i}\hat{Z}_{j}$-term can be changed by adding a
$R_{Z}(t,0)$-gate before or after this sequence. Changing the
$\hat{X}_{i}\hat{X}_{j}$ or $\hat{Y}_{i}\hat{Y}_{j}$-term requires
interleaving $R_{Z}(t,0)$-gates between $\frac{\pi}{2}$-pulses. Using such
operations, an $SU(2)$-invariant $XXX$-Heisenberg model unitary operator can
be obtained at leading order through the sequence,

$\displaystyle(R^{+}_{X}R_{Z}(\epsilon,0)R^{+}_{X})(R^{+}_{Y}R_{Z}(\epsilon,0)R^{+}_{Y})$
$\displaystyle=\exp{-\frac{i\pi}{2}\sum_{j}\hat{Z}_{j}}\exp{-i2\epsilon\sum_{i,j}J_{i,j}\left(\hat{X}_{i}\hat{X}_{j}+\hat{Y}_{i}\hat{Y}_{j}+\hat{Z}_{i}\hat{Z}_{j}\right)+\mathcal{O}(\epsilon^{2})}\
\ .$ (27)

##  Appendix B Analog Trotter-Suzuki Formulas for Adiabatic State Preparation

The pulse sequences introduced in Appendix A can be used to adiabatically
prepare the ground state of the Heisenberg model by approximating the time
evolution generated by the Hamiltonian in Eq. (16). The evolution according to
this time dependent Hamiltonian can be approximated by a sequence of time
independent steps i.e.,

$\displaystyle\mathcal{T}\exp{-i\int_{0}^{T}dt\ \hat{H}(t)}$
$\displaystyle\approx\prod_{m=1}^{M}\exp\left\\{-i\Delta t\hat{H}(m\Delta
t)\right\\}\ \ .$ (28)

The individual terms in this product can be approximated with pulses from the
analog gate set (A). Explicitly we have,

$\displaystyle\exp\left\\{-i\Delta t\hat{H}(m\Delta t)\right\\}$
$\displaystyle=\exp{-i\Delta
t\left(\sum_{i,j}J_{i,j}\left(\frac{m}{M}\left(\hat{X}_{i}\hat{X}_{j}+\hat{Y}_{i}\hat{Y}_{j}\right)+\hat{Z}_{i}\hat{Z}_{j}\right)+h_{P}\left(1-\frac{m}{M}\right)\sum_{i}(-1)^{i}\hat{Z}_{i}\right)}$
$\displaystyle=R_{Z}\left(\Delta t,h_{P}\left(1-\frac{m}{M}\right)\Delta
t\right)\ R_{X}^{-}(\epsilon)\ R_{Z}\left(\frac{m}{M}\Delta t,0\right)\
R_{X}^{+}(\epsilon)\ R_{Y}^{-}(\epsilon)\ R_{Z}\left(\frac{m}{M}\Delta
t,0\right)\ R_{Y}^{+}(\epsilon)+\mathcal{O}(\Delta t^{2},\epsilon)\ \ .$ (29)

By starting in the Néel state, $\ket{N\acute{e}el}$, and applying this
Trotterized evolution operator, a state close to the ground state of the XXX
Heisenberg Hamiltonian (12) will be prepared.

##  Appendix C Minimal Time-Step Second Order Trotter-Suzuki Formulas for
Adiabatic State Preparation

The accuracy of the Trotterized approximation to the time-evolution operator
can be improved by using shorter Trotter steps. However, for Trotter steps
comparable to the pulse length, $\Delta t\approx\epsilon$, it becomes
important to compensate for the ${\cal O}(\epsilon)$-contributions to the
error. As was noted in Appendix A, $\pi$ pulses in this analog gate set have
$\mathcal{O}(\epsilon)$ contributions proportional to terms present in the
Heisenberg Hamiltonian, and can be used to approximate the $XXX$-Heisenberg
model time evolution with errors that are $\mathcal{O}(\epsilon^{2})$ as in
Eq. (27).

It is useful to consider Trotter steps that depend upon the location in the
time sequence. Starting with Eq. (27), a sequence of time intervals that
eliminates leading-order corrections, consistent with the total number of
steps $M$ in Eq. (28), is

$\displaystyle t_{m}=\frac{2\pi}{\Omega}\sqrt{mM}\ \ .$ (30)

The explicit gate sequence that cancels the leading order errors is thus

$\displaystyle\mathcal{T}\exp{-i\int_{0}^{T}dt\ \hat{H}(t)}$
$\displaystyle\approx\prod_{m=1}^{M}U_{2}^{\epsilon}(\delta_{m},\tau_{m})\ \ \
,$ $\displaystyle U_{2}^{\epsilon}(\delta_{m},\tau_{m})$
$\displaystyle=R_{Z}\left(\frac{\delta_{m}-\epsilon}{2},\frac{\kappa_{m}}{2}\right)R^{-}_{Y}R^{+}_{X}R_{Z}\left(\epsilon,0\right)R^{+}_{X}R^{+}_{Y}R_{Z}\left(\frac{\delta_{m}-\epsilon}{2},\frac{\kappa_{m}}{2}\right)$
$\displaystyle=\exp{-\frac{i\pi}{2}\textstyle{\sum_{i}}\hat{Z}_{i}}\exp\left\\{-i\delta_{m}\hat{H}(\tau_{m})+\mathcal{O}(\delta_{m}^{2})\right\\}\
\ \ ,$ (31)

where

$\displaystyle\tau_{m}=\frac{t_{m}+t_{m-1}}{2}\
,\qquad\delta_{m}=t_{m}-t_{m-1}\
,\qquad\kappa_{m}=\delta_{m}h_{P}(1-\tau_{m}/T)=h_{P}(\delta_{m}-2\epsilon)\
.$ (32)

As the device time required for step $m$ is given by
$\theta_{m}=\delta_{m}+4\epsilon=\delta_{m}(1+2\tau_{m}/T)$, the total
required device time to perform the (ideal) Trotterized evolution from the
Néel state to the approximate ground state of the Heisenberg model is
$\Theta=\sum_{m=1}^{M}\theta_{m}=T+4\epsilon M=2T$.

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