State Preparation in the Heisenberg Model through Adiabatic Spiraling

2025-04-15 1 0 1.33MB 22 页 10玖币

State Preparation in the Heisenberg Model through

Adiabatic Spiraling

Anthony N. Ciavarella , Stephan Caspar , Marc Illa , and Martin J. Savage

InQubator for Quantum Simulation (IQuS), Department of Physics, University of Washington, Seattle, Washington 98195-1550,

USA

June 5, 2023

An adiabatic state preparation technique, called the adiabatic spiral, is proposed for the Heisen-

berg 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 compari-

son to Trotterized time evolution is performed and it is shown that the adiabatic spiral is able to

outperform Trotterized adiabatics.

1 Introduction

Quantum simulations of lattice gauge theories, such as quantum chromodynamics (QCD) and quantum electro-

dynamics (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 as-

trophysical 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 to-

ward these objectives in the last decade. This includes the development of techniques to simulate abelian gauge

theories [1–56], non-abelian gauge theories [57–97], fermionic ﬁeld theories [98–101], and scalar ﬁeld theo-

ries [102–108]. There has also been the development of techniques to extract observables of interest to nuclear

physics [109–115], scattering processes in high energy physics [116–123] and methods to mitigate errors on

noisy quantum hardware [124–129]. 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 [66,71,73,83,85,91–93]. Qualitative insights can be gained from simula-

tions 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 [130]. 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

Anthony N. Ciavarella : aciavare@uw.edu

Stephan Caspar : caspar@uw.edu

Marc Illa : marcilla@uw.edu

Martin J. Savage : mjs5@uw.edu

Accepted in Quantum 2023-03-26, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.04965v7 [quant-ph] 1 Jun 2023

low-energy continuum physics can be reproduced from simulations that are tuned near a 2nd order critical

point [131–133]. As an example, it has been proposed that universality assures that the continuum physics of

the

1 + 1

d O(3) NLσM, 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 θvacua, can be recovered from simulations of an anti-ferromagnetic

Heisenberg model [134–141]. 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. One proposal for preparing low energy states in both digital

and analog quantum simulation is adiabatic switching. This works by beginning in the ground state of a known

Hamiltonian and slowly varying the Hamiltonian through a path in parameter space where the energy gap

does not close. Implementation of adiabatic switching in a quantum simulation requires the ability to prepare

the eigenstate of the initial Hamiltonian and simulate time evolution. Schemes for simulating the Heisenberg

model’s time evolution have been proposed using digital quantum simulation [142,143], hybrid digital-analog

simulation [144], periodically driven trapped ions [145], global microwave pulses on Rydberg atoms [146],

nuclear spins [147,148], and adding strong single qubit terms to systems described by Ising interactions [149].

In this work, we present an argument that the eigenstates of the Heisenberg model are approximate eigen-

states of the Ising model with a strong external ﬁeld pointed in an appropriate direction. This is used to develop

an analogue quantum simulation technique, called the adiabatic spiral, to adiabatically prepare the ground state

of the Heisenberg model by adiabatically varying the direction of the external ﬁeld in the Ising model. The fea-

sibility of implementing the adiabatic spiral on Rydberg atoms and D-Wave’s quantum annealer is investigated.

It is found that current Rydberg atom experiments have sufﬁcient coherence time and drive ﬁeld strengths to

implement the adiabatic spiral, while the D-Wave quantum annealer suffers from some limitations.

2 Adiabatic Spirals: Spiraling Toward Ground States

(a) Laboratory (b) Interaction Picture

Figure 1: Paths on the unit sphere taken by (a)

⃗

(

)(the time-dependent drive ﬁeld deﬁned in Eq. (3)) and (b)

⃗e

(

)(deﬁning the time dependence of the

operator in the interaction picture) during the course of the adiabatic

spiral.

(

)is taken to be

(

) =

arccos 1

√3t

and

varies from

= 0 to

. During the evolution,

⃗e

(

)is

precessing around

⃗

(

)while the opening angle

changes adiabatically, resulting in a spiral path on the unit sphere

in the interaction picture. Vectors from the origin indicate the direction at the end of the spiral evolution.

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 [149]. This generalizes

previous work showing the Ising model with a large transverse ﬁeld approximates the dynamics of the XY

model [150–154], and is related to techniques used to study pre-thermalization [155–158]. Explicitly, if a

Accepted in Quantum 2023-03-26, click title to verify. Published under CC-BY 4.0. 2

quantum simulator evolves under the Hamiltonian

HIsing =X

i,j

Ji,j ˆ

Ziˆ

Zj+X

Ω

2cos θˆ

Zi+ sin θˆ

Xi,(1)

then the time evolution of

HHeis. =X

i,j

Ji,j cos2θˆ

Ziˆ

Zj+sin2θ

2ˆ

Xiˆ

Xj+ˆ

Yiˆ

Yj(2)

will be approximated at times that are integer multiples of t

2π

Ω, up to a change of basis and corrections that

are O1

Ω. In this work, ˆ

X,ˆ

Yand ˆ

Zrefer to the respective Pauli operators.

We now observe that if the time evolution was reproduced exactly at these time intervals, it would guarantee

that ˆ

HHeis. and ˆ

HIsing share the same eigenstates and that their energy levels agree up to integer multiples of

Ω

. This would suggest that by beginning with θ

= 0

and adiabatically increasing θ, it should be possible to

prepare an eigenstate of the Heisenberg model from an eigenstate of the Pauli ˆ

Zoperators. When viewed in the

interaction picture where the free part of the Hamiltonian is taken to be the single spin driving terms

H0(t) = Ω

⃗

h(t)·⃗σi,(3)

the local ˆ

Zoperator becomes ˆ

j(t) = U†

0(t)ˆ

ZjU0(t) = ⃗e(t)·⃗σj,(4)

where ⃗σjis a vector of Pauli matrices and ⃗e

(

)

is a unit vector. For the Hamiltonian in Eq. (1), ⃗e

(

)

rotates

in a circle about a vector pointing in the cos θˆz

sin θˆxdirection. As θis adiabatically varied to prepare the

ground state of a Heisenberg model, ⃗e

(

)

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. It is important to note

that the implementation of the adiabatic spiral does not require the switching time to be an integer multiple of

2π

Ω. Studying the evolution of the Ising model at periodic times was only necessary to argue that the eigenstates

of the Heisenberg model are also eigenstates of the Ising model up to O1

Ωcorrections. Typically in quantum

simulation, adiabatic switching is done between ground states of gapped Hamiltonians. In contrast, the adiabatic

spiral can be understood as performing adiabatic switching in the middle of the spectrum of the Ising model to

prepare eigenstates of the Heisenberg model.

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 times that are integer multiples of t=2π

Ω, the time evolution generated by

HIsing =X

i,j

Ji,j ˆ

Ziˆ

Zj+X

Ω

2cos θˆ

Zj+ sin θˆ

Xj+hP(j)

2ˆ

Zj,(5)

can be approximated by the Floquet operator,

UF=ˆ

U†

Bexp (−i2π

ΩX

Jij cos2θˆ

Ziˆ

Zj+sin2θ

2ˆ

Xiˆ

Xj+ˆ

Yiˆ

Yj

−i2π

ΩX

2cos θ hP(j)ˆ

Zj+O1

Ω2)ˆ

UB,

(6)

where ˆ

UBis a local change of basis given by ˆ

Qjeiθ

2ˆ

Yj. 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. With these additional single-qubit terms present, the same arguments at large

Ω

can be used to justify the applicability of the adiabatic spiral.

Accepted in Quantum 2023-03-26, click title to verify. Published under CC-BY 4.0. 3

As an explicit demonstration, we consider the preparation of the ground state of the anti-ferromagnetic

Heisenberg model on a 1D chain, with a Hamiltonian of the form

HHeis. =JX

jhˆ

Xjˆ

Xj+1 +ˆ

Yjˆ

Yj+1 +ˆ

Zjˆ

Zj+1i.(7)

Preparing the ground state of this system with the adiabatic spiral will require beginning in an eigenstate of the

Ising model that is adiabatically connected to the ground state of the Heisenberg model. Equation 2reproduces

Eq. 7when θ

arccos 1

√3and would suggest that the ground state of Eq. 7is connected to the ground state

of Eq. 2at other values of θ. The ground state of the Heisenberg Hamiltonian with θ

= 0

is a state with spins

alternating up and down in the ˆzdirection (a Néel state), e.g., |↑↓↑↓↑↓ ...⟩. This ground state is degenerate and

the degeneracy can be split by adding a single qubit term to the Hamiltonian with alternating signs. Therefore,

the ground state of the full Heisenberg model can be prepared by beginning in a Néel state and applying a

time-dependent Hamiltonian of the form

H(t) = X

jJˆ

Zjˆ

Zj+1 +Ω

21

√3

Zj+f(t)ˆ

Xj+hP(t)

2(−1)jˆ

Zj,(8)

for a time T, where hP

(0)

,hP

(

) = 0

(0) = 0

, and f

(

) =

3. Explicitly, if |V ac⟩is the ground state

of the anti-ferromagnetic Heisenberg model given in Eq. (7) and |N´eel⟩is a Néel state, then

|V ac⟩=Te−iRT

0dt ˆ

H(t)|N´eel⟩,(9)

up to O1

Ωand ﬁnite time corrections where ˆ

H(t)is the time-dependent Hamiltonian deﬁned in Eq. (8).

Typically, analog quantum simulators are initialized with all qubits in their ground state, e.g., | ↓⟩⊗n. How-

ever, to apply the adiabatic spiral to the anti-ferromagnetic Heisenberg model, relevant for simulations of the

O(3) NLσM, 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

is deﬁned to be a product

of Pauli X’s on every other site such that |N´eel⟩

|↓⟩⊗nwhere |↓⟩⊗nis the state with all spins down, then

Eq. (9) can be written as

|V ac⟩=Te−iRT

0dt ˆ

H(t)X|↓⟩⊗n.(10)

Multiplying both sides of this equation by Xyields

X|V ac⟩=XTe−iRT

0dt ˆ

H(t)X|↓⟩⊗n=Te−iRT

0dt ˜

H(t)|↓⟩⊗n,(11)

where

H(t) = X

j−Jˆ

Zjˆ

Zj+1 +Ω

2(−1)j

√3

Zj+f(t)ˆ

Xj+hP(t)

2ˆ

Zj.(12)

If the quantum simulator can directly implement ˜

(

)

, then an adiabatic spiral can be used to adiabatically

prepare the Heisenberg ground state from the |↓⟩⊗nstate 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 −˜

(

)

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.

3 A Numerical Example

To implement the adiabatic spiral in Eqs. (8) and (12), speciﬁc choices have to be made for hP

(

)

and f

(

)

. As

an example, we consider the Heisenberg comb that has recently been shown to reproduce the O(3) NLσM in

Accepted in Quantum 2023-03-26, click title to verify. Published under CC-BY 4.0. 4

the continuum and inﬁnite-volume limits. It has modest qubit requirements, making it a good candidate for near

term quantum simulations [137,138]. The Hamiltonian is given by

H=X

xhJ⃗

Sx,1·⃗

Sx+1,1+Jp⃗

Sx,1·⃗

Sx,2i,(13)

where ⃗

Sx,y

2⃗σx,y is the vector of Pauli matrices divided by

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

H(t) = 1

xhJˆ

Zx,1ˆ

Zx+1,1+Jpˆ

Zx,1ˆ

Zx,2i+X

x,y Ω

21

√3

Zx,y +f(t)ˆ

Xx,y+hP(t)

2(−1)x+yˆ

Zx,y.

(14)

The simplest choices for f

(

)

and hP

(

)

are linear functions of t. Once the functional forms of f

(

)

and hP

(

)

have been chosen, implementing the adiabatic spiral further requires choices of

Ω

,T, and hP

(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,

Ω

and Tshould 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

Ω

for a Heisenberg comb of length

with J

= 1

and a switching time of T

= 25

. In this calculation, hP

(

) = 0

for all times and f

(

)

was

taken to be a linear function. Evolution under ˆ

(

)

was evaluated numerically by computing QN

n=1 e−iT

Nˆ

H(n

NT)

and increasing Nuntil convergence. At small values of

Ω

, the eigenstates of the Ising model and Heisenberg

model are not close and the adiabatic spiral fails. At large

Ω

, 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. 2saturates 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.

0 2 4 6 8 10 12 14

-3.0

-2.5

-2.0

Energy

Figure 2: The energy of the ﬁnal-state obtained after implementing an adiabatic spiral as a function Ωfor a comb

of length 4 with

= 1 and a switching time of

= 25, starting from a Néel state. The spiral utilized the

Hamiltonian given in Eq. (14), with

(

) = 0 and

(

)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 hP

(0)

Unlike with

Ω

and T, the energy of the state prepared by an adiabatic spiral is not monotonic in hP

(0)

. If hP

(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 hP

(0)

can be found

variationally. Fig. 3shows the energy obtained from the adiabatic spiral with

Ω=8

and hP

(0) = 0

. This

value for hP

(0) = 0

was selected by minimizing the energy obtained by performing the adiabatic spiral with

Ω=8and T= 25.

The adiabatic spiral’s performance can be improved further by optimizing the path taken through parameter

Accepted in Quantum 2023-03-26, click title to verify. Published under CC-BY 4.0. 5

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StatePreparationintheHeisenbergModelthroughAdiabaticSpiralingAnthonyN.Ciavarella,StephanCaspar,MarcIlla,andMartinJ.SavageInQubatorforQuantumSimulation(IQuS),DepartmentofPhysics,UniversityofWashington,Seattle,Washington98195-1550,USAJune5,2023Anadiabaticstatepreparationtechnique,calledtheadiabaticspi...

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State Preparation in the Heisenberg Model through Adiabatic Spiraling

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