Time Evolution of Uniform Sequential Circuits Nikita Astrakhantsev1Sheng-Hsuan Lin2Frank Pollmann2 3and Adam Smith4 5 1Department of Physics University of Zurich Winterthurerstrasse 190 CH-8057 Z urich Switzerland

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Time Evolution of Uniform Sequential Circuits

Nikita Astrakhantsev,1, ∗Sheng-Hsuan Lin,2Frank Pollmann,2, 3 and Adam Smith4, 5

1Department of Physics, University of Zurich, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland

2Technical University of Munich, TUM School of Natural Sciences, Physics Department, 85748 Garching, Germany

3Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 M¨unchen, Germany

4School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK

5Centre for the Mathematics and Theoretical Physics of Quantum

Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK

Simulating time evolution of generic quantum many-body systems using classical numerical ap-

proaches has an exponentially growing cost either with evolution time or with the system size. In

this work, we present a polynomially scaling hybrid quantum-classical algorithm for time evolving a

one-dimensional uniform system in the thermodynamic limit. This algorithm uses a layered uniform

sequential quantum circuit as a variational ansatz to represent inﬁnite translation-invariant quan-

tum states. We show numerically that this ansatz requires a number of parameters polynomial in

the simulation time for a given accuracy. Furthermore, this favourable scaling of the ansatz is main-

tained during our variational evolution algorithm. All steps of the hybrid optimization are designed

with near-term digital quantum computers in mind. After benchmarking the evolution algorithm

on a classical computer, we demonstrate the measurement of observables of this uniform state using

a ﬁnite number of qubits on a cloud-based quantum processing unit. With more eﬃcient tensor

contraction schemes, this algorithm may also oﬀer improvements as a classical numerical algorithm.

I. INTRODUCTION

Performing time evolution of quantum states far-from-

equilibrium represents a challenging problem for the con-

temporary study of quantum matter. Beyond rare ana-

lytically tractable settings [1–3], exact numerical meth-

ods scale with the Hilbert space, whose dimension scales

exponentially with the number of degrees of freedom.

Approximate methods based on tensor networks have re-

leased this constraint with the cost of exponential scaling

with bipartite entanglement entropy [4,5]. While this

oﬀers dramatic advances for area-law entangled ground

states [6], the simulation of non-equilibrium states is

generically still limited to short time due to the fast

entanglement growth [7], although in certain cases non-

equilibrium phenomena are accessible even at long times

using additional approximation [8–10].

Recent developments of programmable quantum com-

puters and simulators allow for large-scale studies of

quantum many-body systems [11,12]. The simulation of

non-equilibrium dynamics is one of the tasks where quan-

tum advantage is anticipated in the near term, as its com-

plexity scales linearly in the system size and time [13,14]

on quantum devices. In case of a ﬁnite system, several al-

gorithms for current Noisy Intermediate Scale Quantum

(NISQ) devices were developed to simulate quantum dy-

namics on a ﬁnite system [15–20].

Simulation of dynamics for formally inﬁnite transla-

tionally invariant systems facilitates understanding of

physics in the thermodynamic limit. However, the scal-

ing of complexity for quantum algorithms in this limit is

subtler than in the ﬁnite system case [13,14]. Recent

∗nikita.astrakhantsev@physik.uzh.ch

works have shown that is possible to simulate inﬁnite sys-

tems with a ﬁnite number of qubits [21,22]. It is now

crucial to address the scalability and stability of quan-

tum algorithms working in the thermodynamic limit. In

this work, we present a hybrid quantum-classical algo-

rithm for time-evolving translation invariant systems in

one dimension and demonstrate both the expressibility

and scalability of our algorithm.

We consider layered uniform sequential circuits (l-

USC) ansatz, which is a generalization to the single-

layer USC ansatz introduced in Ref. [21]. The ansatz

l-USC forms a subclass of dense USC (d-USC), which are

equivalent to matrix-product states [23,24]. Moreover,

we propose a gradient-based algorithm for time evolving

quantum states within the manifold spanned by l-USC.

This includes a routine for computing the transfer matrix

and environments of the uniform states that does not re-

quire tomography or post-selection that would lead to an

exponential scaling.

To benchmark the proposed algorithm, we simulate it

on a classical computer. We show that the number of

variational parameters required to accurately time-evolve

a quantum state for the time twith the l-USC ansatz,

scales only polynomially in t. Lastly, having obtained

the time-evolved l-USC state representation on a classical

computer, we compute physical observables on a cloud-

based quantum processing unit (QPU) and demonstrate

agreement with quasi-exact results obtained with inﬁnite

time-evolution block-decimation (iTEBD) algorithm at a

large bond dimension [25].

This article is organized as follows. In Section II, we in-

troduce the layered uniform sequential circuit ansatz and

the gradient-based variational time-evolution algorithm.

In Section III we present the simulation results and an-

alyze the eﬀect of the layered decomposition on the ac-

curacy of the time-evolved quantum state representation

arXiv:2210.03751v4 [quant-ph] 21 Aug 2023

and ﬁxed points of the transfer matrix. In addition, we

show physical observables obtained from the classically

optimized circuit, measured on real quantum hardware.

In Section IV we discuss the obtained data and outline

the prospects for future work.

II. METHODOLOGY

In this section, we introduce and provide motivation

for the l-USC ansatz, which is a subclass of d-USC where

the dense unitary is replaced by the layered decompo-

sition. We present the necessary entities, e. g., transfer

matrices and the environments, to measure the physical

observables with l-USC. We then turn to the algorithm

for time-evolving the l-USC and the routines to perform

the variational time evolution.

A. The Layered Uniform Sequential Circuit Ansatz

The main motivation for our ansatz stems from the

great success of classical simulation for quantum sys-

tems with tensor network methods [26], especially with

matrix-product states (MPS) applied to one-dimensional

systems [27]. The classical simulation methods are so

eﬃcient that it is argued that there might be no expo-

nential quantum advantage with quantum algorithms for

ground state problems in quantum chemistry [28]. In

contrast, the fast growth of entanglement in quantum

non-equilibrium dynamics makes classical tensor network

methods ineﬃcient due to the exponentially growing ten-

sor size with respect to the evolution time. However, for

ﬁnite systems it has been shown that these tensors have

a simple structure that can be eﬃciently represented as

quantum-circuit ans¨atze [16,29]. In this work, we pro-

pose the l-USC for translationally invariant inﬁnite sys-

tems.

An MPS can be equivalently represented as a sequen-

tial quantum circuit [23,24], which we call a d-USC,

shown in Fig. 1(a) for an inﬁnite chain. The d-USC de-

ﬁne the wave functions

|ψR⟩=

+∞

i=−∞

R(θ)|0⟩,|ψL⟩=−∞

i=+∞

L(θ′)|0⟩,(1)

where ˆ

R(θ) and ˆ

L(θ′) are i–independent unitaries act-

ing on Nqconsecutive qubits i, i +1, . . . , i+Nq−1. The

‘R’ index denotes the right representation and similarly

the left representation ‘L’ is deﬁned by a diﬀerent uni-

tary ˆ

UL(θ′) acting in the opposite order. We show in

Appendix Athat the d-USC ansatz in left and right rep-

resentations over Nqqubits are MPS in left and right

isometric forms with the bond dimension χ= 2Nq−1re-

spectively.

The l-USC ansatz is deﬁned as a speciﬁc form of

Eq. (1), where each unitary ˆ

UR/Lis parameterized by a

sequential circuit of MUlayers, as shown in Fig. 1(b).

FIG. 1. (a) Circuit representing ⟨ψR(θ)|ˆ

O|ψL(θ′)⟩with

|ψL(θ′)⟩and |ψR(θ)⟩being the same state in left and right

representations. The shaded region singles out the repeated

circuit element, i.e., the transfer matrix. (b) Decomposition

of a state-unitary into MUlayers of sequential 2-qubit gates.

T{AB}

{ab}(θ,θ′) between the left and right

representations |ψL(θ′)⟩and |ψR(θ)⟩.(d) Circuit represen-

tation of ⟨l, 0|ˆ

U†

Rˆ

Oˆ

UL|0, r⟩on ﬁnite number of qubits (e) De-

composition of an environment-unitary into MElayers of se-

quential 2-qubit gates.

Each layer consists of a consecutive application of 2-

qubit gates between neighboring qubits in the direction

shown in Fig. 1(a-b). While for any d-USC state in

the right representation, there exists an exact d-USC of

the same size in the left representation, this does not al-

ways hold for l-USC with the same Nqand MUunless

the state is inversion symmetric. We note that l-USC

ansatz belongs to the broad class of quantum circuit ten-

sor network ans¨atze [29] [30], where the dense unitaries

in the isometric tensor networks are replaced by various

kinds of local circuits, e. g., brick-wall circuits or sequen-

tial circuits. The d-USC and l-USC wave functions

are both universal: if one allows arbitrary Nq, MU, all

translationally-invariant quantum many-body states can

be approximated to arbitrary accuracy in either d-USC

or l-USC form. The required Nq, MUindicates the com-

plexity of the quantum many-body state. As an example,

the Greenberger–Horne–Zeilinger (GHZ) state [31] can

be represented exactly with Nq= 2, MU= 1. Rigorous

studies of the scaling properties of the quantum circuit

ansatz could give us more insight into the properties of

quantum states, for example, the recent work for ground

states [32]. In this work, we focus on studying the ex-

pressivity of the ansatz applied to time-evolution with

both a free-fermion and a non-integrable Hamiltonian.

The l-USC with local circuits acting on Nqqubits de-

ﬁnes a subclass of states within the manifold of d-USC, or

equivalently uniform MPS of bond dimension χ= 2Nq−1.

As a generic 2-qubit gate, up to a global phase, requires

15 parameters [33], the l-USC ansatz is parametrized by

at most 15(Nq−1)MUoptimization parameters [34], as

compared to 22Nq+1 parameters necessary for the dense

parametrization in the d-USC ansatz. In the previous

works, it has been shown that similar ans¨atze on a ﬁ-

nite system are polynomially more eﬃcient in represent-

ing ground states [29] and exponentially more eﬃcient in

representing time-evolved states [16]. Previous works

studying the dynamics of inﬁnite systems have been fo-

cused on the speciﬁc case Nq= 2, MU= 1. The question

remains on whether there is also an exponential advan-

tage in the thermodynamics limit in expressing quantum

states produced under non-equilibrium dynamics. Later

in the Section III, we will demonstrate that l-USC forms

a physically relevant subset of the d-USC states with the

corresponding bond dimension, and allows for eﬃcient

time-evolution of quantum states. We begin here with

the description of the tools to acquire physical observ-

able from the l-USC state representation.

a. Transfer matrix. Computation of physical ob-

servables and other operations of an inﬁnite system can

be performed on a ﬁnite number of qubits using the trans-

fer matrix and its dominant eigenvectors, known as en-

vironments in the context of tensor networks. Utilizing

the left and right representation of l-USC, we always con-

sider the (mixed) transfer matrix deﬁned between the

states in left representation, |ψL⟩, and in right represen-

tation, |ψR⟩, as shown as the shaded area in Fig. 1(a).

In Fig. 1(c), we explicitly write down the transfer matrix

T{AB}

{ab}(θ,θ′), with {AB}forming a united out-index and

{ab}forming a united in-index. The arrow directions in-

dicate the ﬂow of time of the quantum circuit execution.

With this construction, the transfer matrix is a linear

operator T:Vab →VAB mapping a pure state in Hilbert

space Vab to a pure state in Hilbert space VAB. The

linear map is realized by a combination of unitary oper-

ators with the post-selection on one qubit, as shown in

Fig 1(c). The transfer matrix is therefore generally non-

Hermitian and non-unitary. In Appendix B 4, we show

that the post-selection probability is close to unity for

cases considered in this work. This formalism comes from

the construction of the transfer matrix using simultane-

ously left and right representations. This is diﬀerent from

Ref. [21,22], where the transfer matrix is deﬁned with

the inner product of states in the same representation,

and the transfer matrix is a quantum channel mapping

between density matrices.

The left and right environments |l⟩and |r⟩are the dom-

inant eigenvectors of the transfer matrix ˆ

Tsatisfying the

ﬁxed point equations ˆ

T|r⟩=λ|r⟩,ˆ

T†|l⟩=λ∗|l⟩, where λ

is the eigenvalue of ˆ

Twith the maximum absolute magni-

tude. The absolute value of the eigenvalue |λ|⩽1 deﬁnes

the overlap density between the two states, and |λ|= 1

if and only if the states are identical. In such case, the

left and right environments are identical up to complex

conjugation, as we prove in Appendix I.

From the construction of the transfer matrix, these

environments are of dimension 22Nq−2. To translate the

environments into variational quantum circuits, we in-

troduce two 22Nq−2×22Nq−2parametrized environment

unitaries ˆ

Erand ˆ

El, such that |r⟩=ˆ

Er(φr)|0⟩and

|l⟩=ˆ

El(φl)|0⟩, as shown in Fig. 1(d). Ultimately, we

also consider the decomposition of environment unitaries

in the form of the sequential circuits decomposition with

MElayers, as shown in Fig. 1(e). We discuss the method

of obtaining the environments in the next section.

b. Evaluating local observables. We evaluate the ex-

pectation value of an local observables utilizing the mixed

representation,

⟨ˆ

O⟩ =⟨ψ|ˆ

O|ψ⟩

⟨ψ|ψ⟩=⟨ψR(θ)|ˆ

O|ψL(θ′)⟩

⟨ψR(θ)|ψL(θ′)⟩.(2)

In Fig. 1(a), we show the circuit representation of the

numerator ⟨ψR(θ)|ˆ

O|ψL(θ′)⟩, where ˆ

Ois a local observ-

able that is Hermitian and unitary. Using the deﬁnition

of the environments, the expectation reduces to

⟨ˆ

O⟩ =⟨l, 0|ˆ

U†

Rˆ

Oˆ

UL|0, r⟩

⟨l, 0|ˆ

U†

Rˆ

UL|0, r⟩=⟨l, 0|ˆ

U†

Rˆ

Oˆ

UL|0, r⟩

λ⟨l|r⟩.(3)

Therefore, the expectation value of local observables can

be evaluated by measuring ﬁnite circuits, which can be

implemented on a quantum computer. The projective

measurement on |00 . . . 0⟩at the end of the circuit in

Fig. 1(d) has the probability equal to the squared mag-

nitude of the expectation value |⟨l, 0|ˆ

U†

Rˆ

Oˆ

UL|0, r⟩|2. The

same applies for the denominator. Combining this to-

gether, one can measure the squared magnitude of the

expectation value |⟨ ˆ

O⟩|2. In Appendix B, we provide the

derivation of the above equations. In the next section, we

will describe the procedure to measure the expectation

⟨ˆ

O⟩, including both real and imaginary parts.

We note that the outlined procedure can be gener-

alized to evaluating correlation functions of the form

⟨ψ|ˆ

Aiˆ

Bi+δ|ψ⟩, where the operators ˆ

Ai,ˆ

Bi+δact on single

qubits and are separated by δsites.

B. Translationally-invariant Trotterization

The time evolution of an initial wave function |ψ0⟩

under the action of a Hamiltonian ˆ

His given by ap-

plication of the evolution operator to the initial state

|ψ(t)⟩=ˆ

Ut|ψ0⟩= exp(−it ˆ

H)|ψ0⟩. Here, we consider a

Hamiltonian acting on a one-dimensional inﬁnite spin–

1/2 chain. When ˆ

His local, i. e., can be written as

H=Piˆ

hiwith all terms ˆ

hihaving a ﬁnite support, we

can approximate the evolution operator ˆ

UTusing a se-

quential Trotter decomposition. A ﬁrst-order sequential

Trotterization can be written as

UT= (ˆu(δt))k+O(kδt2),(4)

where ˆu(δt) = Qjˆuj(δt), ˆuj(δt) = exp(iδtˆ

hj) and δt =

T/k. A single sequential evolution operator ˆu(δt) is

shown in Fig. 2(a). Due to the sequential decomposition,

ˆu(δt) and hence ˆ

UTare translationally invariant with a

single site unit cell. Starting with a translationally in-

variant state, we always need only a single unitary ˆ

U(θ)

parameterizing the state as in Eq. (1) [35]. All these con-

siderations can be generalized to cases with a larger unit

cell.

C. The time evolution algorithm

We now introduce a hybrid quantum-classical algo-

rithm to the perform time evolution of the l-USC rep-

resentation. At the time t, we parametrize the state-

unitary ˆ

UR/L(t) by a set of variational parameters θt.

The gradients of the parameters are measured on a quan-

tum computer and the update is performed on a classical

computer. Here, for the sake of concrete notation, we

present the even time steps of the algorithm. In these

steps, representation of the wave function ﬂips from left

to right. The odd steps are done similarly, but with ﬂip-

ping from right representation to left.

To perform the time evolution at an even step,

one is required to ﬁnd the closest state |ψR(θt+δt)⟩

in right representation approximating the time-evolved

state ˆu(δt)|ψL(θt)⟩. The direct measure of the close-

ness is the ﬁdelity, i. e., squared overlap, between the two

states,

|ξ(t, δt)|2=|⟨ψR(θt+δt)|ˆu(δt)|ψL(θt)⟩|2.

It is the probability of measuring the state |. . . 000 . . .⟩at

the end of the circuit shown in Fig. 2(a). This quantity

is either 1 or 0 in the thermodynamic limit and cannot

be used for posing the optimization problem. Instead, we

deﬁne the mixed transfer matrix between the two states

over indices {abc},{ABC}as shown in Fig. 2(a-b) with

one additional index coming from the trotterized unitary.

To ﬁnd the closest state, we maximize the absolute value

of the overlap density |λ|with respect to the parame-

ters at the next time step θt+δt. The squared magnitude

of the overlap density |λ|2is the probability of measur-

ing the state |00 . . . 0⟩at the end of the circuit shown in

Fig. 2(c).

We solve the maximization problem with a gradient

ascent algorithm which requires the knowledge of envi-

ronments |l⟩and |r⟩and the leading eigenvalue from the

mixed transfer matrix. To obtain the environments |l⟩

and |r⟩, we employ the modiﬁed power method. We de-

scribe the procedure of obtaining the right environment

|r⟩, while the procedure for the left environment is simi-

lar, apart from the replacement T→T†. The idea of the

power method is to take an initial state |ψ0⟩and project

it onto the leading eigenvector of ˆ

Tby repeated appli-

cation of ˆ

T, because limp→∞(T/λ)p→ |r⟩⟨l|. Here, we

consider an iterative algorithm, which is a slight mod-

iﬁcation of the power method: at each step, we ﬁnd

the new vector |r(φ′

r)⟩by performing only a single gra-

dient descent step maximizing the overlap magnitude

|λ|2=|⟨r(φ′

r)|ˆ

T|r(φr)⟩|2with respect to φ′

r. Namely,

φ′

r←φ′

r+η∇φ′

r|λ|2, where ηis the learning rate. Alter-

natively, a gradient-free method, such as Rotosolve [36–

39], could be used. The environment vector is then up-

dated |r(φr)⟩←|r(φ′

r)⟩and is used in the next itera-

tion. At each step, |r(φr)⟩has a strictly increasing over-

lap with the leading eigenvector of ˆ

Tprovided a small

enough step size η. The method is presented in Algo-

rithm 1.

Algorithm 1 The power method for an environment

1: procedure Environment(ˆ

T , φp

2: φr,φ′

r←φp

r▷from the previous step

3: while not converged do

4: Measure λ=⟨0|ˆ

E†

r(φ′

r)ˆ

Tˆ

Er(φr)|0⟩

5: Measure ∇φ′

rλ ▷ See Eq. (5)

6: ∇φ′

r|λ|2= 2Re λ∗∇φ′

rλ

7: φ′

r←φ′

r+η∇φ′

r|λ|2

8: φr←φ′

9: return |r(φr)⟩=ˆ

Er(φr)|⃗

0⟩

Next, we show in Algorithm 2how to perform a

time evolution step using gradient ascent method with

the environments we obtained. The algorithm uses a

nested variational approach, in which left- and right-

environments are variationally optimized between the

consecutive gradient descent steps. Both algorithms are

run until the change of |λ|2between two consecutive it-

erations becomes smaller than 10−12.

Algorithm 2 Time evolution algorithm for l-USC

1: procedure Evolution step(θt,φt

r,φt

2: θt+δt ←θt

3: φr,φl←φt

r,φt

4: while not converged do

5: φr←Environment( ˆ

T(θt,θt+δt ),φr);

6: φl←Environment( ˆ

T†(θt,θt+δt ),φl);

7: |r⟩ ← ˆ

Er(φr), |l⟩ ← ˆ

El(φl);

8: λ=⟨l|ˆ

T(θt,θt+δt )|r⟩/⟨l|r⟩;

9: ∇θt+δt |λ|2= 2 Re λ∗∇θt+δt λ;▷See Eq. (5)

10: θt+δt ←θt+δt +η∇θt+δt |λ|2;

11: return θt+δt

Note that we can use the algorithm to ﬁnd the opposite

representation of the same wave function if the time evo-

lution operator is taken to be the identity. The algorithm

proposed here resembles the time evolution algorithm

for a ﬁnite size system [16,40] and for an inﬁnite sys-

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TimeEvolutionofUniformSequentialCircuitsNikitaAstrakhantsev,1,∗Sheng-HsuanLin,2FrankPollmann,2,3andAdamSmith4,51DepartmentofPhysics,UniversityofZurich,Winterthurerstrasse190,CH-8057Z¨urich,Switzerland2TechnicalUniversityofMunich,TUMSchoolofNaturalSciences,PhysicsDepartment,85748Garching,Germany3Muni...

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Time Evolution of Uniform Sequential Circuits Nikita Astrakhantsev1Sheng-Hsuan Lin2Frank Pollmann2 3and Adam Smith4 5 1Department of Physics University of Zurich Winterthurerstrasse 190 CH-8057 Z urich Switzerland

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