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Computational power of one- and two-dimensional

dual-unitary quantum circuits

Ryotaro Suzuki1,2, Kosuke Mitarai1,3,4, and Keisuke Fujii1,3,5

1Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531,

Japan

2Dahlem Center for Complex Quantum Systems, Freie Universit¨

at Berlin, Berlin 14195, Germany

3Center for Quantum Information and Quantum Biology, Institute for Open and Transdisciplinary Research

Initiatives, Osaka University, Osaka 560-8531, Japan

4JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan

5Center for Emergent Matter Science, RIKEN, Wako Saitama 351-0198, Japan

Quantum circuits that are classically simulatable tell us when quantum computa-

tion becomes less powerful than or equivalent to classical computation. Such classi-

cally simulatable circuits are of importance because they illustrate what makes uni-

versal quantum computation diﬀerent from classical computers. In this work, we pro-

pose a novel family of classically simulatable circuits by making use of dual-unitary

quantum circuits (DUQCs), which have been recently investigated as exactly solvable

models of non-equilibrium physics, and we characterize their computational power.

Speciﬁcally, we investigate the computational complexity of the problem of calculat-

ing local expectation values and the sampling problem of one-dimensional DUQCs

whose initial states satisfy certain conditions, and we generalize them to two spatial

dimensions. We reveal that a local expectation value of a DUQC is classically sim-

ulatable at an early time, which is linear in a system length. In contrast, in a late

time, they can perform universal quantum computation, and the problem becomes a

BQP-complete problem. Moreover, classical simulation of sampling from a DUQC

turns out to be hard.

1 Introduction

Quantum computation is widely believed to be intractable by classical computers. However, there

also exist certain types of quantum circuits that can be eﬃciently simulated classically despite

being able to generate highly entangled states. Famous examples are quantum circuits which

consist of Cliﬀord gates [1] or matchgates, corresponding to free-fermionic dynamics [2–5]. Such

classically simulatable quantum circuits are of importance because they illustrate what makes

universal quantum computation diﬀerent from classical computers. Moreover, they have practical

applications such as randomized benchmarking [6–8], simulation by stabilizer sampling [9], and

estimation of an error threshold of a quantum error correction code [10].

Classically simulatable or exactly solvable quantum circuits are also important in the study of

dynamics of isolated quantum systems [11–13]. For example, Cliﬀord circuits include quantum

Ryotaro Suzuki: ryotaro.suzuki.2139@gmail.com

Kosuke Mitarai: mitarai@qc.ee.es.osaka-u.ac.jp

Keisuke Fujii: fujii@qc.ee.es.osaka-u.ac.jp

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arXiv:2103.09211v2 [quant-ph] 18 Jan 2022

dynamics which are both integrable and non-integrable 1, and they have been investigated from

the perspective of quantum thermodynamics, and especially thermalization [14–16]. Moreover,

physical quantities, such as entanglement entropy and out-of-time-ordered correlators, of an en-

semble of one and higher dimensional Haar random unitary circuits are calculated exactly [17–19].

Because, in general, it is notoriously diﬃcult to treat quantum dynamics analytically except for

one-dimensional integrable systems [20], above quantum circuit representations of quantum dy-

namics are powerful tools to analyze their physical properties. In addition, since two-dimensional

quantum dynamics are not explored in comparison with one-dimensional cases, solvable models

in two dimensions are in great demand.

Recently, a new class of quantum gates called “dual-unitary gates” has been introduced [21,

22]. Dual-unitary gates are unitary gates which remain unitarity under reshuﬄing their indices.

The dynamics consisting of dual-unitary gates can be either integrable or non-integrable. In

Refs. [22,23], it has been shown that dynamical correlation functions and time evolution of op-

erator entanglement entropy under dual-unitary quantum circuits (DUQCs) are calculated exactly.

Because these quantum circuits have one spatial dimension, we call them one-dimensional (1D)

DUQCs. Moreover, it has been shown that when the system size is inﬁnite, time evolution of local

observables, correlation functions, and entanglement entropy of 1D DUQCs arising from certain

initial states can be calculated exactly [24]. As we will deﬁne later, initial states are described by

matrix product states (MPSs) whose matrices satisfy certain conditions. They are called solvable

initial states [24]. The simplest example is a chain of EPR pairs, and the simplest counter-example

is a product state.

Interestingly, despite the above property, dual-unitary gates contain arbitrary single-qubit gates

and a certain class of two-qubit entangling gates. Note that these gates form a universal gate set if

we can apply them freely [25]. Nevertheless, the carefully constructed initial states and the inﬁnite

size limit allow us to compute the expectation values eﬃciently. Here, we ask whether they are

classically simulatable or allows universal quantum computation if the system size is ﬁnite. The

ﬁniteness of the system size enables us to consider the circuit depth which scales with the system

size and characterize their computational power.

In this paper, we investigate quantum computational power of 1D and two-dimensional (2D)

DUQCs. Speciﬁcally, we characterize the computational complexity of the problem of calculating

expectation values of local observables and the sampling problem of 1D and 2D DUQCs with

ﬁnite system sizes. Additionally, we study classical simulatability of correlation functions of 2D

DUQCs. Here, 2D DUQCs takes certain initial states which are product states of solvable initial

states. Note that the generalization to the 2D lattice is of interest not only from the viewpoint of

quantum computing but also from the viewpoint that exactly solvable quantum dynamics in two

spatial dimensions is limited.

Our results are summarized in Fig. 1. For the ﬁrst problem, we show that expectation values

of local operators Oof 1D and 2D DUQCs are exponentially close to Tr(O) until time, or cir-

cuit depth, t∼1

2Nand t∼N, respectively, where Nis a system length. In other words, local

expectation values do not depend on speciﬁc choices of dual-unitary gates in those time regions,

and they are classically simulatable. On the other hand, in later time, local expectation values of

DUQCs can depend on dual-unitary gates, and we ﬁnd that DUQCs can simulate universal quan-

tum computation after time poly(N). It means that the problem becomes BQP-complete after

time poly(N). As we will discuss later, the depth overhead for simulating quantum circuits con-

sisting of nearest-neighbor CZ gates and single-qubit gates with DUQCs is O(N). This contrasts

to conventional classically simulatable quantum circuits with a ﬁxed gate set, such as Cliﬀord or

matchgate circuits, where classical simulatability does not change depending on the circuit depth.

1Throughout this paper, we say that quantum dynamics are integrable if there exist an extensive number

of conserved quantities.

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1D DUQC

𝑡~𝑁

in P

𝑡 = 𝑝𝑜𝑙𝑦(𝑁)

BQP-complete

𝑡

Local expectation values

If classically simulatable, then PH collapses

𝑡

𝑡 = 𝑁 − 𝑁

Sampling problem

2D DUQC

𝑡~2𝑁 𝑡 = 𝑝𝑜𝑙𝑦(𝑁) 𝑡

If classically simulatable, then PH collapses

𝑡

𝑡 = 4

in P BQP-complete

1D DUQC

2D DUQC

Figure 1: Summary of our main results. Computational complexity of each problem depending on the

circuit depth in one and two spatial dimensions is shown.

In addition, we show that sampling of the output of 1D and 2D DUQCs is intractable for

classical computers after time t∼1

2(N−√N)and t≥4, respectively, unless polynomial

hierarchy (PH) collapses to its third level. This result is based on the fact that if a quantum

circuit with post-selection can simulate universal quantum computation, the output probability

distribution cannot be sampled by a classical computer eﬃciently unless PH collapses to its third

level [26]. It implies that, especially in the case of two dimensions, the sampling problem of

constant-depth DUQCs is as hard as that of general constant-depth quantum circuits.

Moreover, we ﬁnd that correlation functions of 2D DUQCs along a special direction, which

is determined by the initial state and deﬁned later, become the trace of operators in linear depth,

and hence they are classically simulatable. In contrast, the value of correlation functions along

the other direction can depend on a choice of unitary gates, and they do not seem to be classically

simulatable. We leave as an open problem whether or not the problem of calculating correlation

functions of 2D DUQCs in linear depth is BQP-hard. Finally, we also show a suﬃcient condition

of 2D lattices on which local observables of 2D DUQCs at an early time are classically simulat-

able. For instance, the lattices satisfying this condition include honeycomb lattices. In summary,

we reveal that the computational power of DUQCs strongly depend on their circuit depth and

problem settings. This provides a novel quantum computational model to investigate both clas-

sical simulatability of quantum computation and physical properties of non-equilibrium quantum

systems.

The rest of the paper is organized as follows. In Sec. 2, we introduce and review 1D DUQCs.

In Sec. 3, we characterize the complexity of the problem calculating local expectation values

and the sampling problem of 1D DUQCs. In Sec. 4, we generalize 1D DUQCs to two spatial

dimensions, and characterize the complexity as with the 1D case. After that, we discuss classical

simulatability of correlation functions of 2D DUQCs and a generalization of lattices of qubits.

Sec. 5is devoted to conclusion and discussion.

2 One-dimensional DUQCs

In this section, we review 1D DUQCs and solvable initial states, which have been introduced in

Refs. [21,22,24], and have been studied in Refs. [23,27–38].

2.1 Dual-unitary gates

We consider a 2N-qubit system. Its computational basis is denoted by |i1i2. . . i2Ni, where ij=

0,1indicates a state of the j-th qubit. A dual-unitary gate is a two-qubit gate in the following

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form:

U=eiφu1⊗u2SWAP(CZ)αv1⊗v2,(1)

where SWAP is the swap gate,

SWAP =

1 0 0 0

0 0 1 0

0 1 0 0

0 0 0 1

,(2)

CZ is the controlled-Z gate,

CZ =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 −1

,(3)

u1,u2,v1and v2are arbitrary single-qubit gates, and both φand αare arbitrary real numbers.

Alternatively, Eq. (1) is rewritten as

eφ0u1⊗u2e−iπ

4(X⊗X+Y⊗Y+JZ ⊗Z)v0

1⊗v0

2,(4)

with φ0=φ−π

4α,J=α+ 1,v0

1=eiπ

4αZ v1, and v0

2=eiπ

4αZ v2. It has been shown that these

gates can describe both integrable and non-integrable periodically driven quantum systems, or

Floquet systems [22]. For example, the time evolution operator of a periodically driven quantum

system with XXZ interaction,

e−iπ

4(X⊗X+Y⊗Y+JZ ⊗Z),(5)

is one of the dual-unitary gates. The time evolution operator of a self-dual kicked Ising chain can

also be written in terms of dual-unitary gates as follows:

Te−iR1

0(π

4Z⊗Z+hZ⊗I+π

4δ(t−1)X⊗I)dt

=e−iπ

4e−ihZ eiπ

4X⊗eiπ

4X·e−iπ

4Y⊗e−iπ

4Y

·e−i(π

4X⊗X+π

4Y⊗Y)·eiπ

4Z⊗eiπ

4Z·eiπ

4Ye−ihZ ⊗eiπ

4Y,

(6)

where Tis the time ordered product, δ(t)is the Dirac delta function, and his a real number.

Dual-unitary gates have the following nice property. Let Ube a nearest-neighbor two-qubit

gate. Deﬁne ˜

U, called the dual gate of U, such that

hk|hl|˜

U|ii|ji=hj|hl|U|ii|ki.(7)

Then, ˜

Uis a unitary gate if and only if Uis a dual-unitary gate [22]. This property can be

expressed graphically by using a tensor-network representation of quantum circuits. We represent

a two-qubit gate Uand U†as

→

ij

k l

!!,#

$, %

,

ij

k l

!!

",$

%,&

←

,(8)

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where (i, j)-legs and (k, l)-legs of Uand U†serve as inputs and outputs, respectively. The unitar-

ity, UU †=U†U=I, can be written as

→ ←

＝

→←

＝

.(9)

Similarly, the dual gate of Uand its Hermitian conjugate are represented as

ij

k l

＝

𝑈

"!,#

$,%

→

,

ij

kl

＝

(𝑈

#∗)",$

%,&

←

,(10)

where (i, j)-legs and (k, l)-legs of ˜

Uand ˜

U†serve as input and outputs, respectively. The property

of a dual-unitary gate, namely ˜

U˜

U†=˜

U†˜

U=I, can be written as

→ ←

＝

,

→ ←

＝

.(11)

2.2 Dual-unitary quantum circuits

1D DUQCs are quantum circuits with 2Nqubits which consist of nearest-neighbor dual-unitary

gates. They are deﬁned as follows:

U1D(t) =

t/2

Y

τ=1

U(e)(2τ)U(o)(2τ−1),(12)

where

U(o)(2τ−1) =

N

Y

i=1

U2i,2i+1(2τ−1),(13)

U(e)(2τ) =

N

Y

i=1

U2i−1,2i(2τ),(14)

tis an even number, and Ui,j (τ)is a dual-unitary gate acting on qubit iand jat time τ, or

graphically,

U1D(t) = .(15)

We note that dual-unitary gates can diﬀer from each other, that is, the quantum dynamics can be

inhomogeneous in space and time. In Eqs. (12) to (15), we assume a periodic boundary condition

(PBC) in space, that is, U2N,2N+1 =U2N,1.

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2.3 Solvable initial states

In a 1D DUQC, when an initial state satisﬁes certain conditions and a system size is inﬁnite, it

has been shown that time evolution of local observables, correlation functions and entanglement

entropy can be calculated exactly [24]. Such an initial state is called a solvable initial state and

can be described in terms of a matrix product state (MPS) [24,39].

Here we describe two conditions that make a two-site shift invariant MPS,

|ΨN(A)i=X

{ij}

Tr A(i1,i2)A(i3,i4). . . A(i2N−1,i2N)|i1i2. . . i2Ni,(16)

where A(i,j)is a χ-dimensional square matrix, a solvable initial state. |ΨN(A)ican alternatively

be represented by a tensor-network as,

,(17)

…

.(18)

The ﬁrst condition is that its transfer matrix has a unique eigenvector with a maximum eigenvalue

λ0. A transfer matrix associated with |ΨN(A)iis deﬁned as

Eβ0β,α0α=X

i,j

(A(i,j)∗)β0,α0⊗(A(i,j))β,α ,(19)

or graphically,

.(20)

Note that hΨN|ΨNi= Tr(EN)≈λN

0for large N, which implies λ0= 1 is needed in order to

normalize |ΨN(A)i, namely hΨN|ΨNi= 1 in the limit of N→ ∞. The second condition is that

Asatisﬁes the following condition:

2

X

k=1

A(i,k)(A(j,k))†=1

2δi,j,(21)

or graphically,

,(22)

where δi,j is Kronecker’s delta. Eq. (21) implies

2

X

k=1

(A(i,k))†A(j,k)=1

2δi,j.(23)

Furthermore, Eqs. (21) and (23) imply that the transfer matrix has

|Ii=1

√χ

χ

X

α=1 |ααi,(24)

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as right and left eigenvectors with an eigenvalue 1. Strictly speaking, the second condition consid-

ered here is more restrictive than that of Ref. [24]. However, they are equivalent in the thermody-

namic limit (see Theorem 1 in Ref. [24]), and we adopt Eq. (22) for clarity.

The simplest example of solvable states is a chain of EPR pairs |EPRi⊗N, where |EPRi=

1

√2(|00i+|11i), or graphically,

…

=

,(25)

whose χis zero. This example has been studied to analyze the entanglment dynamics of self-dual

kicked Ising chains [40].

2.4 Local expectation values of 1D DUQCs

Let us brieﬂy describe how expectation values of local observables can be calculated for dual

unitary circuits with solvable initial states [24]. We consider a time-evolved transfer matrix E(t)

deﬁned as

𝐴∗

𝐴

←

←

←

←

→

→

→

→

,(26)

where each number of the right-side 2t+ 2 legs indicates the input space which E(t)acts on. It

can be shown by Eqs. (11) and (22) that

|I(t)i=

t+1

O

j=2 1

√d

2

X

i=1 |iij|ii2t−j+3!⊗ 1

√χ

χ

X

α=1 |αi1|αi2t+2!(27)

is both right and left eigenvectors of E(t)with eigenvalue 1. In fact, |I(t)iis the unique eigenvec-

tor with maximum eigenvalue. This leads to the following equality:

lim

N→∞ E(t)N=|I(t)ihI(t)|.(28)

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By virtue of Eq. (28), one can calculate an expectation value of a local observable Oas follows:

𝐴∗𝐴∗𝐴∗𝐴∗𝐴∗

𝐴∗𝐴∗𝐴∗

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴

→

→

→

→

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

←

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

→

𝐴∗

𝐴

→←

←

←

←

→

→

→

,(29)

where |ΨN

ti=U1D(t)|ΨN(A)iis a solvable initial state evolved to time t. It means that an

expectation value of a local observable does not depend on a speciﬁc choice of dual-unitary gates.

From a similar argument, it has been also shown that correlation functions of 1D DUQCs are

classically simulatable [24].

3 computational power of 1D DUQC

3.1 Local expectation values

Although local observables in a 1D DUQC are calculated exactly when the system size is inﬁnite,

a dual-unitary gate can contain arbitrary single-qubit gates and the CZ gate, which can form a

universal gate set in principle [41]. Thus, it is natural to ask whether or not DUQCs can perform

universal quantum computation when their system size is ﬁnite. Finiteness of the system size

enables us to consider the circuit depth which is scaling with the system size and characterize their

computational power.

In this section, we answer the above question aﬃrmatively and characterize computational

complexity of the problem of calculating local expectation values for dual-unitary circuits. More

precisely, we consider a local observable with length l

O=

l−1

Y

i=0

Oi0+i,(30)

where i0is an integer, and Oi0+iis an observable on qubit i0+i. Local expectation values of O

at time thO(t)iis hO(t)i=hΨN

t|O|ΨN

ti

hΨN

t|ΨN

ti. Here, we note that |ΨN(A)iis not generally normalized

for a ﬁnite N. Then, we deﬁne the following decision problem, which has a parameter t.

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Problem 1 (local expectation values of 1D DUQCs).Consider a 1D DUQC in time t

U1D(t), a solvable 2N-qubit initial state |ΨN(A)iwith χ=O(1), and a local observable

Owhose operator norm is 1 with length l=O(1).hO(t)iis promised to be either ≥aor

≤b, where a−b≥1

poly(N). The problem is to determine whether hΨN

t|O|ΨN

tiis ≥aor

≤b.

We show that Problem 1with time t≤ b(1 −δ)Nc − l/2, for an arbitrary 0< δ < 1, is

in P. In contrast, with time t= poly(N), it becomes BQP-complete. We ﬁrst prove the above

statements in the case where the initial state is a chain of EPR pairs Eq. (25), which is normalized

for any N, for simplicity. Then, we extend to general solvable initial states.

3.1.1 1D DUQC is classically simulatable at an early time

First, we prove the former statement. In this case, an expectation value hΨN

t|O|ΨN

tiwith time

t≤N−l/2can be written as

→

→→

→

→

→

←

←

←

←

←

←

𝑂

→

→

→

←

←

←

←

←

←

←

←

→

→

→

→

→

=

,(31)

where land i0are respectively assumed to be even for clarity, and the normalization coeﬃcient

which arises from Eq. (25) is omitted. To derive Eq. (31), we remove unitary gates outside of the

causal-cone by using Eq. (9). Although we ﬁxed parity of land i1, expectation values with other

combinations of parity can be written likewise. If t≥1

2l+ 1, we obtain hΨN

t|O|ΨN

ti= Tr(O)by

adapting Eq. (11) to the both sides of Eq. (31) sequentially and removing leftover unitary gates

by adapting Eq. (11). In other words, an expectation value hO(t)iis identical to the expectation

value of a maximally mixed state regardless of components of dynamics. The detail of the proof

is in Appendix A. For t < l/2+1, in general, the local expectation value is not equal to Tr(O)

as the dual-unitary gates cannot be cancelled. However, local expectation values at time less than

l/2 + 1 are classically simulatable because only constant number of unitary gates are involved in

calculation of local expectation values.

We can easily generalize the above results to general solvable initial states. Let us note that a

transfer matrix of a solvable MPS to the power Mcan be written as

EM=|IihI|+εM,(32)

where εMis a matrix such that leading order of non-zero elements are O(|λ1|M)and λ1is the

second largest eigenvalue of E. By using Eq. (32), an expectation value of Owith t≤ b(1 −

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δ)Nc − l/2can be calculated as follows:

𝐴∗𝐴∗𝐴∗𝐴∗𝐴∗𝐴∗

𝐴∗𝐴∗𝐴∗𝐴∗

𝐴 𝐴 𝐴 𝐴 𝐴 𝐴

𝐴 𝐴 𝐴 𝐴

＝

→

→→

→

→

→

←

←

←

←

←

←

𝑂

→

→

→

←

←

←

←

←

←

←

←

→

→

→

→

→

＝

𝐴∗𝐴∗𝐴∗𝐴∗

𝐴∗𝐴∗

𝐴 𝐴 𝐴 𝐴

𝐴 𝐴

←

←

←

←

←

←

←

←

←

←

←

←

←

←

→

→→

→

→

→→

→

→

→

→

→

→

→

𝑂

,(33)

where , deriving from the matrix εM, is O|λ1|bδN c. Then, by applying Eqs. (9), (11), (21),

and (23) to Eq. (33), we obtain the expectation value hO(t)i= Tr(O) + . The detail of the

calculation is in Appendix B. Because is exponentially suppressed with respect to N,Problem

1with t≤ b(1 −δ)Nc − l/2is still in P. Here, we note that if an initial state is a chain of EPR

pairs, δcan be chosen as zero, that is, hO(t)iwith t≤N−l/2is classically simulatable.

On the other hand, for time t>N−l/2, the expectation value can be written as

→

→→

→

→

→

←

←

←

←

←

←

𝑂

→

→

→

←

←

←

←

←

←

←

←

→

→

→

→

→

=

.(34)

In contrast to Eq. (31), we cannot use Eqs. (9) and (11) to calculate Eq. (34) eﬃciently. It could

be possible that we can calculate Eq. (34) eﬃciently by another method. In the following section,

we exclude the possibility by showing Problem 1with t= poly(N)is BQP-complete.

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=

.

Figure 2: Graphical representation of Eq. (35). The indices in this ﬁgure correspond to those in Eq.

(35).

3.1.2 1D DUQC is universal in late time

The inclusion of the problem in BQP is trivial; we can just execute the dual-unitary circuit to

obtain an expectation value. What is left to be shown is that the problem is BQP-hard. To do

that, we consider a BQP-complete problem calculating whether an local expectation value cn=

h0n|U†I+Z1

2U|0niis ≥a, where cnis promised to be either ≥aor ≤b, where a−b≥1

poly(N),

and Uis a 1D quantum circuit consisting of poly(n) nearest-neighbor two-qubit gates. Then, we

show that Problem 1is as hard as the BQP-complete problem after time poly(N).

Firstly, we construct the CZ gate acting on an even numbered site and an odd numbered site

by cancelling swap gates from a DUQC, as follows:

CZ2i−2k,2i+2k+1

=

N−k−1

Y

t2=1

(SWAP2SWAP1)SWAP2CZ2i,2i+1·

k

Y

t1=1

(SWAP2SWAP1),(35)

CZ2i−2k−1,2i+2k+2

=

N−k−1

Y

t2=1

(SWAP2SWAP1)CZ2i−1,2iSWAP1·

k

Y

t1=1

(SWAP2SWAP1),(36)

where

SWAP1=

N

Y

i=1

SWAP2i,2i+1,(37)

SWAP2=

N

Y

i=1

SWAP2i−1,2i,(38)

CZ2i,2i+1=

i

Y

j=1

SWAP2j,2j+1

CZ2i,2i+1

N

Y

j=i+1

SWAP2j,2j+1

,(39)

CZ2i−1,2i =

i

Y

j=1

SWAP2j−1,2j

CZ2i−1,2i

N

Y

j=i+1

SWAP2j−1,2j

.(40)

Remember that the index 2N+ 1 is treated as 1according to a PBC. Graphically, Eq. (35) can

be written as Fig. 2. Additionally, one can apply CZ gates in parallel by substituting CZ ·SWAP

gates for some of SWAP gates in Eqs. (39) and (40).

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Secondly, it can be easily shown that an arbitrary product of single-qubit gates can be imple-

mented by substituting suitable u1⊗u2·SWAP gates for some of SWAP gates in Eqs. (35)

and (36). A quantum circuit consisting of arbitrary one-qubit gates and CZ gates has capability

to eﬃciently simulate arbitrary quantum circuit consisting of poly(n) nearest-neighbor two-qubit

gates [41]. Therefore, we can construct a DUQC U1D with poly(N) depth such that

hEPR|⊗NU†

1D I+Z1

2U1D |EPRi⊗N=cn(41)

holds, which means that Problem 1with time t=poly(N)and input a chain of EPR pairs is a

BQP-complete problem. Here, we remark that, from our construction, if Uis written as a depth-d

quantum circuit consisting of nearest-neighbor CZ gates and single-qubit gates, the depth of U1D

which simulates Uis O(dN). This is because U1D requires O(N)simulation cost for CZ gates

in Uin each time. It means that the depth overhead for simulating Uwith U1D is O(N).

3.2 Sampling problem

We now move on to discuss the sampling complexity of 1D DUQCs. We show that classical

simulation of sampling from linear depth 1D DUQCs is hard.

Let {pz}be the probability distribution with pzbeing the probability of obtaining output z∈

{0,1}2Nwhen |Ψtiis measured in computational basis. We deﬁne that a probability distribution

{pz}is sampled by classical computers eﬃciently with a multiplicative error cif there exists a

classical probabilistic polynomial-time algorithm that outputs zwith probability qzsuch that |pz−

qz| ≤ cpz, for all z. It has been shown that if an n-qubit quantum circuit with post-selection can

simulate universal quantum computation, the measurement output of nqubits cannot be sampled

by classical computers eﬃciently with a multiplicative error unless polynomial hierarchy collapses

to its third level [26]. On the other hand, measuring a square lattice cluster state in an arbitrary

measurement basis with post-selection can perform universal quantum computation [42]. A square

lattice cluster state is deﬁned as

|CSi=Y

(i,j)∈E

CZi,j |+i⊗N,(42)

where |+iis H|0i, we assigned a qubit in |+istate to each vertex of a square lattice, and Eis the

set of all edges of the lattice.

Here, we prove that a 1D DUQC can generate a square lattice cluster state with an EPR-

chain initial state in time N−√2N/2 + 1, which implies that sampling from U1D |EPRi⊗Nis

classically hard.

We assume that 2Nis a square of an even number 2m, which enables us to make a one-to-

one correspondence between the 1D qubits and ones on square lattice. Note that the following

equation holds:

I⊗H|EPRi= CZ |++i,(43)

where His the Hadamard gate. With this in mind, we obtain a square lattice cluster state by

applying the following dual-unitary circuit to the initial state |EPRi:

Ucluster =

t=N−m+1

Y

t=1

Ucluster(t),(44)

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Figure 3: A DUQC which generate a square lattice cluster state mapped to one dimension. The DUQC

in Eq.(45), (46), (47), and (48) can be written as (a), and it is equal to a quantum circuit (b), which

include quantum gates acting on two distant qubits. The ﬁnal state of (b) is equivalent to a square

lattice cluster state (c) by deviding a 1D qubit array into 2mand rearranging it to a square lattice.

where,

Ucluster(t= 1) =

N

Y

i=1

(SWAP ·CZ ·H⊗I)2i,2i+1,(45)

Ucluster(t=m+ 1) =

2m

Y

j=1

m−1

Y

i=1

(SWAP ·CZ)2i+m+j,2i+m+1+j·SWAP3n+j,3n+i+j,(46)

Ucluster(t=N−m) =

N

Y

i=1

(SWAP ·CZ)2i,2i+1,(47)

Ucluster(t=N−m+ 1) =

2m

Y

i=1

(SWAP ·CZ ·H⊗I)N+m+j,N+m+1+j,(48)

and Ucluster(t)at other odd and even times are SWAP1and SWAP2, respectively. Graphically,

the above DUQC can be written as Fig. 3(a), and it is equivalent to Fig. 3(b).

As shown in Fig. 3(b) and (c), by rearranging the 1D qubit array (b) to the square lattice

(c), the ﬁnal state of the DUQC is equivalent to the square lattice cluster state. Moreover, because

dual-unitary gates include arbitrary single-qubit gates, we can measure the ﬁnal state in an arbitrary

measurement basis. Therefore, sampling from depth-(N−√2N/2 + 1) 1D DUQCs is unlikely

to be classically simulatable.

4 Generalization to 2D DUQCs

In this section, we generalize DUQCs to two spacial dimensions and characterize their computa-

tional power.

4.1 Deﬁnition of 2D DUQCs

We consider a 2N×2M-qubit system on a 2N×2Msquare lattice. Its computational basis

is denoted by |i(1,1)i(1,2) ···i(1,2M)i(2,1) ···i(2N,2M)i, where i(j,k)= 0,1indicates a state of the

(j, k)-th qubit.

We deﬁne U(1),U(2) ,U(3), and U(4) as

U(1) =

N

Y

j=1

2M

Y

k=1

UD

(2j,k),(2j+1,k),(49)

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U(2) =

2N

Y

j=1

M

Y

k=1

U(j,2k−1),(j,2k),(50)

U(3) =

N

Y

j=1

2M

Y

k=1

UD

(2j−1,k),(2j,k),(51)

U(4) =

2N

Y

j=1

M

Y

k=1

U(j,2k),(j,2k+1),(52)

where UD

(i,j),(k,l)is a dual-unitary gate acting on qubit (i, j)and qubit (k, l), and U(i,j),(k,l)is an ar-

bitrary two-qubit unitary gate acting on qubit (i, j)and qubit (k, l). We also deﬁne U{2,4}as a ma-

trix which is an arbitrary product of U(2) and U(4), for example, U(2),U(2)U(4) , or U(2)U(4)U(2) .

We note that the fact that a unitary gate in U(2) or U(4) can be an arbitrary unitary gate is a sig-

niﬁcant diﬀerence between one and two spatial dimentions. In Eqs. (49) to (52), we assumed a

PBC in space, that is, U(2N,k),(2N+1,k)=U(2N,k),(1,k)and U(j,2M),(j,2M+1) =U(j,2M),(j,1) for all

kand j. Then, we deﬁne 2D DUQCs are quantum circuits with 2N×2Mqubits as follows:

U2D(t) =

t

4

Y

τ=1

U{2,4}(τ+ 3)U(3)(τ+ 2)U{2,4}(τ+ 1)U(1) (τ),(53)

where tis a multiple of four and Ui(τ),i= 1,{2,4},3, is a unitary Uiat time τ.

In the following subsections, we consider initial states |ΨAiwhich are product states of solv-

able initial states |ΨN

Aialigned in rows:

|ΨAi=|ΨN

Ai⊗2M,(54)

or graphically,

…

…

…

…

,(55)

where ij,k indicates the state of (j, k)-th qubit. We call these initial states 2D solvable initial

states in the sense that, as we show in the next section, local expectation values are classically

simulatable at an early time.

The simplest example of 2D solvable initial states is rows of chains of EPR pairs

…

…

…

…

.(56)

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For concreteness, in this section, we ﬁx U{2,4}(τ+3) and U{2,4}(τ+1) in Eq. (53) as U(4) and

U(2), respectively, but one can discuss the argument in this section similarly in the case of other

U{2,4}(τ+ 3) and U{2,4}(τ+ 1). Besides, as with 1D cases, dynamics U2D also contain a 2D

periodically-driven XXZ and a 2D self-dual kicked Ising model (The detail is shown in Appendix

C.).

4.2 Local expectation values

We characterize computational complexity of the problem of calculating local expectation values

for 2D DUQCs. We consider a local observable in an l×lsquare region

O=

l−1

Y

i,j=0

Oi0+i,j0+j,(57)

where Oi0+i,j0+jis an observable of qubit (i0+i, j0+j)for some integers i0and j0. Hereafter,

for clarity, we assume l,i0, and j0to be odd number, even number, and even number, respectively.

Note that the discussion of the following subsections are not limited to this choce, and similar

argument can be applied to other choice of parities. We denote by |ΨA(t)ia 2D solvable initial

state Eq. (55) evolved at time t, that is, |ΨA(t)i=U2D(t)|ΨAi. Then, we deﬁne the following

decision problems as with the 1D case.

Problem 2 (local expectation values of 2D DUQCs).Consider a 2D DUQC in time

tU2D(t)with time t, a 2D solvable 2N×2M−qubit initial state |ΨA(t)iwith M=

O(poly(N)), and a local observable Owhose operator norm is 1 with length l=O(1).

hΨA(t)|O|ΨA(t)iis promised to be either ≥aor ≤b, where a−b≥1

poly(N)The problem

is to determine whether hΨA(t)|O|ΨA(t)iis ≥aor ≤b.

Similary to the 1D case, we show that Problem 2is in Pwith time t≤ b2(1 −δ)Nc − l, for

0< δ < 1, and BQP-complete with time t= poly(N).

4.2.1 2D DUQC is classically simulatable at an early time

The way to calculate local expectation values is similar to the 1D case except for contractions of

unitary gates at even time. The procedure is to contract dual-unitary gates at odd time by using

Eq. (9) and unitary gates at even time by using Eq. (11). As a result, if t≤ b2(1 −δ)Nc − land

t≥l+ 1, we obtain

hΨt|O|Ψti=1

2l2Tr(O) + OM· |λ1|bδN c.(58)

Origins of the conditions t≥lare the same as those of 1D cases. We explain the procedure

graphically in the case that an initial state is rows of EPR pairs (Fig. 4).

Firstly, we remove unitary gates out-side of the causal-cone. Then, the local expectation value

can be represented as Fig. 4(a). A tensor network in dotted line of Fig. 4(a) is depicted in further

detail in Fig. 4(b). In the ﬁrst equality in Fig. 4(b), we use the deﬁnition of dual-unitary gates Eq.

(11), and remove them. After that, unitary gates, which are contracted with removed dual-unitary

gates (unitary gates painted in red and orange in Fig. 4), can be removed by the deﬁnition of

unitary gates Eq. (9). This leads to the second equation in (b). By repeating this procedure, one

obtain that local expectation values are Tr(O). Moreover, local expectation values at time less

than l+ 1 are classically simulatable because only constant number of unitary gates are involved

in calculation of local expectation values. One can straightforwardly generalize the above results

to general 2D solvable initial states similar to the 1D case and Appendix B. Altogether, we obtain

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= =

(a) (b)

𝑂

=

(c)

==

Figure 4: Calculation of a local observable Oin a 2D DUQC . After removing unitary gates outside of

the causal-cone, remaining unitary gates can be represented as (a). Contractions of dual-unitary gates

and unitary gates in the dotted line of (a) are represented in (b). Here, a unitary gate painted in red is

a harmitian conjugate of one painted in orange. Contractions of dual-unitary gates and unitary gates,

in the case that U(2) of (b) are replaced by U(4)U(2), are represented in (c).

Problem 2with t≤ b2(1 −δ)Nc − lis in P. Note that this is true even if U(2) and U(4) are

replaced by U{2,4}. For example, in the case of U{2,4}=U(4)U(2) , dual-unitary gates and unitary

gates are contracted and removed as Fig. 4(c).

We note that classical simulatability of 2D DUQC are also interesting as a solvable model

of quantum dynamics because analytical research on quantum dynamics in 2D systems is much

more diﬃcult than 1D cases. In fact, all local expectation values of 2D DUQCs become Tr(O)

in the thermodynamic limit, which means that a local density matrix of a 2D solvable initial state

evolved by 2D DUQCs is identical to a thermal equilibrium state at inﬁnite temperature. Therefore,

thermalization of solvable initial states can be shown analytically in the thermodynamic limit.

Understanding conditions when thermalization happens is one of the most important problems

in non-equilibrium physics [11–13,43], and it means that 2D DUQCs are rare toy models of 2D

non-equilibrium quantum physics.

4.2.2 2D DUQC is universal in late time

Based on the fact that Problem 1with time t= poly(N)is BQP-complete , BQP-completeness

of Problem 2with time t= poly(N)becomes trivial by noticing that U2D(t)acting on any row

of 2N-qubit can be regarded as 1D DUQCs when both U(2) and U(4) are identity operators.

4.3 Sampling problem

We now move on to discuss the sampling complexity of 2D DUQCs. We show that a constant

depth 2D DUQC can generate a square lattice cluster state.

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𝑂

𝑂

(a)

EPR chains

𝑂

𝑂

(b)

EPR chains

𝑂

𝑂

(c)

Figure 5: Calculation of correlation functions. After removing unitary gates outside of the causal-cone,

remaining unitary gates of C1(r, t)and C2(r, t)can be represented as (a) and (b), respectively, in the

case that C2(r, t)can take a nonzero value. EPR chains in (a) and (b) are aligned along arrows in (a)

and (b), respectively. After removing dual-unitary at odd time by using Eq. (9) and unitary gates at

even time by using Eq. (11), remaining unitary gates of C2(r, t)can be represented as (c).

We obtain a square lattice cluster state |CSiby four depth 2D DUQCs as follows:

|Ψ1i=

2N

Y

i=1

N

Y

j=1

(SWAP ·CZ ·H⊗I)(i,2j),(i,2j+1) |EPRi,(59)

|Ψ2i=

N

Y

i=1

2N

Y

j=1

CZ(2i−1,j),(2i,j)|Ψ1i,(60)

|Ψ3i=

2N

Y

i=1

N

Y

j=1

SWAP(i,2j−1),(i,2j)|Ψ2i,(61)

|CSi=

N

Y

i=1

2N

Y

j=1

CZ(2i,j),(2i+1,j)|Ψ3i.(62)

Therefore, sampling the output of constant depth 2D DUQCs is unlikely to be classically simulat-

able.

4.4 Correlation functions

Let us discuss classical simulatability of two-point correlation functions. For simplicity, we as-

sume that the initial state is |EPRi, but one can easily generalize the following argument to

general solvable initial states. In such a case, we expect two-point correlation functions to be

anisotropic because solvable initial states are anisotropic. We consider two types of correlation

functions, one of which is classically simulatable, and the other does not seem to be classically

simulatable.

First, we consider the following correlation function, which is classically simulatable:

C1(r, t) = hEPRt|Oi,jOi+r,j|EPRti − Tr(Oi,j)Tr(Oi+r,j ),(63)

where |EPRtiis |EPRievolved at time t, and Oi,j is an observable of qubit (i, j). When

2N≥2t−rholds, C1(r, t), which can be graphically represented as Fig. 5(a), can be straight-

forwardly calculated as with calculation of expectation values shown in Fig. 4. As a result, we

have C1(r, t) = 0.In other words, qubit (i, j )and qubit (i+r, j)cannot be correlated for an

arbitrary 2D DUQC in the time region. Second, we consider the following correlation function,

which is expected to be classically intractable:

C2(r, t) = hEPRt|Oi,jOi,j+r|EPRti − Tr(Oi,j )Tr(Oi,j+r).(64)

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(a) (b)

Figure 6: Equality of a honeycomb lattice and a square lattice up to longitudinal edges. As illustrated

by (a), a honeycomb lattice can be deformed into a lattice which is made by rectangles. This deformed

lattice is equal to a square lattice up to longitudinal edges, as illustrated by (b).

We also assume that 2N≥2t−rholds. C2(r, t)can take a nonzero value only in the case of

r=t+ 1 and odd j,r=t, or r=t−1and even j. In those cases, we conjecture that C2(r, t)is

unlikely to be classically simulatable because of the following reason. C2(r, t)can be graphically

represented as Fig. 5(b). After contracting unitary and dual-unitary gates of Fig. 5(b) by using

Eq. (9) and Eq. (11), the remaining gates of it can be represented as Fig. 5(c). This is similar to

correlation functions of 1D DUQCs (see Fig. 3 of Ref. [24]), but important diﬀerence is that un-

contracted gates form a 2D tensor-network in 2D DUQCs. Because of this, calculating correlation

functions in 2D DUQCs seems to be hard for a classical computer. Therefore, classical simulata-

bility of correlation functions seems to depend on the relative position of two local observables.

It is still an open problem whether or not calculating C2(t, t)with the condition 2N≥2t−ris

BQP-hard.

4.5 General lattice pattern

So far, we only consider dynamics in a 2D square lattice. It is natural to consider a generalization

to other lattices, such as a honeycomb lattice. We note that unitaries at even-time of dynamics

deﬁned in Sec. 4.1 can be chosen as identity gates.

If a unitary gate U(j,k),(j,k+1) is an identity gate at all time, the edge between qubit (j, k)and

qubit (j, k + 1) can be eﬀectively removed. So, if we construct lattices by eliminating edges in k-

direction, 2D DUQCs with such a lattice can be treated as with ones with a 2D square lattice. We

name such lattices solvable lattices. Such lattices include, for example, a honeycomb lattice. This

is illustrated by Fig. 6. One can construct an arbitrary solvable lattice in the same way. Therefore,

local expectation values and correlation functions C1of 2D DUQCs with solvable lattices are

classically simulatable at an early time as same as those with square lattices.

5 Conclusion and discussion

We have investigated computational complexity of the problems calculating physical properties

of 1D and 2D DUQCs. First, we have shown that the complexity of calculating local expectation

values of dual-unitary quantum circuits highly depends on their circuit depth. Second, we have

shown that classical simulation of sampling from 1D and 2D DUQCs after linear and constant

depth, respectively, is hard.

The ﬁrst result is in contrast to conventional classically simulatable quantum circuits with ﬁxed

gate sets such as Cliﬀord circuits and matchgates, whose classical simulatability does not change

depending on the circuit depth. The dual-unitary quantum computational model is the ﬁrst example

of the model, where quantum computational power makes a transition between O(N) time and

poly(N) time. It is reminiscent of dynamical phase transitions of computational complexity which

have been recently studied in other contexts [44–46]. It would be interesting to investigate whether

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or not local expectation values of DUQCs at an early time slightly later than that considered

in this paper are also classically simulatable. Another future direction would be to study the

computational power of DUQCs in linear depth with non-solvable initial states. In this context,

it is well known that Cliﬀord circuits and matchgates circuits with certain initial states, so-called

magic states, have potential to outperform those without magic states [47,48]. Thus, it is natural

to expect that there exist magic states enhancing the computational power of DUQCs at an early

time, and we leave it for future work.

In the second result, we have considered classical sampling from a DUQC with a multiplicative

error. Here, we note that classical simulation of sampling with an additive error from certain

quantum computing models, such as linear optical circuits [49], IQP circuits [50], and the DQC1

model [51], is known to be hard under plausible assumptions. With this in mind, because of

the computational universality of DUQCs, sampling with an additive error from DUQCs which

simulates IQP circuits is also intractable for classical computers. Another sampling problem,

which attracts much attention from both theorists and experimentalists, is random circuit sampling

[52,53]. It would be interesting to investigate whether or not classical sampling with an additive

error from a random DUQC, where every gate is chosen randomly from the set of dual-unitary

gates, is hard.

Moreover, we remark that our argument on the computational universality and sampling com-

plexity of one- and two-dimensional DUQCs can be straightforwardly extended to the case of

open boundary conditions (OBCs). However, under the condition, classical simulation of DUQCs

at an early time becomes harder since, in general, unitary gates at the boundary of causal-cone

cannot be cancelled. We discuss this point in some more detail in Appendix D, and we leave a full

characterization of their computational power for future work.

Besides, because analytic research on quantum dynamics in 2D systems is much more diﬃcult

than 1D cases, generalization of DUQC to two spatial dimensions are also interesting as a solvable

model of quantum dynamics. It would be important to generalize DUQCs to higher than two

spatial dimensions and construct more general solvable initial states, for example, using higher-

dimensional tensor-network states such as projected entangled pair states (PEPSs) [54] . We expect

that a high-dimensional DUQC will deepen our understanding of a non-equilibrium phenomenon

in a high-dimensional quantum system, such as a 2D self-dual kicked Ising model.

6 ACKNOWLEDGMENTS

This work is supported by MEXT Quantum Leap Flagship Program (MEXT QLEAP) Grant Num-

ber JPMXS0118067394 and JPMXS0120319794. KM is supported by JST PRESTO Grant No.

JPMJPR2019 and JSPS KAKENHI Grant No. 20K22330. KF is supported by JSPS KAKENHI

Grant No. 16H02211, JST ERATO JPMJER1601, and JST CREST JPMJCR1673.

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A A calculation procedure for local expectation values of a chain of EPR

pairs in 1D DUQCs

In this appendix, we show that local expectation values for 2N≥l+ 2(t−1):

→

→→

→

→

→

←

←

←

←

←

←

𝑂

→

→

→

←

←

←

←

←

←

←

←

→

→

→

→

→

=

(65)

is equal to 1

2lTr(O). The procedure is similar to that in Refs. [21,22,24].

To begin with, Eq. (65) is equal to

←

←

←

←

←

←

←

←

←

𝑂

→

→

→

→

→

→

→

→

→

←

←

←

←

←

→

→

→

→

→

.(66)

By adapting Eq. (11) to the leftmost and rightmost unitaries, Eq. (66) becomes

←

←

←

←

←

←

←

←

←

𝑂

→

→

→

→

→

→

→

→

→

←

←

←

→

→

→

.(67)

By repeating the process, Eq. (67) becomes

←

←

←

𝑂

→

→

→

.(68)

Finally, by using Eq. (9) repeatedly, Eq. (68) becomes

𝑂

.(69)

Taking the renormalization coeﬃcient into account, one obtains hΨN

t|O|ΨN

ti=1

2lTr(O).

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B Local expectation values of general solvable initial states.

In this appendix, we show that local expectation values at time twith general solvable states are

approximated by 1

2lTr(O). Local expectation values without normalization can be written as

＝

𝐴∗𝐴∗𝐴∗𝐴∗𝐴∗𝐴∗

𝐴∗𝐴∗𝐴∗𝐴∗

𝐴𝐴𝐴𝐴𝐴𝐴

𝐴𝐴𝐴𝐴

𝑂

𝑈

𝑈!

,(70)

where Nand Mare the total number of tensors Aand ones which are outside of the causal-cone,

respectively. By inserting identity operators I=Pχ

α=1 |αihα|, one obtains

𝐴∗𝐴∗𝐴∗𝐴∗𝐴∗𝐴∗

𝐴 𝐴 𝐴 𝐴 𝐴 𝐴

𝐴∗𝐴∗𝐴∗𝐴∗

𝐴 𝐴 𝐴 𝐴

𝑂

𝑈

𝑈!

.(71)

The second factor of Eq. (71) is equal to a matrix element of EM, where Eis the transfer matrix

deﬁned in Eq. (20). By the assumption that |Iiis the unique eigenvector of the matrix Ewith the

latgest eigenvalue 1, the Jordan canonical form of Ecan be written as

E=|IihI|+S D

X

i=1

(λiPi+Ni)!S−1,(72)

where Dis the number of Jordan blocks, Piis the diagonal part, Niis the nilpotent part, and λiis

less than 1and ordered in descending order. Let εbe Pi(λiPi+Ni). Then, the n-th power of a

transfer matrix EMcan be written as EM=|IihI|+SεMS−1.

Now we evaluate local expectation values. First, in the same way as the calculation in Ap-

pendix A, we obtain that

𝐴∗𝐴∗𝐴∗

𝐴 𝐴 𝐴

𝑂

𝑈

𝑈!

(73)

is equal to 1

2lTr(O), which arises from |IihI|and is a dominant term of Eq. (71).

Next, we calculate the error term =hΨN

t|O|ΨN

ti − 1

2lTr(O), which arises from SεMS−1.

First, we evaluate the second factor of Eq. (71), namely, hβ0β|SεMS−1|α0αi. We denote the

L2-norm of the vector |aiby k|aik and the L2operator norm of a matrix Bby kBk. Because of

max

k|xik=k|yik=1 |hy|B|xi| =kBk,(74)

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where Bis a matrix, the following holds:

hβ0β|SεMS−1|α0αi≤kSk·kS−1k·kεMk.(75)

By the binomial theorem, εMis expanded as

εM=

D

X

i=1

di−1

X

j=0

MCjλM−j

iPiNj

i

,(76)

where diis the dimension of the i-th Jordan block, that is, the dimension of the space on which Pi

acts nontrivially, and we used the fact that Ndi

iis the zero matrix. Then, kεMkis upper bounded

as follows:

kεMk ≤ v

u

u

u

t

χ2

X

α,β=1 (εM)αβ

2(77)

≤rχ4max

i,j (MCj)2λ2(M−j)

i(78)

=χ2max

i,j MCjλM−j

i,(79)

for j= 0,1,··· , di, where the ﬁrst inequality follows from the fact that for a matrix B,kBkis

less than its Frobenius norm kBkF=qPi,j=1 B2

ij [55], and the second inequality follows from

the fact that the maximum value of matrix elements of εMis maxi,j MCjλM−j

idue to Eq. (76)

. From Eqs. (75) and (79), the following holds:

hβ0β|SεMS−1|α0αi=O(λM

1),(80)

where we used the inequality di< χ2and the assumption χ=O(1).

Second, we evaluate the ﬁrst factor of Eq. (71), namely,

𝐴∗𝐴∗𝐴∗

𝐴𝐴𝐴

𝑂

𝑈

𝑈!

.(81)

Let |Ψ(α,β)ibe,

𝐴 𝐴 𝐴

.(82)

Then, Eq. (81) can be written as

hΨ(α0,β0)|U†OU |Ψ(α,β)i,(83)

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and this is upper bounded as follows:

hΨ(α0,β0)|U†OU |Ψ(α,β)i(84)

=kΨ(α0,β0)k·kΨ(α,β )k · hΨ(α0,β0)|

kΨ(α0,β0)kU†OU |Ψ(α,β )i

kΨ(α,β)k(85)

≤ kΨ(α0,β0)k·kΨ(α0,β 0)k(86)

=hα0α0|EN−M|β0β0i1/2hαα|EN−M|ββi1/2(87)

=hα0α0|(|IihI|+SεMS−1)|β0β0i1/2hαα|(|Ii hI|+SεMS−1)|ββi1/2(88)

≤1 + χ2max

i,j MCjλM−j

i(89)

=O(1) (90)

where in the ﬁrst inequality, we used Eq. (74) and kOk= 1, and in the second inequality, we used

Eqs. (75), (79), and hkk|Ii=1

χ, for any k∈ {α, α0, β, β0}.

From Eqs. (80) and (90), the following holds:

ε=O(λM

1).(91)

Because of the condition that M= 2N−l−2tin Eq. (33), εbecomes O(λ2N−l−2t

1). In addition,

because hΨN

t|ΨN

tiis 1 + O(λN

1), normalized local expectation values hO(t)iis

hO(t)i= Tr(O) + O(λ2N−l−2t

1) + O(λN

1).(92)

Then, if tsatisﬁes t≤ b(1 −δ)Nc−l/2for some 0< δ < 1, the error term O(λ2N−l−2t

1)become

O(λbδN c

1). Therefore, local expectation values with t≤ b(1 −δ)Nc − l/2, for 0< δ < 1, are

classically simulatable.

C Quantum circuit representation of a 2D self-dual kicked Ising model

In this appendix, we show that a 2D self-dual kicked Ising model is represented as a 2D DUQC.

Let us consider a 2D self-dual kicked Ising model, associated with a 2N ×2N square lattice:

H2DKI(t) = HI+P∞

n=−∞ δ(t−n)HK,(93)

HI=P2N

j,k=1 {J(Zj,kZj+1,k +Zj,kZj,k+1) + hZj,k },(94)

HK=bP2N

j,k=1 Xj,k,(95)

where δ(t)is the Dirac delta function, |J|and |b|are equal to π

4,his an arbitrary real number, and

we adopted PBCs, that is, Zj,2N+1 =Zj,1and Z2N+1,k =Z1,k. The Floquet operator associated

to Eq. (93) can be written as

UKI BTe−R1

0dtH(t)=UKUI1 UI2UI3UI4,(96)

where Tdenotes a time ordered product, and we deﬁne UK,UI1,UI2,UI3, and UI4 as follows:

Accepted in Quantum 2022-01-12, click title to verify. Published under CC-BY 4.0. 23

UKBe−P2N

j,k=1 bXj,k ,(97)

UI1 Be−PN

j=1 P2N

k=1(J Z2j,k Z2j+1,k +hZ2j,k),(98)

UI2 Be−P2N

j=1 PN

k=1 JZj,2k−1Zj,2k,(99)

UI3 Be−PN

j=1 P2N

k=1(J Z2j−1,k Z2j,k +hZ2j−1,k),(100)

UI4 Be−P2N

j=1 PN

k=1 JZj,2kZj,2k+1 .(101)

Then, the integer powers of the Floquet operator have forms:

U2t

KI =UKUI3UI2UI4UKI1 (UI2UI4UKI3 UI2UI4 UKI1)t−1UI2UI4 UI3,(102)

U2t+1

KI =UKUI1(UI2UI4UKI3 UI2UI4UKI1 )tUI2UI4 UI3,(103)

where we deﬁne

UKI1 =UI1UKUI1,(104)

UKI3 =UI3UKUI3.(105)

Using the fact that UKI1 and UKI3 are written as dual-unitary gates [22], it follows that the quantum

circuit (UI2UI4UKI3UI2 UI4UKI1)tis one of 2D DUQCs. We note that this is even true if inter-

action strength Jin Eqs. (99) and (101) are replaced by arbitrary real numbers. This is because

unitary gates of 2D DUQCs in k-direction can be chosen arbitrarily.

D Local observables of DUQCs with open boundary conditions

In this appendix, we discuss classical simulatability of DUQCs under OBCs. First, we show that

local expectation values of 1D DUQCs with OBCs at an early time become dependent on dual-

unitary gates but still classically simulatable. We deﬁne 1D DUQCs on 2N qubits with OBCs as

the following:

V(t) =

→

→

→

→

→

→

→

→

→

→

→

→

→

→

.(106)

Solvable initial states with OBCs are deﬁned as with Eq. (82). Then, expectation values of local

observables O, after removing dual-unitary gates outside of the causal-cone, can be written as

hΨ(α∗,β∗)|V(t)†OV(t)|Ψ(α,β)i=

𝐴∗𝐴∗𝐴∗𝐴∗𝐴∗𝐴∗

𝐴∗𝐴∗

𝐴 𝐴 𝐴 𝐴 𝐴 𝐴

𝐴 𝐴

→

→→

→

→

→

←

←

←

←

←

←

𝑂

→

→

→

←

←

←

←

←

←

←

←

→

→

→

→

→

…

…

𝐴∗

𝐴

𝐴∗

𝐴

,(107)

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where without loss of generality we assume that a local observable is supported on the left half of

chain. From the argument in Appendix Cand by using Eqs. (11) and (22), it can be easily shown

that Eq. (107) is exponentially close to

𝐴∗

𝐴

←

←

←

←

→

→

→

→

𝑂

𝐴∗

𝐴

.(108)

It is dependent only on dual-unitary gates on the boundary of the causal-cone, and therefore it is

classically simulatable.

However, when we generalize the above argument to two-dimensional cases, local expectation

values do not seem to be classically simulatable in linear depth because uncontracted unitary gates

on the boundary of the causal-cone form 2D tensor-networks. This situation is similar to that

of correlation functions for 2D DUQCs discussed in Sec. 4.4 of the main text. Besides, it is

reminiscent of matchgate circuits, where classical simulatability depends on their connectivity

[56]. As written in Sec. 5, it would be interesting future work to characterize the computational

power of DUQCs with various connectivity.

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