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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:
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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.
References
[1] D Gottesman. Stabilizer codes and quantum error correction. PhD thesis, California Institute
of Technology, 1997.
[2] Leslie G Valiant. Quantum circuits that can be simulated classically in polyno-
mial time. SIAM Journal on Computing, 31(4):1229–1254, 2002. doi:10.1137/
S0097539700377025.
[3] Barbara M Terhal and David P DiVincenzo. Classical simulation of noninteracting-fermion
quantum circuits. Physical Review A, 65(3):032325, 2002. doi:10.1103/PhysRevA.65.
032325.
[4] Sergey Bravyi. Lagrangian representation for fermionic linear optics. Quantum information
&computation, 5, 05 2004. doi:10.26421/QIC5.3-3.
[5] Richard Jozsa and Akimasa Miyake. Matchgates and classical simulation of quantum cir-
cuits. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,
464(2100):3089–3106, 2008. doi:10.1098/rspa.2008.0189.
[6] Easwar Magesan, Jay M Gambetta, and Joseph Emerson. Scalable and robust randomized
benchmarking of quantum processes. Physical review letters, 106(18):180504, 2011. doi:
10.1103/PhysRevLett.106.180504.
[7] Easwar Magesan, Jay M. Gambetta, and Joseph Emerson. Characterizing quantum gates via
randomized benchmarking. Phys. Rev. A, 85:042311, Apr 2012. doi:10.1103/PhysRevA.
85.042311.
[8] Jonas Helsen, Sepehr Nezami, Matthew Reagor, and Michael Walter. Matchgate bench-
marking: Scalable benchmarking of a continuous family of many-qubit gates, 2020. arXiv:
2011.13048.
Accepted in Quantum 2022-01-12, click title to verify. Published under CC-BY 4.0. 25
[9] Mark Howard and Earl Campbell. Application of a resource theory for magic states to fault-
tolerant quantum computing. Physical review letters, 118(9):090501, 2017. doi:10.1103/
PhysRevLett.118.090501.
[10] Yasunari Suzuki, Keisuke Fujii, and Masato Koashi. Efficient simulation of quantum error
correction under coherent error based on the nonunitary free-fermionic formalism. Physical
review letters, 119(19):190503, 2017. doi:10.1103/PhysRevLett.119.190503.
[11] Anatoli Polkovnikov, Krishnendu Sengupta, Alessandro Silva, and Mukund Vengalattore.
Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod.
Phys., 83:863–883, Aug 2011. doi:10.1103/RevModPhys.83.863.
[12] Jens Eisert, Mathis Friesdorf, and Christian Gogolin. Quantum many-body systems out of
equilibrium. Nature Physics, 11(2):124–130, 2015. doi:10.1038/nphys3215.
[13] Luca D’Alessio, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol. From quantum chaos
and eigenstate thermalization to statistical mechanics and thermodynamics. Advances in
Physics, 65(3):239–362, 2016. doi:10.1080/00018732.2016.1198134.
[14] Anushya Chandran and CR Laumann. Semiclassical limit for the many-body localization
transition. Physical Review B, 92(2):024301, 2015. doi:10.1103/PhysRevB.92.024301.
[15] Sarang Gopalakrishnan and Bahti Zakirov. Facilitated quantum cellular automata as sim-
ple models with non-thermal eigenstates and dynamics. Quantum Science and Technology,
3(4):044004, 2018. doi:10.1088/2058-9565/aad759.
[16] David Berenstein and Jiayao Zhao. Exotic equilibration dynamics on a 1-d quantum cnot
gate lattice, 2021. arXiv:2102.05745.
[17] Adam Nahum, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. Quantum entanglement
growth under random unitary dynamics. Physical Review X, 7(3):031016, 2017. doi:10.
1103/PhysRevX.7.031016.
[18] Adam Nahum, Sagar Vijay, and Jeongwan Haah. Operator spreading in random unitary
circuits. Physical Review X, 8(2):021014, 2018. doi:10.1103/PhysRevX.8.021014.
[19] CW Von Keyserlingk, Tibor Rakovszky, Frank Pollmann, and Shivaji Lal Sondhi. Opera-
tor hydrodynamics, otocs, and entanglement growth in systems without conservation laws.
Physical Review X, 8(2):021013, 2018. doi:10.1103/PhysRevX.8.021013.
[20] Fabian HL Essler and Maurizio Fagotti. Quench dynamics and relaxation in isolated in-
tegrable quantum spin chains. Journal of Statistical Mechanics: Theory and Experiment,
2016(6):064002, 2016. doi:10.1088/1742-5468/2016/06/064002.
[21] Sarang Gopalakrishnan and Austen Lamacraft. Unitary circuits of finite depth and infinite
width from quantum channels. Physical Review B, 100(6):064309, 2019. doi:10.1103/
PhysRevB.100.064309.
[22] Bruno Bertini, Pavel Kos, and Toma ˇ
z Prosen. Exact correlation functions for dual-unitary
lattice models in 1 + 1 dimensions. Phys. Rev. Lett., 123:210601, Nov 2019. doi:10.1103/
PhysRevLett.123.210601.
[23] Bruno Bertini, Pavel Kos, and Tomaz Prosen. Operator Entanglement in Local Quantum
Circuits I: Chaotic Dual-Unitary Circuits. SciPost Phys., 8:67, 2020. doi:10.21468/
SciPostPhys.8.4.067.
[24] Lorenzo Piroli, Bruno Bertini, J. Ignacio Cirac, and Toma ˇ
z Prosen. Exact dynamics in dual-
unitary quantum circuits. Phys. Rev. B, 101:094304, Mar 2020. doi:10.1103/PhysRevB.
101.094304.
Accepted in Quantum 2022-01-12, click title to verify. Published under CC-BY 4.0. 26
[25] Michael J Bremner, Christopher M Dawson, Jennifer L Dodd, Alexei Gilchrist, Aram W Har-
row, Duncan Mortimer, Michael A Nielsen, and Tobias J Osborne. Practical scheme for quan-
tum computation with any two-qubit entangling gate. Physical review letters, 89(24):247902,
2002. doi:10.1103/PhysRevLett.89.247902.
[26] Michael J Bremner, Richard Jozsa, and Dan J Shepherd. Classical simulation of commuting
quantum computations implies collapse of the polynomial hierarchy. Proceedings of the
Royal Society A: Mathematical, Physical and Engineering Sciences, 467(2126):459–472,
2011. doi:10.1098/rspa.2010.0301.
[27] Tianci Zhou and Adam Nahum. Entanglement membrane in chaotic many-body systems.
Phys. Rev. X, 10:031066, Sep 2020. doi:10.1103/PhysRevX.10.031066.
[28] Boris Gutkin, Petr Braun, Maram Akila, Daniel Waltner, and Thomas Guhr. Exact local cor-
relations in kicked chains. Phys. Rev. B, 102:174307, Nov 2020. doi:10.1103/PhysRevB.
102.174307.
[29] Pieter W. Claeys and Austen Lamacraft. Maximum velocity quantum circuits. Phys. Rev.
Research, 2:033032, Jul 2020. doi:10.1103/PhysRevResearch.2.033032.
[30] Bruno Bertini and Lorenzo Piroli. Scrambling in random unitary circuits: Exact results.
Phys. Rev. B, 102:064305, Aug 2020. doi:10.1103/PhysRevB.102.064305.
[31] Pavel Kos, Bruno Bertini, and Toma ˇ
z Prosen. Correlations in perturbed dual-unitary cir-
cuits: Efficient path-integral formula. Phys. Rev. X, 11:011022, Feb 2021. doi:10.1103/
PhysRevX.11.011022.
[32] Pieter W. Claeys and Austen Lamacraft. Ergodic and nonergodic dual-unitary quantum cir-
cuits with arbitrary local hilbert space dimension. Phys. Rev. Lett., 126:100603, Mar 2021.
doi:10.1103/PhysRevLett.126.100603.
[33] Bruno Bertini, Pavel Kos, and Tomaz Prosen. Random matrix spectral form factor of dual-
unitary quantum circuits. Communications in Mathematical Physics, 387:597–620, Jul 2021.
doi:10.1007/s00220-021-04139-2.
[34] Suhail Ahmad Rather, S. Aravinda, and Arul Lakshminarayan. Creating ensembles of dual
unitary and maximally entangling quantum evolutions. Phys. Rev. Lett., 125:070501, Aug
2020. doi:10.1103/PhysRevLett.125.070501.
[35] S. Aravinda, Suhail Ahmad Rather, and Arul Lakshminarayan. From dual-unitary to quan-
tum bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hi-
erarchy. Phys. Rev. Research, 3:043034, Oct 2021. doi:10.1103/PhysRevResearch.3.
043034.
[36] Felix Fritzsch and Toma ˇ
z Prosen. Eigenstate thermalization in dual-unitary quan-
tum circuits: Asymptotics of spectral functions. Phys. Rev. E, 103:062133, Jun 2021.
URL: https://link.aps.org/doi/10.1103/PhysRevE.103.062133,doi:10.1103/
PhysRevE.103.062133.
[37] Isaac Reid and Bruno Bertini. Entanglement barriers in dual-unitary circuits. Phys. Rev. B,
104:014301, Jul 2021. doi:10.1103/PhysRevB.104.014301.
[38] Ryusuke Hamazaki. Exceptional dynamical quantum phase transitions in periodically driven
systems. Nature communications, 12(1):1–7, 2021. doi:10.1038/s41467-021-25355-3.
[39] D. Perez-Garcia,