Crystalline Quantum Circuits Grace M. Sommers1David A. Huse1and Michael J. Gullans2 1Department of Physics Princeton University Princeton NJ 08544 USA

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Crystalline Quantum Circuits

Grace M. Sommers,1David A. Huse,1and Michael J. Gullans2

1Department of Physics, Princeton University, Princeton, NJ 08544, USA

2Joint Center for Quantum Information and Computer Science,

NIST/University of Maryland, College Park, Maryland 20742, USA

(Dated: August 2, 2023)

Random quantum circuits continue to inspire a wide range of applications in quantum infor-

mation science and many-body quantum physics, while remaining analytically tractable through

probabilistic methods. Motivated by an interest in deterministic circuits with similar applications,

we construct classes of nonrandom unitary Cliﬀord circuits by imposing translation invariance in

both time and space. Further imposing dual-unitarity, our circuits eﬀectively become crystalline

spacetime lattices whose vertices are SWAP or iSWAP two-qubit gates and whose edges may contain

one-qubit gates. One can then require invariance under (subgroups of) the crystal’s point group.

Working on the square and kagome lattices, we use the formalism of Cliﬀord quantum cellular

automata to describe operator spreading, entanglement generation, and recurrence times of these

circuits. A full classiﬁcation on the square lattice reveals, of particular interest, a “nonfractal good

scrambling class” with dense operator spreading that generates codes with linear contiguous code

distance and high performance under erasure errors at the end of the circuit. We also break unitar-

ity by adding spacetime-translation-invariant measurements and ﬁnd a class of such circuits with

fractal dynamics.

I. INTRODUCTION

Random quantum circuits are a model system of

many-body quantum physics, in which the degrees of

freedom are qubits or qudits and the evolution under

a local Hamiltonian is modeled by local unitary gates.

Random unitary circuits thus provide a platform for

analytic computation of, for example, out-of-time-order

correlators and entanglement growth [1–4]. They also

have numerous applications to quantum complexity the-

ory [1, 5–9], tomography [10, 11], benchmarking [12, 13],

and circuit complexity bounds [14, 15]. A particular mo-

tivation for this work comes from the ﬁeld of quantum

error correction, where random circuits have also played

an important role [16, 17]. For example, random ﬁnite-

rate stabilizer codes have linear code distance and reach

channel capacity, and their performance under erasure

errors can be modeled by random matrix theory [18].

Randomness has also proven useful for improving the

error threshold and logical error rates of surface codes

under biased noise, through random Cliﬀord-gate defor-

mations [19].

While randomness is a valuable theoretical tool for

studying quantum circuit dynamics, ultimately, there is

a need for deterministic circuits with similar applica-

tions. For example, the behavior of practically relevant

algorithms may not be well captured by random cir-

cuits. Indeed, in the case of the variational quantum

eigensolver (VQE), initializing the solver with random

circuits leads to barren plateaus in the gradient [20, 21].

Nonrandom circuits are likely to be more natural for

many applications and avoid these barren plateaus. In

the context of quantum simulation algorithms, one may

question whether generic Hamiltonian evolution dis-

plays the same phenomena as random circuits. The

growth of quantum circuit complexity with evolution

time is not understood outside random circuits [22]. In

addition, speciﬁc circuit families with more identiﬁable

structure have been necessary to boost the performance

of gate-set tomography in practical use cases [23], and

are likely to play a crucial role in the eﬃcient veriﬁcation

of quantum advantage on near-term devices [24]. Even

addressing these questions from a conceptual point of

view or providing a route towards future progress can

be useful. From a theoretical computer science perspec-

tive, this research avenue has echoes of “derandomiza-

tion”. In classical complexity theory, this term refers

to the process of turning probabilistic algorithms into

deterministic ones as part of the quest to prove that

the latter are just as powerful (i.e., BP P =P) [25].

Similarly, in the theory of expander graphs and error-

correcting codes, derandomization refers to the art of

ﬁnding explicit constructions for objects only known to

exist from probabilistic arguments [26].

Here we take a less formal, more physical view of

the problem by analyzing a class of deterministic cir-

cuits with “translational” invariance in both time and

space. These spacetime translation-invariant (STTI)

circuits are endowed with two special features that en-

able an analytic treatment while still allowing for er-

godic dynamics. First, all the gates are dual-unitary,

namely, unitary when viewed in the spatial direction

as well as the usual time direction. As a nontrivial

model of quantum chaos with certain exactly solvable

correlation functions, dual-unitary circuits are the sub-

ject of a rich, rapidly developing literature on which we

build [27–38]. Second, the gates in our circuits are Clif-

ford. Cliﬀord circuits hold appeal because they can be

classically simulated in polynomial time [39, 40], yet are

physically relevant in the sense that the n-qudit uniform

Cliﬀord ensemble is a unitary 2-design for the n-qudit

Haar ensemble [41] if the qudit dimension is a prime

power [42] (and in fact a 3-design if the qudit dimen-

sion is a power of 2 [42, 43]). Analytically, Cliﬀord

arXiv:2210.10808v3 [quant-ph] 1 Aug 2023

circuits with spacetime randomness obey eﬀective hy-

drodynamic equations [2, 3, 44], while spatially random

Floquet Cliﬀord circuits can exhibit strong localization

in 1+1D [45–47]. In the present work, with spacetime

translation invariance, our circuits can be interpreted

as quantum cellular automata (QCA) [48, 49], and re-

stricting to Cliﬀord gates allows us complement the ex-

act methods for treating dual-unitary circuits with the

tools of symplectic cellular automata [50–52].

Cliﬀord quantum cellular automata (CQCA) on

prime-dimensional qudits with spatial period a= 1 have

received a thorough treatment in earlier work, but to

our knowledge there is no systematic classiﬁcation of

automata with a > 1 and beyond. Our primary focus in

this work is on brickwork dual-unitary Cliﬀord circuits,

which naturally are expressed as qubit CQCA with

a= 2 and exhibit richer behavior than a= 1. We high-

light several physical properties of these circuits that can

be gleaned from the symplectic automaton representa-

tion, including fractality in operator spreading and re-

currence times. In addition to classifying and situating

these circuits within the broader context of CQCA, we

extend the concept of “self-dual-unitary” gates—gates

such as the SWAP gate whose spacetime rotation is not

only unitary, but in fact invariant [28, 53]—to all the

point group symmetries of the lattice, associated with

dual-unitarity, time reversal, and reﬂection. We further

generalize to (self-) tri-unitary [54] automata using the

kagome lattice, for which a= 4 and we can deﬁne 3 axes

of time with unitary evolution.

On the quantum information side, we focus in this

work on the applications to quantum error correction.

We highlight a class of CQCA on the square lattice

in which initially local operators scramble and spread

densely within the lightcone, which can serve as en-

coding circuits for ﬁnite-rate codes with high perfor-

mance under erasure errors and whose quasicyclic struc-

ture [55, 56] could provide a path toward eﬃcient decod-

ing under more general noise [57–61]. More broadly, our

results on these speciﬁc classes of quantum dynamics

have potential applications in the same areas as ran-

dom circuits, including benchmarking, quantum chaos,

and complexity theory.

A. Outline

The paper proceeds as follows. Sec. II provides a

high-level overview of our results. As a case study

in the most novel class of circuits discovered in our

work, Sec. III details the behavior of the “dense good

scrambling class” on the square lattice Sec. III. Taking

a step back, in Sec. IV, we deﬁne the general models

in detail and demonstrate how the symmetry transfor-

mations are enacted at the level of the one- and two-

qubit gates. To gain greater insight into these symme-

tries, we introduce the CQCA formalism and show how

the corresponding matrices transform under rotations

and reﬂections of the lattice, in Sec. V. Sec. VI spe-

FIG. 1: The convention for the dual gate used in this

paper [31, 34]: given a unitary gate Uβ1β2

α1α2(left), we rotate

the spacetime axes by π/2 (center) to obtain the dual

Uβ2α2

β1α1(right). If Uis dual-unitary, then ˜

Uis unitary (as is

the spacetime rotation in the opposite direction).

cializes to the square lattice, classifying the SWAP-core

and iSWAP-core a= 2 automata including the nonfrac-

tal good scrambling class. In Sec. VII, we turn to the

kagome lattice, where the circuits are described by a= 4

CQCA. Returning to the square lattice, in Sec. VIII we

describe the fractal structure that arises when we in-

troduce projective measurements. Finally, we conclude

in Sec. IX with a discussion of future research avenues.

II. OVERVIEW

Before presenting our methods and results in de-

tail, we begin with an overview of our ﬁndings. The

two common features of the STTI circuits considered

in this work—dual-unitarity and Cliﬀordness—provide

complementary avenues for study.

A. Symmetries, dual-unitarity, and tri-unitarity

The circuits we consider are all crystalline lattices, in

which vertices correspond to gates and edges correspond

to qubits, possibly dressed with single-qubit gates. Fo-

cusing our attention on two-qubit gates, we choose lat-

tices with coordination number z= 4. In addition to

spacetime translation invariance, the bare lattices are

invariant under the rotations and reﬂections that com-

prise their point group [62]. In the circuit perspective,

however, vertices are no longer pointlike objects, and

edges have a directionality imposed by the single-qubit

gates. We can therefore ask which of the symmetries of

the lattice are also symmetries of the circuit.

One main thrust of this work is organizing and classi-

fying these symmetries for two such lattices, square and

kagome. Implicit in this analysis is that the transformed

gates are unitary. For two-qubit gates, this imposes

dual-unitarity: rotating the gate by π/2 in spacetime

yields another unitary gate (Fig. 1). From the parame-

terization of dual-unitary gates in Ref. [28], restricting

to the Cliﬀord group, the dual-unitary operator Uim-

plemented by the gate can be written as either a SWAP

core (non-entangling) or iSWAP core (maximally entan-

gling), with single-qubit Cliﬀord gates on each of the

four legs.

Our main model is the brickwork circuit shown

in Fig. 2, a square lattice of SWAP or iSWAP cores,

FIG. 2: STTI dual-unitary brickwork circuit represented

as a rotated square lattice. Black squares are (i)SWAP

cores. Edges are decorated with single-qubit Cliﬀord gates,

represented as red and blue circles. One time step is

deﬁned as two layers of the brickwork circuit.

with single-qubit gates on each edge. The bare SWAP

and iSWAP cores are “self-octa-unitary” since they are

invariant under all eight point group transformations of

the square. With the inclusion of single-qubit gates,

the resulting STTI circuit can have some, all, or none

of these symmetries. This is the focus of Sec. IV.

On the kagome lattice, whose point group is D6in-

stead of D4, we can deﬁne three axes (six arrows)

of time, making these circuits (self)-tri-unitary. In

Ref. [54], where tri-unitarity is ﬁrst introduced, tri-

unitary gates are deﬁned on three qubits and tiled on

a triangular lattice. However, as the authors note, the

family of tri-unitary gates considered in that paper can

be decomposed into three two-qubit gates, and the re-

sulting circuit can then be expressed on the kagome

lattice. The three axes of time restrict the two-point

correlations between traceless one-site operators aver-

aged over all states to vanish except at x1−x2= 0 and

at |x1−x2|=v|t1−t2|where vis the velocity of the

lightcone.

B. Classiﬁcation of CQCA

Because our circuits are both STTI and Cliﬀord, we

can represent them as Cliﬀord quantum cellular au-

tomata (CQCA), which is the primary analytic tech-

nique used in this work. For a more detailed intro-

duction to the CQCA formalism, the reader is referred

to Sec. V and to Refs. [50–52].

The circuit in Fig. 2 is translation-invariant with a

unit cell of T= 1/2, a= 2, composed with a shift

by 1 site, so it can be treated as an “a= 2 automa-

ton.” In Sec. VI, we classify all iSWAP-core automata

on the square lattice into six classes, where members of

each class are related by a reﬂection about the center

of the gate, and/or a change of basis. The point group

transformations exchange members of the same class.

A similar classiﬁcation scheme can be applied on the

kagome lattice, where a= 4, but in Sec. VII we focus

our attention on those with a high amount of symmetry,

the “self-tri-unitary” circuits.

Since the Cliﬀord group normalizes the Pauli group,

the dynamics under a Cliﬀord circuit with spatial pe-

riod ais fully encoded (modulo phases) by the image

of Xiand Zion each site i= 1, ..., a of the unit cell.

Leveraging this translation invariance, a CQCA with a

unit cell containing aqudits is described by a 2a×2a

matrix M, whose entries are Laurent polynomials in the

variable uwhich labels the unit cell [63].

We adapt and extend to a > 1 the techniques pre-

sented in foundational works [50–52], which focus on

prime q,a= 1 automata [64]. a= 1 CQCA have

determinant u2dwhere d∈Z. Factoring out a shift

of udmakes a centered symplectic cellular automaton

(CSCA) with determinant 1, whose characteristic poly-

nomial is uniquely determined by Tr(M) [51, 52]:

χM(y) = y2+ Tr(M)y+ 1.(1)

While this simple relationship between Tr(M) and

χM(y) no longer holds for a > 1, the characteristic

polynomial remains inextricably linked to three related

properties of the automaton: entanglement generation,

operator spreading, and the recurrence time in a ﬁnite

system.

The recurrence time of the unitary, up to a phase, on

a system of Lqubits, or m=L/a unit cells (with pe-

riodic boundary conditions) is denoted τ(m), the mini-

mum power such that Mτ= mod (um−1) up to global

shifts. Under the evolution of the automaton, any sta-

bilizer group, mixed or pure, repeats modulo signs and

shifts after an interval that divides τ(m). The scal-

ing of τ(m) divides the six square lattice classes into

two groups: three for which τ(m)≤3mfor all m, and

three for which τ(m) is linear in mfor m= 2k, but

grows much faster for generic m. We also demonstrate

a sharp distinction between these two groups with re-

spect to the entanglement generation for a random ini-

tial product state. The ﬁrst group consists of “poor

scramblers,” for which the resulting Page curve [65] has

a slope less than 1, i.e. the total entropy of a subsystem

of length |A|< L/2 is f|A|, where 0 < f < 1. This sub-

maximal entanglement generation can be attributed, at

least in part, to the presence of conserved Zcharges, or

“gliders.” In particular, we ﬁnd a close connection be-

tween the “bare iSWAP class” (all single-qubit gates are

the identity) and the standard glider automaton with

a= 1 [52].

C. Fractality, dense operator spreading, and

quantum error correction

The second group of iSWAP-core automata on the

square lattice is comprised of “good scramblers”, which,

when acting on random initial product states, gener-

ate Page curves of slope 1 at times away from the re-

currences. The three classes within this group exhibit

diﬀerent fractal behavior. The fractal in question is

the footprint of an initially local Pauli operator which

spreads within the lightcone. We deﬁne the fractal di-

mension through the scaling of the cumulative number

of non-identity sites within this footprint vs. the depth

of the circuit, so that df≤2 for CQCA deﬁned in

1+1D. In the limit of inﬁnite time, the fractal structure

of the footprint depends only on the minimal polyno-

mial µMof the automaton M[66]. The minimal poly-

nomial is the lowest-degree monic polynomial µMfor

which µM(M) = 0, thus encoding a recursion relation

for M.

We refer to one class as the self-dual kicked Ising

(SDKI) class, a representative of which maps to the

SDKI model via a “boundary” circuit [28]. Without in-

voking this direct mapping at the level of gates, the con-

nection to SDKI is clear from the automata, which both

have the minimal polynomial µ(y) = y2+(u−1+1+u)y+

1. Initially local operators spread in this class of cir-

cuits with a fractal dimension df= log2[(3 + √17)/2] =

1.8325... [66]. A second good scrambling class has frac-

tal dimension df∼

=1.9, a pattern not seen in a= 1

automata [67].

Special attention is paid to the third “good scram-

bling” class, the subject of a case study in Sec. III.

We describe its operator spreading as “nonfractal” or

“dense”, because the number of X,Y, and Zsites

within a spreading operator are all a ﬁnite fraction of

the lightcone volume (df= 2). On one hand, as with

all of these dual-unitary CQCA, this nonfractal class has

large amounts of structure not seen in random Cliﬀord

circuits. In fact, a representative of this class, which

has π/2Xrotations on each leg, is self-octa-unitary.

On the other hand, it shares important features with

random circuits, including dense operator spreading. It

also has promise for error correction. Namely, when a

random initial product state with nonzero entropy den-

sity is fed into this circuit, the logical operators spread

linearly in time, so that at late times the contiguous

length of the shortest logical operator—the contiguous

code distance [68]—is linear in m. Since operators also

spread densely, we expect their weight to scale propor-

tionally to their length, which then implies a linear code

distance. Indeed, quasicyclic codes generated from ini-

tial periodic product states perform well under erasure

errors applied at the end of the circuit. Under more

general noise, the crystalline symmetries of the encod-

ing circuit could be beneﬁcial for ﬁnding eﬃcient op-

timal decoders. Note that we have not addressed the

overhead needed to make these codes or the circuits

fault-tolerant, which we leave as a problem for future

work.

D. Adding measurements

Finally, in Sec. VIII we break unitarity by adding

measurements in a STTI fashion. With one measure-

ment per doubled spacetime unit cell of the square lat-

tice, in most cases an initial fully mixed state reaches a

steady state (mixed or pure) after O(1) time steps, but

for the df∼

=1.9 good scrambling class in the appropriate

measurement basis, a fully mixed initial state puriﬁes

in mtime steps for m= 2k. During the initial tran-

sient, the state acquires volume-law entanglement, but

loses it before reaching the steady state, which has zero

entanglement. A perturbation to this product steady

state spreads as a Sierpinski gasket, a pattern not seen

on the square-lattice dual-unitary circuits without mea-

surements. We present this as just one example of the

rich menagerie of hybrid STTI circuits, deferring an

extended discussion of the hierarchical classiﬁcation of

such circuits, including those whose steady state is a

high-performing ﬁnite-rate code, to a future paper [69].

III. CASE STUDY OF THE DENSE GOOD

SCRAMBLING CLASS

As motivation for the broader classiﬁcation program

undertaken in the rest of this paper, consider a realiza-

tion of Fig. 2 in which all of the two-qubit gates (black

squares) are the iSWAP gate:

iSWAP = e−iπ

4(XX+Y Y )=





1 0 0 0

0 0 −i0

0−i0 0

0 0 0 1





(2)

and all of the single-qubit gates (red and blue circles)

are rotations by π/2 about the Xaxis on the Bloch

sphere:

RX[π/2] = e−iπ

4X.(3)

This circuit is a Cliﬀord quantum cellular automa-

ton (CQCA) with unit cell a= 2 composed solely

of dual-unitary gates, thus lending it a high degree

of structure. In fact, in addition to being spacetime

translation-invariant (STTI), the class to which this

(RX[π/2], RX[π/2]) circuit belongs is the only one, be-

sides the “bare iSWAP class” (in which all the single-

qubit gates are the identity), that contains circuits

left invariant under the 8 rotations and reﬂections of

the unit cell of the square lattice. We call this prop-

erty “strong self-octa-unitarity” and deﬁne it formally

in Sec. IV.

On the other hand, the dynamics under this circuit

is in many ways reminiscent of random Cliﬀord cir-

cuits, with local operators spreading densely rather than

as fractals, and with initial product states evolving to

volume-law-entangled states whose Page curve has slope

1. In this section, we explore this dichotomy between

structure and scrambling and discuss the application of

these circuits to developing codes with linear distance.

We will revisit these concepts in more general settings

throughout the paper.

A. Recurrence times

An immediate diﬀerence from random circuits is the

presence of recurrences: since the dynamics are Floquet,

Cliﬀord, and unitary, any initial state on a ﬁnite system

must eventually repeat under the action of the circuit.

To wit, there are QL−1

k=0 (2L−k+ 1) = O(2cL2) unique

stabilizer groups (modulo signs) on Lqubits [40], which

places an upper bound on the recurrence time.

In fact, for all m, where m=L/a is the number

of unit cells with periodic boundary conditions, the re-

currence time τ(m) is well below this bound. Of spe-

cial note are system sizes m= 2k, for which τ(m)

grows linearly. This linear trend in τ(m) for STTI cir-

cuits has been proven for m= 2kin a= 1 CQCA

over qubits [4, 70] as well as for m=qkin a class

of dual-unitary circuits known as perfect permutation

maps, where the odd prime qis the dimension of the

qudits [34].

What distinguishes this circuit and the other good

scrambling classes from the “poor scrambling” classes

discussed in Sec. VI C is the trend in m̸= 2k. As the

example of Ref. [34] indicates, the sensitivity in our good

scrambling circuits to the power of 2 is related to the on-

site Hilbert space dimension q= 2. As shown in Fig. 3,

τ(m) is strongly nonmonotonic in m. A curious trend,

left for the interested reader to ponder, is that if we

write m=j2k, then τ(m)/2kis either 2p+ 2 or 2p−2

for some p, where pis a function of jalone. If this

trend holds for all m, then τ(29) ≥224 −2 (indicated

as the lower bound on an error bar in Fig. 3). Specula-

tively, the upper envelope of τ(m) grows exponentially

in mbut no faster than O(2m) (gray line), which is still

exponentially smaller than the generic upper bound of

O(2cm2).

B. Entanglement generation for pure product

states

The second deﬁning feature of this class, along with

the other good scrambling classes, is in the generation

of entanglement for initial pure product states. In this

aspect it behaves like a random circuit: starting from a

random product state, the subsystem entropy averaged

over all contiguous regions of the same length increases

linearly in time before saturating at a near-Page curve

with slope 1 (Fig. 4) [2]. However, the initial product

state does eventually recur. Since τ(m) is linear in m

for m= 2k, on those system sizes, the system spends

a ﬁnite fraction of its evolution in a state of suppressed

entanglement. For the time evolution on m= 64 unit

cells shown in Fig. 4, the initial product state recurs

(modulo signs) with a period of τ(m) = 128, but the

FIG. 3: Recurrence time τ(m) of the unitary, modulo

signs and shifts, for a brickwork circuit of iSWAP cores and

π/2Xrotations (Eq. (2) and Eq. (3)), acting on m=L/2

unit cells with periodic boundary conditions. Gray line is

τ= 2m+1, which appears to be an upper bound on τ(m).

0 20 40 60 80 100 120

|A|

S(|A|, t0+t)

t0= 0

0 20 40 60 80 100 120

|A|

t0= 45

FIG. 4: Entanglement generation on a random pure

product state on L= 128 qubits, or m= 64 unit cells, for a

brickwork circuit of iSWAP cores and π/2Xrotations (Eq.

(2) and Eq. (3)). For t < 20, the subsystem entropy

increases at a near-maximal rate before reaching a Page

curve with slope 1 (left). The state remains near-maximally

entangled until t∼

=45, before the subsystem entropy starts

to decrease until reaching an area-law state at t= 64

(right). In both panels, the entropy ⟨S(|A|, t0+ ∆t)⟩is

averaged over all contiguous regions of length |A|with

periodic boundary conditions, with darker (lighter) curves

corresponding to later (earlier) times ∆twith respect to t0.

state returns to area-law entanglement twice per pe-

riod. For generic large m, the recurrence time generally

satisﬁes τ(m)≫m, so the state spends most of its time

near-maximally entangled.

C. Operator content

The two above properties—superlinear recurrence

times for generic mand generation of slope-1 Page

curves starting from a pure product state—are also seen

in two other classes of good scrambling automata, dis-

cussed in Sec. VI. What makes this class unique among

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CrystallineQuantumCircuitsGraceM.Sommers,1DavidA.Huse,1andMichaelJ.Gullans21DepartmentofPhysics,PrincetonUniversity,Princeton,NJ08544,USA2JointCenterforQuantumInformationandComputerScience,NIST/UniversityofMaryland,CollegePark,Maryland20742,USA(Dated:August2,2023)Randomquantumcircuitscontinuetoinspi...

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Crystalline Quantum Circuits Grace M. Sommers1David A. Huse1and Michael J. Gullans2 1Department of Physics Princeton University Princeton NJ 08544 USA

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