From Dual Unitarity to Generic Quantum Operator Spreading Michael A. Rampp Roderich Moessner and Pieter W. Claeys Max Planck Institute for the Physics of Complex Systems 01187 Dresden Germany

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From Dual Unitarity to Generic Quantum Operator Spreading

Michael A. Rampp, Roderich Moessner, and Pieter W. Claeys

Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany

(Dated: May 4, 2023)

Dual-unitary circuits are paradigmatic examples of exactly solvable yet chaotic quantum many-

body systems, but solvability naturally goes along with a degree of non-generic behaviour. By

investigating the eﬀect of weakly broken dual unitarity on the spreading of local operators we study

whether, and how, small deviations from dual unitarity recover fully generic many-body dynamics.

We present a discrete path-integral formula for the out-of-time-order correlator and recover a but-

terﬂy velocity smaller than the light-cone velocity, vB< vLC , and a diﬀusively broadening operator

front, two generic features of ergodic quantum spin chains absent in dual-unitary circuit dynamics.

The butterﬂy velocity and diﬀusion constant are determined by a small set of microscopic quantities

and the operator entanglement of the gates has a crucial role.

The dynamical behaviour of strongly correlated quan-

tum many-body systems out of equilibrium is notoriously

hard to describe. Quantum many-body dynamics is in-

timately related to questions of thermalization, informa-

tion scrambling, quantum chaos, and the emergence of

hydrodynamics [1–8]. Simple model systems that can

be solved analytically are highly desirable since they of-

fer an invaluable window into these questions. In re-

cent years dual-unitary circuits (DUCs) have emerged

as paradigmatic examples of exactly solvable yet chaotic

many-body systems [9–27] in which a variety of dynam-

ical quantities are analytically accessible [10–15,17,18].

However, solvability comes at a cost in genericity, and

in several respects DUCs display behaviour that diﬀers

strikingly from the phenomenology observed numerically

in more generic models [4–8,28,29]. In particular, the

space-time duality present in DUCs enforces that corre-

lations and operators spread with the maximum possible

velocity. This nongeneric maximal spreading has also

been observed experimentally on Google’s quantum pro-

cessor [30].

It is natural to ask whether and in what sense DUCs

might serve as a starting point to understand the behav-

ior of more general systems. This question is especially

relevant given recent advances in noisy intermediate scale

quantum devices. While much is now known about DUCs

themselves, there are only few results about deviations

from dual unitarity [31–33] and many questions remain

open. In Ref. [31] the behavior of local correlation func-

tions in circuits close to dual unitarity has been investi-

gated. It was found that in most instances local corre-

lators acquire a more generic spatio-temporal structure,

not being exclusively supported on the light cone any-

more.

In this paper we investigate operator dynamics by

considering out-of-time-order correlators (OTOCs) in a

broad class of chaotic quantum circuits in which dual uni-

tarity is weakly broken. We show that the OTOC can be

expressed as a sum over all possible paths resulting from

scattering on individual dual-unitarity-breaking gates,

acting as defects in an otherwise dual-unitary circuit. We

ﬁnd that after an initial period in which the dual-unitary

form is approximately preserved, even a weakly broken

FIG. 1. (a) The unitary time evolution operator is con-

structed as a brickwork circuit composed out of identical

two-site gates. One row of gates corresponds to a time step

∆t= 1. (b) Graphical depiction of processes contributing to

the OTOC. When dual unitarity is broken, the edge of the

operator string can scatter into the light cone. The OTOC

is then given by a weighted sum of all paths. ’Folded’ dual-

unitary gates are depicted in red and dual-unitary-breaking

gates in purple.

duality leads the OTOC to recover a nonmaximal but-

terﬂy velocity and a diﬀusive broadening of the operator

front, hallmarks of generic one-dimensional chaotic quan-

tum many-body systems absent from pure DUCs. The

operator front at late times takes a universal form and

its parameters are microscopically determined by the en-

tangling properties of the gate. Our manipulations are

controlled in the limit where the dual-unitarity-breaking

gates are dilute along the direction of the light cone, but

we argue that they capture the relevant characteristics

of the OTOC on intermediate to long timescales even in

the case of a Floquet circuit with dense perturbations.

The developed framework is expected to be applicable to

diﬀerent probes of operator spreading.

Dual-unitary circuits. We consider circuits composed

of unitary gates Uacting on two sites with local Hilbert

space dimension q, with matrix elements Uab,cd graphi-

cally expressed as

(1)

In this notation each leg corresponds to an index in the lo-

cal Hilbert space, and connecting two indices corresponds

arXiv:2210.13490v3 [quant-ph] 3 May 2023

to a tensor contraction (see e.g. [34]). Unitary gates ar-

ranged in a brickwork geometry [Fig. 1(a)] provide simple

models for local, unitary quantum many-body dynamics

on a one-dimensional lattice [3–6,35], with the number

of discrete time steps tcorresponding to the number of

rows in the circuit. A gate Uis called dual-unitary if the

associated dual gate ˜

Udeﬁned by ˜

Uab,cd ≡Udb,ca is also

unitary [10,13].

Let us review the computation of OTOCs in dual-

unitary circuits [17]. We consider a basis of local opera-

tors σαnormalized according to tr σ†

ασβ=qδαβ. Set-

ting σ0= 1, the remaining operators are traceless, and we

take σα(x) to act as σαon site xand as the identity ev-

erywhere else. Denoting the time evolution operator as

U(t), we write σα(x, t) = U(t)†σα(x)U(t) and consider

the OTOC

Cαβ(x, t) = hσα(0, t)σβ(x, 0)σα(0, t)σβ(x, 0)i,(2)

with respect to the maximally mixed state, hOi ≡

tr[O]/tr[ ]. This function quantiﬁes the spreading of op-

erators and the scrambling of information into nonlocal

degrees of freedom [36–38], and is experimentally accessi-

ble in (digital) quantum simulation platforms [30,39–43].

The OTOC’s propagation speed is called butterﬂy veloc-

ity vBand sets the maximal speed at which information

can spread in the circuit [44,45].

The OTOC exhibits a strong parity dependence. Here,

we focus on C+(x, t) for which (t−x)∈2Z, but the

derivation is analogous for C−(x, t), for which (t−x)∈

2Z+ 1. In general circuits, Eq. (2) can be graphically

represented as the contraction of a two-dimensional ten-

sor network, the size of which is set by the light-cone

coordinates n= (t−x+ 2)/2, m = (t+x)/2,

C+

αβ(n, m) = 1

qn+m,

= (Ln(σα)|(Tn)m|R+

n(σβ)),(3)

where we have introduced the ‘folded’ gate acting on four

copies of the local Hilbert space,

= ⊗!⊗2

,(4)

as well as the following vectors,

q,=1

(5)

The tensor network (3) can be understood as the con-

traction of powers of a transfer matrix

Tn=1

| {z }

.(6)

This transfer matrix is a contracting map, i.e. kTnvk ≤

kvk. Hence, all its eigenvalues lie inside or on the bound-

ary of the complex unit disk. In the limit m→ ∞ the

OTOC is completely determined by the eigenvectors of

Tnwith leading eigenvalue (i.e., modulus 1). If the gate

is dual-unitary there exist n+1 leading eigenvectors that

are independent of any further characteristics of the gate

and can be constructed explicitly [17]. If these vectors

exhaust the set of leading eigenvectors, the gate is called

maximally chaotic [15].

While the existence of a local conserved quantity im-

plies the existence of further leading eigenvectors, the set

of maximally chaotic gates is dense in the set of dual-

unitary gates [15]. We call the subspace spanned by

this generic set of vectors the maximally chaotic subspace

(MCS). Its construction and properties are elaborated in

the supplemental material [46].

For circuits composed of maximally chaotic dual-

unitary gates the following holds [17]: After an initial

transient regime, the OTOC for (t−x)∈2Zis only non-

vanishing on the light cone edge, |x|=t, where it takes

the universal value −1/(q2−1), resulting in a maximal

butterﬂy velocity vB=vLC = 1. For (t−x)∈2Z+ 1

the OTOC decays exponentially inside the light cone.

This behavior is to be contrasted with the OTOC in

generic unitary dynamics, where vB<1 and the ballis-

tic spreading is accompanied by a diﬀusively broadening

front [5,6].

Breaking dual unitarity. For concreteness we consider

gates of the form U=V eiεW where Vis a maximally

chaotic dual-unitary gate, Wis Hermitian, and εis taken

to be small (ε1).

How does the breaking of dual unitarity aﬀect the

OTOC? In the absence of dual unitarity and with-

out further constraints, the transfer matrix has only

a single, ‘trivial’, leading eigenvector that leads to

limm→∞ C+

αβ(n, m) = 1 for ﬁnite n[17]. Signaling that

the butterﬂy velocity in such a circuit is nonmaximal,

vB<1, the butterﬂy velocity then needs to be deter-

mined from the subleading eigenvectors of the transfer

matrix.

The contribution of a subleading eigenvector with

eigenvalue |λ|<1 decays on a timescale τ∼(−log|λ|)−1.

For weak perturbations from dual unitarity, the manifold

of leading eigenvectors is split by an amount O(ε2). If in

the limit m→ ∞ a ﬁnite spectral gap to the remaining

spectrum exists, then for suﬃciently small εthe largest

subleading eigenvectors are predominantly composed out

of vectors in the MCS, and the behavior of the OTOC on

long times t≥1/ε2is determined by those eigenvectors.

In the absence of symmetries that force a gap closing, we

expect this to be the generic scenario.

To proceed, we project the transfer matrix to the MCS,

similar in spirit to degenerate perturbation theory, and

compute the OTOC with the projected transfer matrix.

If the perturbed gates are dilute along the light-cone, this

approximation is controlled [46]. However, for perturba-

tions that are suﬃciently small compared to the spectral

gap we expect this description to remain valid in the

dense limit.

Notably, the MCS only grows linearly with the size of

the transfer matrix compared to the full operator space,

which grows exponentially. Reducing the dynamics to

the MCS presents a signiﬁcant computational advantage.

Moreover, the resulting transfer matrix can be eﬃciently

truncated, allowing for analytic evaluation (see below).

The matrix elements of the transfer matrix in the MCS,

(`|Tn|k), can be readily calculated [46]. These depend on

the properties of the gates through a set of quantities Bk,

k= 1, . . . , n. Graphically

Bk=1

qk+1 .(7)

Deﬁning z1:= (B1−1)/(q2−1) and zk:= (Bk−

Bk−1)/(q2−1) for k > 1 we ﬁnd that (0|Tn|0) = 1 and

(0|Tn|k) = pq2−1zk

qk−1, k ≥1 (8a)

(`|Tn|k) = 









q2zk−l−zk−l+1

qk−l, k > l,

1−z1, k =l,

0,otherwise.

(8b)

They have the following structure: (i) the matrix is upper

triangular, as a direct consequence of unitarity [46]. (ii)

Except in the ﬁrst row, the k-th side diagonal has the

same entry everywhere. This is only the case if all gates

are identical and follows from translational invariance.

Taken together, these imply that the eigenvalues are λ0=

1 with algebraic multiplicity 1, and λ1= 1 −z1<1 with

algebraic multiplicity n.

All matrix elements can be given a quantum-

information theoretic interpretation. Inserting the

Schmidt decomposition of the gate, U=Pj√σjXj⊗Yj,

into z1reveals that this quantity is equivalent to the lin-

ear operator entanglement of the gate E(U) [47], with

z1= 1 −q2

q2−1E(U). The properties of the operator en-

tanglement imply 0 ≤z1≤1. As dual unitarity of U

is equivalent to Uhaving maximal operator entangle-

ment [48], z1= 0 iﬀ Uis dual unitary, and z1hence

quantiﬁes proximity to dual unitarity.

The subleading eigenvalues follow from Eq. (8b) as 1−

z1, such that the timescale τfor deviations from dual

unitarity to become apparent follows as

τ−1∼ −log (1 −z1) = −log q2

q2−1E(U).(9)

All Bkfor k > 1 can analogously be given the inter-

pretation as operator entanglements of k-fold diagonally

composed gates on enlarged Hilbert spaces [46]. Because

the diagonal composition preserves dual unitarity [24],

zk= 0 for all kiﬀ Uis dual-unitary. The zkare bounded

by 0 ≤zk≤(1 −z1)k−1, implying that zk→0 for

k→ ∞ [46]. In general, knowledge of all higher-order op-

erator entanglements is necessary to compute the OTOC,

however they are increasingly less important.

While it is not possible to calculate arbitrary powers

of the transfer matrix exactly, the structure of Eq. (8)

allows for a systematic expansion of the OTOC using a

path-integral approach. We write,

Tn=D+

j=1|uj)(j|,(10)

D= diag (1,1−z1,...,1−z1),(11)

where the |uj) are vectors containing the oﬀ-diagonal ma-

trix elements, (`|uk)=(`|Tn|k) for k > ` and (`|uk)=0

otherwise.

The oﬀ-diagonal terms are of order zk∼2, and we

expand the OTOC in powers of (Tn−D). Each such

power can be expressed as

(Tn−D)ν=

j1,j2,...,jν=1

(|uj1)(j1|)(|uj2)(j2|). . . (|ujν)(jν|)

j1<j2···<jν|uj1)(j1|uj2). . . (jν−1|ujν)(jν|(12)

The sum consists of products of oﬀ-diagonal matrix ele-

ments indexed by sets {j1, . . . , jν}. These indices can be

interpreted as nodes of a path and the oﬀ-diagonal ma-

trix elements (`|uk) act as propagators determining the

amplitude of jumping k−`steps inside the light cone.

Crucially, these only depend on the diﬀerence k−`and

the amplitudes for negative step sizes vanish, making sure

only causal paths contribute to the OTOC [Fig. 1(b)].

From Eq. (8) it follows that jumps of size kare controlled

by zk/qk, such that jumps with a large kare exponen-

tially suppressed.

We note that having maximal operator entanglement

implies that all oﬀ-diagonal matrix elements vanish.

Hence, in dual-unitary circuits the edges of an operator

string can only move along the edges of the light cone

and we recover the previous result.

Although the mathematical origin is diﬀerent, the pic-

ture presented here is qualitatively similar to the one

presented in Ref. [31] for the two-point functions. The

skeleton diagrams appearing there resemble the scatter-

ing paths introduced above. The similarity of the two

results might merely be a reﬂection of the underlying

physics of dual-unitary circuits. In dual-unitary circuits

all excitations move with maximal velocity. Breaking

dual unitarity then allows processes that violate this rule,

suggesting expansions in orders of such processes.

0.0

0.5

1.0

OTOC

a) ε= 0.15

t= 450

t= 1000

t= 1800

−40 −20 0 20 40

x−vBt

0.0

0.5

1.0

OTOC

t= 1200, ε = 0.25

projection to MCS

error function ﬁt

1-step approx.

2-step approx.

FIG. 2. (a) Broadening of the OTOC front as a function

of time. (b) Comparison of the asymptotic front for diﬀerent

approximations (the shift in the center of the front is corrected

for better comparability).

Truncation of the path integral. To make analytical

progress, we restrict the sum over paths. In the large-

qlimit the path integral is dominated by those paths in

which single steps are at most of size 1, since higher-order

steps are suppressed by powers of q[see Eq. (8)].

We now approximate the OTOC for general qby only

considering these paths, and call this the one-step ap-

proximation. On the level of scattering amplitudes this

approximation corresponds to letting only z1be nonzero.

However, even for qubits (q= 2) the one-step approxi-

mation already produces the correct functional form of

the asymptotic proﬁle close to the center of the front

[Fig. 2(c)].

Within this approximation the path integral can be

evaluated exactly, leading to

C+

(1) =q2Fz1(n, m)−Fz1(n−1, m)

q2−1,(13a)

Fz1(n, m)≡nm

nBz1(n, m −n+ 1),(13b)

where Bz1(a, b) denotes the incomplete β-function. An

asymptotic expansion yields a butterﬂy velocity vB,1=

(1 −z1)/(1 + z1) and a front that takes the form

C+

(1) ≈1

21 + erf x−vB,1t

√2D1t.(14)

This is the form expected from a diﬀusively broadening

front with diﬀusion constant D1=vB,1(1 −v2

B,1). Both

the form of the front and the scaling of the diﬀusion con-

stant for vB,1→1 agree with results from Haar random

circuits [5,6].

To obtain a better quantitative agreement for q= 2,

we consider the two-step approximation. An exact cal-

culation yields [46]

C+

(2)(x, t)≈C+

(1) x+t−x

2ξ, t −t−x

2ξ,(15)

100102104

−0.5

0.0

0.5

1.0

C+(t, t)

ε= 0.00

ε= 0.01

ε= 0.03

ε= 0.10

FIG. 3. Early time relaxation of the dual-unitary form to

the generic shape on the geometric light cone for a range of

perturbation strengths. The timescale is indicated by vertical

dotted lines.

0.90

0.95

untruncated

1-step approximation

2-step approximation

0.10 0.15 0.20 0.25

perturbation strength ε

0.1

0.2

0.3

FIG. 4. Plots of (a) butterﬂy velocity and (b) diﬀusion con-

stant as function of perturbation strength and comparison to

the analytical predictions of the one- and two-step approxi-

mations.

with ξ=z2/(q2z1). This result can be understood by

noting that the path integral is asymptotically domi-

nated by the typical path. If z2z1the typical fraction

of steps of size two is approximately given by the ratio

of scattering amplitudes ξ=z2/(q2z1). The steps of

size two therefore eﬀectively shift the proﬁle [Eq. (13b)]

deeper into the light cone, n→n(1 −ξ).

Close to the shifted front the shape of the proﬁle is

preserved, with renormalized parameters

vB,2=vB,1−δ

1−δ, D2=D1

1−(1 −vB,2)ξ

(1 −δ)2,(16)

where δ= (1 + vB,1)ξ/2. By taking into account longer

steps, operator strings can move into the light cone

faster, diminishing the butterﬂy velocity and, since this

increases the variance of the distribution of the endpoints

of operators, enhancing diﬀusion.

Numerical results. Restricting the dynamics to the

MCS (of linear size in n) presents an enormous simpli-

ﬁcation compared to the full exponentially large space

when probing the long time limit. This restriction allows

the eﬃcient numerical evaluation of the OTOC under the

assumptions stated above.

First, we study the behavior of the OTOC on the geo-

metric light cone for a Floquet circuit consisting of iden-

tical perturbed dual-unitary gates with q= 2. We ﬁnd

that the main features of dual unitarity persist up to the

timescale (9) [Fig. 3]. At this timescale, the deviation of

the OTOC on the geometric light cone C+(t, t) from the

dual-unitary prediction, −1/(q2−1) = −1/3, becomes of

order 1. This indicates that for earlier times most of the

operator strings still travel at maximal velocity.

In the late-time regime the operator front slows down,

moving ballistically with vB<1, and shows approximate

diﬀusive broadening [see Fig. 2(a,b)]. At late times the

shape of the operator front is well described by an error

function of the form described in Eq. (14), indicating

that higher k-steps only serve to further renormalize the

arguments of the obtained proﬁle.

We extract the butterﬂy velocity and diﬀusion con-

stant and compare them to the analytic prediction ob-

tained by truncating the path integral [Fig. 4]. For a spe-

ciﬁc but randomly selected perturbation we ﬁnd that the

two-step approximation is in good agreement with the

full path integral result for low to intermediate perturba-

tion strengths. For the diﬀusion constant the discrepancy

is larger, but the qualitative behavior is captured. We

attribute this discrepancy to paths which contain large

steps. For both quantities, the two-step approximation

signiﬁcantly improves the one-step approximation, and

the accuracy of this approximation is expected to increase

for larger qor by including higher steps.

Discussion.

The observation of diﬀusively broadening fronts in

non-random systems far away from the dual-unitary

limit [8,29] hints at the presence of a more general mech-

anism. Numerical results indicate that the degeneracy of

the subleading eigenvalue can remain stable far from dual

unitarity, suggesting that a similar path-integral descrip-

tion remains possible beyond the perturbative regime.

Moreover, the non-Hermiticity of the transfer matrix

might play a central role – Ref. [49] previously observed

that non-Hermiticity can strongly inﬂuence the behavior

of OTOCs.

Our work shows that dual-unitary circuits can

serve as a starting point to investigate more generic

settings. We hope that this work opens up further

studies on perturbed dual-unitary cicuits, e.g., on

entanglement dynamics, the relation to transport, or

spectral properties. The developed framework can

be directly applied to more general probes of oper-

ator dynamics in perturbed dual-unitary dynamics

involving multiple replicas of the circuit, e.g. R´enyi

(operator) entanglement [15,32], spectral form factors

[16,50], or in studies of ‘deep thermalization’ [33,51,52].

ACKNOWLEDGMENTS

We are grateful to Dominik Hahn, Pavel Kos, Chris

R. Laumann, David M. Long, Frank Pollmann, Tomaˇz

Prosen, and Philippe Suchsland for useful discus-

sions. This work was in part funded by the Deutsche

Forschungsgemeinschaft (DFG) via the cluster of excel-

lence ct.qmat (EXC 2147, project-id 390858490).

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FromDualUnitaritytoGenericQuantumOperatorSpreadingMichaelA.Rampp,RoderichMoessner,andPieterW.ClaeysMaxPlanckInstituteforthePhysicsofComplexSystems,01187Dresden,Germany(Dated:May4,2023)Dual-unitarycircuitsareparadigmaticexamplesofexactlysolvableyetchaoticquantummany-bodysystems,butsolvabilitynaturall...

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