Quantum State Complexity in Computationally Tractable Quantum Circuits
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PRX QUANTUM 2, 010329 (2021)
Quantum State Complexity in Computationally Tractable Quantum Circuits
*
Jason Iaconis
Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder,
Colorado 80309, USA
(Received 28 September 2020; revised 29 December 2020; accepted 26 January 2021; published 23 February 2021)
Characterizing the quantum complexity of local random quantum circuits is a very deep problem
with implications to the seemingly disparate fields of quantum information theory, quantum many-body
physics, and high-energy physics. While our theoretical understanding of these systems has progressed in
recent years, numerical approaches for studying these models remains severely limited. In this paper, we
discuss a special class of numerically tractable quantum circuits, known as quantum automaton circuits,
which may be particularly well suited for this task. These are circuits that preserve the computational basis,
yet can produce highly entangled output wave functions. Using ideas from quantum complexity theory,
especially those concerning unitary designs, we argue that automaton wave functions have high quantum
state complexity. We look at a wide variety of metrics, including measurements of the output bit-string
distribution and characterization of the generalized entanglement properties of the quantum state, and find
that automaton wave functions closely approximate the behavior of fully Haar random states. In addition
to this, we identify the generalized out-of-time ordered 2k-point correlation functions as a particularly use-
ful probe of complexity in automaton circuits. Using these correlators, we are able to numerically study
the growth of complexity well beyond the scrambling time for very large systems. As a result, we are
able to present evidence of a linear growth of design complexity in local quantum circuits, consistent with
conjectures from quantum information theory.
DOI: 10.1103/PRXQuantum.2.010329
I. INTRODUCTION this concept to gain insight into how closed quantum sys-
tems reach equilibrium and thermalize under a generic
Understanding the evolution of a quantum wave func-
Hamiltonian dynamics [8].
tions from a simple initial state to a generic vector in
Two of the main tools that have been used to under-
an exponentially large Hilbert space is a notoriously dif-
stand information scrambling are the entanglement entropy
ficult problem in modern theoretical physics. Aspects of
of the quantum state and the evolution of the out-of-
this evolution underlie important open problems in quan-
time-ordered (OTO) correlation function. It can be shown
tum information theory, quantum many-body physics, and
that the entanglement entropy in these systems grows lin-
high-energy physics. Great progress has been made in
early with time until it reaches a near maximal value [1],
recent years by focusing on local random circuit mod-
and a decay of the out-of-time ordered 4-point correlator
els, which provide a relatively clean system where these
has been shown to be equivalent to the Hayden-Preskill
dynamics can be studied [1–5]. A particularly important
definition of scrambling [9]. While such measurements are
element of a generic quantum dynamics is the concept
useful, it has become clear that these relatively simple
of information scrambling. Originally studied in the con-
measures cannot capture all the fine-grained aspects of the
text of black holes [6,7], scrambling defines the process
random unitary evolution. Two states may look maximally
whereby initially local information spreads throughout the
scrambled according to these two measures and yet have
system and becomes stored in the many-body nonlocal
important differences in the precise way the information is
entanglement of the state. Similar works have since used
stored nonlocally.
Quantum state complexity theory has been suggested as
a means to quantify these differences [10–12]. Roughly
*
jason.iaconis@colorado.edu speaking, the complexity of a quantum state is the depth
of the smallest local unitary circuit that can create the state
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license. Fur- from an initial product state. In random circuit models,
ther distribution of this work must maintain attribution to the the growth of quantum state complexity directly corre-
author(s) and the published article’s title, journal citation, and sponds to an increased difficulty in distinguishing the pure
DOI. quantum state from the maximally mixed state [10]. This
2691-3399/21/2(1)/010329(19) 010329-1 Published by the American Physical SocietyJASON IACONIS PRX QUANTUM 2, 010329 (2021)
is a physical property whereby initially local information the level spacing distribution of the entanglement spec-
is more effectively hidden in high complexity states. trum. We will see that, by these measures, the automaton
It is known that a generic Haar random state will have wave functions behave like highly complex states.
a complexity that is exponentially large in system size N . In a dynamical context, the generalized k-point OTO
As a result, almost all quantum states cannot be efficiently correlation functions can describe the growth of quantum
simulated, even with a quantum computer [13]. A state state complexity beyond the scrambling time [11]. Again,
that is the output of a depth D random circuit composed according to this metric, complexity in automaton circuits
from a universal gate set will have a complexity that is appears to grow in the same way as in generic Haar random
conjectured to grow linearly with D [14,15]. Ensembles of circuits. Furthermore, using our efficient quantum Monte
these wave functions form what is known as an approx- Carlo algorithm, we are able to numerically study the
imate projective unitary k-design [16]. Measurements on growth of these OTO correlation functions in this poorly
k-designs can approximate, for large enough k, arbitrar- understood “beyond scrambling regime” for very large cir-
ily high moments of measurements on fully Haar random cuits. By doing this, we are able to identify specific k-point
states. On the other hand, states that are output from OTO correlation functions that appear to track the pre-
Clifford circuits in general form only a unitary 2-design cise rate of complexity growth in local random circuits
[17]. Although these wave functions display volume law and give results that are consistent with the linear growth
entanglement and information scrambling, they are still conjectured in the literature [10,14].
of relatively low complexity and only approximate a few The rest of this paper is organized as follows. In Sec.
moments of the Haar random states. II, we introduce and describe key properties of the quan-
In this paper, we show that high complexity quantum tum automaton circuits. We also describe the quantum
states can be prepared from a special type of nonuniver- Monte Carlo algorithm we use to simulate these wave
sal local quantum circuit. These circuits, which we call functions. In Sec. III, we review the concept of quan-
“automaton” quantum circuits, consist of any quantum tum state complexity, and describe several measurements
gate that preserves the computational basis. These automa- that we use to distinguish between high and low complex-
ton circuits have very recently started to be used as a ity states. We see that, by these metrics, automaton states
tool for studying dynamics in quantum systems [18–20]. behave like high complexity Haar random states. We con-
Specifically, in Ref. [20], it was realized that the opera- trast these results to those of low complexity Clifford wave
tor entanglement and OTO correlator properties of such functions. In Sec. IV, we discuss the generalized k-point
circuits appear to give results that are identical to that of out-of-time-ordered correlator as a probe of complexity
a generic chaotic dynamics. We go beyond this and show growth in dynamic systems. We see that automaton cir-
that, when acting on initial product states not in the compu- cuits can make use of these correlation functions to give us
tational basis, automaton circuits produce highly entangled new insights into complexity growth beyond scrambling in
wave functions in which the quantum state complexity local quantum circuits. In Sec. V we summarize our results
grows with circuit depth in the same way as in univer- and discuss potential applications of this work.
sal local random circuits. Furthermore, the evolution of
these wave functions can be efficiently simulated clas- II. AUTOMATON QUANTUM CIRCUITS
sically using a quantum Monte Carlo algorithm that we
describe. This may be appreciated in the context of several A. Definitions and review of previous results
other results in quantum information theory that demon- In this paper, we define automaton dynamics simply as
strate that the presence of entanglement in a quantum state any unitary evolution of a quantum system that does not
is not enough to show that a quantum algorithm that simu- generate any entanglement when applied to product states
lates the state achieves a speedup over a classical algorithm in an appropriate basis (which we choose to be the com-
[21–23]. Our results imply that complexity of the wave putational basis). As stated in Ref. [20], an automaton
function is also not a sufficient condition for such purposes. unitary operator U acting on an appropriate set of product
We do not attempt to provide a rigorous proof that states in a d-dimensional Hilbert space—labeled |m, with
automaton circuits output states of high complexity. m ∈ {0, . . . , d − 1}—permutes these states up to a phase
Instead, we characterize the complexity of the automaton factor, i.e.,
states using a series of measurements that were developed
to probe the fine-detailed structure of wave functions. We U|m = eiθm |π(m), (1)
consider metrics such as the generalized kth Renyi entropy
[12,24] and the sampled output bit-string distribution [25], where π ∈ Sd is an element of the permutation group on d
which can be used to differentiate between high and low elements.
complexity states that both have near maximal bipartite Similar unitary circuits with sparse output distributions
entanglement entropy. We also consider other measure- have been studied in the quantum information literature,
ments such as the fluctuation of entanglement entropy and where it was shown that efficient classical simulation
010329-2QUANTUM STATE COMPLEXITY IN COMPUTATIONALLY. . . PRX QUANTUM 2, 010329 (2021)
methods exist [26,27]. These circuits were first studied in of a quantum evolution describes the time it takes for an
a condensed matter context in integrable models in Refs. initial wave function to return to a nearby quantum state so
[18,19]. In Ref. [20], it was realized that a generic automa- that ψ0 |U|ψ0 ∼ O(1). For automaton circuits, the recur-
ton evolution leads to dynamics that appear to show “quan- rence time of an initial state (not necessarily a product state
tum chaotic” behavior. The out-of-time ordered correlators in the computation basis) corresponds to the order of a ran-
propagate ballistically and saturate to the consistent values dom element of thepermutation group Sd and on average
for a Haar scrambled operator. While automaton circuits gives trec → exp[λ d/ log(d)] as d → ∞.
do not generate entanglement in the computational basis, We also note that in Ref. [20] it was found that the oper-
a key property is that they do generically generate a high ator spreading in automaton circuits, as quantified by the
degree of operator entanglement. That is, the evolution 4-point out-of-time-ordered correlation function, behaves
identically to that of a Haar random chaotic circuit. In
O → U† OU (2) particular, the operator weights spread ballistically with a
wave front that broadens with a power law that is consis-
can be very complex and shows many of the generic tent with the universal exponents of a generic local chaotic
features of a Haar random unitary evolution. dynamics [2].
One important example of such an automaton gate is a In what follows, we take a complementary approach
quantum version of the controlled-controlled-NOT (CCNOT) and study the evolution of quantum states that are initially
gate product states in a basis orthogonal to the computational
basis. We refer to the output of such circuits as automa-
T(θ)123 = 1 − 12 + 12 eiθ X3 , (3)
ton wave functions. This approach allows us to focus on
where 12 = |0000| is the projection onto the |00 state the entanglement and complexity of the resulting wave
on sites 1 and 2. When θ = 0, this is the classical Toffoli function, and lets us compare our algorithm with known
gate that is known to be universal for classical reversible variational Monte Carlo techniques.
computation and can therefore implement any permuta-
tion π ∈ Sd on the computational basis states |m, m ∈ B. A variational Monte Carlo algorithm
{0, . . . , d − 1}. When θ = 0, such a gate also includes a
The defining feature of automaton circuits, that com-
state-dependent phase.
putational basis states only evolve to other computational
A second important automaton gate set is the set
basis states, is what allows us to simulate automaton wave
{CNOT, SWAP, Rz (θ )}, (4) functions on a classical computer. Despite their apparent
simplicity, such an evolution produces highly nontrivial
where Rz (θ) = eiθ Z implements a single-qubit rotation wave functions when applied to initial wave functions that
about the Z axis. At θ = π/2, all three gates belong to are not product states in the computational basis.
the Clifford group. The set of Clifford gates is capable We start with an initial ansatz wave function
of generating volume law entanglement when applied to
an appropriate initial product state, and the dynamics can |ψ0 = cm |m, (5)
be exactly simulated classically [21,22]. Therefore, the m
automaton gate set generalizes the above Clifford group by
allowing single-qubit rotations by arbitrary angles. where we assume that we know the coefficients cm exactly.
Note that both sets of gates defined above are universal Throughout this paper, we often choose |ψ0 to be a prod-
for quantum computation if supplemented by any single- uct state in the X basis, cm = (−1)m·σ /d, where m is a
qubit gate that does not preserve the computational basis binary vector representation of the integer m, and σ is a
[28]. vector of Pauli-Xi eigenvalues of |ψ0 . However, this need
We first review a few important analytic results derived not be the case, and we can choose any initial state |ψ0 for
in Ref. [20], in the case that the automaton circuit is which we have a variational ansatz cm .
composed of T(θ = 0). First, an initially local diagonal We then time evolve the wave function by applying the
operator Odiag will evolve into a superposition over O(d) quantum circuit
other diagonal operators (where d = 2N for qubits) and
will have a near maximal average operator entanglement.
T
Second, initially off-diagonal operators will evolve into
all elements of the conjugacy class of Sd , which implies Uλ = U(t)
j ,j +1 U(t)
j +1,j +2 , (6)
that an initial operator can evolve into O(dd ) possible off- t=1 j j
diagonal operators. That is, a generic operator can evolve,
under automaton dynamics, into a superexponential num- where λ are the variational parameters that represent the
ber of other possible operators. Finally, the recurrence time precise set of gates {Utj ,j +1 } that are applied. The resulting
010329-3JASON IACONIS PRX QUANTUM 2, 010329 (2021)
wave function is then time O(NT2 ). On the other hand, if O is a diagonal oper-
ator then x = x and we can get an estimate for the entire
|ψ(t) = Uλ |ψ0 = cm eiθm |πλ (m). (7) time evolution in a time that scales like O(NT).
m This approach can be straightforwardly adapted to mea-
sure operators that contain multiple copies of the uni-
Again πλ (m) is the permutation on the computational basis tary, U. For example, we can evaluate the k-point OTO
states, |m, which is implemented by Uλ . Therefore, we can correlation functions, ψ0 |A1 (0)B1 (t) · · · Ak (0)Bk (t)|ψ0 ,
exactly calculate the coefficients of the final wave function by running the forward and backward time evolu-
|ψ(t) = x ψλ (x, t)|x as tion k times. We simply act the unitary circuit Utot =
A1 U† B1 U · · · Ak U† Bk U on the sampled basis states |m. We
ψλ (x, t) = x|ψ(t) = cπ −1 (x) exp[iθπ −1 (x) ]. (8) can also consider a wave function on the Hilbert space H⊗k
λ λ
that consists of k tensor copies of the time-evolved wave
For a circuit with N qubits and depth T, this can be cal- function
culated in a time that scales like O(NT). That is, since |m
only evolves to a simple product state, |π(m), instead of ⊗k = U|ψ0 ⊗ U|ψ0 ⊗ · · · ⊗ U|ψ0 . (12)
a superposition over basis states, we can simply classically
sample the initial states |m and track their time evolu- Then we can evaluate the estimator for any operator in H⊗k
tion. Nevertheless, as long as |ψ0 is not a product state in as
the computational basis, |ψ(t) will generally evolve into k
a volume law entangled state. In this way we are able to ⊗k ⊗k 1 ∗ −i(θx −θxij )
classically simulate the circuit evolution of highly entan- |A = {cx cxij e ij f (xij )} ,
M i j =1
ij
gled quantum wave functions in a way that is equivalent to
the well-known variational Monte Carlo methods. (13)
We can therefore efficiently calculate estimates for sim-
where xij is the basis state xi in the j th tensor copy of
ple operator expectation values as
the Hilbert space. Examples of such observables are the
kth-order SWAP operators that are used to evaluate the kth
O = ψ0 |U† OU|ψ0
Renyi entropy. Expectation values of this form are impor-
tant in this work as they can be used to distinguish between
∗ †
= cy cx y Ut O Ut x
approximate unitary k-designs of different order.
x,y t t
We finally note that using this approach to study quan-
= ψλ∗ [π(y), t]ψλ [π(x), t]o[π(x), π(y)], (9) tum circuit dynamics allows us to make use of other tools
x,y developed in the context of variational Monte Carlo algo-
rithms. For example, one may incorporate Jastrow factors
where o[π(x), π(y)] = π(y)|O|π(x). [29], Lanczos steps [30,31], or other perturbative correc-
Since U is an automaton circuit then, if O is a simple tions to the quantum wave function [32]. Furthermore, a
Pauli operator, we have o[π(x), π(y)] = f [π(x)]δ(y, x ) promising direction for future work may involve applying
with
x = πλ−1 [πO (πλ (x)]. (10)
Therefore, we can write
Time
1 ∗
M
O ≈ ψ [π(xi ), t]ψλ [π(xi ), t]f [π(xi )]
M x =1 λ
i
1 ∗
M
i(θx −θx )
= cx cxi e i i f (xi ). (11)
M x =1 i
i
Note that, for a generic off-diagonal operator O, to FIG. 1. The local random circuit architecture used throughout
determine which state x (t) has a nonzero overlap with this paper. Each two-site gate is chosen randomly to be one of
O|x(t), we perform the full forward and backward time three basic automaton gates: the SWAP gate, the CNOT gate, or
evolution in Eq. (9) for each time step independently. Esti- the single-site rotation about the z axis Rz (θ) = eiθ Ẑ (applied
mating the full time dependence of O(t) therefore takes a independently to each site with a random angle θ).
010329-4QUANTUM STATE COMPLEXITY IN COMPUTATIONALLY. . . PRX QUANTUM 2, 010329 (2021)
automaton circuits to restricted Boltzman machine or other This is a very useful operational definition of complexity.
neural network wave functions. Such models were studied It is directly related to an experimental property of |ψ,
for a subset of automaton gates in Ref. [33]. the probability of distinguishing |ψ from the maximally
In the rest of this work, we focus on a specific one- mixed state with some fidelity (1 − δ), given a measure-
dimensional random circuit model consisting of two-site ment implemented on a circuit of size at most r. As δ → 0,
gates in alternating layers, as shown in Fig. 1. The gates in this definition of complexity implies the weaker condi-
this circuit are randomly chosen to be either the two-site tion, that |ψ requires a minimum circuit of depth r to be
SWAP or CNOT gate or a single-site rotation by a random prepared, but the converse is not in general true.
angle θ , Rz (θ) = eiθ Ẑ . We also compare the results to those Theoretically, complexity in random unitary circuits can
of a random Clifford circuit, where we randomly choose be understood using another important concept, namely
the gates to be either the two-site SWAP or CNOT gate or the that of unitary designs [16]. An ensemble of quantum gates
single-site Hadamard gate. E = {pi , Ui } acting on H is said to form an approximate
unitary k-design if the average over all such operators
approximates the first k moments of the Haar measure on
III. QUANTUM STATE COMPLEXITY
all d-dimensional unitary operators.
A. Background A similar concept applies to ensembles of quantum
Quantum complexity theory quantifies the difficulty of states. An ensemble ν of pure states, ψ, forms a complex
particular tasks for a quantum computer, in terms of the projective k-design if
minimum number of basic quantum gates a computation
requires. Interestingly, in contrast to classical complexity Eν [p(ψ)] = dψ p(ψ) for all p ∈ Hom(k,k) (Cd ),
theory, in the quantum setting one can also meaningfully νHaar
discuss the complexity of a quantum state. Roughly speak- (16)
ing, the complexity of a quantum state is the size of the
smallest k-local quantum circuit required to prepare the where p is the space of polynomials homogeneous of
state from an initial simple reference state. Unlike with degree k both in the coordinates of vectors in Cd and in
classical bit strings, creating a given quantum state from a their complex conjugates [24]. In other words, for a com-
given initial state may in general require an exponentially plex projective k-design, all expectation values that can be
long quantum circuit. In fact, since the number of possible written as a polynomial of degree k in the wave function
quantum circuits is exponential in gate number, while the coefficients must be equal to the expectation value of a
number of quantum states is superexponential in system random quantum state chosen from the Haar measure. In
size, one can show that almost all wave functions require fact, in most cases it suffices for the expectation value to be
an exponentially long circuit to prepare. only approximately equal to the Haar random value. Such
Importantly, the quantum state complexity of a wave distributions are known as -approximate unitary designs.
function can be directly related to measurable physical These two seemingly different ideas, complexity and
properties. This can be seen in the strong notion of com- design, are in fact very closely related. Since almost all
plexity put forward in Ref. [10]. In their work, the authors states in the Hilbert space have exponentially high com-
defined the complexity of a quantum state |ψ as the size plexity, one may guess that relatively high complexity
of the smallest local circuit, U, which, when combined states are required to approximate distributions on the
with measurement M in the computational basis, can dis- Haar measure. In Ref. [10] such a rigorous connection is
tinguish |ψ from the maximally mixed state ρ = (1/d)I. made between unitary designs and quantum state complex-
Mathematically, we define ity. It was shown that an -approximate unitary k-design
has, with high probability, a complexity approximately
equal to O(Nk). More precisely, it was shown that, for an
βr = max Tr[M (|ψψ| − ρ0 )]
M -approximate k-design in a (d = qN )-dimensional Hilbert
(14)
subject to M ∈ Mr (d), space formed from a set of |G| basic gates,
k
16k 2
where Mr (d) is the set of generalized measurements com- Pr[Cδ (|ψ) ≤ r] ≤ 2(1 + )dN |G| r r
, (17)
posed of a unitary circuit of depth r acting on a Hilbert d(1 − δ)2
space of size d, which is followed by a projective mea- which qualitatively remains very small until r ≈ k[N −
surement in the computational basis. We say that |ψ has 2 log(k)]/ log(N ). In other words, with high proba-
strong δ-state complexity less than r, Cδ (|ψ) < r, if bility, such a k-design has state complexity at least
O[kN / log(N )].
1 Unitary k-designs define a fine-grained hierarchy of
βr ≥ 1 − − δ. (15)
d quantum states of increasing complexity. This concept is
010329-5JASON IACONIS PRX QUANTUM 2, 010329 (2021)
referred to in the literature as complexity by design and is Sec. IV, we study measures of complexity that can be effi-
explored, for example, in Refs. [10–12]. ciently implemented using our Monte Carlo algorithm, and
This idea allows us to bridge the gap between local uni- therefore can be estimated with a classical complexity that
versal unitary gates, which form the basis of local quantum grows linearly in both circuit depth and the number of
circuits, and generic d-dimensional unitary operators U, qubits, O(ND).
which a random circuit tries to emulate. Characterizing the
rate of complexity growth in local random circuits is an B. Deviations from the maximally mixed state
important open question. In Ref. [25] it was shown that, Like normal random variables, fluctuations in the
with high probability, a local random circuit composed of matrix elements of random unitaries must satisfy strict
universal gates of depth O(Nk 11 ) forms at least a unitary bounds. For fully Haar random unitaries, these bounds
k-design. In other words, the design order of a local ran- imply that probability amplitudes of randomly sampled
dom circuit grows polynomially with circuit depth. It is bit strings follow the well-known “Porter-Thomas” dis-
expected, however, that this bound is not very tight. In tribution, p(xj ) = |xj |ψ|2 ∼ de−dp(xj ) . Such an output
Ref. [14] it was argued that the average complexity of local distribution is a signature of quantum chaos, and sampling
circuits in fact grows linearly with circuit depth. random bit strings from this distribution for universal local
Conversely, there are certain ensembles of quantum random gates is expected to be a hard problem to simulate
gates that are known to form only a fixed finite k-design. classically [34].
α
The set of Pauli strings, S = Ni=1 σi i , forms an exact 1- If a unitary matrix U is drawn instead only from a
design. The set of Clifford gates on q-dimensional qudits unitary k-design, fluctuations of matrix elements can be
are known to form a unitary 2-design in general, a 3-design shown [25] to satisfy a weaker bound. In this case, one
for q = 2, and never form a 4-design [17]. While wave finds that, for any two unit vectors |α and |β,
functions resulting from Clifford circuits are sufficient to
see properties such as volume law entanglement and infor- γ
mation scrambling, we will see that there exists a range Pr |β|U|α|2 ≥ ≤ (1 + )e− min(k,γ ) . (18)
U d
of observable properties that they do not possess and that
are characteristic of the higher complexity regime. In a If we let |α = |ψ0 and |β be any basis vector, this
sense, quantum state complexity generalizes the notion of bounds the fluctuations of the coefficients |cn |2 of |ψ(t).
information scrambling. The degree to which information Indeed, if we let k N , as we expect for a universal local
is spread nonlocally in a quantum state can be quantified random circuit at late times, and assume that the fluctu-
by the difficultly of recovering such information. ations saturate this bound, we see that k-designs approx-
In the rest of this section, we proceed in the follow- imate the Porter-Thomas distribution arbitrarily well for
ing way. We first identify several observable properties of sufficiently large k.
quantum states that have been explored in the literature and For automaton gates, the bit-string distribution in the
can be used to diagnose complexity beyond scrambling. computational basis remains constant. Therefore, for an
Strict bounds on these measurements can be formulated initial state orthogonal to the computational basis, sam-
when they are averaged over a unitary design. We measure pling the computational basis bit strings is equivalent to
these properties in automaton wave functions. The results sampling from the maximally mixed state. However, we
suggest that automaton wave functions have high state find that bit strings measured in the orthogonal “X ” basis
complexity. Where useful, we also compare these mea- form a nontrivial probability distribution and further that
surements to those of Clifford circuits, which are known this distribution satisfies the strict bounds set for generic
to form a finite low-order unitary design. As a conse- unitary k-designs.
quence, we show that, while a universal local gate set is To see this, we simulated the exact quantum circuit
sufficient for creating wave functions of high complex- dynamics of an initial product state with all spins oriented
ity, it is not in fact necessary. Indeed, wave functions of perpendicular to the computational basis,
high complexity can be formed by acting with an automa-
ton circuit, and therefore such a circuit evolution can be 1
|ψ0 = ⊗|+x = |m, (19)
simulated efficiently with a classical computer in the man- 2N m
ner described in the previous section. We note, however,
that the specific measurements used in the rest of this which is evolved with gates chosen randomly from our
section cannot generally be implemented efficiently with automaton gate set. To compare, we also simulated ran-
a Monte Carlo algorithm and so we instead simulate the dom Clifford circuits, with gates chosen randomly from
exact automaton and Clifford dynamics on relatively small {CNOT, SWAP, H } and acting on an initial product state,
system sizes. This exact simulation method has a clas- with spins oriented in a randomly chosen direction. We
sical computational complexity that grows like O(D2N ) should emphasize that, since we are interested in the distri-
and is therefore exponential in system size. However, in bution of a many-bit output, we cannot use the polynomial
010329-6QUANTUM STATE COMPLEXITY IN COMPUTATIONALLY. . . PRX QUANTUM 2, 010329 (2021)
Automaton
H = HA ⊗ HB can be shown to be
−1
10 Clifford
Porter-Thomas dist.
1 dA
SvN ≥ log(dA ) − , (20)
P (d|an|2)
2 ln(2) dB
10−3
where dA ≤ dB are the dimensions of HA and HB , respec-
tively.
10−5 Our definition of quantum state complexity implies
that high complexity states cannot easily be distinguished
from the maximally mixed state. This property necessar-
10−7
0 5 10 15 ily requires the state to be nearly maximally entangled, so
that the reduced density matrix ρA is close to the max-
2
d |an| imally mixed state for all subregions |A| < L/2. There-
fore, the process of scrambling requires that initially local
information becomes stored in the nonlocal many-body
FIG. 2. The probability distribution of bit strings P(2N |an |2 ) entanglement of the wave function. However, the converse
as measured in the x basis for automaton and Clifford wave func- statement is not always true. States of high entanglement
tions on N = 16 sites. The fluctuations of bit-string amplitudes are not necessarily always of high complexity. To distin-
in the automaton wave functions obey the strict bound for unitary
guish between states of different complexity, we need to
γ designs given by Eq. (18) up to at least γ ∼ N . The Clifford
wave function, on the other hand, only obeys this bound up to develop more fine-grained measures of entanglement.
γ ∼ 3. In Fig. 3(a), we show the time evolution of the bipar-
tite von Neumann entanglement entropy SvN (t) for a single
circuit realization of both automaton and Clifford circuit
time classical algorithm to simulate either the automaton types, for the same initial state as the previous section.
or Clifford circuits [35]. Instead, we are forced to track the In both cases, we observe a short regime of linear entan-
evolution of the entire wave function for a small system glement growth followed by a late time regime where the
size. We simulated a circuit with L = 16 sites and circuit entanglement saturates near the volume law Page value
depth D = 100. A histogram of the final projective mea- SvN = L/2 − 1/[2 ln(2)]. The main difference between the
surement outcomes for both cases is shown in Fig. 2. For two cases is that, for the automaton circuit, after reaching
the automaton circuit, the state is initialized with all spins saturation, SvN remains very close to the exact Page value
oriented perpendicular to the computational basis, and the at all times, while in the Clifford circuit there are relatively
final output bit strings are measured in the x basis. The large fluctuations in SvN (t). We argue that these fluctua-
results are averaged over 100 different circuit realizations. tions in the entanglement entropy are a sign that a state is
We see that the probability of different basis strings not drawn from a sufficiently high unitary design.
decays exponentially, up to the resolution we are able The parameter SvN measures the entropy of the reduced
to measure. For the Clifford circuits, the Porter-Thomas density matrix ρA , which encodes all information about
bound is satisfied only up to γ = 3, but is violated for observables that can be measured locally in region A. Fluc-
γ > 3. The implication is that, for automaton circuits, tuations in SvN (t) therefore imply that there are fluctuations
measurements in the orthogonal basis are extremely uni- in the value of some measurement in region A. In Brandao
form in the same way as for high complexity Haar random et. al. [10], it was shown that, for a unitary k-design, the
states. This is evidence that the automaton circuits at high higher-order moments of a generic expectation value are
enough depth form an -approximate k-design for arbitrar- bounded by
ily large k. We examine the evolution of the design error
for this measurement in the Appendix. k/2
k2
E|ψ ({Tr(M |ψψ|) − E|ψ [Tr(M |ψψ|)]} ) ≤ k
.
d
C. Entanglement and complexity (21)
The pattern of entanglement in quantum states is very
closely related to the quantum state complexity. We will For a highly complex state, which forms a large-k unitary
see that the entanglement in states drawn from a unitary design, the higher-order fluctuations on all measurements
k-design must satisfy certain constraints. As shown by become very small. If we partition our lattice into regions
Page [7], nearly all quantum states chosen from the Haar A and B, and let M be any projective measurement imple-
measure will have a nearly maximal amount of entangle- mented on the spins in subsystem A, then this should also
ment. More precisely, the bipartite von Neumann entangle- bound fluctuations of the entanglement entropy. Therefore,
ment entropy of a random quantum state with Hilbert space the temporal fluctuations in the entanglement entropy are
010329-7JASON IACONIS PRX QUANTUM 2, 010329 (2021)
(a) 8 We show the histogram of these entropies in Fig. 3(b) for
both automaton and Clifford circuits. We see that indeed,
6 for automaton circuits, almost all bipartitions of the state
SvN (t) have the same entanglement entropy, which is very close to
4 the Page entropy. However, for Clifford circuits, while the
Clifford
average entanglement entropy is equal to the Page entropy,
2 Automaton there are significant, O(1), variations in this measurement
Haar depending on which bipartition is selected. This implies
0 that the Clifford states are much less uniform than the
0 100 200 300 automaton wave functions. Therefore, the variance in mea-
Circuit depth (time) surements in automaton states should satisfy Eq. (21) for a
much higher value of k compared to Clifford states.
(b) Perhaps the most direct connection between entangle-
P (SvN) ment and unitary design can be made by studying the
generalized Renyi entanglement entropies. In Ref. [12], it
was shown that the higher-order αth Renyi entropies can
be used as a direct probe of the design order. The α-Renyi
entropy is defined as
P (SvN)
1 1
S α (ρA ) = log(Tr[ρAα ]) = log λαi ,
1−α 1−α i
(22)
0.1
⟨σ ⟩
0.01 where the λi are the eigenvalues of the reduced density
0.001 matrix ρA . As α → ∞, S α (ρA ) = Smin (ρA ) = − log(λmax )
approaches the min entropy. Here Smin (ρA ) simply probes
the largest eigenvalue of ρA , and bounds all other Renyi
entropies S α (ρA ) ≥ Smin (ρA ) for all α. In Ref. [12], it was
FIG. 3. (a) The bipartite entanglement entropy SvN as a func-
tion of time for both Clifford and automaton circuits. In both shown that the α-Renyi entropy averaged over a unitary
cases, the late time entanglement averages to the Haar random α-design is nearly maximal. Therefore, the higher-order
“Page entropy”; however, the temporal fluctuations are signifi- Renyi entropies can be seen as a probe of higher-order
cant in the Clifford circuit, while they appear negligibly small in complexity in the wave function. It was shown that
the automaton circuit. (b) This uniformity of entanglement can
be seen in a single realization of an automaton wave function, if Eνk [S k (ρA )] ≥ dA + O(1), (23)
we measure the entanglement in all possible bipartitions of the
lattice. We show the probability distribution of the entanglement where Eνk is the average over the k-design distribution of
entropy across the different partitions for the automaton (top) and unitary matrices. Furthermore, they showed that
Clifford (middle) wave functions. The standard deviation of these
distributions (bottom), σ = (SvN − SvN )2 , decays expo- Eνk [Tr{ρAk }] = EHaar [Tr{ρAk }]. (24)
nentially with system size for automaton circuits and appears to
saturate to a constant value for Clifford circuits.
In other words, the kth Renyi entropies are all nearly max-
imal up to an O(1) constant for a unitary k-design, and
the trace of ρAk exactly equals the Haar random value.
evidence that the Clifford circuit is of lower complexity This exact equality does not hold in general for the Renyi
than the automaton circuit. entropies since the log of an average does not in general
Using this intuition, we can develop an entanglement equal the average of a log.
measure that acts on a quantum wave function at a sin- We measure the different Renyi entropies for Haar
gle time and quantifies the degree of the entanglement random local circuits, automaton circuits, and Clifford
fluctuations. This measure is simply the full probability circuits. In all cases, we again perform the measure-
distribution of bipartite entanglement entropies measured ments on small circuits where we can track the evolu-
across all NN/2 bipartitions of the lattice. Comparing the tion exactly. In principle, we could measure these Renyi
entropy across many different lattice partitions effectively entropies for larger automaton circuits using our classi-
measures the multipartite entanglement of the wave func- cal algorithm, using observables in the form of Eq. (13).
tion [36], similar to the entanglement measure developed However, this involves measuring higher-order “SWAP”
by Meyer and Wallach [37]. operators, which have a value that is exponentially small
010329-8QUANTUM STATE COMPLEXITY IN COMPUTATIONALLY. . . PRX QUANTUM 2, 010329 (2021)
Clifford D. Entanglement spectrum
−4
10 Automaton Entanglement spectrum is the name given to the statis-
Eν [ Tr(ρkA ) ] Haar tical distribution of the eigenvalues of a reduced density
10−9
matrix [38,39]. The spacing between these eigenvalues
10−14 form a distribution that is known for different ensembles
of random matrices [40] and generically follows a Wigner-
10−19 Dyson distribution. For a random U(N ) unitary matrix,
10−24
the spacing between eigenvalues follows the Gaussian uni-
tary ensemble (GUE). These Wigner-Dyson distributions
10−29 have the special property that there is repulsion between
0 5 10 15
neighboring eigenvalues. On the other hand, the reduced
Renyi index k density matrix of wave functions that result from inte-
grable dynamics do not, in general, form a random matrix.
FIG. 4. The average trace of ρAk for different values of the In such a case, the eigenvalues of ρA do not show the same
Renyi index k. We find that the expectation value Eν [Tr(ρAk )] degree of level repulsion and may follow a simple Poisson
over the ensemble of automaton circuits, ν, is equal to the Haar
random value, EHaar [Tr(ρAk )] for all k that we tested. For Clifford
distribution.
circuits, which form a unitary 3-design, ECliff [Tr(ρAk )] are only To measure the entanglement spectrum, we first rewrite
constrained to match the Haar random value up to k = 3, and the wave function |ψ using the Schmidt decomposition.
show significant deviation above k ≈ 5.
|ψ = λi |αi |βi , (25)
i
in the amount of entanglement. Since the amount of where the λi are real positive numbers.
entanglement of a bipartition grows like a volume law, We can then define the entanglement spacing, si =
this measurement becomes exponentially hard in these λ2i+1 − λ2i , where we order the Schmidt coefficients such
systems. that λ0 ≤ λ1 ≤ · · · ≤ λM . For convenience [41,42], we
In Fig. 4, we show the expectation value for the kth define the level spacing ratio
Renyi entropy, as measured in both automaton and Clif-
ford circuits. Amazingly, the expectation value for the si si+1
automaton wave functions appears to be exactly equal to ri = min , . (26)
si+1 si
the Haar random value for all values of k that we mea-
sured. These measurements are consistent with those of The entanglement spectrum is then the probability distri-
an -approximate unitary k-design with very small error bution of the ri random variables.
. In the Appendix, we measure the evolution of the error In Fig. 5, we show the entanglement spectrum statistics
as a function of circuit depth. For Clifford circuits, on the for wave functions that result from both the automaton cir-
other hand, the expectation value matches the Haar value cuit and Clifford circuits. We see that the spectrum in the
for low Renyi index, but deviates significantly at higher automaton case follows very closely the universal form of
values of k. We denote by Eν [Tr(ρAk )] the expectation the Gaussian unitary ensemble [41,43], while the Clifford
value of the kth Renyi entropy measured over the ensem- states do not show the same level repulsion and appear to
ble of circuits ν. At high Renyi index k, small fluctuations follow a Poisson distribution.
away from this mean will be amplified. Therefore, these The relationship between chaotic dynamics and the
results are again consistent with the hypothesis that fluc- entanglement spectrum has been studied in Refs. [44–
tuations of random measurements are highly suppressed in 46]. However, a complete theoretical understanding of
automaton wave functions, to the extent that such measure- the connection between quantum state complexity and the
ments mimic that of a fully Haar random wave function. entanglement spectrum is still lacking. In Ref. [44], it was
Interestingly, in Ref. [24], it was found that the infinite- noted that dynamics under a universal set of quantum gates
order Renyi entropy S∞ ∼ − log(|λmax |) saturates near its is sufficient to generate GUE statistics of the entanglement
maximal value after only an O(N ) time. Such a state is spectrum, while evolution under Clifford gates results in
known as “max scrambled.” Although the complexity of Poisson statistics. Here, we have shown that this condition
the quantum state continues to grow past the max scram- of a universal set of quantum gates is not necessary to gen-
bling time, all max scrambled states will appear maximally erate GUE statistics. Indeed, we have created a state with
complex according to the Renyi entanglement measures. such statistics using only the automaton gate set, which
Our results strongly imply that automaton wave func- can be simulated classically in the way outlined in Sec. II.
tions will become max scrambled for polynomial depth It is interesting that such signatures of quantum chaos also
circuits. appear in wave functions that can be simulated classically.
010329-9JASON IACONIS PRX QUANTUM 2, 010329 (2021)
Automaton
and therefore the simulation quickly becomes intractable
Clifford for even moderately large design orders t. The automa-
2 GUE
ton circuits, on the other hand, have no restriction of the
Poisson
design order that can be efficiently simulated. It appears
P (r)
that we may use automaton circuits to study wave func-
tions of arbitrarily high design order and that have a near
1 maximal amount of magic.
Furthermore, there exist observables which involve mul-
tiple copies of the unitary U, such as those in the form
0 of Eq. (13), that can be efficiently estimated using our
0.0 0.2 0.4 0.6 0.8 1.0 classical algorithm. These observables may in general dif-
1
fer dramatically from those measured in a circuit that is
r = min (s, s ) a low-order unitary design. The behavior of such observ-
ables are essentially uniquely accessed for large systems
FIG. 5. The level spacing distribution of the entanglement using our automaton circuits. In the following section we
spectrum for automaton wave functions show Wigner-Dyson look at one example from this class of observables, the k-
GUE statistics, while the Clifford states show Poisson-like statis- point OTO correlation function. Also, note that automaton
tics. Wigner-Dyson statistics are expected for the eigenvalue circuits offer the potential to simulate types of gates that
distribution of random matrices and are a signature of quantum cannot easily be accessed in Clifford circuits. For example,
chaos.
various symmetries may be incorporated into the local uni-
taries and long-range diagonal gates may also be applied.
It remains an open question whether there is a concrete In the future it may be interesting to study fast-scrambling
relationship between entanglement statistics and unitary k- models using automaton circuits such as those in Ref. [50].
designs.
IV. MEASURING COMPLEXITY IN AUTOMATON
CIRCUITS
E. Summary of Sec. III
In this section, we have found that measurements from a A. Generalized out-of-time-ordered correlators
series of information theoretic quantities in automata wave Out-of-time-ordered correlators (OTOCs) have recently
functions are consistent with bounds for highly complex - been found to be an important tool for characterizing
approximate projective unitary design wave functions. We operator spreading in quantum circuits. The 4-point OTOC
have measured the observables at very late times and found
results that are indistinguishable from the expected Haar F (4) = A(t)B(0)A(t)B(0) (27)
random results. In the Appendix, we look at the behavior
of the error estimate and see that it decays rapidly with measures the average degree of nonlocality of an oper-
circuit depth. As we previously noted, the measurements ator A(t) = U† AU, and has been extensively studied in
in this section cannot be implemented efficiently using our the context of thermalization and quantum chaos [2,9,51].
classical algorithm. In fact, the distribution of wave func- This quantity will only be small if A(t) evolves into a
tion amplitudes and fine-grained probes of entanglement highly nonlocal operator. In Ref. [9], it was shown that the
discussed below could not even be measured efficiently on information-theoretic definition of scrambling is implied
a quantum computer as they require an exponential num- by the generic decay of this four-point function. Further-
ber of measurements to resolve. Nevertheless, it is amazing more, any initial product state that is evolved by such a
that we can identify a class of simulable wave functions unitary can be shown to be nearly maximally entangled.
that possess the properties of high complexity states. Following the work of Roberts and Yoshida [11], we can
Recent results show that some of these properties, such generalize this operator and define the 2k-point out-of-time
as the GUE entanglement spacing distribution, can also ordered correlators:
be seen in Clifford circuits that are perturbed by a finite
number of non-Clifford unitary gates [47]. These gates F (2k) = A1 (t)B1 (0) · · · Ak (t)Bk (0). (28)
create many-body quantum magic in the Clifford wave
functions [48]. In Ref. [49], it was proven that in order Deep connections have been found between the generic
to form an -approximate t-design, it suffices to inject smallness of the 2k-point functions, unitary k-designs, and
O[t4 log2 (t) log(1/ )] non-Clifford gates. Therefore, these quantum circuit complexity. A generic 2k-point function
simulations may be also be useful for studying proper- contains k copies of U and k copies of U† . Therefore, if
ties of low-order unitary designs. However, the simulation U is sampled from a unitary k-design then the average of
cost is exponential in the number of non-Clifford gates the 2k-point function over the ensemble {U} must equal
010329-10QUANTUM STATE COMPLEXITY IN COMPUTATIONALLY. . . PRX QUANTUM 2, 010329 (2021)
the Haar random value, and therefore will be exponen- time t lower bounds the time required to achieve an /dk -
tially small. Since the four-point OTOC expectation value design. Furthermore, these structured OTOCs with local
is quadratic in the U and U† operators, we see that only operators are the slowest to decay and therefore we can
a unitary 2-design is necessary for scrambling. We know, reasonably use them to establish an upper bound for all
however, that the complexity of the wave function will expectation values in Eq. (29).
continue to grow well past this scrambling time. We proceed as follows. We first identify a class of
These higher-order correlators are therefore an impor- k-order OTOCs that have a special recursive structure that
tant tool for understanding complexity beyond scrambling can be physically motivated and can be easily general-
in quantum dynamics. Crucially, the generalized OTOC ized. We then also perform a brute-force search over all
functions give us a probe that is insensitive to the onset of 2k-point OTOCs for a fixed value of k. These correlators
lower-order forms of complexity. For example, the 4-point are not as easily generalizable, but give a more complete
function in local circuits takes on an O(1) value through- picture of complexity growth in local random circuits. We
out the “thermalization” regime, before decaying to an can reasonably expect that the maximum OTOC value that
exponentially small value after a time t∗ ∼ O(L) for local we find in this search serves as an upper bound on all
circuits. This is in contrast to the entanglement entropies, k-order OTO correlation functions. In both cases, we are
which can also be used to diagnose scrambling and com- able to efficiently measure the correlation function in high
plexity, but which always require an exponentially hard depth automaton circuits with a large number sites. We
measurement to implement. This feature of the OTOCs is find that in both cases the correlators eventually decay to
both useful experimentally and, critically, allows us to use an exponentially small value in automaton circuits, provid-
automaton circuits to numerically probe the onset of com- ing strong evidence in very large systems that automaton
plexity efficiently in large systems. The k-point OTOC is circuits produce high complexity wave functions. Further-
therefore a concrete example of an interesting observable more, our brute-force search is able to identify a large,
of high complexity wave functions that can be uniquely linear in k, regime where the quantum wave function
probed numerically with automaton circuits. appears scrambled but the higher-order OTOCs have not
We can therefore use these higher-order correlation yet decayed. This gives us an unprecedented ability to
functions to probe the structure of the wave functions out- numerically study complexity growth in local quantum
put from quantum circuits. If we can find a 2k-point OTOC circuits.
that is nonzero, this implies that the unitary ensemble is
not a k-design and the wave function is likely of lower
B. Recursive k-point functions
complexity.
In Ref. [11], it was shown that the average value of We begin by studying a special instructive class of
the 2k-point correlation function can directly give a lower k-point OTOC functions that often retain an O(1) expec-
bound for the quantum circuit complexity of a unitary tation value beyond the scrambling time tsc . In these corre-
ensemble {U} = E , lators, the time-evolved Heisenberg operators Õ = U† OU
can be treated as a generalized unitary operator U(n) :
C(E ) ≥ (2k − 1)2N − log A1 (t)B1 · · · Ak (t)Bk .
A1 ···B1 ··· U(0) = U, (30)
(29) (1)
UO = U† OU, (31)
This expression is useful for showing that a generic decay ....
of the higher-order OTOCs implies a growth in circuit †
complexity. Unfortunately, it is not very useful for numer- U(n)
O ,O
= U(n) (n)
O O UO , (32)
ically calculating a bound on circuit complexity since the
main contribution comes from calculating a sum over an The higher-order OTOCs can be interpreted as assessing
exponentially large number of operators, each of which is the scrambling properties of U(n) . For example, we can
in general exponentially small. As we explain below, in write
this work, we take an alternative route by identifying spe-
(1)† (1)†
cial structured OTO correlators that have an O(1) value for ÃBÃCÃBÃC = UA B U(1) (1)
A C UA B UA C
low complexity dynamics. This gives a k-order generaliza-
tion of the notion of scrambling, which measures the decay = B(t)CB(t)C, (33)
of the local k-point OTO correlation functions from 1 at
t = 0 to some O( ) value. In the Appendix, we show how where, following the notation of Ref. [11], we let à =
this can be related to the design error for -approximate k- U† AU. Therefore, this 8-point function under U can be
designs and therefore can be used to estimate the circuit thought of as a 4-point function under U(1) . These recur-
complexity. Finding any k-order OTOC with value at sive OTOCs can always be interpreted as 4-point OTOCs
010329-11JASON IACONIS PRX QUANTUM 2, 010329 (2021)
with additional local operators hiding in the generalized (4)
Here FL,0 is simply the usual 4-point OTOC that mea-
unitaries. (4)
sures operator scrambling, so that FL,0 = 1 if and only
Under a fully Haar random U(2N ) dynamics, all k- (8)
point correlation functions will decay to an exponentially if [X̃L , X0 ] = 0. On the other hand, FL,1,0 measures the
small value. Therefore, not only does U have high quan- scrambling of X1 under a time evolution by X̃L = U† XL U.
tum complexity, but so do the operators U(1) = U† AU, In this case, we have FL,1,0(8)
= 1 if either [X̃L , X0 ] = 0 or
U(2) = U(1)† BU(1) , etc. [X̃L , X1 ] = 0. Under the approximation that these commu-
Conversely, for the known examples of exact unitary tators always take a value of either 0 or 1, so that the
designs, such as the ensemble of Pauli strings and Clifford operators either fully commute or are fully scrambled, we
circuits, the higher-order generalized unitaries are of lower have
complexity than the original operator.
Consider the case where {U} is an ensemble of Clifford
(8) (4)
circuits. These are known to form a unitary 2-design in E[FL,1,0 ] ≥ E[FL,0 ]. (36)
general, and a 3-design when the local Hilbert space is
qubits, but never form a 4-design [17]. Therefore, when
averaged over the ensemble of all Clifford circuits, all These higher-order OTOCs are a more strict measure of
4-point functions are found to be exponentially small, complexity, can easily be generalized, and retain a sim-
ple interpretation as measuring the scrambling properties
ÃBÃB = 4−N . However, the defining feature of Clifford
of the generalized unitary operators.
circuits is that they evolve Pauli strings to other Pauli
(1) We measure these recursively defined operators for cir-
(1)
αi operator U is
strings. Therefore, the generalized unitary
cuits that act on a state that is again initialized with spins
simply a Pauli string, U = S = i σi . The ensemble
polarized in the +x direction. We show the results in Fig. 6
of Pauli strings {S} are known to merely form a 1-design,
for an automaton circuit with L = 100 sites. The results are
and so the 4-point functions under {U(1) } do not decay to
averaged over many different random circuit realizations.
zero. The 4-point function under U(1) is an 8-point function
We point out several important features of this data.
under U. Therefore, there always exist 8-point functions
First, we see that, for automaton circuits, all generalized
for Clifford circuits that do not decay to the Haar ran-
OTOC functions do eventually decay to an exponentially
dom value. Therefore, the fact that Clifford circuits merely
small value. We take this as important further evidence that
scramble is demonstrated by the fact that while {U} scram-
automaton circuits have high quantum circuit complexity
bles, the ensemble of unitary operators {A(t) = U† AU}
and the resulting wave functions have a high quantum state
do not. In this way, the higher-order OTOCs expose a
complexity. Again, this should be seen as a stark contrast
hierarchical structure of unitary designs.
to other examples of numerically tractable quantum cir-
With this understanding, we use the higher-order OTOC
cuits such as Clifford circuits, for which we can always
to probe the dynamics of automaton circuits in the “beyond
find higher-order OTOCs that do not decay at all.
scrambling” regime. Since automaton circuits apply non-
Second, we see that in these circuits, the higher-order
trivial dynamics in the direction perpendicular to the
OTOCs are nonzero at later times than the usual 4-point
computational basis, we further define a set of recursive (4)
unitaries that are composed of only single-site X Pauli function FL,0 = X̃L X0 X̃L X0 . This concretely demon-
operators: strates that in such local random circuits there exists a
well-defined regime beyond the scrambling time where
U(0) = U, information about the original state |ψ0 is not completely
lost. In these “intermediate complexity states,” local infor-
U(1)
i1 = U Xi1 U,
†
mation from |ψ0 can still be probed using these spe-
U(2)
(1)† (1) cial measurements. Furthermore, note that the expectation
i1 i2 = Ui1 Xi2 Ui1 ,
value of the higher-order OTOCs at late times is gener-
(m−1)†
U(m) (m−1)
i1 i2 ···im = Ui1 i2 ···im−1 Xm Ui1 i2 ···im−1 .
ally much greater than twice the previous order, yet only
requires twice the computational effort to measure.
We then write down the special class of generalized Finally, we see that the “scrambling time” t∗ for this
OTOCs class of higher-order OTOCs appears to only increase
k (k−1)† logarithmically with order k. In particular, we find that
Fi(21 ,...,i
)
k−1 ,0
= Ui1 ···ik−1 X0 U(k−1)
i1 ···ik−1 X0 . (34)
Then, for example, t∗ = vB L + vk log2 (k). (37)
(4)
FL,0 = X̃L X0 X̃L X0 ,
(35) In the next subsection, we see that this is not a generic
(8)
FL,1,0 = X̃L X1 X̃L X0 X̃L X1 X̃L X0 . feature of the higher-order OTOCs.
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