Quantum State Complexity in Computationally Tractable Quantum Circuits

PRX QUANTUM 2, 010329 (2021)

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 useful 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 systems 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 *[email protected] 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 Society JASON 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 θ to probe the fine-detailed structure of wave functions. We U|m = ei m |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

010329-2 QUANTUM 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 the permutation group Sd and on average for a Haar scrambled operator. While automaton circuits gives trec → exp[λ d/ log(d)]asd →∞. 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 θ basis. We refer to the output of such circuits as automa- T(θ) = 1 − + ei X ,(3) 123 12 12 3 ton wave functions. This approach allows us to focus on the entanglement and complexity of the resulting wave where 12 = |0000| is the projection onto the |00 state 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- π ∈ | ∈ tion 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 functions on a classical computer. Despite their apparent {CNOT, SWAP, Rz(θ)},(4) simplicity, such an evolution produces highly nontrivial iθZ where Rz(θ) = e 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 c exactly. Note that both sets of gates defined above are universal m Throughout this paper, we often choose |ψ0 to be a prod- for quantum computation if supplemented by any single- m·σ uct state in the X basis, cm = (−1) /d, where m is a qubit gate that does not preserve the computational basis binary vector representation of the integer m,andσ is a [28]. vector of Pauli-X eigenvalues of |ψ . However, this need We first review a few important analytic results derived i 0 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 c . composed of T(θ = 0). First, an initially local diagonal m O ( ) We then time evolve the wave function by applying the operator diag 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 (t) (t) Uλ = U + U + + ,(6) all elements of the conjugacy class of Sd, which implies j ,j 1 j 1,j 2 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 { t } ber of other possible operators. Finally, the recurrence time precise set of gates Uj ,j +1 that are applied. The resulting

010329-3 JASON IACONIS PRX QUANTUM 2, 010329 (2021) wave function is then time O(NT2). On the other hand, if O is a diagonal operator then x = x and we can get an estimate for the entire iθm |ψ(t) = Uλ|ψ0 = cme |πλ(m).(7)time evolution in a time that scales like O(NT). m This approach can be straightforwardly adapted to measure operators that contain multiple copies of the uni- π ( ) Again λ 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) = ψλ(x, t)|x as x tion k times. We simply act the unitary circuit Utot = † † A1U B1U ···AkU BkU on the sampled basis states |m.We ψλ(x, t) =x|ψ(t)=c −1 exp[iθ −1 ]. (8) H⊗k πλ (x) πλ (x) can also consider a wave function on the Hilbert space 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 ⊗k only evolves to a simple product state, |π(m), instead of = 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 − (θ −θ ) a volume law entangled state. In this way we are able to ⊗ ⊗ 1 ∗ i x k| k = { xij ij ( )} A c cx e f xij , classically simulate the circuit evolution of highly entan- xij ij M = gled quantum wave functions in a way that is equivalent to i j 1 the well-known variational Monte Carlo methods. (13) We can therefore efficiently calculate estimates for sim- where x is the basis state x in the j th tensor copy of ple operator expectation values as ij i the Hilbert space. Examples of such observables are the kth-order SWAP operators that are used to evaluate the kth O=ψ |U†OU|ψ 0 0 Renyi entropy. Expectation values of this form are important in this work as they can be used to distinguish between = c∗c y U† O U x y x t t 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

−1 x = πλ [πO(πλ(x)]. (10)

Therefore, we can write Time M 1 ∗ O≈ ψλ [π(x ), t]ψλ[π(x ), t]f [π(x )] M i i i xi=1 M 1 ∗ i(θx −θ ) = i xi ( ) c cx e f xi . (11) M xi i xi=1

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 iθZˆ evolution in Eq. (9) for each time step independently. Esti- the single-site rotation about the z axis Rz(θ) = e (applied mating the full time dependence of O(t) therefore takes a independently to each site with a random angle θ).

010329-4 QUANTUM 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.Asδ → 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. iθZˆ angle θ, Rz(θ) = e . 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 ν ψ Quantum complexity theory quantifies the difficulty of states. An ensemble of pure states, , 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 d requires. Interestingly, in contrast to classical complexity Eν[p(ψ)] = dψ p(ψ) for all p ∈ Hom(k,k)(C ), ν theory, in the quantum setting one can also meaningfully Haar 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 Cd state from an initial simple reference state. Unlike with degree k both in the coordinates of vectors in 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 β = max Tr[M(|ψψ|−ρ )] r 0 -approximate k-design in a (d = qN )-dimensional Hilbert M (14) space formed from a set of |G| basic gates, subject to M ∈ Mr(d), 2 k r r 16k where M (d) is the set of generalized measurements com- Pr[Cδ(|ψ) ≤ r] ≤ 2(1 + )dN |G| , (17) r ( − δ)2 posed of a unitary circuit of depth r acting on a Hilbert d 1 space of size d, which is followed by a projective mea- which qualitatively remains very small until r ≈ k[N − |ψ surement in the computational basis. We say that has 2 log(k)]/ log(N). In other words, with high proba- δ C (|ψ)< strong -state complexity less than r, δ r,if bility, such a k-design has state complexity at least O[kN/ log(N)]. 1 β ≥ 1 − − δ. (15) Unitary k-designs define a fine-grained hierarchy of r d quantum states of increasing complexity. This concept is

010329-5 JASON 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 ( 11) universal gates of depth O Nk 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 2 −dp(xj ) tribution, p(xj ) =|xj |ψ| ∼ de . 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. α classically [34]. = N σ i The set of Pauli strings, S i=1 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 |β| |α|2 ≥ ≤ ( + ) − min(k,γ) mation scrambling, we will see that there exists a range Pr U 1 e . (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 2 sense, quantum state complexity generalizes the notion of bounds the fluctuations of the coefficients |cn| 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- 1 ton circuit, and therefore such a circuit evolution can be |ψ =⊗|+x = |m, (19) simulated efficiently with a classical computer in the man- 0 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

010329-6 QUANTUM STATE COMPLEXITY IN COMPUTATIONALLY. . . PRX QUANTUM 2, 010329 (2021)

H = H ⊗ H Automaton A B can be shown to be 10−1 Cliﬀord Porter-Thomas dist. 1 dA v ≥ ( ) − ) S N log dA , (20)

2 2ln(2) d | −3 B n 10 a | ≤ H H

d where dA dB are the dimensions of A and B, respec- ( −5 tively. P 10 Our definition of quantum state complexity implies that high complexity states cannot easily be distinguished 10−7 from the maximally mixed state. This property necessar- 0 5 10 15 ily requires the state to be nearly maximally entangled, so ρ 2 that the reduced density matrix A is close to the max- 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 ( N | |2) FIG. 2. The probability distribution of bit strings P 2 an 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 bipartite 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. 2 k/2 k k E|ψ({Tr(M|ψψ|) − E|ψ[Tr(M|ψψ|)]} ) ≤ . 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

010329-7 JASON 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

) have the same entanglement entropy, which is very close to t ( 4 the Page entropy. However, for Clifford circuits, while the vN average entanglement entropy is equal to the Page entropy, S Clifford 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 (S ) vN 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 (S ) vN 1 1 Sα(ρ ) = log(Tr[ρα]) = log λα , A 1 − α A 1 − α i i (22) 0.1 〈σ 〉 0.01 where the λi are the eigenvalues of the reduced density α 0.001 matrix ρA.Asα →∞, S (ρA) = Smin(ρA) =−log(λmax) approaches the min entropy. Here Smin(ρA) simply probes the largest eigenvalue of ρA, and bounds all other Renyi α entropies S (ρA) ≥ Smin(ρA) for all α.InRef.[12], it was FIG. 3. (a) The bipartite entanglement entropy SvN as a func- α tion of time for both Clifford and automaton circuits. In both shown that the -Renyi entropy averaged over a unitary α cases, the late time entanglement averages to the Haar random -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 E k(ρ ) ≥ + O( ) νk [S A ] dA 1 , (23) we measure the entanglement in all possible bipartitions of the lattice. We show the probability distribution of the entanglement E where νk 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 = (Sv −Sv )2, decays expo- k k N N Eν [Tr{ρ }] = E [Tr{ρ }]. (24) nentially with system size for automaton circuits and appears to k A Haar A saturate to a constant value for Clifford circuits. In other words, the kth Renyi entropies are all nearly maximal up to an O(1) constant for a unitary k-design, and ρk the trace of A 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- N across all N/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

010329-8 QUANTUM 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-

] Haar tical distribution of the eigenvalues of a reduced density ) 10−9 k A matrix [38,39]. The spacing between these eigenvalues ρ

( 10−14 form a distribution that is known for diﬀerent ensembles

Tr of random matrices [40] and generically follows a Wigner- [ 10−19 ν Dyson distribution. For a random U(N) unitary matrix, E 10−24 the spacing between eigenvalues follows the Gaussian unitary ensemble (GUE). These Wigner-Dyson distributions 10−29 0 5 10 15 have the special property that there is repulsion between neighboring eigenvalues. On the other hand, the reduced Renyi index k density matrix of wave functions that result from integrable dynamics do not, in general, form a random matrix. ρk FIG. 4. The average trace of A for different values of the ρ k In such a case, the eigenvalues of A do not show the same Renyi index k. We find that the expectation value Eν [Tr(ρ )] A degree of level repulsion and may follow a simple Poisson over the ensemble of automaton circuits, ν, is equal to the Haar E (ρk ) distribution. random value, Haar[Tr A ]forallk that we tested. For Clifford circuits, which form a unitary 3-design, E [Tr(ρk )] are only To measure the entanglement spectrum, we first rewrite Cliff A | 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 λ2 − λ2 i+1 i , 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 ri = min , . (26) automaton wave functions appears to be exactly equal to si+1 si the Haar random value for all values of k that we measured. 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 E (ρk ) values of k. We denote by ν[Tr A ] 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.

010329-9 JASON 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 Poisson ton circuits, on the other hand, have no restriction of the design order that can be eﬃciently simulated. It appears )

r that we may use automaton circuits to study wave func- ( tions of arbitrarily high design order and that have a near

P 1 maximal amount of magic. Furthermore, there exist observables which involve multiple 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- fer dramatically from those measured in a circuit that is r = min (s, 1) s a low-order unitary design. The behavior of such observables 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 unitaries 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 (2k) number of non-Clifford unitary gates [47]. These gates F =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

010329-10 QUANTUM 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 O = U†OU (n) can be treated as a generalized unitary operator U : C(E) ≥ (2k − 1)2N − log A1(t)B1 ···Ak(t)Bk . U(0) = U, (30) A1···B1··· (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 (n) = (n)† O (n) complexity. Unfortunately, it is not very useful for numer- UO,O UO 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 special structured OTO correlators that have an O(1) value for ˜ ˜ ˜ ˜ = (1)† (1) (1)† (1) ABACABAC UA B UA 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 = 0tosomeO( ) value. In the Appendix, we show how where, following the notation of Ref. [11], we let A˜ = 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

010329-11 JASON 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. ( ) sures operator scrambling, so that F 4 = 1 if and only Under a fully Haar random U(2N ) dynamics, all k- L,0 ˜ = (8) point correlation functions will decay to an exponentially if [XL, 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 XL = U XLU. tum complexity, but so do the operators U(1) = U†AU, (8) = ˜ = In this case, we have FL,1,0 1 if either [XL, X0] 0or U(2) = U(1)†BU(1),etc. ˜ [XL, 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 E (8) ≥ E (4) circuits. These are known to form a unitary 2-design in [FL,1,0] [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 AB˜ AB˜ =4 N . However, the defining feature of Clifford of the generalized unitary operators. circuits is that they evolve Pauli strings to other Pauli ( ) We measure these recursively defined operators for cir- strings. Therefore, the generalized unitary operator U 1 is ( ) α cuits that act on a state that is again initialized with spins simply a Pauli string, U 1 = S = σ i . The ensemble i i 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 XLX0XLX0 . This concretely demon- operators: strates that in such local random circuits there exists a well-defined regime beyond the scrambling time where (0) U = U, information about the original state |ψ0 is not completely ( ) lost. In these “intermediate complexity states,” local infor- U 1 = U†X U, i1 i1 mation from |ψ0 can still be probed using these spe- (2) (1)† (1) cial measurements. Furthermore, note that the expectation U = U Xi U , i1i2 i1 2 i1 value of the higher-order OTOCs at late times is gener- (m) = (m−1)† (m−1) ally much greater than twice the previous order, yet only Ui i ···im Ui i ···i − XmUi i ···i − . 1 2 1 2 m 1 1 2 m 1 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 (2k) (k−1)† (k−1) logarithmically with order k. In particular, we find that F =U ··· X U ··· X . (34) i1,...,ik−1,0 i1 ik−1 0 i1 ik−1 0 Then, for example, ∗ = v + v ( ) t BL k log2 k . (37) (4) =˜ ˜ FL,0 XLX0XLX0 , (35) In the next subsection, we see that this is not a generic (8) =˜ ˜ ˜ ˜ FL,1,0 XLX1XLX0XLX1XLX0 . feature of the higher-order OTOCs.

010329-12 QUANTUM STATE COMPLEXITY IN COMPUTATIONALLY. . . PRX QUANTUM 2, 010329 (2021)

1.0 that design order (and therefore the complexity) in these circuits grows at least like the polynomial D(1/11),was 0.8 proven rigorously. Linear complexity growth can also be rigorously shown in the limit of a large local Hilbert space ) t ( 1 0.6 dimension [10]. However, there is no known proof of lin- ,i 2 ear complexity growth for the more diﬃcult case of a local k = 4 ,...,i random circuit composed of qubits. m 0.4 ,i k = 8 ) Quantum automaton circuits allow us to test this conjec- k ( L/2 k = 16

F ture numerically in large systems by using the generalized 0.2 k = 32 OTOC operators as a measure of complexity. To do this, k = 64 we should search over the full set of out-of-time-ordered 0.0 0 100 200 300 400 operators Circuit depth (time) (k) = E ˜ ··· ˜ OTOCmax max ν[ A1B1 AkBk ], (38) A1,B1,...,Ak,Bk 100

where Ai, Bi are any Pauli string operators. However, this Nk 10−1 would require computing 4 diﬀerent expectation values,

) which is clearly intractable even for small values of N t ( 1 and k. ,i 2 10−2 k = 4 Instead, we again restrict the operators Ai, Bi to be only ,...,i

m single-site Pauli-X operators and deﬁne

,i k = 8 ) k ( L/2 −3 k = 16 ( )

F k 10 = Eν ˜ ··· ˜ k = 32 Fmax max [ Xi1 Xi2 Xi2k−1 Xi2k ]. (39) i1,...,ik k = 64 10−4 In practice, we find a lower bound for this operator by 0 100 200 300 400 searching only over operators with support on a small Circuit depth (time) fixed number of sites. Even with these practical restric- tions, F(k) (t) gives an upper bound on the design order ( ) max FIG. 6. The recursive kth-order OTOCs, F k , L/2,im,...,i2,i1 of the circuit at time t. defined in Eq. (34), for a system with L = 100 sites with For k up to k = 16, we search over all possible cor- periodic boundary conditions. These OTOCs are com- relators with Ai, Bi ∈{X0, X1, XL/2−1, XL/2}.Fork = 4, the posed of single-site Pauli-X operators acting on sites {L/2, (4) = ... }={ / ( / ) ... } maximum OTOC is the usual 4-point function Fmax im, , i3, i2, i1 L 2, log2 k 2 , ,2,1,0 .SeeEq.(35) for ˜ ˜ = the forms of the k = 4andk = 8 operators. We can measure XL/2X0XL/2X0 .Atk 8, we find that the maximum these correlators for very high values of k. Top: the higher-order expectation value occurs when the OTOC takes the form OTOCs decay at progressively later times. The highest-order (8) =O˜ O˜ OTOCs remain nonzero well past the scrambling time. Bottom: Fmax , at late times, all correlators eventually decay at an exponential (40) ˜ ˜ ˜ rate. O = X0XL/2X0X0.

To calculate F(16), we must measure approximately 32 000 C. Higher-order OTOCs and complexity max different correlation functions. In this case, we find that the A more complete picture of complexity growth in our maximum expectation value occurs for local random automaton circuits can be found by studying more general classes of k-point OTOCs. (16) =O˜ O˜ Fmax , For local random circuits, it was shown in Ref. [25] that (41) ˜ ˜ ˜ ˜ ˜ the unitary design order will continue to grow far beyond O = X0XL/2X0X0X0XL/2XL/2X0, the scrambling time. It is expected that the complexity in these local circuits grows linearly with circuit depth, D, plus special equivalent permutations of these operators that for an exponentially long time [14]. This linear complex- are related by symmetry. ity growth is of great interest in the high-energy literature. Note that we find this correlation function appears to In the context of the anti-de Sitter space - conformal field “peak” only at special values of k. That is, we find that (16)( ) (8) ( ) (4) ( ) theory (AdS-CFT) correspondence, the linearly growing at late times, Fmax t Fmax t Fmax t , but that this is complexity of the dual CFT is related to the growth of not true for other values of k that we tested (up to at least an AdS wormhole [52,53]. In Ref. [25] a weaker result, k = 24).

010329-13 JASON IACONIS PRX QUANTUM 2, 010329 (2021)

1.0 k = 4 0.8 k = 8 k = 16 ) ) t

t 0.6 ( ( ) ) k k ( max ( max 0.4 F F

0.2

0.0 200 400 600 800 Circuit depth (time) Circuit depth (time)

(k) ( ) = = FIG. 8. The maximum OTOC Fmax t for k 4andk 16 for diﬀerent system sizes L = 100, 400, 1600 (shown left to right in separate shaded regions). The gap between the k = 4andk = 16 wave fronts grows with system size. This gap size versus L is shown in the inset to grow like ∼ Lα with α = 0.48(1). ) t (

) 4 k ( max 3

F ∗ 2 we reach the scrambling time tsc = t , when the wave func- 4 ∗ 1 tion forms an approximate 2-design. At some later time t , Gap ratio 8 (8) < 0 we will have Fmax , and the wave function will form an 0.0001 0.001 0.01 0.1 1.0 = ∗ − ∗ F (k) approximate 4-design. This difference t8 t4 defines max the rate of complexity growth. The inset in Fig. 7(b) shows that t∗ − t∗ ≈ 2 for sufficiently small . In Fig. 8,we Circuit depth (time) 16 8 find that as the system size in increased,√ the size of the “complexity gap” grows like (L). Therefore, at least FIG. 7. A brute-force search over a wide class of kth-order for these three values of k, the generalized scrambling time OTOCs gives the maximum OTOC F(k) (see the text for a pre- max appears to follow the form cise definition). We see that the k = 16 point function decays at a much later time in this case compared to the recursive OTOCs of √ the previous section. In the bottom plot, we show the same data t∗ = v L + Lk. (42) ∗( ) B on a logarithmic scale and define the scrambling time tk as the (k) < time beyond which Fmax . In the inset we see that the ratio = ( ∗ − ∗)/( ∗ − ∗) between scrambling times t16 t8 t8 t4 approaches The linear growth with k is consistent with a linear = 2as → 0. growth of design order (and therefore complexity) with circuit depth. That is, we show that, for the values of k studied here, there exists a regime beyond scrambling in local ran- The results for the 4-, 8-, and 16-point functions are dom circuits where the wave function is definitively not a shown in Fig. 7. Again, we see that all correlation func- (k/2)-design, and the size of this regime appears to grow tions we measured decay to an exponentially small value linearly with k. at late times. The dramatic difference is that the (k = 16)- point function remains nonzero for a significantly longer period of time. V. DISCUSSION We see that the complexity growth in local random cir- In this paper, we study in detail the quantum state com- cuits is subject to the so-called switchback effect [53], plexity of wave functions that are the output of local whereby there is a delay in the onset of linear complex- automaton circuits. These gates act very simply on com- ity growth for initially local operators. This occurs due to putational basis states, which allows us to simulate these the exact cancelation of unitary gates outside the lightcone circuit efficiently using a classical computer. Despite this, in the operator evolution O˜ = U†OU. when acting on an initial product state with spins oriented ∗( ) We define the time tk as the circuit depth beyond perpendicular to the computational basis, local random (k) < which Fmax . Because of the switchback effect, we automaton circuits generate very complex highly entan- expect that the linear complexity growth begins only after gled quantum wave functions.

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We can quantify the complexity of the wave functions serves to highlight the difference between high and low using tools from quantum information theory. We specifi- complexity states. cally relate the quantum state complexity to the difficulty In the second section, we study the 2k-point OTOC of distinguishing the wave function from the maximally functions. These are a generalization of the popular 4- mixed state. We consider several different measurements point OTOC, which is known to characterize the onset in order to argue that the ensemble of automaton wave of scrambling in quantum systems. Unlike the previous functions form an approximate projective unitary design set of measurements that require an exponential effort to of high order. Based on known connections between uni- compute using our quantum Monte Carlo algorithm, the tary designs and quantum complexity, these results imply generalized OTOC functions are capable of probing com- that automaton wave functions have a high quantum state plexity beyond the scrambling regime using computational complexity. resources that scale linearly with system size. Therefore, In the first section, we consider four basic measures we can use these as a tool to efficiently study complexity of complexity. First, we see that in quantum automaton growth in very large circuits. circuits, the distribution of bit strings (which result from We first identify a set of recursively defined 2k-point many-qubit projective measurements in a basis orthogo- OTOCs whose precise form is physically motivated and nal to the computational basis) follows the well-known that can be easily generalized to very high values of k. “Porter-Thomas” distribution up to the resolution of our We argue that in low complexity unitary circuits, such as numerics. Second, we characterize von Neumann entan- Clifford circuits, these higher-order recursive OTOCs do glement entropy across all possible bipartitions of the not generically decay. In automaton circuits, however, we lattice and find that it is always nearly exactly equal to find that at some time t∗ beyond the scrambling time these the Page entropy. This is in contrast to Clifford circuits, correlation functions always decay to an exponentially which show significant fluctuations between different par- small value. This is both further evidence that automaton titions. We also study the level spacing distribution in the wave functions form projective unitary designs and proof entanglement spectrum and see a convergence to the Gaus- that generalized OTOCs can be practically used to probe sian unitary ensemble Wigner-Dyson distribution, a result complexity beyond scrambling in large systems. that implies that the reduced density matrices of automa- Importantly, using automaton circuits, we are also able ton wave functions behave like random matrices. Finally, to study more generic forms of out-of-time ordered corre- we study the generalized kth Renyi entropies, which were lation functions. By searching over thousands of possible shown in Ref. [12] to be a direct probe of unitary design k-point OTOCs up to k = 16, we are able to identify complexity. We find that, for automaton wave functions, special orderings of operators for which the average cor- the kth Renyi entropy is exactly equal to the Haar ran- relators remain nonzero for a far longer time even than the dom value for all values of k we measured. This suggests recursive OTOCs discussed above. Measuring these spe- that design complexity in automaton wave functions will cial OTOCs gives us an unprecedented ability to numer- grow until the infinite-order limit of the Renyi entropy is ically characterize the rate of complexity growth in large saturated, a condition known as “max scrambling.” local random circuits. In particular, we find that the scram- By every metric, the automaton circuits exhibit the bling time for the kth-order OTOC appears to increase same properties as a wave function from high depth local linearly with k. Notably, this result is consistent with the random circuit composed of universal basic gates. Such linear growth of complexity that is conjectured in the circuits are known to form an approximate polynomial literature. unitary design and therefore at sufficient depth approxi- Taken together, all of the above results are very strong mate the fully Haar random wave functions to arbitrary evidence that the automaton circuits are capable of produc- accuracy. All the above results suggest that fluctuations ing wave functions with high quantum state complexity. in automaton wave functions are highly suppressed in the Our results therefore imply that automaton circuits are a same way as in universal random circuits. In fact, the bit- rare example of a numerically tractable system that can string distribution and generalized Renyi entropies follow generically simulate quantum chaotic dynamics on a clas- the constraints imposed for large-k projective k-designs, sical computer. We expect there to be a wide variety implying that automaton wave functions approximate at of applications across a wide range of fields. Already, least the first k moments of the fully random Haar measure. this technique has been applied to a range of quantum Furthermore, convergence of the bit-string distribution to circuit models in the context of understanding quantum the Porter-Thomas form and the entanglement level spac- dynamics in systems with different symmetries [20,54– ing to the Wigner-Dyson distribution are often cited as 56]. We predict that it will become a prominent tech- key signatures of quantum chaos. Throughout, we com- nique for studying generic chaotic circuit dynamics and pare the results for automaton circuits to those of Clifford may be an important tool for understanding how the circuits that are known to form only a finite low-order uni- growth of complexity beyond scrambling plays a role in tary design, even for high depth circuits. This comparison thermalizing condensed matter systems. There also exist

010329-15 JASON IACONIS PRX QUANTUM 2, 010329 (2021) obvious applications to quantum information theory, where In Ref. [25], it was shown that in order to form an - characterizing the complexity of local circuits is a key approximate k-design, it is sufficient to form a depth D uni- problem. Beyond this, our work may be useful for practical versal local random circuit where D ∼ N log2(k)k9.5[Nk + quantum information processing tasks such as randomized log(1/ )]. Therefore, we expect that in the automaton cir- benchmarking [57,58] and decoupling [59], where unitary cuits the error will also decrease exponentially with some designs play a key role. The ability of OTOCs to identify polynomial power of the circuit depth D. In Sec. III,we unique observables in high depth random circuits that are compared measurements in the automaton circuits at very fully scrambled may be useful experimentally for charac- late times to the Haar random results. The information terizing the fidelity of non-Clifford quantum circuits. In theoretic results for the bit-string distribution and Renyi high-energy physics, characterizing the rate of complex- entropy were also derived for the more general case of ity growth in quantum systems is conjectured to be related, approximate unitary designs. In the main text, we looked at through the holographic principle, to the growth of the vol- high depth circuits where we expect to be much smaller ume in the bulk geometry beyond the event horizon in than we could resolve. In what follows, we see how mea- black holes. Future work in identifying the specific form surements of these quantities violate these bounds in lower of higher-order OTOCs that best characterize the complex- depth circuits. Once again, we perform these simulations ity growth may therefore shed important insight into these on small systems where we can exactly simulate the full problems. evolution of the wave function. In Ref. [25], it was shown that a small error in the k- ACKNOWLEDGMENTS design distribution allows a small multiplicative violation of the Porter-Thomas distribution given by Eq. (18).Foran J.I. thanks Rahul Nandkishore, Xiao Chen, Itamar Kim- approximate k-design, the probability of a fluctuation with chi, Roger Melko, and Zi-Wen Liu for useful discus- −γ d|a |2 >γ is less than (1 + )e up to γ = k. We plot sions. This material is based upon work supported by n this distribution in Fig. 9 for automata circuits with a range the Air Force Office of Scientific Research under Grant of circuit depths between D = 26 and D = 48. We see that No. FA9550-20-1-0222. J.I. is supported by a Simons the probability amplitudes follow the Porter-Thomas dis- Investigator Award to Leo Radzihovsky from the Simons tribution very closely for small k. At larger k, there is a Foundation. noticeable violation of this bound at low depth, but the distribution once again approaches the k-design bound as the APPENDIX: APPROXIMATE UNITARY DESIGNS depth is increased. 1. Error analysis From this we can find an estimate of the design error . Using Eq. (18), we lower bound the error by counting the Throughout this paper we have seen that quantum number of bit strings in excess of what we expect from the automata circuits of sufficient depth are able to approx- Haar random result. That is, imate measurements on d copies of the Haar random distribution for 2N -dimensional unitary matrices. The difference between these measurements can be used to quan- >ek P(x)dx − 1, (A3) tify the closeness of the automata ensemble to the Haar x>k/d distribution. Specifically, we define the metric 2 where P(|an| ) is the probability density function for 2 amplitudes |an| of bit strings in the x basis. In Fig. 9, = F (k) − F (k) : E Haar (A1) we see that at fixed k, error decreases exponentially with circuit depth D. On the other hand, at fixed depth, the = ⊗k ⊗ ( †)⊗k − ⊗k ⊗ ( †)⊗k piUi Ui dU Ui Ui , error increases as we increase the bound cutoff k. When i 2 (k) ∼ O(1), our ensemble is not close to the Haar dis- (A2) tribution and we should not think of E as an approximate k-design. (k) where FE is the kth-order frame potential. We say that We can also estimate the design error using our mea- E forms an -approximate k-design if < . There are surement of the Renyi entropies. In the main text we saw several different ways of defining the closeness of an that at high depth, all Renyi entropies are nearly maximal E ρk approximate unitary design, which vary mainly in the and the trace ν[ A] exactly equals the Haar random value. definition of the operator norm. In Ref. [25], the results In Ref [12], it was further shown that, for an - were derived using the ·∞ norm, while in Ref. [12] the approximate k-design, the reduced density matrix obeys ·1 norm was used. See Ref. [60] for the relationship the relationship between different norms. The different choices of norms 1 give sightly different definitions for the error . We assume E (ρk ) ≤ (ρk ) + ν[Tr A ] dU Tr A . (A4) these differences are negligible for our analysis. Haar dA

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103 −1 10 30 2 10 32 −3 26 101 34

) 10

2 28 36 | 0 n 30 10 38

a −5 | 10 32 40 ( 10−1 36 42 P −2 10−7 40 10 44 −3 Haar 10 −9 10 10−4 0 5 10 15 0 5 10 15 2 k = d |an| Renyi index k

103 26 FIG. 10. The design-error from measurements of the Renyi 102 28 entanglement. We estimate using Eq. (A5). At large Renyi 30 1 index k it becomes harder to achieve maximum entanglement. 10 32 36 However, the automaton circuits become exponentially close to 0 10 40 the Haar random value as the circuit depth is increased. 10−1 44 48 10−2 2. OTOCs and design error 10−3 Finally, we note that we expect the same general behavior of the design error to carry over to larger systems. 10−4 0 5 10 15 However, the information theoretic results we used above 2 k = d|an| cannot be easily calculated in larger circuits. Instead, we may follow the approach of Sec. IV and infer the behavior of the error from the behavior of the OTO correlation FIG. 9. Top: the probability distribution of bit strings in the x functions. For an exact k-design, we expect the expectation basis in automaton circuits at different depths. Lower depth cir- value of all OTO correlation functions with nonidentity cuits have large fluctuations above the Porter-Thomas bound at operators to be exponentially small. For -approximate high k. Bottom: we quantify the cumulative error given by Eq. unitary designs, the error can bound the value of the (A3).Atfixedk, the error decreases exponentially with circuit OTOCs. depth. At fixed circuit depth, the error increases exponentially with fluctuation size k = d|a |2. In Ref. [11], the frame potential of Eq. (A1) is related to n the OTOCs via the expression

F (k) Therefore, the diﬀerence between the Renyi trace at order k 1 ˜ ˜ 2 E |A B ···A B E | = . (A6) compared to the Haar random result gives us a lower bound 4k+2 1 1 k k 2k+2 d ··· d estimate of the k-design error: A1B1 AkBk

We follow the analysis of Ref. [61] for 4-point functions, and make the simplifying assumption that only correlation k k > Eν (ρ ) − (ρ ) ˜ dA [Tr A ] dU Tr A . (A5) functions containing two copies of all operators Ai and Bi Haar are nonzero. This ensures all correlation functions contain ˜ both the operators Ai, Bi and their inverse. In Fig. 10, we show this error estimate as a function of Under this assumption, we ﬁnd that, for an - Renyi index k and circuit depth D. Again, we see that the approximate k-design, the typical OTO correlation func- error appears to increase exponentially with k. We can also tion has a value clearly see an exponential decrease in the error with cir- < −1 | ˜ ··· ˜ |≈ cuit depth. We expect Eq. (A5) to break down for dA , A1B1 AkBk E (A7) which we already reach for the range of k we study around D ∼ 42. for nonidentity operators Ai and Bi. This expression applies Using these two metrics, we therefore clearly see that the for both local and nonlocal operators. Since there are dk design error decays exponentially with circuit depth. For such OTO expectation values, we would need E to form our local circuits with N = 16 sites at depth D ∼ 50, we an /dk-approximate design in order to guarantee that all estimate that the circuit forms an -approximate k-design correlation functions have a value less than . Therefore, if for k = 10 with <0.001. we ﬁnd any OTO correlation function with value at time

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