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Quantum State Complexity in Computationally Tractable Quantum Circuits

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Quantum State Complexity in Computationally Tractable Quantum Circuits 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 *[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.
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