PHYSICAL REVIEW D 101, 026021 (2020) Chaos and complexity in quantum mechanics † ‡ ∥ Tibra Ali,1,* Arpan Bhattacharyya ,2,3, S. Shajidul Haque,4, Eugene H. Kim,5,§ Nathan Moynihan,4, and Jeff Murugan4,¶ 1Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada, 2Indian Institute of Technology, Gandhinagar, Gujarat 382355, India 3Center for Gravitational Physics, Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan 4The Laboratory for Quantum Gravity & Strings, Department of Mathematics & Applied Mathematics, University of Cape Town, Private Bag, Rondebosch 7701, South Africa 5Department of Physics, University of Windsor, 401 Sunset Avenue, Windsor, Ontario N9B 3P4 Canada (Received 16 June 2019; published 29 January 2020) We propose a new diagnostic for quantum chaos. We show that the time evolution of complexity for a particular type of target state can provide equivalent information about the classical Lyapunov exponent and scrambling time as out-of-time-order correlators. Moreover, for systems that can be switched from a regular to unstable (chaotic) regime by a tuning of the coupling constant of the interaction Hamiltonian, we find that the complexity defines a new time scale. We interpret this time scale as recording when the system makes the transition from regular to chaotic behavior. DOI: 10.1103/PhysRevD.101.026021 I. INTRODUCTION expðλLtÞ, where λL is the system’s largest Lyapunov exponent [2]. This does not happen in quantum mechanics: Quantum chaos is intrinsically difficult to characterize. two nearly identical states, i.e., states with a large initial Consequently, a precise definition of quantum chaos in overlap, remain nearly identical for all time (as their overlap many-body systems remains elusive and our understanding is constant under unitary evolution). It has been argued of the dynamics of quantum chaotic systems is still [3,4] that a quantum analog of “sensitive dependence on inadequate. This lack of understanding is at the heart of initial conditions” is to consider evolving identical states a number of open questions in theoretical physics such as ˆ ˆ δ ˆ ˆ thermalization and transport in quantum many-body sys- with slightly different Hamiltonians, H and H þ H.IfH is tems, and black hole information loss. It has also precipi- the quantization of a (classically) chaotic Hamiltonian, the tated the renewed interest in quantum chaos from various states will evolve into two different states whose inner branches of physics from condensed matter physics to product decays exponentially in time. quantum gravity [1]. Traditionally, chaos in quantum systems has been Chaotic classical systems on the other hand are charac- identified by comparison with results from random matrix terized by their sensitive dependence on initial conditions: theory (RMT) [5]. Recently however, other diagnostics two copies of such a system, prepared in nearly identical have been proposed to probe chaotic quantum systems – initial states (namely, two distinct points in phase space, [6 8]. One such diagnostic is out-of-time-order correlators separated by a very small distance), will evolve over time (OTOCs) [9,10] from which both the (classical) Lyapunov – into widely separated configurations. More precisely, the exponent as well as the scrambling time [11 13] may be distance between the two points in phase space grows as extracted. However, recent work in mass-deformed Sachdev-Ye-Kitaev (SYK) models [14] have revealed some tension between the OTOC and RMT diagnostics that arise, *[email protected] in part, through the nature of the probes. The OTOC † [email protected] captures early-time quantum mechanical features of the ‡ [email protected] system while RMT diagnostics typically capture late-time §[email protected] ∥ statistical features. Evidently, a deeper understanding of [email protected] ¶[email protected] probes of quantum chaos is required. In this light, it is interesting therefore to ask whether one can characterize Published by the American Physical Society under the terms of chaos in quantum systems using quantum-information- the Creative Commons Attribution 4.0 International license. theoretic measures.1 In this work, we propose a new Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. 1Some progress in this direction was made in Refs. [15–21]. 2470-0010=2020=101(2)=026021(11) 026021-1 Published by the American Physical Society TIBRA ALI et al. PHYS. REV. D 101, 026021 (2020) diagnostic/probe of quantum chaos using the notion of II. THE MODEL – ’ circuit complexity [22 34], adopting Nielsen s geometric We are interested in comparing the complexity of a – approach [35 37]. More specifically, we study the regular system with that of an unstable/chaotic system. To circuit complexity of a particular target state obtained from that end, we consider the Hamiltonian a reference state by performing a forward evolution followed by a backward evolution with slightly different 1 Ω2 2 2 Ω2 2 − λ Hamiltonians. Then we demonstrate how this enables one H ¼ 2 p þ 2 x where ¼ m : ð1Þ to probe/characterize chaotic quantum systems, giving information beyond what is contained in the OTOC. For λ <m2, Eq. (1) describes a simple harmonic oscillator; Note that instead of using the complexity for a target state for λ >m2, we have an inverted oscillator. The λ ¼ m2 that is forward and then backward evolved from a reference case, of course describes a free particle. Our inverted state as mentioned above, one can as well study the oscillator model can be understood as a short-time approxi- complexity of a different circuit where both the target mation for unstable/chaotic systems. In particular, this and reference states are obtained from time evolution model captures the exponential sensitivity to initial con- (once) by applying slightly different Hamiltonians from ditions exhibited by chaotic systems. Let us start with the some common state. For circuit complexity from the following state at t ¼ 0: correlation matrix method that we will be using in this ω x2 paper, the authors of Ref. [29] concretely showed that the ψðx; t ¼ 0Þ¼N ðt ¼ 0Þ exp − r ; ð2Þ time evolution of complexity in these two scenarios is 2 identical. To establish out testing method, we consider a simple, where exactly solvable system—the inverted oscillator, described ω by the Hamiltonian H ¼ p2=2 − ω2x2=2 [38]—which r ¼ m: ð3Þ captures the exponential sensitivity to initial conditions exhibited by chaotic systems [39]. Classically, the inverted Evolving this state in time by the Hamiltonian (1) produces [43] oscillator has an unstable fixed point at ðx ¼ 0;p¼ 0Þ;a particle accelerates exponentially away from the fixed point ωðtÞx2 when perturbed. Though the phase-space volume of the ψ N − ðx; tÞ¼ ðtÞ exp 2 ; ð4Þ inverted oscillator is unbounded, our results are relevant to systems with a bounded phase space in that such a system where N ðtÞ is the normalization factor and would be described by an inverted oscillator up to a certain time. The two systems produce the same results over the Ω − iω cotðΩtÞ time of interest (but would not be analytically solvable ωðtÞ¼Ω r : ð5Þ ω − iΩ Ωt beyond that time). In what follows, we include the analysis r cotð Þ for a regular oscillator as a reference for what arises in a We will be computing the complexity for this kind of time- nonchaotic system and explore also a many-body system evolved state (4) with respect to Eq. (2) and (quantum field theory) where the inverted oscillator appears. It is worth noting that the inverted quantum ωðt ¼ 0Þ¼ω : ð6Þ oscillator is not just a toy model; it has been realized r experimentally [40] and has even played a role in math- ematics, in attacking the Riemann hypothesis [41]. It also III. COMPLEXITY FROM THE provides important insights into the bound of the Lyapunov COVARIANCE MATRIX exponents [42]. We will start this section with a quick review of circuit The rest of the paper is organized as follows. In Sec. II, complexity and then conclude with a computation of the we present the model and states considered in this work. In circuit complexity for a single oscillator. For circuit Sec. III, we review the ideas behind circuit complexity, and complexity we will use the covariance matrix method. compute the circuit complexity for our system. Section IV Note that a similar analysis can be done for circuit demonstrates how quantum chaos can be detected and complexity from the full wave function. quantified using circuit complexity while Sec. V discusses the OTOC and its relation to the results obtained from the circuit complexity. In Sec. VI, we discuss a many-body A. Review of circuit complexity system (quantum field theory) where the inverted oscillator Here we will briefly sketch the outline of the compu- arises. Finally, we summarize and present concluding tation of circuit complexity. Details of this can be found in remarks in Sec. VII. Refs. [22,24]. We will highlight only the key formulas and 026021-2 CHAOS AND COMPLEXITY IN QUANTUM MECHANICS PHYS. REV. D 101, 026021 (2020) interested readers are referred to Refs. [22,24] and citations minimizing the circuit depth. There are various possible _ thereof. The problem is simple enough to state: given a set choices for these functions FðU;˜ U˜ Þ. For further details, of elementary gates and a reference state, we want to build we refer the reader to the extensive literature in the most efficient circuit that starts at the reference state and Refs. [22–24,35–37]. In this paper, we will choose terminates at a specified target state. Formally, rXffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 2 ψ ˜ τ 1 ψ F 2ðU; YÞ¼ ðY Þ : ð14Þ j τ¼1i¼Uð ¼ Þj τ¼0i; ð7Þ I where For this choice, one can easily see that, after minimization Z ˜ τ the CðUÞ defined in Eq.