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
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. (13) corresponds to the geodesic ˜ ⃖ UðτÞ¼P exp i dτHðτÞ ; ð8Þ distance on the manifold of unitaries. Note also that we can 0 reproduce our analysis done in the following sections with is the unitary operator representing the quantum circuit, other choices of the cost functional. We will, however, ψ leave this for future work. which takes the reference state j τ¼0i to the target state ψ τ j τ¼1i. parametrizes a path in the space of the unitaries and given a particular basis (elementary gates) MI, B. Circuit complexity for a single oscillator For our case, the covariance matrix corresponding to the τ I τ Hð Þ¼Y ð ÞMI: target state (4) will take the form, 0 1 In this context, the coefficients fYIðτÞg are referred to as 1 − ImðωðtÞÞ “control functions.” The path ordering in Eq. (8) is B Re ω t Re ω t C τ¼1 @ ð ð ÞÞ ð ð ÞÞ A ’ G ¼ 2 ; ð15Þ necessary as all the MI s do not necessarily commute with − ImðωðtÞÞ jωðtÞj each other. ReðωðtÞÞ ReðωðtÞÞ Now, since the states under consideration (2) and (4) are Gaussian, they can be equivalently described by a covari- where ωðtÞ is defined in Eq. (5). For the reference state (2) ance matrix as follows: it will take the following form: ab ψ ξaξb ξbξa ψ 1 0 G ¼h ðx; tÞj þ j ðx; tÞi; ð9Þ τ 0 ω G ¼ ¼ r : ð16Þ 0 ω where ξ ¼fx; pg. This covariance matrix is typically a real r symmetric matrix with unit determinant. We will always Next we change the basis as follows: transform the reference covariance matrix such that [27,29] G˜ τ¼1 ¼ S · Gτ¼1 · ST; G˜ τ¼0 ¼ S · Gτ¼0 · ST; ð17Þ G˜ τ¼0 ¼ S · Gτ¼0 · ST ð10Þ with where G˜ τ¼0 is an identity matrix and S is a real symmetric T pffiffiffiffiffi matrix whose transpose is denoted by S . Similarly, the ωr 0 reference state will transform as S ¼ 1 ; ð18Þ 0 pffiffiffiffi ωr G˜ τ¼1 ¼ S · Gτ¼1 · ST: ð11Þ such that G˜ τ¼0 ¼ I is an identity matrix. For the case under ˜ τ The unitary Uð Þ acts on this transformed covariance study, the reference frequency ωr is real. We will choose the matrix as, following three generators:
˜ τ¼1 ˜ τ ˜ τ¼0 ˜ −1 τ G ¼ Uð Þ · G · U ð Þ: ð12Þ → i → i 2 → i 2 M11 2 ðxp þ pxÞ;M22 2 x ;M33 2 p : ˜ ˜_ Next we define a suitable cost function FðU; UÞ and define ð19Þ [22,35–37] Z These will serve as our elementary gates and satisfy the 1 _ 2 CðU˜ Þ¼ FðU;˜ U˜ Þdτ: ð13Þ SLð ;RÞ algebra, 0 ½M11;M22 ¼2M22; ½M11;M33 ¼−2M33; Minimizing this cost functional gives us the optimal set of YIðτÞ, which in turn give us the most efficient circuit by ½M22;M33 ¼M11: ð20Þ
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Next, if we parametrize the U˜ ðτÞ as, cosðμðτÞÞ coshðρðτÞÞ − sinðθðτÞÞ sinhðρðτÞÞ − sinðμðτÞÞ coshðρðτÞÞ þ cosðθðτÞÞ sinhðρðτÞÞ U˜ ðτÞ¼ ð21Þ sinðμðτÞÞ coshðρðτÞÞ þ cosðθðτÞÞ sinhðρðτÞÞ cosðμðτÞÞ coshðρðτÞÞ þ sinðθðτÞÞ sinhðρðτÞÞ and set the boundary conditions as,
G˜ τ¼1 ¼ U˜ ðτ ¼ 1Þ · G˜ τ¼0 · U˜ −1ðτ ¼ 1Þ; G˜ τ¼0 ¼ U˜ ðτ ¼ 0Þ · G˜ τ¼0 · U˜ −1ðτ ¼ 0Þ; ð22Þ we find that [29], ω2 þjωðtÞj2 ω2 − jωðtÞj2 fcoshð2ρð1ÞÞ; tanðθð1Þþμð1ÞÞg ¼ r ; r ; fρð0Þ; θð0Þþμð0Þg ¼ f0;cg: ð23Þ 2ωrReðωðtÞÞ 2ωrImðωðtÞÞ
Here c is an arbitrary constant. For simplicity we choose ω2 − jωðtÞj2 μðτ ¼ 1Þ¼μðτ ¼ 0Þ¼0; θðτ ¼ 0Þ¼θðτ ¼ 1Þ¼c ¼ tan−1 r : ð24Þ 2ωrImðωðtÞÞ
From Eq. (8) we have,
I ˜ ˜ −1 I T Y ¼ Trð∂τUðτÞ · UðτÞ · ðM Þ Þ; ð25Þ where TrðMI:ðMJÞTÞ¼δIJ. Using this we can define the metric
2 I J ds ¼ GIJdY dY ; ð26Þ
1 where the GIJ ¼ 2 δIJ is known as a penalty factor. Given the form of UðsÞ in Eq. (21) we will have,
ds2 ¼ dρ2 þ coshð2ρÞcosh2ρdμ2 þ coshð2ρÞsinh2ρdθ2 − sinhð2ρÞ2dμdθ; ð27Þ and the complexity functional defined in Eq. (13) will take IV. QUANTIFYING CHAOS USING COMPLEXITY the form, The goal of this paper is to explore whether we can implement the notion of quantum circuit complexity as a Z qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 diagnostic of a system’s chaotic behavior. Classically, C ˜ τ _i _j ðUÞ¼ d gijx x : ð28Þ chaos is diagnosed by studying trajectories in the phase 0 space of some dynamical systems, a notion that is not well defined in quantum systems, essentially because of the The simplest solution for the geodesic is again a straight uncertainty principle. It is important to keep in mind that line on this geometry [22,29]. when we speak of geodesics in the context of circuit complexity, we will mean trajectories defined on the space of unitaries. ρðτÞ¼ρð1Þτ: ð29Þ Now we propose a new diagnostic for chaotic behavior based on circuit complexity. We consider a target state jψ 2i obtained by evolving a reference state jψ 0i forward in time Evaluating Eq. (28) we simply get with Hˆ and then backward in time with Hˆ þ δHˆ , 2 2 iðHˆ þδHˆ Þt −iHtˆ 1 ω ω jψ 2i¼e e jψ 0i: ð31Þ C ˜ ρ 1 −1 r þj ðtÞj ðUÞ¼ ð Þ¼2 cosh 2ω ω : ð30Þ rReð ðtÞÞ ˆ We would like to compute the complexity CðU˜ Þ of this target state jψ 2i with respect to the reference state jψ 0i This is the geodesic distance in the space of SLð2;RÞ [29]. For a chaotic quantum system, even if the two ˆ ˆ ˆ unitaries with the end points anchored at the two points Hamiltonians H and H þ δH are arbitrarily close, jψ 2i determining the boundary conditions (23). will be quite different from jψ 0i,
026021-4 CHAOS AND COMPLEXITY IN QUANTUM MECHANICS PHYS. REV. D 101, 026021 (2020) 1 ω2 ωˆ 2 complexity grows linearly for a very short period and Cˆ ˜ −1 r þj ðtÞj ðUÞ¼ cosh ; ð32Þ Cˆ ˜ 2 2ωrReðωˆ ðtÞÞ reaches a saturation with some fluctuations. However, ðUÞ for the inverted oscillator tells a completely different story. ˆ where now The overall behavior of CðU˜ Þ for the inverted oscillator appears to be some complicated monotonically growing 1 ˆ 2 function. However, a closer look at Fig. 2, reveals that it ψ 2ðx; tÞ¼N ðtÞ exp − ωˆ ðtÞx ; ð33Þ 2 takes a small amount of time for the complexity to pick up and
Ω02 ωˆ ðtÞ¼iΩ0 cotðΩ0tÞþ : ð34Þ sin2ðΩ0tÞðωðtÞþiΩ0 cotðΩ0tÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In this last expression, Ω0 ¼ m2 − λ0 is the frequency 0 1 2 Ω02 2 associated with the Hamiltonian H ¼ 2 p þ 2 x and λ0 ¼ λ þ δλ with δλ being very small. The time dependence of this complexity demonstrates that there is a clear qualitative difference between a regular oscillator and an inverted oscillator as evident from Figs. 1 and 2. For the regular oscillator we get oscillatory behavior [29–31]; the
ˆ FIG. 3. CðU˜ Þ vs time for different values of λ (with δλ ¼ 0.01, m ¼ 1).
ˆ FIG. 1. CðU˜ Þ vs time for regular oscillator (m ¼ 1, λ ¼ 1.2, δλ ¼ 0.01).
ˆ FIG. 4. Complexity vs λ for different times. (a) For all t, CðU˜ Þ starts to increase before the critical of λ namely λ ¼ m2. (b) We ˆ can observe that near t ¼ 40 there is a sharp increase in CðU˜ Þ at ˆ FIG. 2. CðU˜ Þ vs time for inverted oscillator (m ¼ 1, λ ¼ 15, the critical value of λ (λ ¼ 1, for the choice of the parameter). We δλ ¼ 0.01). have set δλ ¼ 0.01 and m ¼ 1 in both panels.
026021-5 TIBRA ALI et al. PHYS. REV. D 101, 026021 (2020) after which it displays a linear ramp with time. For a different choice of coupling (λ > λc) we get similar behavior with different pick-up time and slope (ϕ) for the linear ramp. These features are displayed for different values of the coupling in Fig. 3. As we increase λ (beyond the critical value), we are in effect making the model more unstable and consequently from our very specific circuit model we expect a larger complexity and a smaller pick-up time. Therefore, the slope and pick-up time scale are natural candidates for measuring the unstable nature of the inverted oscillator. When we ϕ ˆ explore the slope of the linear region (as in Fig. 3) for FIG. 6. Dependence of CðU˜ Þ on δλ (m ¼ 1, λ ¼ 10). different values of coupling λ we find the behavior shown in Fig. 5. In the following section, we will argue that this slope is similar to the Lyapunov exponent. time. For this oscillator model (1), we can show it Note that the linear growth kicks in near a certain time analytically following Ref. [44]. It is shown in the next section. t ¼ ts (as in the Fig. 2) which depends on the choice of the parameters, fm; λg. We plot this pick-up time as function of Now we will study how this complexity changes with the λ λ in Fig. 5(b). We believe that this time scale is equivalent to coupling for a fixed time. Figure 4 shows how the λ the scrambling time which frequently appears in the chaos complexity changes with the coupling for various times. λ 2 λ literature. One way to confirm this is to compute the out- We denote c ¼ m as the critical value of , after which the of-time-order four-point correlator. The time when the system becomes an inverted oscillator. We find that for Λ − smaller values of t the complexity start to increase for OTOC ∼ e ðt t Þ becomes Oð1Þ, is called the scrambling λ < λc. Around t ¼ 40 the complexity sharply increases at λ ¼ λc. We call this time the critical time tc. Figure 4 shows that, it takes a certain amount of time for the system to “know” that it has become chaotic; tc marks when this occurs. We further check the sensitivity of our results to the ˆ magnitude of δλ. We plot CðU˜ Þ for the inverted oscillator for a fixed value of λ but for different δλ. We find that while the slope of the linear region remains the same, the pick-up time is sensitive to δλ as exhibited in Fig. 6. We will conclude this section by highlighting the fact that we get the same result regarding diagnosing chaos when we explore the circuit complexity (using the corre- lation matrix method), where both the target and reference states are evolved by slightly different Hamiltonians from some other state. The equality between these two complex- ities was concretely shown in Ref. [29].
V. OTOC, LYAPUNOV EXPONENT AND SCRAMBLING TIME The exponential behavior of the four-point OTOC has recently emerged as a popular early-time diagnostic for quantum chaos.2 In Ref. [44] the authors explicitly calcu- lated the OTOCs for a harmonic oscillator. For our model, the OTOC for the x and p operators (after reinstating the factor of ℏ)gives[44]
2An alternative to the OTOC, FðtÞ¼hA†ðtÞB†ð0ÞAðtÞBð0Þi, is the thermally averaged commutator squared CðtÞ¼ h½AðtÞ;Bð0Þ 2i with the two being related through ϕ λ δ 0 01 1 λ FIG. 5. (a) Slope vs ( ¼ . , m ¼ ). (b) ts vs CðtÞ¼2 − 2ReðFðtÞÞ. Unless there is an explicit ambiguity, (δ ¼ 0.01, m ¼ 1). we will refer to them both as the OTOC.
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2 ℏ2 2Ω 4 2 h½xðtÞ;p i¼ cos t; ð35Þ ts ¼ logð Þt : ð39Þ where Ω is defined in Eq. (1). When Ω is imaginary, we can Also, the λ dependence of the Lyapunov exponent is shown write the above expression as an exponential function in Fig. 7(a). Again the nature of the graph is in agreement with Fig. 5(a). After fitting the data we get for the slope ϕ of 2 2 2 Ω h½xðtÞ;p i ≈ ℏ e j jt þ : ð36Þ the linear region of the graph in Fig. 5(a),
2λ − Rewriting the above expression as e Lðt t Þ, with the ϕ ¼ 2jΩj¼2λL: ð40Þ Lyapunov exponent λL allows us to immediately read off λ Ω that for our system, L ¼j j while the scrambling (or We have also checked the m dependence of the slope ϕ Ehrenfest) time is given by and the pick-up time ts and they are in agreement with the λ 1 1 m dependence of L and t respectively. t ¼ λ log ℏ : ð37Þ Before we end this section we would like to highlight the L following interesting point. Mathematically, it is evident 1 The λ dependence of this time scale (in units of log ℏ)is from Eqs. (35) and (36) that the exponential decay of the shown in Fig. 7(b). The nature of the graph is in agreement OTOC is a consequence of a simple analytic continuation. with Fig. 5(b). In fact from Fig. 7(b) after doing a data This is due to the change of cos Ωt to cosh Ωt, when Ω fitting we get for the pick-up time, becomes imaginary. This implies that a simple analytic continuation is essentially capturing the scrambling and 4 logð2Þ chaotic behavior in this quantum system. t ¼ : ð38Þ s jΩj
1 VI. TOWARDS A FIELD THEORY ANALYSIS In the scale of log ℏ this is related to the scrambling By using the single oscillator model we have illustrated time t as, how complexity can capture chaotic behavior. In this section we will explore a possible field theory model in which the inverted oscillator appears naturally. Consider two free scalar field theories [(1 þ 1)-dimensional c ¼ 1 conformal field theories] deformed by a marginal coupling. The Hamiltonian is given by
H ¼ H0 þ H Z I 1 Π2 ∂ ϕ 2 Π2 ∂ ϕ 2 2 ϕ2 ϕ2 ¼ 2 dx½ 1 þð x 1Þ þ 2 þð x 2Þ þ m ð 1 þ 2Þ Z
þ λ dxð∂xϕ1Þð∂xϕ2Þ: ð41Þ
We can discretize this theory by putting it on a lattice. Then by using the definitions
xðn⃗Þ¼δϕðn⃗Þ;pðn⃗Þ¼Πðn⃗Þ=δ; ω ¼ m; 1 m Ω ; λˆ λδ−4; mˆ ; ¼ δ2 ¼ ¼ δ ð42Þ
we get
δ X 2 2 Ω2 − 2 H ¼ 2 ½p1;n þ p2;n þð ðx1;nþ1 x1;nÞ n Ω2 − 2 ˆ 2 2 2 þ ðx2;nþ1 x2;nÞ þðm ðx1;n þ x2;nÞ λˆ − − þ ðx1;nþ1 x1;nÞðx2;nþ1 x2;nÞÞ : ð43Þ
λ λ 1 λ 1 FIG. 7. (a) L vs (m ¼ ). (b) t vs (m ¼ ). Next we perform a series of transformations,
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−1 1 XN 2πik x1 ¼ pffiffiffiffi exp a x˜1 ; ;a N ;k N k¼0 −1 1 XN 2πik p1 ¼ pffiffiffiffi exp − a p˜ 1 ; ;a N ;k N k¼0 −1 1 XN 2πik x2 ¼ pffiffiffiffi exp a x˜2 ; ;a N ;k N k¼0 −1 1 XN 2πik p2 ¼ pffiffiffiffi exp − a p˜ 2 ; ;a N ;k N k¼0 ps;k þ pa;k ps;k − pa;k p˜ 1 ¼ pffiffiffi ; p˜ 2 ¼ pffiffiffi ; ;k 2 ;k 2 ˆ ˜ − FIG. 8. CðUÞ vs time for the inverted oscillator (δ ¼ 0.1, xs;k þ xa;k xs;k pa;k 1 1000 λ 10 δλ 0 01 x˜1 ¼ pffiffiffi ; x˜2 ¼ pffiffiffi ; ð44Þ m ¼ , N ¼ , ¼ , ¼ . ). ;k 2 ;k 2 that lead to the Hamiltonian where
2 δ XN−1 Ω0 2 2 2 2 2 2 ωˆ Ω0 Ω0 k Ω¯ Ω kðtÞ¼i cotð tÞþ 2 ; H ¼ ½ps;k þ kxs;k þ pa;k þ kxa;k ; ð45Þ k k Ω0 ω Ω0 Ω0 2 sin ð ktÞð kðtÞþi k cotð ktÞÞ k¼0 πk Ω0 2 ¼ mˆ 2 þ 4ðΩ2 − λˆ − δλˆÞsin2 ð49Þ where k N π 2 ¯ 2 2 2 ˆ 2 k ω ω Ω ¼ mˆ þ 4ðΩ þ λÞsin ; and kðtÞ, r;k are defined as, k N Ω − ω Ω πk k i r;k cotð ktÞ Ω2 ˆ 2 4 Ω2 − λˆ 2 ω ðtÞ¼Ω ; ð50Þ k ¼ m þ ð Þsin : ð46Þ k k ω − Ω Ω N r;k i k cotð ktÞ
It is immediately clear that by tuning the value of λˆ, the and frequencies Ωk can be made arbitrarily negative resulting in Ω¯ πk coupled inverted oscillators. Note that k will always be ω2 ¼ mˆ 2 þ 4Ω2sin2 : ð51Þ positive. Therefore, one can view Eq. (45) as a sum of r;k N regular and inverted oscillators for each value of k. With regards to studying the unstable behavior, the regular Using our testing method (outlined in Sec. IV), once oscillator part is not very interesting. Hence, we will simply again for the inverted oscillator we can immediately read investigate the inverted oscillator part with the Hamiltonian off the scrambling time and Lyapunov exponent from the ˆ time evolution of CðU˜ Þ as shown in Fig. 8. −1 δXN πk H˜ ðm;Ω;λˆÞ¼ p2 þ mˆ 2 þ 4ðΩ2 − λˆÞsin2 x2 : 2 k N k k¼0 VII. DISCUSSION ð47Þ In this paper we used a harmonic oscillator model that converts to an inverted oscillator for large coupling of the Note that by tuning λˆ for this Hamiltonian one can get both interaction Hamiltonian. The coupling behaves as a regu- regular and inverted oscillators. At t ¼ 0 we start with the lator and by tuning it we can switch between regular and ˆ ground state of H˜ ðm; Ω; λˆ ¼ 0Þ. Then we compute CðU˜ Þ as inverted regimes. Our motivation was to use this inverted oscillator as a toy model to study quantum chaos. In this before by considering two Hamiltonians H˜ and H˜ 0 with two context, the regular oscillator serves as a reference system. λˆ λˆ0 λˆ δλˆ δλˆ slightly different couplings, and ¼ þ , where is We developed a new diagnostic for quantum chaos by small. We get constructing a particular quantum circuit and computing vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u the corresponding complexity. Our diagnostic can extract 1 uXN−1 ω2 ωˆ 2 2 equivalent information as the out-of-time-order correlator ˆ t −1 r;k þj kðtÞj CðU˜ Þ¼ cosh ; ð48Þ 2 2ω Reðωˆ ðtÞÞ with the additional feature that complexity can detect when k¼0 r;k k the system switches from the regular to the chaotic regime.
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We considered a target state which is first forward As a final point of motivation, we note that by virtue of evolved and then backward evolved with slightly different the recent “complexity = action” [50] and “complexity = Hamiltonians and found that the behaviors for the regular volume” [51] conjectures, the computational complexity of and inverted oscillators are completely different in this a holographic quantum system has a well-defined (if not case. For the regular oscillator we got some oscillatory entirely unambiguous) dual. This opens up tantalizing new behavior as in Refs. [28,29,31]. However, for the inverted possibilities in the study of quantum chaos in strongly oscillator we got an exponential-type function with two coupled quantum systems. We leave these issues for distinct features: for an initial period the complexity is future work. nearly zero, after which it exhibits a steep linear growth. By comparing with the operator product expansion, we dis- ACKNOWLEDGMENTS covered the small time scale and slope of the linear portion to be equivalent to the scrambling time and the Lyapunov We would like to thank Dario Rosa and Jon Shock for exponent respectively. To elaborate further, we note that the useful comments. A. B. thanks Aninda Sinha, Pratik only difference between the scrambling time [as in Nandy, Jose Juan Fernandez-Melgarejo and Javier Eq. (37)] and pick-up time [as in Eq. (38)] is just Oð1Þ Molina Vilaplana for many discussions and ongoing [as evident from Eq. (39)] (constant in natural units). collaborations on complexity. A. B. also thanks the Hence, these definitions capture the same scrambling Department of Physics, University of Windsor, Perimeter physics. Institute and the FISPAC group (especially Jose Juan To give a proof-of-principle argument for complexity Fernandez-Melgarejo and E. Torrente-Lujan) of the as a chaos diagnostic, we have used the inverted oscillator Department of Physics, University of Murcia for their as a toy model. This is, however, a rather special example warm hospitality during this work. A. B. is supported by and, by no means, a realistic chaotic system. To put JSPS Grant-in-Aid for JSPS fellows (17F17023). J. M. is complexity on the same footing as, say the OTOC as a supported by the NRF of South Africa under Grant probe of quantum chaos will take much more work, with No. CSUR 114599. N. M. is supported by the South more “realistic” systems like the maximally chaotic SYK African Research Chairs Initiative of the Department of 3 model and its many variants (see, for example, Refs. [46– Science and Technology and the National Research 48] and references therein) in the (0 þ 1)-dimensional Foundation of South Africa. Any opinion, finding and quantum mechanical context, or the Murugan-Stanford- conclusion or recommendation expressed in this material is 1 1 Witten class of ( þ )-dimensional (nonmaximally) cha- that of the authors and the NRF does not accept any liability otic conformal field theories [49]. in this regard. Research at Perimeter Institute is supported by the Government of Canada through the Department of 3It is worth pointing out that as this manuscript was being Innovation, Science, and Economic Development, and by prepared, we were made aware of the recent article [45] also the Province of Ontario through the Ministry of Research advocating for the idea of complexity as a tool for the study of quantum chaotic systems. and Innovation.
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