Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator

Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator Selected for a Viewpoint in Physics PHYSICAL REVIEW X 7, 031011 (2017) Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator 1 2,3 3 3 4,5,2,* 2,6,† 7,8,9,‡ 7,8 Jun Li, Ruihua Fan, Hengyan Wang, Bingtian Ye, Bei Zeng, Hui Zhai, Xinhua Peng, and Jiangfeng Du Beijing Computational Science Research Center, Beijing 100193, China Institute for Advanced Study, Tsinghua University, Beijing 100084, China Department of Physics, Peking University, Beijing 100871, China Department of Mathematics and Statistics, University of Guelph, Guelph N1G 2W1, Ontario, Canada Institute for Quantum Computing, University of Waterloo, Waterloo N2L 3G1, Ontario, Canada Collaborative Innovation Center of Quantum Matter, Beijing 100084, China Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Synergetic Innovation Centre of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China College of Physics and Electronic Science, Hubei Normal University, Huangshi, Hubei 435002, China (Received 30 December 2016; published 19 July 2017) The idea of the out-of-time-order correlator (OTOC) has recently emerged in the study of both condensed matter systems and gravitational systems. It not only plays a key role in investigating the holographic duality between a strongly interacting quantum system and a gravitational system, it also diagnoses the chaotic behavior of many-body quantum systems and characterizes information scrambling. Based on OTOCs, three different concepts—quantum chaos, holographic duality, and information scrambling—are found to be intimately related to each other. Despite its theoretical importance, the experimental measurement of the OTOC is quite challenging, and thus far there is no experimental measurement of the OTOC for local operators. Here, we report the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator. We observe that the OTOC behaves differently in the integrable and nonintegrable cases. Based on the recent discovered relationship between OTOCs and the growth of entanglement entropy in the many-body system, we extract the entanglement entropy from the measured OTOCs, which clearly shows that the information entropy oscillates in time for integrable models and scrambles for nonintgrable models. With the measured OTOCs, we also obtain the experimental result of the butterfly velocity, which measures the speed of correlation propagation. Our experiment paves a way for experimentally studying quantum chaos, holographic duality, and information scrambling in many-body quantum systems with quantum simulators. DOI: 10.1103/PhysRevX.7.031011 Subject Areas: Quantum Physics, Quantum Information, Statistical Physics ˆ ˆ I. INTRODUCTION iHt −iHt ˆ ˆ Hamiltonian, BðtÞ¼ e Be , and h  i denotes aver- aging over a thermal ensemble at the temperature The out-of-time-order correlator (OTOC), given by 1=β ¼ k T. For a many-body system with local operators † † ˆ ˆ ˆ ˆ ˆ ˆ A and B, the exponential deviation from unity of a FðtÞ¼hB ðtÞA ð0ÞBðtÞAð0Þi ; ð1Þ λ t normalized OTOC, i.e., FðtÞ ∼ 1 − #e , gives rise to the Lyapunov exponent λ . is proposed as a quantum generalization of a classical Quite remarkably, it was found recently that the OTOC measure of chaotic behaviors [1,2]. Here, H is the system also emerges in a different system that seems unrelated to chaos, that is, the scattering of shock waves nearby the zengb@uoguelph.ca horizon of a black hole and the information scrambling hzhai@tsinghua.edu.cn there [3–5]. A Lyapunov exponent of λ ¼ 2π=β is found xhpeng@ustc.edu.cn there. Later it was also found that the quantum correction Published by the American Physical Society under the terms of from string theory always makes the Lyapunov exponent the Creative Commons Attribution 4.0 International license. smaller [5]. Thus, it leads to a conjecture that 2π=β is an Further distribution of this work must maintain attribution to upper bound of the Lyaponuv exponent, which was later the author(s) and the published article’s title, journal citation, and DOI. proved for generic quantum systems [6]. This is a profound 2160-3308=17=7(3)=031011(12) 031011-1 Published by the American Physical Society LI, FAN, WANG, YE, ZENG, ZHAI, PENG, and DU PHYS. REV. X 7, 031011 (2017) efficient with practical algorithms proposed [25]. Here, the theoretical result. If a quantum system is exactly holo- graphic dual to a black hole, its Lyapunov exponent will quantum computer we use is liquid-state nuclear magnetic resonance (NMR) with molecules. In this work, we report saturate the bound; and a more nontrivial speculation is that measurements of OTOCs on a NMR quantum simulator. if the Lyapunov exponent of a quantum system saturates the We stress that, on one hand, our approach is universal and bound, it will possess a holographic dual to a gravity model can be applied to any system that has full local quantum with a black hole. A concrete quantum mechanics model, control, including a superconducting qubit and trapped ion; now known as the Sachdev-Ye-Kitaev model, has been on the other hand, this experiment is currently limited to a shown to fulfill this conjecture [2,7,8]. This establishes a small size not because of our scheme but because of the profound connection between the existence of holographic scalability issue of the quantum computer. duality and the chaotic behavior in many-body quantum systems [9]. II. NMR QUANTUM SIMULATION OF THE OTOC Recent studies also reveal that the OTOC can be applied to study physical properties beyond chaotic systems. The The system we simulate is an Ising spin chain model, decay of the OTOC is closely related to the delocalization whose Hamiltonian is written as of information and implies the information-theoretic def- z z z inition of scrambling. In the high- temperature limit (i.e., H ¼ ð−σ ˆ σ ˆ þ gσ ˆ þ hσ ˆ Þ; ð2Þ i iþ1 i i β ¼ 0), a connection between the OTOC and the growth of entanglement entropy in quantum many-body systems has x;y;z where σ ˆ are Pauli matrices on the i site. The parameter also been discovered quite recently [10,11]. The OTOC can values g ¼ 1, h ¼ 0 correspond to the traverse field Ising also characterize many-body localized phases, which are model, where the system is integrable. The system is not even thermalized [10,12–15]. nonintegrable whenever both g and h are nonzero. We Despite the significance of the OTOC revealed by recent simulate the dynamics governed by the system Hamiltonian theories, experimental measurement of the OTOC remains H, and measure the OTOCs of operators that are initially challenging. First of all, unlike the normal time-ordered acting on different local sites. The time dynamics of the correlators, the OTOC cannot be related to conventional OTOCs are observed, from which entanglement entropy of spectroscopy measurements, such as angle-resolved photo- the system and butterfly velocities of the chaotic systems emission spectroscopy (ARPES) and neutron scattering, are extracted. through the linear response theory. Secondly, direct simu- lation of this correlator requires the backward evolution in A. Physical system time, that is, the ability to completely reverse the The physical system to perform the quantum simulation Hamiltonian, which is extremely challenging. One exper- is the ensemble of nuclear spins provided by iodotrifluro- imental approach closely related to time reversal of quantum systems is the echo technique [16], and the echo has been ethylene (C F I), which is dissolved in d chloroform; see 2 3 studied extensively for both noninteracting particle systems Fig. 1(a) for the sample’s molecular structure. For this and many-body systems to characterize the stability (a) (b) of quantum evolution in the presence of perturbations [17–19], and the physics is already quite close to OTOC. Recently it has been proposed that the OTOC can be measured using echo techniques [20]. In addition, there also exists several other theoretical proposals based on the interferometric approaches [21–23]. However, none of them (c) have been experimentally implemented thus far. Here, we adopt a different approach to measure the OTOC. To make our approach work, some extent of “local control” is required. A universal quantum computer fulfills this need by having “full local control” of the system—that is, a universal set of local evolutions can be realized, and this set of local evolutions can build up any unitary FIG. 1. Illustration of the physical system, the Ising model, and evolution of the many-body system, both forward and the experimental scheme. (a) The structure of the C F I molecule 2 3 backward evolution in time. That is to say, we use a used for the NMR simulation. (b) The four site Ising spin chain. quantum computer to perform the measurement of the A and B label two subsystems for the later discussion of the OTOC. In fact, historically, one of the key motivations to entanglement entropy. (c) Quantum circuit for measuring the OTOC for the general N-site Ising chain when β ¼ 0 (in our case, develop quantum computers is to simulate the dynamics of z x ˆ ˆ ˆ many-body quantum systems [24], and quantum simulation N ¼ 4). Here, R ¼ 1, R ð−π=2Þ, R ðπ=2Þ for A ¼ σ ˆ , σ ˆ , σ ˆ , x y 1 1 1 respectively. of many-body dynamics has been theoretically shown to be 031011-2 MEASURING OUT-OF-TIME-ORDER CORRELATORS ON A … PHYS. REV. X 7, 031011 (2017) ˆ ˆ ˆ ˆ ˆ ˆ 13 19 19 −iHmτ −iH τ=2 −iH τ=2 −iH τ −iH τ=2 −iH τ=2 m x z zz z x molecule, the C nucleus and the three F nuclei ( F , 1 e ≈ ðe e e e e Þ ð5Þ 19 19 F , and F ) constitute a four-qubit quantum simulator. 2 3 Each nucleus corresponds to a spin site of the Ising chain, for small enough τ. Here, the dynamics is divided into m as shown in Fig. 1(b). In experiment, the sample is placed in pieces with t ¼ mτ, and a static magnetic field along the z ˆ direction, resulting in the following form of the system Hamiltonian, H ¼ gσ ˆ ; ð6aÞ x i 4 4 X X πJ 0i ij X z z z H ¼ − σ ˆ þ σ ˆ σ ˆ ; ð3Þ NMR i i j ˆ H ¼ hσ ˆ ; ð6bÞ 2 2 z i¼1 i<j;¼1 where ω =2π is the Larmor frequency of spin i and J is 0i ij z z H ¼ −σ ˆ σ ˆ : ð6cÞ zz i iþ1 the coupling strength between spins i and j. The values of these system parameters are given in Appendix A. The system is controlled by radio-frequency (rf) pulses, and the Each propagator inside the bracket of Eq. (5) corresponds corresponding control Hamiltonian is to either single-spin operation or coupled two-spin oper- ation, and can be implemented through manipulating H NMR H ðtÞ¼ −ω ðtÞfcos½ϕðtÞσ ˆ þ sin½ϕðtÞσ ˆ g; ð4Þ rf 1 i i with rf control H : single-spin operation terms are global rf rotations around the x or z axis, which can be easily done where ω ðtÞ and ϕðtÞ denote the amplitude and the −iH τ zz through hard pulses; the two-spin operation term e can emission phase of the rf field, respectively. The control be generated through some suitably designed pulse pulse shape can be elaborately monitored to realize the sequence based on the NMR refocusing techniques [27]. desired dynamic evolution. Actually, complete controllabil- More details of the method are described in Appendix B. ity of such a system has been demonstrated [26], which iHt The reversal of Ising dynamics e can be done in a similar guarantees that arbitrary system evolution can be imple- mented on it. Our experiments are carried out on a Bruker manner. Note that in the case we consider here, B is a local AV 400 MHz spectrometer (9.4 T) at tempera- ˆ unitary operator on the site-N spin and B ¼ σ ˆ with γ ¼ x, ture T ¼ 305 K. y, z that can be implemented by a selective rf π pulse on the site-N spin. Hence, for any given t, the total unitary ˆ ˆ B. Experimental procedure iHt −iHt evolution e Be can be simulated. As schematically illustrated in Fig. 1(c), here we focus 3. Readout. The OTOC is obtained by measuring on the β ¼ 0 case, and measuring the OTOC mainly the expectation value of the observable O ¼ ˆ ˆ ˆ ˆ iHt −iHt iHt −iHt ˆ ˆ ˆ ˆ consists of the following parts. e Be Ae Be A. For the infinite temperature 1. Initial state preparation. This step aims at preparing an β ¼ 0, the equilibrium state of the many-body system ˆ 4 initial state with density matrix ρ ∝ A ¼ σ ˆ , α ¼ x, y,or z. ˆ i 1 H is the maximally mixed state 1=2 .Since 1.1. The natural system is originally in the thermal ˆ † equilibrium state ρ populated according to the Boltzmann ˆ ˆ ˆ ˆ eq hOi ¼ Tr½UðtÞρ ˆ U ðtÞA; ð7Þ β¼0 distribution. In the high-temperature approximation, 4 4 ρ ˆ ≈ 1=2 ð1 þ ϵ σ ˆ Þ, where 1 is the identity and † eq i ˆ ˆ ˆ i¼1 i when B is unitary, UðtÞρ ˆ U ðtÞ is a density matrix ρðtÞ −5 ϵ ∼ 10 denotes the equilibrium polarization of spin i. i ˆ evolved from ρ by UðtÞ, as simulated in step 2. Finally, Because there is no observable and unitary dynamical effect measuring the expectation value of A under ρðtÞ gives 4 z on 1, effectively we write ρ ˆ ¼ ϵ σ ˆ . eq i i¼1 i hOi . Because the NMR detection is performed on a z β¼0 1.2. We engineer the system from ρ ˆ into ρ ˆ ¼ σ .This is eq 0 bulk ensemble of molecules, readout is an ensemble– accomplished in two steps: first we remove the polarizations averaged macrosopic measurement. When the system is of the spins except for that of F by using selective saturation prepared at state ρðtÞ, the expectation value of A can then pulses, and then we transfer the polarization from F to C. be directly obtained from the spectrum. See Appendix B Details of the method are described in Appendix B. for details. 1.3. For the initial state ρ ˆ with α ¼ x, y, we need to further rotate the spin at site 1 by a π=2 pulse around the y C. Results of OTOC or −x axes, respectively. 2. Implementation of unitary evolution of UðtÞ¼ Two sets of typical experimental results of the OTOC at ˆ ˆ iHt −iHt e Be . The key point is that according to the Trotter β ¼ 0 are shown in Fig. 2. Here, we normalize the OTOC † † † −iHt ˆ ˆ ˆ ˆ ˆ ˆ formula [25], the time evolution e of the Ising spin by hB ð0ÞBð0ÞihA ð0ÞAð0Þi, and because A and B com- mute at t ¼ 0, the initial value of this normalized OTOC is chain of Eq. (2) can be approximately simulated through the decomposition unity. The experimental data (red points) agree very well 031011-3 LI, FAN, WANG, YE, ZENG, ZHAI, PENG, and DU PHYS. REV. X 7, 031011 (2017) (a) (b) ˆ ˆ FIG. 2. Experimental results of OTOC measurement for an Ising spin chain. (a) A ¼ σ ˆ at the first site and B ¼ σ ˆ at the fourth site. 1 4 ˆ ˆ (b) A ¼ σ ˆ at the first site and B ¼ σ ˆ at the fourth site. The three columns correspond to g ¼ 1, h ¼ 0; g ¼ 1.05, h ¼ 0.5; and g ¼ 1, 1 4 h ¼ 1 of model Eq. (2), respectively. The red points are experimental data, the blue curves are theoretical calculation of OTOC with model Eq. (2) for four sites. with the theoretical results (blue curves). The sources of However, measuring entanglement entropy is always chal- experimental errors include imperfections in state prepa- lenging [28]. ration, control inaccuracy, and decoherence. See The OTOC opens a new door for entropy measurement. Appendix C for more details. We also measure OTOC An equivalence relationship between OTOCs at equilib- ˆ rium and the growth of the second Rényi entropy after a for other operators (A ¼ σ ˆ , B ¼ σ ˆ , with α, γ ¼ x, y, z) 1 4 quench has recently been established [10], which states that and they all behave similarly. The experimental results are in Appendix B. ð2Þ ˆ ˆ ˆ ˆ In both the integrable case (the first column in Fig. 2) and expð−S Þ¼ hMðtÞVð0ÞMðtÞVð0Þi : ð8Þ β¼0 the nonintegrable cases (the second and third columns in M∈B Fig. 2), the early time behaviors look similar. That is, the ð2Þ In the left-hand side of Eq. (8), S is the second Rényi OTOC starts to deviate from unity after a certain time (for A entropy of the subsystem A, after the system is quenched the unit of time t, see Appendix D for details.). However, ð2Þ the long time behaviors are very different between the by an operator O at time t ¼ 0. That is, S ¼ − log ρ ˆ and A A ˆ ˆ −iHt iHt † integrable and nonintegrable cases. In the integrable case, ˆ ˆ ˆ ρ ˆ ¼ Tr ðe Ve Þ, and V ¼ OO , up to a certain A B after the decreasing period, the OTOC revives and recovers normalization condition (see Appendix E). The right-hand unity. This reflects that the system has a well-defined side of Eq. (8) is a summation over OTOCs at equilibrium. quasiparticle. And there exists an extensive number of M is a complete set of operators in the subsystem B. integrals of motion, which is related to the fact that an In our experiment, we choose the quench operator O ∝ integrable system does not thermalize. While in the non- x ð1 þ σ ˆ Þ at the first site, and we take the first three sites as integrable case, the OTOC decreases to a small value and the subsystem A and the fourth site as the subsystem B,as oscillates, which will not revive back to unity in a practical ð2Þ marked in Fig. 1(b). In this setting, S measures how time scale. This relates to the fact that the information does much the quench operation induces additional correlation scramble in a nonintegrable system [11]. between the subsystems A and B. We take a complete set of operators in the subsystems B III. ENTROPY DYNAMICS as σ ˆ (up to a normalization factor), where α ¼ 0;x;y;z 0 † x ˆ ˆ and σ ˆ ¼ 1. Since V ¼ OO ∝ ð1 þ σ ˆ Þ, the right-hand To better illustrate the different behaviors of the infor- side of Eq. (8) becomes a set of OTOCs that are given by mation dynamics in the two cases of integrable and non- integrable systems, we reconstruct the entanglement α x α x hσ ˆ ðtÞð1 þ σ ˆ Þσ ˆ ðtÞð1 þ σ ˆ Þi : ð9Þ β¼0 entropy of a subsystem from the measured OTOCs. 4 1 4 1 Entanglement entropy has become an important quantity α x α α α x Notice that Tr½σ ˆ ðtÞσ ˆ σ ˆ ðtÞ ¼ Tr½σ ˆ ðtÞσ ˆ ðtÞσ ˆ ¼ 0; the not only for quantum information processing but also for 4 1 4 4 4 1 nonzero terms in Eq. (9) are nothing but OTOCs with B ¼ describing a quantum many-body system, such as quantum α x phase transition, topological order, and thermalization. σ ˆ (α ¼ x, y, z) and A ¼ σ ˆ , which are exactly what we 4 1 031011-4 MEASURING OUT-OF-TIME-ORDER CORRELATORS ON A … PHYS. REV. X 7, 031011 (2017) (a) (b) ð2Þ FIG. 3. The second Rényi entropy S after a quench. A quench operator ð1 þ σ ˆ Þ (up to a normalization factor) is applied to the system at t ¼ 0, and the entropy is measured by tracing out the fourth site as the subsystem B. Different colors correspond to FIG. 4. Measurement of the butterfly velocity. (a) The OTOCs different parameters of g and h in the Ising spin model. The points z x for A ¼ σ ˆ and B ¼ σ ˆ ,with i ¼ 4 (blue), i ¼ 3 (green), and are experimental data, the curves are theoretical calculations. 1 i ˆ ˆ i ¼ 2 (red). (b) The OTOCs for A ¼ σ ˆ and B ¼ σ ˆ , with i ¼ 4 1 i (blue), i ¼ 3 (green), and i ¼ 2 (red). The insets of (a) and measure. That is to say, with the help of the relationship (b) show the time for the onset of chaos t for the OTOCs versus between OTOCs and entanglement growth, we can extract the distance between two operators. The slope gives 1=v . Here, the growth of the entanglement entropy after the quench g ¼ 1.05 and h ¼ 0.5. from the experimental data. ð2Þ The results of the second Rényi entropy S are shown in Fig. 3. At short time, all three curves start to grow become more and more important and some terms fail to significantly after a certain time. This demonstrates that commute with A, at which the normalized OTOC starts to it takes a certain time for the perturbation applied at the first drop. Thus, the larger the distance between sites for A and site to propagate to the subsystem B at the fourth site (see B, the later the time the OTOC starts deviating from unity. the discussion of butterfly velocity below). Then, for all In general, the OTOC behaves as ð2Þ three cases, S ’s grow roughly linearly in time. This λ ðt−jxj=v Þ L B indicates that the extra information caused by the initial FðtÞ¼ a − be þ  ; ð11Þ quench starts to scramble between subsystems A and B. The differences lie in the long-time regime. For the where a and b are two nonuniversal constants and jxj ð2Þ denotes the distance between two operators. Here, v integrable model, the S oscillates back to around its defines the butterfly velocity [5,11,29–31]. It quantifies initial value after some time, which means that this extra the speed of a local operator growth in time and defines a information moves back to the subsystem A around that light cone for chaos, which is also related to the Lieb- time window. As a comparison, such a large amplitude Robinson bound [31,32]. oscillation does not occur for the two nonintegrable cases ˆ ˆ ð2Þ In our experiment, we fix A at the first site, and move B and the S s saturate after growing. This supports the from the fourth site to the third site, and to the second site. physical picture that the local information moves around in From the experimental data, we can phenomenologically the integrable model, while it scrambles in the nonintegr- determine a characteristic time t for the onset of chaos in able models [11]. each OTOC, i.e., the time that the OTOC starts departing from unity. By comparing the three different OTOCs in IV. BUTTERFLY VELOCITY Fig. 4, it is clear that the closer the distance between A and The OTOC also provides a tool to determine the speed B, the smaller t . In the insets of Figs. 4(a) and 4(b), we plot ˆ ˆ for correlation propagating. At t ¼ 0, A and B commute t as a function of the distance, and extract the butterfly with each other since they are operators at different sites. velocity from the slope. We find that, for the OTOC with As time grows, the higher-order terms in the Baker- z x ˆ ˆ A ¼ σ ˆ and B ¼ σ ˆ , v ¼ 2.10, and for the OTOC with 1 i Campbell-Hausdorff formula, ˆ ˆ A ¼ σ ˆ and B ¼ σ ˆ , v ¼ 2.22. The butterfly velocity is 1 i ∞ nearly independent of the choice of local operators, which ðitÞ BðtÞ¼ ½H; …; ½H; B; …; ð10Þ is a kind of manifestation of the chaotic behavior of the k! k¼0 system. 031011-5 LI, FAN, WANG, YE, ZENG, ZHAI, PENG, and DU PHYS. REV. X 7, 031011 (2017) V. OUTLOOK The OTOC provides a faithful reflection of the information scrambling and chaotic behavior of quantum many-body systems. It goes beyond the normal order correlators studied in linear response theory, which only capture the thermal- ization behavior of the system. Measuring the OTOC FIG. 5. Characteristics of iodotrifluroethylene. Molecular struc- functions can reveal how quantum entanglement and infor- ture together with a table of the chemical shifts (on the diagonal) mation scrambles across all of the degrees of freedom in a and J-coupling strengths (lower off diagonal), all in Hz. The system. In the future it will be possible to simulate more chemical shifts are given with respect to base frequency for Cor sophisticated systems that may possess holographic duality, 19 F transmitters on the 400-MHz spectrometer that we use. with larger size and different β, to extract the corresponding Lyapunov exponents such that one can experimentally verify Note added.—Recently, we noticed a related work [40], the connection between the upper bound of the Lyapunov where OTOCs are measured in a trapped-ion quantum exponent and the holographic duality. magnet. We use liquid-state NMR as a quantum simulator for the demonstration of OTOC measurement. NMR provides an excellent platform to benchmark the measurement ideas and APPENDIX A: PARAMETERS OF THE SYSTEM techniques. Our work here represents a first and encouraging HAMILTONIAN step towards further experimentally observing OTOCs on We use iodotrifluroethylene dissolved in d-chloroform large-sized quantum systems. The present method can be [41]. The system Hamiltonian is given by readily translated to other controllable systems. For instance, in trapped-ion systems, high-fidelity execution of arbitrary 4 4 X X ω πJ 0i ij z z z control with up to five atomic ions has been realized [33]. H ¼ − σ ˆ þ σ ˆ σ ˆ ; ðA1Þ NMR i i j 2 2 Superconducting quantum circuits also allow for engineering i¼1 i<j;¼1 on local qubits with errors at or below the threshold [34,35], hence offering another very promising experimental where ω =2π is the Larmor frequency of spin i, andJ are 0i ij approach. The progress in recent years in these two quantum the coupling strength between spins i and j. The values of hardware platforms has been fast and astounding, particu- parameters ω and J are given in Fig. 5. 0i ij larly in the pursuit of fabrication of quantum computing architecture at large scale. It is reckoned that quantum APPENDIX B: EXPERIMENTAL PROCEDURE simulators consisting of tens of or even hundreds of qubits are within reach in the near future [36–39]. Experimentalists 1. Initialization will see the great opportunity of applying these technologies The system is required to be initialized into ρ ˆ ∝ σ ˆ from for studying quantum chaotic behaviors for much more the equilibrium state ρ ˆ . We first exploit the steady-state eq complicated quantum many-body systems. effect when a relaxing nuclear spin system is subjected to multiple-pulse irradiation [42]. To implement this, we ACKNOWLEDGMENTS apply the periodic sequence ½π − d to the system, 1;2;4 where π means simultaneous π rotations on the spins 1;2;4 We thank Huitao Shen, Pengfei Zhang, Yingfei Gu, and C, F , and F , and d is a time delay parameter to be Xie Chen for helpful discussions. B. Z. is supported by 1 3 adjusted; see the first part of the circuit shown in Fig. 6(a). NSERC and CIFAR. H. Z. is supported by MOST (Grant To do π , we use a pulse that is composed of three no. 2016YFA0301604), Tsinghua University Initiative 1;2;4 frequency components, each Hermite-180 shaped in 500 Scientific Research Program, and NSFC Grant segments, with a duration of 1 ms. With increasing the No. 11325418. H. W., X. P., and J. D. would like to thank number of applied cycles, under the joint effects of the following funding sources: NKBRP (2013CB921800 relaxation and π reversions, the equilibrium Zeeman and 2014CB848700), the National Science Fund for magnetizations hσ ˆ i gradually decay to zero. Only the Distinguished Young Scholars (11425523), and NSFC 1;2;4 magnetization σ ˆ is retained at last as it is the fixed point to (11375167, 11227901, and 91021005). J. L. is supported by the National Basic Research Program of China (Grants the periodic driving. We adjust the time interval d between No. 2014CB921403 and No. 2016YFA0301201), National the π pulses to achieve the best-quality steady state. In Natural Science Foundation of China (Grants experiment, we set d ¼ 25 ms and after more than 500 No. 11421063, No. 11534002, No. 11375167, and cycles we find that the system is effectively steered into a No. 11605005), the National Science Fund for steady state ρ ˆ ∝ σ ˆ (in this sample, we do not see ss Distinguished Young Scholars (Grant No. 11425523), observable Overhauser enhancement). Next, with a SWAP and NSAF (Grant No. U1530401). operation we transfer the polarization from the 031011-6 MEASURING OUT-OF-TIME-ORDER CORRELATORS ON A … PHYS. REV. X 7, 031011 (2017) (a) (b) (c) FIG. 6. (a) Quantum circuit that measures the OTOCs. The first part aims to reset an arbitrary state to the desired initial state. Here, the time interval between the π pulses is 25 ms, the number of cycles is l ¼ 500, and G denotes z axis gradient pulse. (b) Sequences for ˆ ˆ −iH τ iH τ zz zz implementing the dynamics of e (left) and e (right). The refocusing circuits are designed to generate the right amount of coupled evolution. (c) C experimental spectrum for equilibrium state (blue) and state ρ ˆ (red) after a readout pulse R ðπ=2Þ. They are 0 y shown at the same scale for comparison. the evolution is also reserved. Hence, by designing a high-sensitivity F nucleus to the low-sensitivity C nucleus. Using the method, we finally get an initial state suitable refocusing scheme, the dynamics of H and zz ρ ˆ ∝ σ ˆ . The resulting experimental spectrum is shown in ˆ 0 −H can be efficiently simulated. zz Fig. 6(c). Although a general and efficient refocusing scheme exists for any σ ˆ σ ˆ -coupled evolution [27], for the present z z task it is possible to find a much simplified circuit 2. Simulating time evolution of Ising spin chain construction. Figure 6(b) shows our ideal circuits. Let According to Eq. (5) of the main text, the key ingredient O and O define the reference frequency for the carbon 1 2 in simulating the evolution of Ising Hamiltonian H is to and fluorine channel, respectively. Consider the refocusing implement −iH τ zz circuit [Fig. 6(b), left] for implementing e ; it auto- matically refocuses the fluorine spins and decouples the ˆ ˆ ˆ ˆ ˆ −iH τ=2 −iH τ=2 −iH τ −iH τ=2 −iH τ=2 x z zz z x e e e e e : ðB1Þ terms J , J , and J , and the evolution of the other terms 31 41 43 should fulfil the following requirements to yield the right −iH τ zz Here, except for e , all of the other four terms are amount of evolution: global rotation around the x (and z) axis, which can be −iH τ zz ðω =2π − O Þð4τ − t þ t Þ¼ 0; ðB2aÞ easily done through hard pulses. e can be generated by 01 1 1 1 2 manipulating the natural physical Hamiltonian H with NMR πJ =2 × 4τ ¼ −τ; ðB2bÞ a suitable refocusing scheme [43]. The basic idea is to 21 1 evolve the system with the J term in H and then to use NMR −πJ =2 × t ¼ −τ; ðB2cÞ 32 2 spin echoes to engineer the evolution. That is to say, for z z instance, for the σ ˆ σ ˆ term, when a transverse π pulse is i j πJ =2 × t ¼ −τ: ðB2dÞ 43 1 applied to reverse the polarization of one of the two spins, 031011-7 LI, FAN, WANG, YE, ZENG, ZHAI, PENG, and DU PHYS. REV. X 7, 031011 (2017) The solution to the above system of equations is given performance, we employ the gradient ascent pulse engi- by O ¼ ω =2π ¼ 15480.0 Hz, t ¼ 0.004935τ, t ¼ neering (GRAPE) technique [46] on the complied sequen- 1 01 1 2 0.009870τ, and τ ¼ 0.000534τ. As to the refocusing ces. Because that compilation procedure has the capability iH τ zz of directly providing a good initial start for subsequent circuit for implementing e , we find that it suffices to gradient iteration, the GRAPE searching quickly finds iH τ zz just make slight changes to the circuit for −e , as shown high-performance pulse controls for the desired propaga- in Fig. 6, and one can then reverse the dynamics of tors. The obtained shaped pulses for different sets of all terms. Hamiltonian parameters ðg; hÞ all have numerical fidelities Now, the whole network for implementing Ising dynam- above 0.999, and have been optimized with practical ics is expressed in terms of single-spin rotations and control field inhomogeneity taken into consideration. evolution of J terms in H . In practice, each single- NMR The Ising dynamics to be simulated is discretized into 20 spin rotation is realized through a selective rf pulse of steps, with each time step of duration τ ¼ 0.35 ms. Gaussian shape, with a duration of 0.5–1 ms. We then ˆ ˆ Choosing different operators for A and B, we experimen- conduct a compilation procedure to the sequence of tally measure the corresponding OTOC. All of the exper- selective pulses to eliminate the control imperfections imental results are given in Fig. 7. The theoretical caused by off-resonance and coupling effects up to the trajectories are plotted for comparison. Although some first order [44,45]. To further improve the control ˆ ˆ FIG. 7. Experimental results for measuring OTOCs for different Ising model parameters and different pairs of A and B. The red points are experimental data, the blue curves are theoretical calculation of OTOCs with the model, and the blue points are theoretical values displayed for comparison. 031011-8 MEASURING OUT-OF-TIME-ORDER CORRELATORS ON A … PHYS. REV. X 7, 031011 (2017) TABLE I. The standard deviations of hAi for the experiments rotation along y ˆ. By fitting the C spectrum, the real part ˆ ˆ and numerical simulations when A ¼ σ ˆ , B ¼ σ ˆ . and the imaginary parts of the peaks are extracted, which 1 4 corresponds to hσ ˆ i and hσ ˆ i, respectively. 1 1 g ¼ 1, h ¼ 1 g ¼ 1.05, h ¼ 0.5 g ¼ 1, h ¼ 0 σ 0.1097 0.0456 0.0308 expt APPENDIX C: EXPERIMENTAL err σ 0.0340 0.0340 0.0340 ini ERROR ANALYSIS err σ 0.0323 0.0150 0.0188 inhomo err σ 0.0461 0.0161 0.0214 The sources of experimental errors include imperfections in initial state preparation, infidelities of the GRAPE pulses, rf inhomogeneity, and decoherences. We make discrepancies between the data and the simulations remain, y ˆ ˆ an analysis to the data set of the case A ¼ σ ˆ , B ¼ σ ˆ 1 4 the experimental results reflect very well how OTOCs to get an understanding of the role of each type of behave differently in the integrable and chaotic cases. error source. We calculate the standard deviations qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20 i i 2 ˆ ˆ 3. Readout σ ≔ ðhAi − hAi Þ =20 for the experimental expt i¼1 expt th All the observations are made on the probe spin C. data, which are presented in Table I. Because we use an unlabeled sample in the real experiment, We run the initialization process 50 times and find that the molecules with a C nucleus are present at a concen- the fluctuation of the initial state polarization of ρ ˆ is tration of about 1%. The NMR signal in the high field is around 3.40%. The fluctuation is due to (i) error in state obtained from the precessing transverse magnetization of preparation and (ii) error in spectrum fitting. The latter can the ensemble of molecules in the sample: be inferred from the signal-to-noise ratio of the spectrum, which is estimated to be ≈2.13%. MðtÞ¼ M ðtÞþ iM ðtÞ ˆ ˆ −iHτ iHτ x y All the GRAPE pulses for implementing e and e X X are of fidelities above 0.999. On such a precise level, if we ¼ Tr ρðtÞ hσ iþ i hσ i : ðB3Þ j j assume no other sources of error and assume that the pulse j j generator ideally generates these pulses, then the exper- As the precession frequencies of different spins are dis- imental results should match the theoretical predictions tinguishable, they can be individually detected; e.g., we almost perfectly. x 13 obtain the measurements of hσ ˆ i and hσ ˆ i at the C Larmor Figure 8 plots the robustness of the GRAPE pulses in the 1 1 z 13 frequency. To measure hσ ˆ i, we need to apply a π=2 presence of imperfections of rf fields in the C channel and FIG. 8. Robustness of the used GRAPE pulses against rf field inhomogeneity. Here, the transverse axis denotes relative error of output 13 19 field power of the C channel and the longitudinal axis is that of the F channel. U denotes the corresponding propagator and f is the sim fidelity function. 031011-9 LI, FAN, WANG, YE, ZENG, ZHAI, PENG, and DU PHYS. REV. X 7, 031011 (2017) F channel. To understand to what extent the rf field inhomogeneity may affect the experimental results, we calculate the deviation of the dynamics based on a simple inhomogeneity model. The model assumes that the output power discrepancy of the rf fields is uniformly distributed err between 3%. The simulated results σ are shown in inhomo Table I. Another major source of error comes from decoherence effects. We compare the experimental data to a simple phenomenological error model, i.e., the system undergoes uncorrelated dephasing channel, parametrized with a set of phase flip error probabilities fp g per evolution time i i¼1;2;3;4 step t . The density matrix ρ ˆ is then, at each evolution step, FIG. 9. Numerical results of OTOCs for the integrable case. subjected to the composition of the error channels E for The arrows denote the revival time, which approximately linearly each qubit [47]: increases with respect to the distance between operators. Here, we 1 n ˆ ˆ choose A ¼ σ ˆ on the first site and B ¼ σ ˆ on the final site. The z z ρ ˆ → E ∘E ∘E ∘E ðρ ˆÞ; ðC1Þ 4 3 2 1 parameters are g ¼ 1, h ¼ 0. where where the summation is taken over a complete set of ˆ ˆ ˆ z z operators in B and V ¼ OO . Here, we should choose E ðρ ˆÞ¼ð1 − p Þρ ˆ þ p σ ˆ ρ ˆσ ˆ : ðC2Þ i i i P i i the following normalization condition: M M ¼ M∈B ij lm −t =T 0 2;i ˆ ˆ ˆ with p ¼ð1−e Þ=2 (see Fig. 5 for the values of T ). δ δ ,Tr½OO ¼ 1. im lj i 2;i The results are presented in Table I. The results indicate Here, we quench the first site and take the first three sites that, with decoherence effects taken into account, the as the subsystem A and the fourth site as the subsystem B, discrepancy between theoretical and experimental data as marked in Fig. 1(b) of the main text. Hence, we choose x ðDþ1Þ=2 for g ¼ 1, h ¼ 0 is expected to be larger than that of the ˆ ˆ O ¼ð1 þ σ ˆ Þ=2 (D ¼ 4 is the total number of sites). other two cases, consistent with the experiment data. The complete set of operators in the subsystems B can be pffiffiffi In summary, we conclude that rf inhomogeneity and α 0 taken as σ ˆ = 2, where α ¼ 0, x, y, z and σ ˆ ¼ 1.By decoherence effects are two major sources of errors. summing over the measured data with the conventions above, we can get the points in Fig. 3 of the main text. The APPENDIX D: UNIT OF TIME t theoretical curves are obtained by directly computing entanglement entropy from the density matrix. Our model Hamiltonian is actually written as H ¼ z z z ð−Jσ ˆ σ ˆ þ gσ ˆ þ hσ ˆ Þ, where we automatically set i i iþ1 i i J ¼ 1 in the main text, and we choose the natural unit APPENDIX F: REVIVAL TIME OF OTOC AND ℏ ¼ 1 throughout. So our time t is in fact in the unit of ℏ=J. THE DISTANCE BETWEEN THE OPERATORS As we see from Fig. 2 of the main text, for the integrable APPENDIX E: NORMALIZATION CONDITION case, the OTOCs will increase back around their initial FOR THE ENTANGLEMENT ENTROPY AND values at some time. The revival time in fact depends on the OTOC RELATION spatial distance between the two operators, as depicted in Fig. 9. That is, the larger the distance, the later the revival The relationship between the growth of the second Rényi happens. From the relationship between the growth of the entropy after a quench and the OTOCs at equilibrium is second Rényi entropy after a quench and the OTOCs at given in Ref. [10]. 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Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator

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Selected for a Viewpoint in Physics PHYSICAL REVIEW X 7, 031011 (2017) Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator 1 2,3 3 3 4,5,2,* 2,6,† 7,8,9,‡ 7,8 Jun Li, Ruihua Fan, Hengyan Wang, Bingtian Ye, Bei Zeng, Hui Zhai, Xinhua Peng, and Jiangfeng Du Beijing Computational Science Research Center, Beijing 100193, China Institute for Advanced Study, Tsinghua University, Beijing 100084, China Department of Physics, Peking University, Beijing 100871, China Department of Mathematics and Statistics, University of Guelph, Guelph N1G 2W1, Ontario, Canada Institute for Quantum Computing, University of Waterloo, Waterloo N2L 3G1, Ontario, Canada Collaborative Innovation Center of Quantum Matter, Beijing 100084, China Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Synergetic Innovation Centre of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China College of Physics and Electronic Science, Hubei Normal University, Huangshi, Hubei 435002, China (Received 30 December 2016; published 19 July 2017) The idea of the out-of-time-order correlator (OTOC) has recently emerged in the study of both condensed matter systems and gravitational systems. It not only plays a key role in investigating the holographic duality between a strongly interacting quantum system and a gravitational system, it also diagnoses the chaotic behavior of many-body quantum systems and characterizes information scrambling. Based on OTOCs, three different concepts—quantum chaos, holographic duality, and information scrambling—are found to be intimately related to each other. Despite its theoretical importance, the experimental measurement of the OTOC is quite challenging, and thus far there is no experimental measurement of the OTOC for local operators. Here, we report the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator. We observe that the OTOC behaves differently in the integrable and nonintegrable cases. Based on the recent discovered relationship between OTOCs and the growth of entanglement entropy in the many-body system, we extract the entanglement entropy from the measured OTOCs, which clearly shows that the information entropy oscillates in time for integrable models and scrambles for nonintgrable models. With the measured OTOCs, we also obtain the experimental result of the butterfly velocity, which measures the speed of correlation propagation. Our experiment paves a way for experimentally studying quantum chaos, holographic duality, and information scrambling in many-body quantum systems with quantum simulators. DOI: 10.1103/PhysRevX.7.031011 Subject Areas: Quantum Physics, Quantum Information, Statistical Physics ˆ ˆ I. INTRODUCTION iHt −iHt ˆ ˆ Hamiltonian, BðtÞ¼ e Be , and h  i denotes aver- aging over a thermal ensemble at the temperature The out-of-time-order correlator (OTOC), given by 1=β ¼ k T. For a many-body system with local operators † † ˆ ˆ ˆ ˆ ˆ ˆ A and B, the exponential deviation from unity of a FðtÞ¼hB ðtÞA ð0ÞBðtÞAð0Þi ; ð1Þ λ t normalized OTOC, i.e., FðtÞ ∼ 1 − #e , gives rise to the Lyapunov exponent λ . is proposed as a quantum generalization of a classical Quite remarkably, it was found recently that the OTOC measure of chaotic behaviors [1,2]. Here, H is the system also emerges in a different system that seems unrelated to chaos, that is, the scattering of shock waves nearby the zengb@uoguelph.ca horizon of a black hole and the information scrambling hzhai@tsinghua.edu.cn there [3–5]. A Lyapunov exponent of λ ¼ 2π=β is found xhpeng@ustc.edu.cn there. Later it was also found that the quantum correction Published by the American Physical Society under the terms of from string theory always makes the Lyapunov exponent the Creative Commons Attribution 4.0 International license. smaller [5]. Thus, it leads to a conjecture that 2π=β is an Further distribution of this work must maintain attribution to upper bound of the Lyaponuv exponent, which was later the author(s) and the published article’s title, journal citation, and DOI. proved for generic quantum systems [6]. This is a profound 2160-3308=17=7(3)=031011(12) 031011-1 Published by the American Physical Society LI, FAN, WANG, YE, ZENG, ZHAI, PENG, and DU PHYS. REV. X 7, 031011 (2017) efficient with practical algorithms proposed [25]. Here, the theoretical result. If a quantum system is exactly holo- graphic dual to a black hole, its Lyapunov exponent will quantum computer we use is liquid-state nuclear magnetic resonance (NMR) with molecules. In this work, we report saturate the bound; and a more nontrivial speculation is that measurements of OTOCs on a NMR quantum simulator. if the Lyapunov exponent of a quantum system saturates the We stress that, on one hand, our approach is universal and bound, it will possess a holographic dual to a gravity model can be applied to any system that has full local quantum with a black hole. A concrete quantum mechanics model, control, including a superconducting qubit and trapped ion; now known as the Sachdev-Ye-Kitaev model, has been on the other hand, this experiment is currently limited to a shown to fulfill this conjecture [2,7,8]. This establishes a small size not because of our scheme but because of the profound connection between the existence of holographic scalability issue of the quantum computer. duality and the chaotic behavior in many-body quantum systems [9]. II. NMR QUANTUM SIMULATION OF THE OTOC Recent studies also reveal that the OTOC can be applied to study physical properties beyond chaotic systems. The The system we simulate is an Ising spin chain model, decay of the OTOC is closely related to the delocalization whose Hamiltonian is written as of information and implies the information-theoretic def- z z z inition of scrambling. In the high- temperature limit (i.e., H ¼ ð−σ ˆ σ ˆ þ gσ ˆ þ hσ ˆ Þ; ð2Þ i iþ1 i i β ¼ 0), a connection between the OTOC and the growth of entanglement entropy in quantum many-body systems has x;y;z where σ ˆ are Pauli matrices on the i site. The parameter also been discovered quite recently [10,11]. The OTOC can values g ¼ 1, h ¼ 0 correspond to the traverse field Ising also characterize many-body localized phases, which are model, where the system is integrable. The system is not even thermalized [10,12–15]. nonintegrable whenever both g and h are nonzero. We Despite the significance of the OTOC revealed by recent simulate the dynamics governed by the system Hamiltonian theories, experimental measurement of the OTOC remains H, and measure the OTOCs of operators that are initially challenging. First of all, unlike the normal time-ordered acting on different local sites. The time dynamics of the correlators, the OTOC cannot be related to conventional OTOCs are observed, from which entanglement entropy of spectroscopy measurements, such as angle-resolved photo- the system and butterfly velocities of the chaotic systems emission spectroscopy (ARPES) and neutron scattering, are extracted. through the linear response theory. Secondly, direct simu- lation of this correlator requires the backward evolution in A. Physical system time, that is, the ability to completely reverse the The physical system to perform the quantum simulation Hamiltonian, which is extremely challenging. One exper- is the ensemble of nuclear spins provided by iodotrifluro- imental approach closely related to time reversal of quantum systems is the echo technique [16], and the echo has been ethylene (C F I), which is dissolved in d chloroform; see 2 3 studied extensively for both noninteracting particle systems Fig. 1(a) for the sample’s molecular structure. For this and many-body systems to characterize the stability (a) (b) of quantum evolution in the presence of perturbations [17–19], and the physics is already quite close to OTOC. Recently it has been proposed that the OTOC can be measured using echo techniques [20]. In addition, there also exists several other theoretical proposals based on the interferometric approaches [21–23]. However, none of them (c) have been experimentally implemented thus far. Here, we adopt a different approach to measure the OTOC. To make our approach work, some extent of “local control” is required. A universal quantum computer fulfills this need by having “full local control” of the system—that is, a universal set of local evolutions can be realized, and this set of local evolutions can build up any unitary FIG. 1. Illustration of the physical system, the Ising model, and evolution of the many-body system, both forward and the experimental scheme. (a) The structure of the C F I molecule 2 3 backward evolution in time. That is to say, we use a used for the NMR simulation. (b) The four site Ising spin chain. quantum computer to perform the measurement of the A and B label two subsystems for the later discussion of the OTOC. In fact, historically, one of the key motivations to entanglement entropy. (c) Quantum circuit for measuring the OTOC for the general N-site Ising chain when β ¼ 0 (in our case, develop quantum computers is to simulate the dynamics of z x ˆ ˆ ˆ many-body quantum systems [24], and quantum simulation N ¼ 4). Here, R ¼ 1, R ð−π=2Þ, R ðπ=2Þ for A ¼ σ ˆ , σ ˆ , σ ˆ , x y 1 1 1 respectively. of many-body dynamics has been theoretically shown to be 031011-2 MEASURING OUT-OF-TIME-ORDER CORRELATORS ON A … PHYS. REV. X 7, 031011 (2017) ˆ ˆ ˆ ˆ ˆ ˆ 13 19 19 −iHmτ −iH τ=2 −iH τ=2 −iH τ −iH τ=2 −iH τ=2 m x z zz z x molecule, the C nucleus and the three F nuclei ( F , 1 e ≈ ðe e e e e Þ ð5Þ 19 19 F , and F ) constitute a four-qubit quantum simulator. 2 3 Each nucleus corresponds to a spin site of the Ising chain, for small enough τ. Here, the dynamics is divided into m as shown in Fig. 1(b). In experiment, the sample is placed in pieces with t ¼ mτ, and a static magnetic field along the z ˆ direction, resulting in the following form of the system Hamiltonian, H ¼ gσ ˆ ; ð6aÞ x i 4 4 X X πJ 0i ij X z z z H ¼ − σ ˆ þ σ ˆ σ ˆ ; ð3Þ NMR i i j ˆ H ¼ hσ ˆ ; ð6bÞ 2 2 z i¼1 i<j;¼1 where ω =2π is the Larmor frequency of spin i and J is 0i ij z z H ¼ −σ ˆ σ ˆ : ð6cÞ zz i iþ1 the coupling strength between spins i and j. The values of these system parameters are given in Appendix A. The system is controlled by radio-frequency (rf) pulses, and the Each propagator inside the bracket of Eq. (5) corresponds corresponding control Hamiltonian is to either single-spin operation or coupled two-spin oper- ation, and can be implemented through manipulating H NMR H ðtÞ¼ −ω ðtÞfcos½ϕðtÞσ ˆ þ sin½ϕðtÞσ ˆ g; ð4Þ rf 1 i i with rf control H : single-spin operation terms are global rf rotations around the x or z axis, which can be easily done where ω ðtÞ and ϕðtÞ denote the amplitude and the −iH τ zz through hard pulses; the two-spin operation term e can emission phase of the rf field, respectively. The control be generated through some suitably designed pulse pulse shape can be elaborately monitored to realize the sequence based on the NMR refocusing techniques [27]. desired dynamic evolution. Actually, complete controllabil- More details of the method are described in Appendix B. ity of such a system has been demonstrated [26], which iHt The reversal of Ising dynamics e can be done in a similar guarantees that arbitrary system evolution can be imple- mented on it. Our experiments are carried out on a Bruker manner. Note that in the case we consider here, B is a local AV 400 MHz spectrometer (9.4 T) at tempera- ˆ unitary operator on the site-N spin and B ¼ σ ˆ with γ ¼ x, ture T ¼ 305 K. y, z that can be implemented by a selective rf π pulse on the site-N spin. Hence, for any given t, the total unitary ˆ ˆ B. Experimental procedure iHt −iHt evolution e Be can be simulated. As schematically illustrated in Fig. 1(c), here we focus 3. Readout. The OTOC is obtained by measuring on the β ¼ 0 case, and measuring the OTOC mainly the expectation value of the observable O ¼ ˆ ˆ ˆ ˆ iHt −iHt iHt −iHt ˆ ˆ ˆ ˆ consists of the following parts. e Be Ae Be A. For the infinite temperature 1. Initial state preparation. This step aims at preparing an β ¼ 0, the equilibrium state of the many-body system ˆ 4 initial state with density matrix ρ ∝ A ¼ σ ˆ , α ¼ x, y,or z. ˆ i 1 H is the maximally mixed state 1=2 .Since 1.1. The natural system is originally in the thermal ˆ † equilibrium state ρ populated according to the Boltzmann ˆ ˆ ˆ ˆ eq hOi ¼ Tr½UðtÞρ ˆ U ðtÞA; ð7Þ β¼0 distribution. In the high-temperature approximation, 4 4 ρ ˆ ≈ 1=2 ð1 þ ϵ σ ˆ Þ, where 1 is the identity and † eq i ˆ ˆ ˆ i¼1 i when B is unitary, UðtÞρ ˆ U ðtÞ is a density matrix ρðtÞ −5 ϵ ∼ 10 denotes the equilibrium polarization of spin i. i ˆ evolved from ρ by UðtÞ, as simulated in step 2. Finally, Because there is no observable and unitary dynamical effect measuring the expectation value of A under ρðtÞ gives 4 z on 1, effectively we write ρ ˆ ¼ ϵ σ ˆ . eq i i¼1 i hOi . Because the NMR detection is performed on a z β¼0 1.2. We engineer the system from ρ ˆ into ρ ˆ ¼ σ .This is eq 0 bulk ensemble of molecules, readout is an ensemble– accomplished in two steps: first we remove the polarizations averaged macrosopic measurement. When the system is of the spins except for that of F by using selective saturation prepared at state ρðtÞ, the expectation value of A can then pulses, and then we transfer the polarization from F to C. be directly obtained from the spectrum. See Appendix B Details of the method are described in Appendix B. for details. 1.3. For the initial state ρ ˆ with α ¼ x, y, we need to further rotate the spin at site 1 by a π=2 pulse around the y C. Results of OTOC or −x axes, respectively. 2. Implementation of unitary evolution of UðtÞ¼ Two sets of typical experimental results of the OTOC at ˆ ˆ iHt −iHt e Be . The key point is that according to the Trotter β ¼ 0 are shown in Fig. 2. Here, we normalize the OTOC † † † −iHt ˆ ˆ ˆ ˆ ˆ ˆ formula [25], the time evolution e of the Ising spin by hB ð0ÞBð0ÞihA ð0ÞAð0Þi, and because A and B com- mute at t ¼ 0, the initial value of this normalized OTOC is chain of Eq. (2) can be approximately simulated through the decomposition unity. The experimental data (red points) agree very well 031011-3 LI, FAN, WANG, YE, ZENG, ZHAI, PENG, and DU PHYS. REV. X 7, 031011 (2017) (a) (b) ˆ ˆ FIG. 2. Experimental results of OTOC measurement for an Ising spin chain. (a) A ¼ σ ˆ at the first site and B ¼ σ ˆ at the fourth site. 1 4 ˆ ˆ (b) A ¼ σ ˆ at the first site and B ¼ σ ˆ at the fourth site. The three columns correspond to g ¼ 1, h ¼ 0; g ¼ 1.05, h ¼ 0.5; and g ¼ 1, 1 4 h ¼ 1 of model Eq. (2), respectively. The red points are experimental data, the blue curves are theoretical calculation of OTOC with model Eq. (2) for four sites. with the theoretical results (blue curves). The sources of However, measuring entanglement entropy is always chal- experimental errors include imperfections in state prepa- lenging [28]. ration, control inaccuracy, and decoherence. See The OTOC opens a new door for entropy measurement. Appendix C for more details. We also measure OTOC An equivalence relationship between OTOCs at equilib- ˆ rium and the growth of the second Rényi entropy after a for other operators (A ¼ σ ˆ , B ¼ σ ˆ , with α, γ ¼ x, y, z) 1 4 quench has recently been established [10], which states that and they all behave similarly. The experimental results are in Appendix B. ð2Þ ˆ ˆ ˆ ˆ In both the integrable case (the first column in Fig. 2) and expð−S Þ¼ hMðtÞVð0ÞMðtÞVð0Þi : ð8Þ β¼0 the nonintegrable cases (the second and third columns in M∈B Fig. 2), the early time behaviors look similar. That is, the ð2Þ In the left-hand side of Eq. (8), S is the second Rényi OTOC starts to deviate from unity after a certain time (for A entropy of the subsystem A, after the system is quenched the unit of time t, see Appendix D for details.). However, ð2Þ the long time behaviors are very different between the by an operator O at time t ¼ 0. That is, S ¼ − log ρ ˆ and A A ˆ ˆ −iHt iHt † integrable and nonintegrable cases. In the integrable case, ˆ ˆ ˆ ρ ˆ ¼ Tr ðe Ve Þ, and V ¼ OO , up to a certain A B after the decreasing period, the OTOC revives and recovers normalization condition (see Appendix E). The right-hand unity. This reflects that the system has a well-defined side of Eq. (8) is a summation over OTOCs at equilibrium. quasiparticle. And there exists an extensive number of M is a complete set of operators in the subsystem B. integrals of motion, which is related to the fact that an In our experiment, we choose the quench operator O ∝ integrable system does not thermalize. While in the non- x ð1 þ σ ˆ Þ at the first site, and we take the first three sites as integrable case, the OTOC decreases to a small value and the subsystem A and the fourth site as the subsystem B,as oscillates, which will not revive back to unity in a practical ð2Þ marked in Fig. 1(b). In this setting, S measures how time scale. This relates to the fact that the information does much the quench operation induces additional correlation scramble in a nonintegrable system [11]. between the subsystems A and B. We take a complete set of operators in the subsystems B III. ENTROPY DYNAMICS as σ ˆ (up to a normalization factor), where α ¼ 0;x;y;z 0 † x ˆ ˆ and σ ˆ ¼ 1. Since V ¼ OO ∝ ð1 þ σ ˆ Þ, the right-hand To better illustrate the different behaviors of the infor- side of Eq. (8) becomes a set of OTOCs that are given by mation dynamics in the two cases of integrable and non- integrable systems, we reconstruct the entanglement α x α x hσ ˆ ðtÞð1 þ σ ˆ Þσ ˆ ðtÞð1 þ σ ˆ Þi : ð9Þ β¼0 entropy of a subsystem from the measured OTOCs. 4 1 4 1 Entanglement entropy has become an important quantity α x α α α x Notice that Tr½σ ˆ ðtÞσ ˆ σ ˆ ðtÞ ¼ Tr½σ ˆ ðtÞσ ˆ ðtÞσ ˆ ¼ 0; the not only for quantum information processing but also for 4 1 4 4 4 1 nonzero terms in Eq. (9) are nothing but OTOCs with B ¼ describing a quantum many-body system, such as quantum α x phase transition, topological order, and thermalization. σ ˆ (α ¼ x, y, z) and A ¼ σ ˆ , which are exactly what we 4 1 031011-4 MEASURING OUT-OF-TIME-ORDER CORRELATORS ON A … PHYS. REV. X 7, 031011 (2017) (a) (b) ð2Þ FIG. 3. The second Rényi entropy S after a quench. A quench operator ð1 þ σ ˆ Þ (up to a normalization factor) is applied to the system at t ¼ 0, and the entropy is measured by tracing out the fourth site as the subsystem B. Different colors correspond to FIG. 4. Measurement of the butterfly velocity. (a) The OTOCs different parameters of g and h in the Ising spin model. The points z x for A ¼ σ ˆ and B ¼ σ ˆ ,with i ¼ 4 (blue), i ¼ 3 (green), and are experimental data, the curves are theoretical calculations. 1 i ˆ ˆ i ¼ 2 (red). (b) The OTOCs for A ¼ σ ˆ and B ¼ σ ˆ , with i ¼ 4 1 i (blue), i ¼ 3 (green), and i ¼ 2 (red). The insets of (a) and measure. That is to say, with the help of the relationship (b) show the time for the onset of chaos t for the OTOCs versus between OTOCs and entanglement growth, we can extract the distance between two operators. The slope gives 1=v . Here, the growth of the entanglement entropy after the quench g ¼ 1.05 and h ¼ 0.5. from the experimental data. ð2Þ The results of the second Rényi entropy S are shown in Fig. 3. At short time, all three curves start to grow become more and more important and some terms fail to significantly after a certain time. This demonstrates that commute with A, at which the normalized OTOC starts to it takes a certain time for the perturbation applied at the first drop. Thus, the larger the distance between sites for A and site to propagate to the subsystem B at the fourth site (see B, the later the time the OTOC starts deviating from unity. the discussion of butterfly velocity below). Then, for all In general, the OTOC behaves as ð2Þ three cases, S ’s grow roughly linearly in time. This λ ðt−jxj=v Þ L B indicates that the extra information caused by the initial FðtÞ¼ a − be þ  ; ð11Þ quench starts to scramble between subsystems A and B. The differences lie in the long-time regime. For the where a and b are two nonuniversal constants and jxj ð2Þ denotes the distance between two operators. Here, v integrable model, the S oscillates back to around its defines the butterfly velocity [5,11,29–31]. It quantifies initial value after some time, which means that this extra the speed of a local operator growth in time and defines a information moves back to the subsystem A around that light cone for chaos, which is also related to the Lieb- time window. As a comparison, such a large amplitude Robinson bound [31,32]. oscillation does not occur for the two nonintegrable cases ˆ ˆ ð2Þ In our experiment, we fix A at the first site, and move B and the S s saturate after growing. This supports the from the fourth site to the third site, and to the second site. physical picture that the local information moves around in From the experimental data, we can phenomenologically the integrable model, while it scrambles in the nonintegr- determine a characteristic time t for the onset of chaos in able models [11]. each OTOC, i.e., the time that the OTOC starts departing from unity. By comparing the three different OTOCs in IV. BUTTERFLY VELOCITY Fig. 4, it is clear that the closer the distance between A and The OTOC also provides a tool to determine the speed B, the smaller t . In the insets of Figs. 4(a) and 4(b), we plot ˆ ˆ for correlation propagating. At t ¼ 0, A and B commute t as a function of the distance, and extract the butterfly with each other since they are operators at different sites. velocity from the slope. We find that, for the OTOC with As time grows, the higher-order terms in the Baker- z x ˆ ˆ A ¼ σ ˆ and B ¼ σ ˆ , v ¼ 2.10, and for the OTOC with 1 i Campbell-Hausdorff formula, ˆ ˆ A ¼ σ ˆ and B ¼ σ ˆ , v ¼ 2.22. The butterfly velocity is 1 i ∞ nearly independent of the choice of local operators, which ðitÞ BðtÞ¼ ½H; …; ½H; B; …; ð10Þ is a kind of manifestation of the chaotic behavior of the k! k¼0 system. 031011-5 LI, FAN, WANG, YE, ZENG, ZHAI, PENG, and DU PHYS. REV. X 7, 031011 (2017) V. OUTLOOK The OTOC provides a faithful reflection of the information scrambling and chaotic behavior of quantum many-body systems. It goes beyond the normal order correlators studied in linear response theory, which only capture the thermal- ization behavior of the system. Measuring the OTOC FIG. 5. Characteristics of iodotrifluroethylene. Molecular struc- functions can reveal how quantum entanglement and infor- ture together with a table of the chemical shifts (on the diagonal) mation scrambles across all of the degrees of freedom in a and J-coupling strengths (lower off diagonal), all in Hz. The system. In the future it will be possible to simulate more chemical shifts are given with respect to base frequency for Cor sophisticated systems that may possess holographic duality, 19 F transmitters on the 400-MHz spectrometer that we use. with larger size and different β, to extract the corresponding Lyapunov exponents such that one can experimentally verify Note added.—Recently, we noticed a related work [40], the connection between the upper bound of the Lyapunov where OTOCs are measured in a trapped-ion quantum exponent and the holographic duality. magnet. We use liquid-state NMR as a quantum simulator for the demonstration of OTOC measurement. NMR provides an excellent platform to benchmark the measurement ideas and APPENDIX A: PARAMETERS OF THE SYSTEM techniques. Our work here represents a first and encouraging HAMILTONIAN step towards further experimentally observing OTOCs on We use iodotrifluroethylene dissolved in d-chloroform large-sized quantum systems. The present method can be [41]. The system Hamiltonian is given by readily translated to other controllable systems. For instance, in trapped-ion systems, high-fidelity execution of arbitrary 4 4 X X ω πJ 0i ij z z z control with up to five atomic ions has been realized [33]. H ¼ − σ ˆ þ σ ˆ σ ˆ ; ðA1Þ NMR i i j 2 2 Superconducting quantum circuits also allow for engineering i¼1 i<j;¼1 on local qubits with errors at or below the threshold [34,35], hence offering another very promising experimental where ω =2π is the Larmor frequency of spin i, andJ are 0i ij approach. The progress in recent years in these two quantum the coupling strength between spins i and j. The values of hardware platforms has been fast and astounding, particu- parameters ω and J are given in Fig. 5. 0i ij larly in the pursuit of fabrication of quantum computing architecture at large scale. It is reckoned that quantum APPENDIX B: EXPERIMENTAL PROCEDURE simulators consisting of tens of or even hundreds of qubits are within reach in the near future [36–39]. Experimentalists 1. Initialization will see the great opportunity of applying these technologies The system is required to be initialized into ρ ˆ ∝ σ ˆ from for studying quantum chaotic behaviors for much more the equilibrium state ρ ˆ . We first exploit the steady-state eq complicated quantum many-body systems. effect when a relaxing nuclear spin system is subjected to multiple-pulse irradiation [42]. To implement this, we ACKNOWLEDGMENTS apply the periodic sequence ½π − d to the system, 1;2;4 where π means simultaneous π rotations on the spins 1;2;4 We thank Huitao Shen, Pengfei Zhang, Yingfei Gu, and C, F , and F , and d is a time delay parameter to be Xie Chen for helpful discussions. B. Z. is supported by 1 3 adjusted; see the first part of the circuit shown in Fig. 6(a). NSERC and CIFAR. H. Z. is supported by MOST (Grant To do π , we use a pulse that is composed of three no. 2016YFA0301604), Tsinghua University Initiative 1;2;4 frequency components, each Hermite-180 shaped in 500 Scientific Research Program, and NSFC Grant segments, with a duration of 1 ms. With increasing the No. 11325418. H. W., X. P., and J. D. would like to thank number of applied cycles, under the joint effects of the following funding sources: NKBRP (2013CB921800 relaxation and π reversions, the equilibrium Zeeman and 2014CB848700), the National Science Fund for magnetizations hσ ˆ i gradually decay to zero. Only the Distinguished Young Scholars (11425523), and NSFC 1;2;4 magnetization σ ˆ is retained at last as it is the fixed point to (11375167, 11227901, and 91021005). J. L. is supported by the National Basic Research Program of China (Grants the periodic driving. We adjust the time interval d between No. 2014CB921403 and No. 2016YFA0301201), National the π pulses to achieve the best-quality steady state. In Natural Science Foundation of China (Grants experiment, we set d ¼ 25 ms and after more than 500 No. 11421063, No. 11534002, No. 11375167, and cycles we find that the system is effectively steered into a No. 11605005), the National Science Fund for steady state ρ ˆ ∝ σ ˆ (in this sample, we do not see ss Distinguished Young Scholars (Grant No. 11425523), observable Overhauser enhancement). Next, with a SWAP and NSAF (Grant No. U1530401). operation we transfer the polarization from the 031011-6 MEASURING OUT-OF-TIME-ORDER CORRELATORS ON A … PHYS. REV. X 7, 031011 (2017) (a) (b) (c) FIG. 6. (a) Quantum circuit that measures the OTOCs. The first part aims to reset an arbitrary state to the desired initial state. Here, the time interval between the π pulses is 25 ms, the number of cycles is l ¼ 500, and G denotes z axis gradient pulse. (b) Sequences for ˆ ˆ −iH τ iH τ zz zz implementing the dynamics of e (left) and e (right). The refocusing circuits are designed to generate the right amount of coupled evolution. (c) C experimental spectrum for equilibrium state (blue) and state ρ ˆ (red) after a readout pulse R ðπ=2Þ. They are 0 y shown at the same scale for comparison. the evolution is also reserved. Hence, by designing a high-sensitivity F nucleus to the low-sensitivity C nucleus. Using the method, we finally get an initial state suitable refocusing scheme, the dynamics of H and zz ρ ˆ ∝ σ ˆ . The resulting experimental spectrum is shown in ˆ 0 −H can be efficiently simulated. zz Fig. 6(c). Although a general and efficient refocusing scheme exists for any σ ˆ σ ˆ -coupled evolution [27], for the present z z task it is possible to find a much simplified circuit 2. Simulating time evolution of Ising spin chain construction. Figure 6(b) shows our ideal circuits. Let According to Eq. (5) of the main text, the key ingredient O and O define the reference frequency for the carbon 1 2 in simulating the evolution of Ising Hamiltonian H is to and fluorine channel, respectively. Consider the refocusing implement −iH τ zz circuit [Fig. 6(b), left] for implementing e ; it auto- matically refocuses the fluorine spins and decouples the ˆ ˆ ˆ ˆ ˆ −iH τ=2 −iH τ=2 −iH τ −iH τ=2 −iH τ=2 x z zz z x e e e e e : ðB1Þ terms J , J , and J , and the evolution of the other terms 31 41 43 should fulfil the following requirements to yield the right −iH τ zz Here, except for e , all of the other four terms are amount of evolution: global rotation around the x (and z) axis, which can be −iH τ zz ðω =2π − O Þð4τ − t þ t Þ¼ 0; ðB2aÞ easily done through hard pulses. e can be generated by 01 1 1 1 2 manipulating the natural physical Hamiltonian H with NMR πJ =2 × 4τ ¼ −τ; ðB2bÞ a suitable refocusing scheme [43]. The basic idea is to 21 1 evolve the system with the J term in H and then to use NMR −πJ =2 × t ¼ −τ; ðB2cÞ 32 2 spin echoes to engineer the evolution. That is to say, for z z instance, for the σ ˆ σ ˆ term, when a transverse π pulse is i j πJ =2 × t ¼ −τ: ðB2dÞ 43 1 applied to reverse the polarization of one of the two spins, 031011-7 LI, FAN, WANG, YE, ZENG, ZHAI, PENG, and DU PHYS. REV. X 7, 031011 (2017) The solution to the above system of equations is given performance, we employ the gradient ascent pulse engi- by O ¼ ω =2π ¼ 15480.0 Hz, t ¼ 0.004935τ, t ¼ neering (GRAPE) technique [46] on the complied sequen- 1 01 1 2 0.009870τ, and τ ¼ 0.000534τ. As to the refocusing ces. Because that compilation procedure has the capability iH τ zz of directly providing a good initial start for subsequent circuit for implementing e , we find that it suffices to gradient iteration, the GRAPE searching quickly finds iH τ zz just make slight changes to the circuit for −e , as shown high-performance pulse controls for the desired propaga- in Fig. 6, and one can then reverse the dynamics of tors. The obtained shaped pulses for different sets of all terms. Hamiltonian parameters ðg; hÞ all have numerical fidelities Now, the whole network for implementing Ising dynam- above 0.999, and have been optimized with practical ics is expressed in terms of single-spin rotations and control field inhomogeneity taken into consideration. evolution of J terms in H . In practice, each single- NMR The Ising dynamics to be simulated is discretized into 20 spin rotation is realized through a selective rf pulse of steps, with each time step of duration τ ¼ 0.35 ms. Gaussian shape, with a duration of 0.5–1 ms. We then ˆ ˆ Choosing different operators for A and B, we experimen- conduct a compilation procedure to the sequence of tally measure the corresponding OTOC. All of the exper- selective pulses to eliminate the control imperfections imental results are given in Fig. 7. The theoretical caused by off-resonance and coupling effects up to the trajectories are plotted for comparison. Although some first order [44,45]. To further improve the control ˆ ˆ FIG. 7. Experimental results for measuring OTOCs for different Ising model parameters and different pairs of A and B. The red points are experimental data, the blue curves are theoretical calculation of OTOCs with the model, and the blue points are theoretical values displayed for comparison. 031011-8 MEASURING OUT-OF-TIME-ORDER CORRELATORS ON A … PHYS. REV. X 7, 031011 (2017) TABLE I. The standard deviations of hAi for the experiments rotation along y ˆ. By fitting the C spectrum, the real part ˆ ˆ and numerical simulations when A ¼ σ ˆ , B ¼ σ ˆ . and the imaginary parts of the peaks are extracted, which 1 4 corresponds to hσ ˆ i and hσ ˆ i, respectively. 1 1 g ¼ 1, h ¼ 1 g ¼ 1.05, h ¼ 0.5 g ¼ 1, h ¼ 0 σ 0.1097 0.0456 0.0308 expt APPENDIX C: EXPERIMENTAL err σ 0.0340 0.0340 0.0340 ini ERROR ANALYSIS err σ 0.0323 0.0150 0.0188 inhomo err σ 0.0461 0.0161 0.0214 The sources of experimental errors include imperfections in initial state preparation, infidelities of the GRAPE pulses, rf inhomogeneity, and decoherences. We make discrepancies between the data and the simulations remain, y ˆ ˆ an analysis to the data set of the case A ¼ σ ˆ , B ¼ σ ˆ 1 4 the experimental results reflect very well how OTOCs to get an understanding of the role of each type of behave differently in the integrable and chaotic cases. error source. We calculate the standard deviations qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20 i i 2 ˆ ˆ 3. Readout σ ≔ ðhAi − hAi Þ =20 for the experimental expt i¼1 expt th All the observations are made on the probe spin C. data, which are presented in Table I. Because we use an unlabeled sample in the real experiment, We run the initialization process 50 times and find that the molecules with a C nucleus are present at a concen- the fluctuation of the initial state polarization of ρ ˆ is tration of about 1%. The NMR signal in the high field is around 3.40%. The fluctuation is due to (i) error in state obtained from the precessing transverse magnetization of preparation and (ii) error in spectrum fitting. The latter can the ensemble of molecules in the sample: be inferred from the signal-to-noise ratio of the spectrum, which is estimated to be ≈2.13%. MðtÞ¼ M ðtÞþ iM ðtÞ ˆ ˆ −iHτ iHτ x y All the GRAPE pulses for implementing e and e X X are of fidelities above 0.999. On such a precise level, if we ¼ Tr ρðtÞ hσ iþ i hσ i : ðB3Þ j j assume no other sources of error and assume that the pulse j j generator ideally generates these pulses, then the exper- As the precession frequencies of different spins are dis- imental results should match the theoretical predictions tinguishable, they can be individually detected; e.g., we almost perfectly. x 13 obtain the measurements of hσ ˆ i and hσ ˆ i at the C Larmor Figure 8 plots the robustness of the GRAPE pulses in the 1 1 z 13 frequency. To measure hσ ˆ i, we need to apply a π=2 presence of imperfections of rf fields in the C channel and FIG. 8. Robustness of the used GRAPE pulses against rf field inhomogeneity. Here, the transverse axis denotes relative error of output 13 19 field power of the C channel and the longitudinal axis is that of the F channel. U denotes the corresponding propagator and f is the sim fidelity function. 031011-9 LI, FAN, WANG, YE, ZENG, ZHAI, PENG, and DU PHYS. REV. X 7, 031011 (2017) F channel. To understand to what extent the rf field inhomogeneity may affect the experimental results, we calculate the deviation of the dynamics based on a simple inhomogeneity model. The model assumes that the output power discrepancy of the rf fields is uniformly distributed err between 3%. The simulated results σ are shown in inhomo Table I. Another major source of error comes from decoherence effects. We compare the experimental data to a simple phenomenological error model, i.e., the system undergoes uncorrelated dephasing channel, parametrized with a set of phase flip error probabilities fp g per evolution time i i¼1;2;3;4 step t . The density matrix ρ ˆ is then, at each evolution step, FIG. 9. Numerical results of OTOCs for the integrable case. subjected to the composition of the error channels E for The arrows denote the revival time, which approximately linearly each qubit [47]: increases with respect to the distance between operators. Here, we 1 n ˆ ˆ choose A ¼ σ ˆ on the first site and B ¼ σ ˆ on the final site. The z z ρ ˆ → E ∘E ∘E ∘E ðρ ˆÞ; ðC1Þ 4 3 2 1 parameters are g ¼ 1, h ¼ 0. where where the summation is taken over a complete set of ˆ ˆ ˆ z z operators in B and V ¼ OO . Here, we should choose E ðρ ˆÞ¼ð1 − p Þρ ˆ þ p σ ˆ ρ ˆσ ˆ : ðC2Þ i i i P i i the following normalization condition: M M ¼ M∈B ij lm −t =T 0 2;i ˆ ˆ ˆ with p ¼ð1−e Þ=2 (see Fig. 5 for the values of T ). δ δ ,Tr½OO ¼ 1. im lj i 2;i The results are presented in Table I. The results indicate Here, we quench the first site and take the first three sites that, with decoherence effects taken into account, the as the subsystem A and the fourth site as the subsystem B, discrepancy between theoretical and experimental data as marked in Fig. 1(b) of the main text. Hence, we choose x ðDþ1Þ=2 for g ¼ 1, h ¼ 0 is expected to be larger than that of the ˆ ˆ O ¼ð1 þ σ ˆ Þ=2 (D ¼ 4 is the total number of sites). other two cases, consistent with the experiment data. The complete set of operators in the subsystems B can be pffiffiffi In summary, we conclude that rf inhomogeneity and α 0 taken as σ ˆ = 2, where α ¼ 0, x, y, z and σ ˆ ¼ 1.By decoherence effects are two major sources of errors. summing over the measured data with the conventions above, we can get the points in Fig. 3 of the main text. The APPENDIX D: UNIT OF TIME t theoretical curves are obtained by directly computing entanglement entropy from the density matrix. Our model Hamiltonian is actually written as H ¼ z z z ð−Jσ ˆ σ ˆ þ gσ ˆ þ hσ ˆ Þ, where we automatically set i i iþ1 i i J ¼ 1 in the main text, and we choose the natural unit APPENDIX F: REVIVAL TIME OF OTOC AND ℏ ¼ 1 throughout. So our time t is in fact in the unit of ℏ=J. THE DISTANCE BETWEEN THE OPERATORS As we see from Fig. 2 of the main text, for the integrable APPENDIX E: NORMALIZATION CONDITION case, the OTOCs will increase back around their initial FOR THE ENTANGLEMENT ENTROPY AND values at some time. The revival time in fact depends on the OTOC RELATION spatial distance between the two operators, as depicted in Fig. 9. That is, the larger the distance, the later the revival The relationship between the growth of the second Rényi happens. From the relationship between the growth of the entropy after a quench and the OTOCs at equilibrium is second Rényi entropy after a quench and the OTOCs at given in Ref. [10]. 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