Probing Operator Spreading via Floquet Engineering in a Superconducting Circuit
S. K. Zhao, Zi-Yong Ge, Zhongcheng Xiang, Guangming Xue, Hongtao Yan, Ziting Wang, Zhan Wang, Huikai Xu, Feifan Su, Z. H. Yang, He Zhang, Yu-Ran Zhang, Xueyi Guo, Kai Xu, Ye Tian, Haifeng Yu, Dongning Zheng, Heng Fan, S. P. Zhao
Operator spreading, often characterized by out-of-time-order correlators (OTOCs), is one of the central concepts in quantum many-body physics. However, measuring OTOCs is experimentally challenging due to the requirement of reversing the time evolution of systems. Here we apply Floquet engineering to investigate operator spreading in a superconducting 10-qubit chain. Floquet engineering provides an effective way to tune the coupling strength between nearby qubits, which is used to demonstrate quantum walks with tunable couplings, reversed time evolution, and the measurement of OTOCs. A clear light-cone-like operator propagation is observed in the system with multiple excitations, and has a nearly equal velocity as the single-particle quantum walk. For the butterfly operator that is nonlocal (local) under the Jordan-Wigner transformation, the OTOCs show distinct behaviors with (without) a signature of information scrambling in the near integrable system.
Operator spreading, often characterized by out-of-time-order correlators (OTOCs), is one of the central concepts in quantum many-body physics. However, measuring OTOCs is experimentally challenging due to the requirement of reversing the time evolution of systems. Here we apply Floquet engineering to investigate operator spreading in a superconducting 10-qubit chain. Floquet engineering provides an effective way to tune the coupling strength between nearby qubits, which is used to demonstrate quantum walks with tunable couplings, reversed time evolution, and the measurement of OTOCs. A clear light-cone-like operator propagation is observed in the system with multiple excitations, and has a nearly equal velocity as the single-particle quantum walk. For the butterfly operator that is nonlocal (local) under the Jordan-Wigner transformation, the OTOCs show distinct behaviors with (without) a signature of information scrambling in the near integrable system.
Measuring out-of-time-ordered correlators (OTOCs) is experimentally challenging, often requiring effective reversal of many-body dynamics. Here, we propose an alternative approach to probing operator spreading through local quantum fluctuations, using only forward evolution and local measurements. Motivated by the Heisenberg uncertainty relation, we introduce a differential local variance and establish a rigorous bound connecting it to operator noncommutativity. For a ferromagnetic Heisenberg chain in the single-magnon regime, we derive an exact relation between the local variance and a squared-commutator OTOC. In integrable and nonintegrable Ising chains, the differential variance and OTOCs reveal qualitatively consistent propagation fronts. Our results establish local quantum fluctuations as an experimentally accessible, state-dependent, and complementary diagnostic of operator spreading.
A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems? We first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone. Next, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed. Finally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
Interactions in quantum systems can spread initially localized quantum information into the exponentially many degrees of freedom of the entire system. Understanding this process, known as quantum scrambling, is key to resolving several open questions in physics. Here, by measuring the time-dependent evolution and fluctuation of out-of-time-order correlators, we experimentally investigate the dynamics of quantum scrambling on a 53-qubit quantum processor. We engineer quantum circuits that distinguish operator spreading and operator entanglement and experimentally observe their respective signatures. We show that whereas operator spreading is captured by an efficient classical model, operator entanglement in idealized circuits requires exponentially scaled computational resources to simulate. These results open the path to studying complex and practically relevant physical observables with near-term quantum processors.
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.
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.
The out-of-time-ordered correlator (OTOC) is a powerful tool for probing quantum information scrambling, a fundamental process by which local information spreads irreversibly throughout a quantum many-body system. Experimentally measuring the OTOC, however, is notoriously challenging due to the need for time-reversed evolution. Here, we present an experimental evaluation of the OTOC on a quantum computer, using three distinct protocols to address this challenge: the rewinding time method (RTM), the weak-measurement method (WMM), and the irreversibility-susceptibility method (ISM). Our experiments investigate the quantum dynamics of an XXZ spin-1/2 chain prepared in a thermal Gibbs state. As a key contribution, we provide the first experimental demonstration of the ISM, using the trapped-ion quantum computer, reimei. We also conduct a detailed comparative analysis of all three methods, revealing method-dependent behaviors in the measured OTOC. This work not only validates these protocols as practical tools for exploring quantum chaos on near-term hardware but also offers crucial insights into their respective advantages and limitations, providing a practical framework for future experimental investigations.