Constructive interference at the edge of quantum ergodic dynamics

TL;DR

The paper investigates how time-reversal Echo protocols elevate the sensitivity of quantum observables to underlying many-body dynamics, using OTOCs as interferometers. By introducing higher-order OTOCs (OTOC^(k)) and nested echoes, it reveals a wavefront of information with constructive interference among Pauli strings, especially in OTOC^(2). The study demonstrates that large-loop, off-diagonal Pauli-path interference makes OTOC^(2) and especially C^(4)_off-diag profoundly hard to simulate classically, enabling measurements beyond current classical capabilities and enabling Hamiltonian learning tasks that approach practical quantum advantage.

Abstract

Quantum observables in the form of few-point correlators are the key to characterizing the dynamics of quantum many-body systems. In dynamics with fast entanglement generation, quantum observables generally become insensitive to the details of the underlying dynamics at long times due to the effects of scrambling. In experimental systems, repeated time-reversal protocols have been successfully implemented to restore sensitivities of quantum observables. Using a 103-qubit superconducting quantum processor, we characterize ergodic dynamics using the second-order out-of-time-order correlators, OTOC$^{(2)}$. In contrast to dynamics without time reversal, OTOC$^{(2)}$ are observed to remain sensitive to the underlying dynamics at long time scales. Furthermore, by inserting Pauli operators during quantum evolution and randomizing the phases of Pauli strings in the Heisenberg picture, we observe substantial changes in OTOC$^{(2)}$ values. This indicates that OTOC$^{(2)}$ is dominated by constructive interference between Pauli strings that form large loops in configuration space. The observed interference mechanism endows OTOC$^{(2)}$ with a high degree of classical simulation complexity, which culminates in a set of large-scale OTOC$^{(2)}$ measurements exceeding the simulation capacity of known classical algorithms. Further supported by an example of Hamiltonian learning through OTOC$^{(2)}$, our results indicate a viable path to practical quantum advantage.

Figures (5)

• Figure 1: OTOCs as interferometers.a, When dynamical protocols involve echoing, the Heisenberg picture of the operator evolution is the natural framework for studying dynamics. b, OTOC and OTOC$^{(2)}$ can be viewed as time interferometers, which highlights their capability of refocusing on desired details and echoing out unwanted dynamics. See text for the definition of parameters.
• Figure 2: Sensitivity of OTOCs toward microscopic details of quantum dynamics.a, Top: Quantum circuit schematic for measuring OTOCs of different orders, OTOC$^{(k)}$. Here, $\ket{\psi_M}$ is an eigenstate of the measurement operator $M$ (realized as $Z$ in this work). The operator $B$ is realized as $X$. Bottom: Implementation of the unitary $U$ as $t$ cycles of single- and two-qubit gates. Each single-qubit (SQ) gate is $e^{-i \frac{\theta}{2} (\cos(\phi) X + \sin(\phi) Y)}$, where $\theta/\pi \in \{0.25, 0.5, 0.75\}$ and $\phi/\pi$ is chosen randomly from the interval $[-1,1]$. Each iSWAP-like gate is equivalent to an iSWAP followed by a CPHASE gate with a conditional phase of $\approx 0.35$ rad. b, The mean ($\overline{\mathcal{C}}^{(4)}$) and standard deviation ($\sigma [ \mathcal{C}^{(4)} ]$) of OTOC$^{(2)}$ ($\mathcal{C}^{(4)}$), measured over 100 circuit instances for $t$ = 6, 12 and 18 cycles. The color at each qubit site indicates data collected with $B$ applied to the given qubit. Purple dot indicates the fixed location of $q_\text{m}$. Cyan lines represent the lightcone of $q_\text{m}$. c, Standard deviation of four quantities, TOC ($\mathcal{C}^{(1)}$), OTOC ($\mathcal{C}^{(2)}$), OTOC$^{(2)}$ ($\mathcal{C}^{(4)}$) and the off-diagonal component of OTOC$^{(2)}$ ($\mathcal{C}^{(4)}_\text{off-diag}$). In the cases of $\mathcal{C}^{(2)}$, $\mathcal{C}^{(4)}$ and $\mathcal{C}^{(4)}_\text{off-diag}$, $q_\text{m}$ has the same fixed location as panel b whereas $q_\text{b}$ is gradually moved further from $q_\text{m}$ as circuit cycles increase, such that an OTOC mean of $\overline{\mathcal{C}}^{(2)} \approx 0.5$ is maintained. $\mathcal{C}^{(1)}$ corresponds to $\braket{Z (t) Z}$ measured at a qubit close to the center of the lattice.
• Figure 3: Quantum interference and classical simulation complexity of OTOC$^{(2)}$.a, In the Heisenberg picture, the time-evolved $B(t)$ branches into a superposition of multi-qubit Pauli strings. For $\mathcal{C}^{(2)}$ in which only two copies of $B (t)$ are present, the final strings $P_\alpha$ and $P_\beta$ need to be identical to contribute. For $\mathcal{C}^{(4)}$, the strings ($P_\alpha$, $P_\beta$, $P_\gamma$, $P_\delta$) contribute a "diagonal" component $\mathcal{C}^{(4)}_\text{diag}$ when $P_\alpha = P_\beta$ and $P_\gamma = P_\delta$, or an "off-diagonal" component $\mathcal{C}^{(4)}_\text{off-diag}$ when $P_\alpha \neq P_\beta \neq P_\gamma \neq P_\delta$. b, Protocol for probing quantum interference: Random Pauli operators are inserted at one circuit cycle, which changes the signs of Pauli string coefficients. c, Relative signal change, characterized by $1 - \rho$, as a function of the cycle at which Paulis are inserted. $\rho$ refers to the Pearson correlation between experimental data from 50 different 40-qubit circuits (t = 22 cycles), obtained with and without Pauli insertion (see insets for data at cycle 11). Error bars denote standard errors estimated from resampling the experimental data. d, Comparison of experimental $\mathcal{C}^{(2)}$ values against exactly simulated $\mathcal{C}^{(2)}$ for a set of 40-qubit circuit instances. Values computed using cached Monte-Carlo (CMC) heuristic algorithms are shown for comparison, achieving an SNR of 5.3 similar to the quantum processor (SNR = 5.4). Inset shows the circuit geometry (red: $q_\text{m}$; blue: $q_\text{b}$) used for the experiments in panels c, d and e. e, Experimental $\mathcal{C}^{(4)}_\text{off-diag}$ values on the same set of 40 qubits, alongside exact and CMC simulations. $\mathcal{C}^{(4)}_\text{off-diag}$ is measured by subtracting the Pauli-averaged $\mathcal{C}^{(4)}$ from the non-averaged $\mathcal{C}^{(4)}$. Here the experimental SNR is 3.9 whereas the SNR from CMC is 1.1. Error bars on experimental data are based on an empirical error model discussed in SI Section II.F.3 and II.F.4.
• Figure 4: Measuring OTOC$^{(2)}$ in the classically challenging regime.a, $\mathcal{C}^{(4)}_\text{off-diag}$ measured on a set of 65-qubit circuits each having $t = 23$ cycles. The qubit geometry is indicated in the inset, where $B$ acts simultaneously on three different qubits. b, Experimental SNRs for circuits measured with system sizes ranging from 18 to 40 qubits. Error bars correspond to the 95% confidence interval of an empirical error model (see SI Section II.F.3 and II.F.4). Error bars in panel a are based on the same empirical error model. c, Estimated time to compute $\mathcal{C}^{(4)}_\text{off-diag}$ of a single circuit in a on the Frontier supercomputer using TN contraction. The estimate is obtained by running a specially designed optimization algorithm boixo2017simulationgray2021hyperKalachev_arxiv_2022Pan_TN_PRL_2022 on 20 Google Cloud virtual machines (totaling 1200 CPUs) up to a period of 24 hours ($x$-axis). Estimates using a publicly available library cotengragray2021hyper lead to costs that are ten times higher after the same optimization time.
• Figure 5: Application to Hamiltonian learning.a, Scheme for applying OTOC$^{(2)}$ toward Hamiltonian learning: OTOC$^{(2)}$ measured in a physical system of interest are compared with quantum simulation of OTOC$^{(2)}$ using a parameterized Hamiltonian of the same system. Hamiltonian parameters are then optimized to minimize the difference between the two data sets. b, Demonstrating a one-parameter learning experiment: Here a collection of classically simulated $\mathcal{C}^{(4)}_\text{off-diag}$ values from 20 circuit instances having a 34-qubit geometry (bottom left panel) are treated as data from a physical system of interest. The goal is to learn a particular phase $\xi / \pi = 0.6$ of the two-qubit gate unitary $U_\text{2Q}$ belonging to one pair of qubits (green bar in the top and bottom left panels). c, Experimentally measured $\mathcal{C}^{(4)}_\text{off-diag}$ (quantum processor data) as a function of $\xi$ for three different circuit instances. Blue lines indicate the ideal values of $\mathcal{C}^{(4)}_\text{off-diag}$ from classical simulation, which intersect all three data sets close to the target value of $\xi$ (vertical dashed line). d, An optimization cost function, corresponding to the RMS difference between quantum processor and classical simulation data of all 20 circuit instances, as a function of $\xi$. The cost function is minimized at the target value of $\xi$.