Observation of constructive interference at the edge of quantum ergodicity

Abstract

The dynamics of quantum many-body systems is characterized by quantum observables that are reconstructed from correlation functions at separate points in space and time^1-3. In dynamics with fast entanglement generation, however, quantum observables generally become insensitive to the details of the underlying dynamics at long times due to the effects of scrambling. To circumvent this limitation and enable access to relevant dynamics in experimental systems, repeated time-reversal protocols have been successfully implemented⁴. Here we experimentally measure the second-order out-of-time-order correlators (OTOC⁽²⁾)^5-18 on a superconducting quantum processor and find that they remain sensitive to the underlying dynamics at long timescales. Furthermore, OTOC⁽²⁾ manifests quantum correlations in a highly entangled quantum many-body system that are inaccessible without time-reversal techniques. This is demonstrated through an experimental protocol that randomizes the phases of Pauli strings in the Heisenberg picture by inserting Pauli operators during quantum evolution. The measured values of OTOC⁽²⁾ are substantially changed by the protocol, thereby revealing constructive interference between Pauli strings that form large loops in the configuration space. The observed interference mechanism also endows OTOC⁽²⁾ with high degrees of classical simulation complexity. These results, combined with the capability of OTOC⁽²⁾ in unravelling useful details of quantum dynamics, as shown through an example of Hamiltonian learning, indicate a viable path to practical quantum advantage.

Fig. 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.

Fig. 2. Sensitivity of OTOCs towards microscopic details of quantum dynamics.

a, Top, quantum circuit schematic for measuring OTOCs of different orders, OTOC^(k). Here, $\mid {\psi }_M⟩$ 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 gate is $\exp \left(-\mathrm{i}\frac{\theta }{2}\left(\cos \left(\varphi \right)X+\sin \left(\varphi \right)Y\right)\right)$, where θ/π ∈ {0.25, 0.5, 0.75} and ϕ/π 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 approximately 0.35 rad. b, The mean (${\overline{\mathcal{C}}}^{\left(4\right)}$) and standard deviation ($\sigma \left[{\mathcal{C}}^{\left(4\right)}\right]$) of OTOC⁽²⁾ (${\mathcal{C}}^{\left(4\right)}$) measured over 100 circuit instances for t = 6, 12 and 18 cycles. The colour at each qubit site indicates data collected with B applied to the given qubit. Purple dots indicate the fixed location of q_m. Cyan lines represent the light cone of q_m. c, Standard deviation of four quantities, TOC (${\mathcal{C}}^{\left(1\right)}$), OTOC (${\mathcal{C}}^{\left(2\right)}$), OTOC⁽²⁾ (${\mathcal{C}}^{\left(4\right)}$) and the off-diagonal component of OTOC⁽²⁾ (${\mathcal{C}}_{\text{off-diag}}^{\left(4\right)}$). For ${\mathcal{C}}^{\left(2\right)}$, ${\mathcal{C}}^{\left(4\right)}$ and ${\mathcal{C}}_{\text{off-diag}}^{\left(4\right)}$, q_m has the same fixed location as in b whereas q_b gradually moves further from q_m as the number of circuit cycles increases, such that the OTOC mean ${\overline{\mathcal{C}}}^{\left(2\right)}\approx 0.5$ is maintained. ${\mathcal{C}}^{\left(1\right)}$ corresponds to $⟨Z\left(t\right)Z⟩$ measured at a qubit close to the centre of the lattice. SQ, single qubit.

Fig. 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 ${\mathcal{C}}^{\left(2\right)}$, in which only two copies of B(t) are present, the final strings P_α and P_β need to be identical to contribute. For ${\mathcal{C}}^{\left(4\right)}$, the strings (P_α, P_β, P_γ, P_δ) contribute a ‘diagonal’ component ${\mathcal{C}}_{\mathrm{diag}}^{\left(4\right)}$ when P_α = P_β and P_γ = P_δ, or an ‘off-diagonal’ component ${\mathcal{C}}_{\text{off-diag}}^{\left(4\right)}$ when P_α ≠ P_β ≠ P_γ ≠ P_δ. b, Protocol for probing quantum interference. Random Pauli operators are inserted at one circuit cycle, which changes the signs of the Pauli string coefficients. c, Relative signal change, characterized by 1 − ρ, as a function of the cycle at which Paulis are inserted. ρ refers to the Pearson correlation between experimental data from 50 different 40-qubit circuits (t = 22 cycles), obtained with and without Pauli insertion. Error bars denote standard errors estimated from resampling the experimental data. Insets, Data at cycle 11. d, Comparison of experimental ${\mathcal{C}}^{\left(2\right)}$ values against exactly simulated ${\mathcal{C}}^{\left(2\right)}$ for a set of 40-qubit circuit instances. Values computed using CMC heuristic algorithms are shown for comparison, achieving an SNR of 5.3, like that of the quantum processor (SNR = 5.4). Inset, circuit geometry (red for q_m and blue for q_b) used for the experiments in c–e. e, Experimental ${\mathcal{C}}_{\text{off-diag}}^{\left(4\right)}$ values on the same set of 40 qubits, alongside exact and CMC simulations. ${\mathcal{C}}_{\text{off-diag}}^{\left(4\right)}$ is measured by subtracting the Pauli-averaged ${\mathcal{C}}^{\left(4\right)}$ from the non-averaged ${\mathcal{C}}^{\left(4\right)}$. 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 Supplementary Information sections II.F.3 and II.F.4. Exp, experiment; MC, Monte Carlo; sim, simulation.

Fig. 4. Measuring OTOC⁽²⁾ in the classically challenging regime.

a, ${\mathcal{C}}_{\text{off-diag}}^{\left(4\right)}$ measured on a set of 65-qubit circuits each having t = 23 cycles. Inset, Qubit geometry. 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 (Supplementary Information sections II.F.3 and II.F.4). Error bars in a are based on the same empirical error model. c, Estimated time to compute ${\mathcal{C}}_{\text{off-diag}}^{\left(4\right)}$ of a single circuit in a on the Frontier supercomputer using tensor-network contraction. The estimate was obtained by running a specially designed optimization algorithm^– on 20 Google Cloud virtual machines (totalling 1,200 CPUs) up to a period of 24 h (x axis). Estimates using a publicly available library cotengra lead to costs that are ten times higher after the same optimization time. TNCO, tensor-network contraction.

Fig. 5. Application to Hamiltonian learning.

a, Scheme for applying OTOC⁽²⁾ in Hamiltonian learning. OTOC⁽²⁾ measured in a physical system of interest is compared with a 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 datasets. b, Demonstrating a one-parameter learning experiment. A collection of classically simulated ${\mathcal{C}}_{\text{off-diag}}^{\left(4\right)}$ 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 ξ/π = 0.6 of the two-qubit gate unitary U_2Q belonging to one pair of qubits (green bar in the top and bottom left panels). c, Experimentally measured ${\mathcal{C}}_{\text{off-diag}}^{\left(4\right)}$ (quantum processor data) as a function of ξ for three different circuit instances. Blue lines indicate the ideal values of ${\mathcal{C}}_{\text{off-diag}}^{\left(4\right)}$ from a classical simulation. These lines intersect all three datasets close to the target value of ξ (vertical dashed line). d, An optimization cost function, corresponding to the root-mean-square difference between the quantum processor data and the classical simulation data of all 20 circuit instances as a function of ξ. The cost function is minimized at the target value of ξ.