Robust multiparameter estimation using quantum scrambling

TL;DR

A versatile and efficient multiparameter quantum sensing protocol, which simultaneously estimates many non-commuting and time-dependent signals that are coherently or incoherently coupled to sensing particles, and opens the door to applications ranging from precise noise benchmarking of quantum dynamics to learning time-dependent Hamiltonians.

Abstract

We propose and analyze a versatile and efficient multiparameter quantum sensing protocol, which simultaneously estimates many non-commuting and time-dependent signals that are coherently or incoherently coupled to sensing particles. Even in the presence of control imperfections and readout errors, our approach can detect exponentially many parameters in the system size while maintaining the optimal scaling of sensitivity. To accomplish this, scrambling dynamics are leveraged to map distinct signals to unique patterns of bitstring measurements, which distinguishes a large number of signals without significant sensitivity loss. Based on this principle, we develop a computationally efficient protocol utilizing random global Clifford unitaries and evaluate its performance both analytically and numerically. Our protocol naturally extends to scrambling dynamics generated by random local Clifford circuits, local random unitary circuits (RUCs), and ergodic Hamiltonian evolution--commonly realized in near-term quantum hardware--and opens the door to applications ranging from precise noise benchmarking of quantum dynamics to learning time-dependent Hamiltonians.

Figures (3)

• Figure 1: (a) Our multiparameter sensing protocol subjects sensor qubits to coherent signals (blue), including external fields or multi-body interactions, as well as incoherent dissipation (red). (b) The system evolves under known random dynamics while accumulating signal, followed by possibly noisy measurement in the computational basis. (c) The resulting empirical bitstring distributions are processed classically using a multivariate least-squares regression procedure (described in the main text) to simultaneously estimate many signal parameters. Estimated amplitudes and their uncertainties are indicated schematically by markers, in close agreement with their true values (bars).
• Figure 2: (a) Quadratic Ramsey protocol and tilted Ramsey protocol for the estimation of commuting signals generated by Pauli $Z$ operators. (b) The change of the output distribution $p(z)$ after signal accumulation with each Ramsey protocol. The effect of readout error is indicated by red bars. In the tilted Ramsey protocol, unlike the quadratic Ramsey protocol, the change in $p(z)$ scales linearly with the strength of the signal, providing weak robustness against readout error. (c) Numerical demonstrations of the quadratic and tilted Ramsey sensing protocols. Left: Reconstructed strengths for all single-body $Z_i$ and nearest-neighbor $Z_iZ_{i+1}$ signals. Bars show the ground truth signal, while purple and red markers indicate estimates at readout error $\gamma_r = 0$ and $\gamma_r = 0.05$, respectively. The simulation is on a system $N=10$ with $M=2000$ samples. Right: Scaling behavior of the root-mean-square (RMS) error for the Ramsey estimators. In quadratic Ramsey sensing (top), $\gamma_r = 0.05$ is shown for only single-body signals. The blue solid line indicates the theoretical prediction for the typical sample complexity (derived in the SM supp), while the horizontal dashed line indicates the bias of the estimator, where RMS error saturates for large enough sample size. In the tilted Ramsey protocol (bottom), the error scaling remains robust even when $\gamma_r=0.05$.
• Figure 3: (a) Schematic of the multiparameter sensing protocol based on random Clifford circuits. (b) Numerical demonstration of our protocol in estimating a large number of time-dependent, coherent and incoherent signals ($K_{c} = K_{ic}=580$ and $T=10$ time steps) using only $N=12$ qubits and $M = 10^4$ measurements. Circuit repetitions $n_c = 10$ and $3$ are used for coherent and incoherent signals, respectively. We choose the signals to be on-site local fields ($X_i$, $Y_i$, and $Z_i$) and nearest-neighbor interaction terms ($X_iX_{i+1}$ and $Z_iZ_{i+1}$). Near-zero signals below a certain threshold are set to zero using standard statistical regularization Wainwright2019 (see SM supp), and error bars on such signals are omitted for clarity. Error bars are smaller than the marker size for incoherent signals. (c) Our protocol robustly achieves multiparameter SQL scaling, even in the presence of noise, and has a typical sample complexity that does not depend on $N$ or the number of signals. Left: RMS error versus $M$ for coherent (top) and incoherent (bottom) signals from (b). Light and dark markers indicate the protocol with ($\gamma_r = 0.05$) and without readout error, respectively. Blue and pink solid lines indicate theory curves for the typical sample complexity (see SM supp), and the horizontal dashed line marks the estimator bias. Right: Scaling of typical sample complexity---defined as the slope of RMS error versus $M$---with the number of signals for different system sizes. The typical sample complexity of our protocol is independent of the number of signals; imposing worst-case performance guarantees leads to the $\log(K_cK_{ic})$ dependence discussed in the main text. We consider a single layer $T = 1$, and in order to simulate a large number of signals, we consider one-, two- and three-body signals which need not be geometrically local. Circuit repetitions $n_c$ = 15 and 3 are used for coherent and incoherent signals. Black solid lines indicate the theory approximation for typical signals. (d) Our protocol maps incoherent signals into random bitstrings, which can be treated as the codewords of a classical error correcting code. This enables classical error correction for readout error per qubit up to $d_{min}/2N$, where $d_{min}$ is the closest Hamming distance between two codewords.