Optimal Low-Depth Quantum Signal-Processing Phase Estimation

TL;DR

The paper introduces Quantum Signal-Processing Phase Estimation (QSPE), a low-depth metrology protocol that robustly learns two-qubit gate parameters by decoupling the swap angle $\theta$ from the drift-prone phase $\varphi$ via quantum signal processing and Fourier analysis. QSPE achieves estimators that saturate classical CRLBs in the pre-asymptotic regime and exhibits favorable variance scaling, including a faster-than-Heisenberg-like behavior for $\varphi$ at small depths, with a transition toward Heisenberg scaling as depth grows. The approach is validated both theoretically—through QFI/QCRLB bounds and polynomial representations—and experimentally on Google Quantum AI superconducting qubits, achieving $10^{-4}$ STD accuracy in $<10$-gate-depth circuits and outperforming prior methods by up to two orders of magnitude. The work demonstrates robustness to time-dependent drift, depolarizing noise, and readout errors, and provides a comprehensive framework for generalizing QSPE to broader two-level subspaces and swap-angle regimes, paving the way for practical, high-precision gate calibration in near-term quantum devices.

Abstract

Quantum effects like entanglement and coherent amplification can be used to drastically enhance the accuracy of quantum parameter estimation beyond classical limits. However, challenges such as decoherence and time-dependent errors hinder Heisenberg-limited amplification. We introduce Quantum Signal-Processing Phase Estimation algorithms that are robust against these challenges and achieve optimal performance as dictated by the Cramér-Rao bound. These algorithms use quantum signal transformation to decouple interdependent phase parameters into largely orthogonal ones, ensuring that time-dependent errors in one do not compromise the accuracy of learning the other. Combining provably optimal classical estimation with near-optimal quantum circuit design, our approach achieves a standard deviation accuracy of $10^{-4}$ radians for estimating unwanted swap angles in superconducting two-qubit experiments, using low-depth ($<10$) circuits. This represents up to two orders of magnitude improvement over existing methods. Theoretically and numerically, we demonstrate the optimality of our algorithm against time-dependent phase errors, observing that the variance of the time-sensitive parameter $\varphi$ scales faster than the asymptotic Heisenberg scaling in the small-depth regime. Our results are rigorously validated against the quantum Fisher information, confirming our protocol's ability to achieve unmatched precision for two-qubit gate learning.

Figures (22)

• Figure 1: Quantum circuit for QSPE with an exemplified two-qubit $U$-gate. The input quantum state is prepared to be Bell state in either $\ket{+_\ell}$ or $\ket{\mathrm{i}_\ell}$ according to the type of experiment. The quantum circuit enjoys a periodic structure of the unknown $U$-gate and a tunable $Z$ rotation.
• Figure 1: Quantum circuit for periodic calibration using the population method.
• Figure 2: Flowchart of main procedures in QSPE. The experimental data are collected from depth $d$ quantum circuit experiments featuring equally-spaced phase modulation angles $\omega$, as shown in the left panels. Probabilities from each experiment of different phase modulations are analyzed using Fourier transformation. As illustrated in the right panels, the Fourier-space data are better structured compared to real-space data. Gate angles are then derived using our QSPE estimators.
• Figure 2: Distribution of run-to-run variation of swap-angle estimation across a device. The swap angles were estimated using periodic calibration on four independent datasets for each CZ gate, with 10,000 samples per circuit and a maximum depth of 30. Due to the behavior of the periodic-calibration estimator for particularly small swap angles, a substantial fraction of swap angles were estimated to be identically 0, leading to the portion of the cumulative distribution function that extends off the plot to the left. Discarding these instances leaves us with a median run-to-run standard deviation of close to $4\times10^{-3}$ radians.
• Figure 3: A nontrivial transition of the optimal variance in solving QSPE. The theoretical analysis of the transition is in Supplementary Note 6. a Phase diagram showing the nontrivial transition of the optimal variance in solving QSPE. QSPE estimators attain the optimal variance in the pre-asymptotic regime. b Cramér-Rao lower bound (CRLB) and the theoretically derived estimation variance. The single-qubit phases are set to $\varphi = \pi/16$ and $\chi = 5\pi/32$. The number of measurement samples is set to $M = 1\times10^5$.

Theorems & Definitions (36)

• Theorem 2
• Theorem 1: Polynomial structure of symmetric QSP
• proof
• Theorem 2: Markov brothers' inequality Markov1890
• Theorem 3: Structure of QSPE
• Definition 4: Building block of QSPE
• Lemma 5
• proof
• Theorem 6
• proof