Optimal low-depth quantum signal-processing phase estimation

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 φ 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.

Fig. 1. Quantum circuit for QSPE with an exemplified two-qubit U-gate.

The input quantum state is prepared to be Bell state in either $∣{+}_{\ell }⟩$ or $∣{\mathrm{i}}_{\ell }⟩$ according to the type of experiment. The quantum circuit enjoys a periodic structure of the unknown U-gate and a tunable Z rotation.

Fig. 2. Flowchart of main procedures in QSPE.

The experimental data are collected from depth d quantum circuit experiments featuring equally-spaced phase modulation angles ω, 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.

Fig. 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 φ = π/16 and χ = 5π/32. The number of measurement samples is set to M = 1 × 10⁵.

Fig. 4. Learning CZ gate with small unwanted swap angle.

Each data point is the average of 10 independent repetitions and the error bars in the top panels stand for the standard deviation across those repetitions. The number of measurement samples is set to M = 1 × 10⁴. These columns display the estimated values of gate angles θ, φ, and circuit fidelity α.

Fig. 5

Comparison of the variance in learning swap angle θ of CZ gates over seventeen pairs of qubits between QSPE and XEB each repeated for 10 times.