Quantum Phaselift

TL;DR

Quantum Phaselift introduces a novel framework for estimating quantum time-series by reconstructing a rank-one matrix, significantly reducing circuit depth requirements and enabling efficient, high-quality recovery on near-term quantum hardware.

Contribution

The paper presents a lifting-based approach that estimates a matrix instead of the time-series directly, with algorithms that are robust and scalable for practical quantum systems.

Findings

Decouples circuit depth from evolution time via band-limited measurements

Achieves exact recovery with constant bandwidth in noiseless settings

Demonstrates effective recovery for large systems with limited measurements

Abstract

Estimating quantum time-series such as the Loschmidt amplitude $f(t)=\langle\psi|\mathrm{e}^{-\mathrm{i}Ht}|\psi\rangle$ is central to spectroscopy, Hamiltonian analysis, and many phase-estimation algorithms. Direct estimation via the Hadamard test requires controlled implementations of $\mathrm{e}^{-\mathrm{i}Ht}$, and the depth of these controlled circuits grows with $t$, making long-time estimation challenging on near-term hardware. We introduce Quantum Phaselift, a lifting-based framework that estimates the rank-one matrix $Z = f f^\dagger$ rather than estimating $f$ directly. We propose simple quantum circuits for estimating the entries of $Z$ and show that measuring only a narrow band of this matrix around the diagonal is sufficient to uniquely recover $f$. Crucially, this reformulation decouples the controlled circuit depth from the maximum evolution time to scale instead with the width of the measured band. We prove that a $O(1)$ bandwidth suffices for generic signals, leading to substantial savings in controlled operations compared to direct estimation methods. We develop three recovery algorithms with provable exact recovery in the noiseless setting and stability under measurement noise. Finally, we numerically demonstrate that high-quality recovery is possible for the 2D Fermi-Hubbard and 2D transverse-field Ising model signals of size exceeding 100 time points using only a few million measurement shots and reasonable post-processing time, making our time-series estimation techniques efficient and effective for near-term implementations.