Quantum Phaselift

TL;DR

This work addresses the challenge of estimating quantum time-series, such as the Loschmidt amplitude $f(t)$, under hardware constraints that make long-time controlled evolutions costly. It introduces Quantum Phaselift, a lifting-based framework that recasts $f$ as the rank-one matrix $Z = f f^{\dagger}$ and uses measurements of a narrow $K$-band around the diagonal to recover $f$ with depth independent of the total evolution time. Three recovery algorithms—block-by-block algebraic, block-by-block eigenvector, and least-squares—are shown to achieve exact recovery in the noiseless setting and to be stable under measurement noise; they offer provable guarantees and favorable resource scaling. Numerical simulations on 2D Fermi-Hubbard and transverse-field Ising models demonstrate accurate reconstruction for signals with more than 100 time points using only a few million shots, highlighting the approach's practicality for near-term quantum devices and spectral estimation.

Abstract

Estimating quantum time-series such as the Loschmidt amplitude $f(t)=\langleψ|\mathrm{e}^{-\mathrm{i}Ht}|ψ\rangle$ 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.

Figures (11)

• Figure 1: Accurate signal recovery for a representative instance of a $4\times 3$ transverse-field Ising model defined in \ref{['eq:tf-ising-model']}, with randomly chosen coupling parameters and a uniform superposition input state. The total number of measurement shots is fixed to approximately one million and distributed uniformly across the measured $K$-band entries of the lifted matrix $Z$. The reconstructed signal is obtained using the block-by-block eigenvector estimator (Algorithm \ref{['algo:eigen-estimator-averaging']}).
• Figure 2: Schematic of the Block-by-Block Algebraic Estimator (shown for $K=2$).
• Figure 3: Schematic of the Block-by-Block Eigenvector Estimator (shown for $K=2$).
• Figure 4: Signal recovery using our three proposed estimators (algebraic, eigenvector, and least-squares) for some example signals corresponding to the $2\times 2$ Fermi-Hubbard Model, $2\times 3$ Fermi-Hubbard Model, and $4 \times 3$ transverse-field Ising Model. For recovering each of the above signals, we use $3000$ shots for estimating each entry of the $6$-band of the corresponding lifted matrix $Z$.
• Figure 5: Signal recovery for the signal of Lemma \ref{['lem:integer-H']} using the algebraic and eigenvector estimators. We plot the average normalized Euclidean error in the recovered signal in grayscale for all values of $K= 2, \ldots T-1$ and $W= 2, \ldots, T-1$

Theorems & Definitions (33)

• Lemma 1: Identifiability; informal Lemma \ref{['lem:identifiability']} statement
• Theorem 2: Informal Theorem \ref{['thm:exact-rec-algebraic-est']} statement
• Theorem 3: Informal Theorem \ref{['thm:stability-base-case']} statement
• Theorem 4: Informal Theorem \ref{['thm:exact-rec-eigen-est']} statement
• Theorem 5: Informal Theorem \ref{['thm:exact-recovery-quantum-phaselift-estimator']} statement
• Theorem 6: Informal Theorem \ref{['thm:stability-proof-least-squares-estimator']} statement
• Definition 7: Signal Recovery Problem
• Lemma 8: Identifiability
• proof
• Lemma 9: Integer Hamiltonian Example