Low-depth amplitude estimation via statistical eigengap estimation

TL;DR

The paper introduces a novel amplitude estimation method that leverages the energy gap of an effective Hamiltonian, achieving optimal tradeoffs and improved empirical performance in low-depth quantum circuits.

Contribution

It presents a new amplitude estimation algorithm based on statistical eigengap estimation, suitable for both Heisenberg-limited and low-depth regimes, with provable guarantees.

Findings

Achieves performance comparable to state-of-the-art in Heisenberg-limited regime.

Obtains optimal query--depth tradeoffs up to polylogarithmic factors in low-depth regime.

Demonstrates improved empirical performance over prior approaches.

Abstract

Amplitude estimation, in its original form, is formulated as phase estimation upon the Grover iterate. Subsequent improvements to the algorithm have eliminated the need for phase estimation and introduced low-depth variants that trade speedups for lower circuit depth. We make the key observation that amplitude estimation is equivalent to estimating the energy gap of an effective Hamiltonian, whereby discrete-time evolution is generated by amplitude amplification. This enables us to develop an amplitude estimation algorithm for both Heisenberg-limited and low-depth circuit regimes, inspired by statistical phase estimation techniques developed for early fault-tolerant ground-state energy estimation. In the Heisenberg-limited regime, our approach achieves performance comparable to state-of-the-art methods while using simplified classical post-processing. In the low-depth regime, it obtains optimal query--depth tradeoffs up to polylogarithmic factors, with provable guarantees and improved empirical performance over prior approaches. The resulting protocol is ancilla-free and requires only standard Grover reflections. Due to its flexibility, generality, and robustness, we expect our approach to be a key enabler for a broad range of early fault-tolerant applications.