Sample-Based Quantum Diagonalization with Amplitude Amplification

TL;DR

The paper introduces SQD-AA, an improved quantum diagonalization method combining sample-based diagonalization with amplitude amplification, significantly reducing query complexity and circuit depth for quantum state computations.

Contribution

It presents the SQD-AA algorithm that enhances sample-based quantum diagonalization with amplitude amplification, achieving substantial efficiency gains over prior methods.

Findings

Over 100-fold reduction in query complexity for certain distributions.

Analytical proof of quadratic advantage for exponential distributions.

Lower T-gate counts and shallower circuits compared to iterative quantum phase estimation.

Abstract

Recently, sample-based quantum diagonalization (SQD) has emerged as a promising approach to compute ground and excited states of problem Hamiltonians.This method classically diagonalizes a Hamiltonian in a subspace that is spanned by samples obtained from a quantum computer. However, by its nature, SQD suffers from a fundamental sampling problem, as some basis states that are required for a targeted accuracy may only be sampled extremely rarely. To alleviate this limitation, we introduce the SQD-AA algorithm that combines SQD with amplitude amplification (AA). SQD-AA uses AA to sequentially reduce probabilities of already measured bitstrings, thus making the observation of new ones more likely. We observe a reduction in the total query complexity of more than a factor 100 for algebraically and exponentially decaying model distributions, and analytically show a quadratic advantage for the latter. Moreover, we evaluate real molecules in an early fault-tolerant scenario and compare SQD-AA to SQD and iterative quantum phase estimation (iQPE). For all considered examples, we observe the lowest total number of T-gates for SQD-AA while only requiring circuits that are 3-4 orders of magnitude shallower than those needed for iQPE. Given this substantial reduction in circuit depth compared to iQPE while saving 2 orders of magnitude in total runtime compared to SQD, we expect a significant regime in early fault-tolerance where SQD-AA runs feasibly, but iQPE circuits are too deep to execute confidently.