Faster Coherent Quantum Algorithms for Phase, Energy, and Amplitude Estimation

Abstract

We consider performing phase estimation under the following conditions: we are given only one copy of the input state, the input state does not have to be an eigenstate of the unitary, and the state must not be measured. Most quantum estimation algorithms make assumptions that make them unsuitable for this 'coherent' setting, leaving only the textbook approach. We present novel algorithms for phase, energy, and amplitude estimation that are both conceptually and computationally simpler than the textbook method, featuring both a smaller query complexity and ancilla footprint. They do not require a quantum Fourier transform, and they do not require a quantum sorting network to compute the median of several estimates. Instead, they use block-encoding techniques to compute the estimate one bit at a time, performing all amplification via singular value transformation. These improved subroutines accelerate the performance of quantum Metropolis sampling and quantum Bayesian inference.

A Grand Unification of Quantum Algorithms

“…Before proceeding, we note that a very similar QSVTbased phase estimation algorithm was independently discovered in [17]. A primary goal of their work is maintaining coherence, and accordingly the paper presents a thorough performance analysis of their algorithm including the derivation of constant factors as well as precise use of the diamond norm.…”

“…Ref. [17] also extends their QSVT-based phase estimation algorithm to coherent energy and amplitude estimation.…”

“…While some of these applications have been covered in recent works on QSVT [12,17], here we aim to present these constructions as pedagogically as possible, providing detailed procedures and intuition for each, and including explicit examples to support abstract ideas where appropriate. We also supply performance bounds and resource requirements for each of the algorithms presented here, which we anticipate will be helpful.…”

“…Nonetheless, we believe that this logarithmic factor could be removed by integrating phase estimation by QSVT with a more sophisticated phase estimation protocol, such as the inference procedures of [38,39], which already attain speedups over conventional phase estimation. Indeed, the QSVT-based phase estimation algorithm of [17] requires only O(n) queries to the oracle, which is achieved by using a procedure very similar to Algorithm 3, but varying the degree of the QSVT polynomial at each iteration (schematically, by increasing ∆ at each iteration).…”

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

“…These algorithms demonstrate the ability of simple circuit ansätze to expressively and efficiently control quantum subsystem dynamics. In addition to improving the performance of many known quantum algorithms [2,3], QSVT has great explanatory utility: unifying the presentation of most major known quantum algorithms [4]. This includes Hamiltonian simulation [3,5], search [1], phase estimation [2], quantum walks [1], fidelity estimation [6], sophisticated techniques for measurement [7], and channel discrimination [8].…”