[Paper Review] Faster Coherent Quantum Algorithms for Phase, Energy, and Amplitude Estimation

Patrick Rall|arXiv (Cornell University)|Mar 17, 2021

Quantum Computing Algorithms and ArchitectureComputer Science64 references34 citations

TL;DR

This paper presents faster, simpler coherent quantum algorithms for phase, energy, and amplitude estimation that avoid the quantum Fourier transform and median amplification. Using block-encoding and singular value transformation, the algorithms estimate one bit at a time with reduced query complexity and ancilla qubit requirements, achieving up to 14x speedup over textbook phase estimation while preserving coherence and enabling applications in quantum Metropolis sampling and Bayesian inference.

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.

Motivation & Objective

• To develop conceptually and computationally simpler quantum algorithms for phase, energy, and amplitude estimation under strict coherence constraints.
• To eliminate reliance on the quantum Fourier transform and quantum sorting networks for median amplification in coherent estimation.
• To reduce query complexity and ancilla qubit overhead while maintaining high accuracy in superposition estimation.
• To enable practical acceleration in quantum Metropolis sampling, Bayesian inference, and quantum matrix inversion by avoiding costly subroutines.

Proposed method

• The algorithms use block-encoding techniques to represent non-unitary matrices, enabling manipulation of singular values via unitary operations.
• Singular value transformation is applied to polynomials of the singular values to amplify estimation probabilities without additional ancillae.
• Estimation proceeds one bit at a time using a coherent version of the Hadamard test, with amplification achieved through controlled singular value transformations.
• The method avoids median amplification by using a rounding promise that ensures deterministic output when the promise holds, and provides reasonable non-deterministic estimates otherwise.
• For amplitude estimation, the algorithm constructs a Hamiltonian whose eigenstate is the input state, enabling non-destructive estimation of a² directly.
• The framework unifies phase, energy, and amplitude estimation under a single block-encoding and singular value transformation framework, minimizing ancilla and query overhead.

Experimental results

Research questions

• RQ1 Can coherent phase estimation be performed without the quantum Fourier transform or median amplification, while maintaining high accuracy and low resource cost?
• RQ2 What is the minimal ancilla footprint and query complexity achievable in coherent phase estimation when only one copy of a non-eigenstate input is available?
• RQ3 How can singular value transformation be leveraged to estimate quantum amplitudes coherently and non-destructively?
• RQ4 What are the fundamental limitations of coherent estimation in superposition, and how can they be addressed via rounding promises or probabilistic guarantees?
• RQ5 Can the framework be extended to enable faster, lower-overhead amplitude estimation without requiring iterative repair or prior knowledge of amplitude bounds?

Key findings

• The proposed algorithms achieve up to 14x to 10x faster performance than textbook phase estimation in terms of query complexity, with significantly reduced ancilla qubit requirements.
• The algorithms avoid the quantum Fourier transform and median amplification, replacing them with block-encoding and singular value transformation for efficient amplitude amplification.
• For non-destructive amplitude estimation, the method requires only O(n + 1) ancilla qubits, compared to O(n log(δ⁻¹)) in prior work, due to elimination of median amplification.
• The algorithm provides a direct estimate of a² with high probability, avoiding the need to coherently compute arcsin(a) and thus reducing ancilla overhead.
• When the rounding promise holds, the algorithm performs the deterministic transformation (1) as required for coherent applications; otherwise, it yields reasonable non-deterministic estimates.
• The framework enables a modest constant-factor speedup over prior methods in amplitude estimation, with improved adaptability and no requirement for lower bounds on the amplitude.