Adaptive Bayesian quantum algorithm for phase estimation

Abstract

Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>N</a:mi><a:mi>tot</a:mi></a:msub></a:math> times, the variance of the estimates can scale as <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>O</b:mi><b:mo>(</b:mo><b:mn>1</b:mn><b:mo>/</b:mo><b:msubsup><b:mi>N</b:mi><b:mi>tot</b:mi><b:mn>2</b:mn></b:msubsup><b:mo>)</b:mo></b:mrow></b:math>, compared to the best “classical” strategy with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>O</c:mi><c:mo>(</c:mo><c:mn>1</c:mn><c:mo>/</c:mo><c:msub><c:mi>N</c:mi><c:mi>tot</c:mi></c:msub><c:mo>)</c:mo></c:mrow></c:math>. The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without interprobe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>O</d:mi><d:mo>(</d:mo><d:mn>1</d:mn><d:mo>/</d:mo><d:msubsup><d:mi>N</d:mi><d:mi>tot</d:mi><d:mn>2</d:mn></d:msubsup><d:mo>)</d:mo></d:mrow></d:math>. Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analizes the results of previous steps. Published by the American Physical Society 2024

Topics & Concepts