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Reductive Quantum Phase Estimation

Nicholas J. C. Papadopoulos, Jarrod T. Reilly, John Drew Wilson, Murray J. Holland

TL;DR

The paper introduces reductive quantum phase estimation (RQPE) as a unifying framework that encompasses Ramsey interferometry (RI) and quantum phase estimation (QPE) for estimating quantum phases. It presents a polynomial-time algorithm to generate RQPE circuits capable of perfectly distinguishing a finite set of rational-phase values with certainty in a single run, using fewer qubits and unitary applications than conventional QPE in some cases. The study formalizes a trade-off between measurement precision and phase distinguishability via resources and Fisher information, and uses Bayesian reconstruction to demonstrate convergence to the Cramér-Rao bound in the large-sample limit, illustrating practical pathways for efficient phase estimation in quantum metrology and computing. The work highlights the potential to tailor phase estimation circuits to specific tasks, including systems with degenerate energy levels or prior phase information, and sets the stage for further optimization and experimental integration.

Abstract

Estimating a quantum phase is a necessary task in a wide range of fields of quantum science. To accomplish this task, two well-known methods have been developed in distinct contexts, namely, Ramsey interferometry (RI) in atomic and molecular physics and quantum phase estimation (QPE) in quantum computing. We demonstrate that these canonical examples are instances of a larger class of phase estimation protocols, which we call reductive quantum phase estimation (RQPE) circuits. Here we present an explicit algorithm that allows one to create an RQPE circuit. This circuit distinguishes an arbitrary set of phases with a fewer number of qubits and unitary applications, thereby solving a general class of quantum hypothesis testing to which RI and QPE belong. We further demonstrate a trade-off between measurement precision and phase distinguishability, which allows one to tune the circuit to be optimal for a specific application.

Reductive Quantum Phase Estimation

TL;DR

The paper introduces reductive quantum phase estimation (RQPE) as a unifying framework that encompasses Ramsey interferometry (RI) and quantum phase estimation (QPE) for estimating quantum phases. It presents a polynomial-time algorithm to generate RQPE circuits capable of perfectly distinguishing a finite set of rational-phase values with certainty in a single run, using fewer qubits and unitary applications than conventional QPE in some cases. The study formalizes a trade-off between measurement precision and phase distinguishability via resources and Fisher information, and uses Bayesian reconstruction to demonstrate convergence to the Cramér-Rao bound in the large-sample limit, illustrating practical pathways for efficient phase estimation in quantum metrology and computing. The work highlights the potential to tailor phase estimation circuits to specific tasks, including systems with degenerate energy levels or prior phase information, and sets the stage for further optimization and experimental integration.

Abstract

Estimating a quantum phase is a necessary task in a wide range of fields of quantum science. To accomplish this task, two well-known methods have been developed in distinct contexts, namely, Ramsey interferometry (RI) in atomic and molecular physics and quantum phase estimation (QPE) in quantum computing. We demonstrate that these canonical examples are instances of a larger class of phase estimation protocols, which we call reductive quantum phase estimation (RQPE) circuits. Here we present an explicit algorithm that allows one to create an RQPE circuit. This circuit distinguishes an arbitrary set of phases with a fewer number of qubits and unitary applications, thereby solving a general class of quantum hypothesis testing to which RI and QPE belong. We further demonstrate a trade-off between measurement precision and phase distinguishability, which allows one to tune the circuit to be optimal for a specific application.
Paper Structure (12 sections, 41 equations, 6 figures)

This paper contains 12 sections, 41 equations, 6 figures.

Table of Contents

  1. Introduction
  2. RQPE Generation Algorithm
  3. Trade-off Between Precision and Distinguishability
  4. Conclusion and outlook
  5. Gates on a Qubit
  6. Estimating θ
  7. Complexity
  8. Building a Circuit from Bit Values
  9. Fisher Information
  10. Examples of Comparing Optimality
  11. Features of Indistinguishability
  12. RQPE as a Gate

Figures (6)

  • Figure 1: (a) Visualization of $\Theta = \{ \frac{\pi x}{64} : x \in \{ 21, 22, 64, 65, 107, 108 \} \}$ around the equator of the Bloch sphere. (b) Application of Algorithm \ref{['CircuitAlgorithm']} to find $G_{i}$ and $A_{i}$ for all iterations. (c) The circuit generated by Algorithm \ref{['CircuitAlgorithm']} by way of Eq. \ref{['eq:AppliedGates']}.
  • Figure 2: (a) Visualization of $\Theta = \{ \frac{\pi x}{70} : x \in \{ 66, 93, 108, 123, 138 \} \}$ around the equator of the Bloch sphere. (b) Application of Algorithm \ref{['CircuitAlgorithm']} to find $G_{i}$ and $A_{i}$ for all iterations. (c) The circuit generated by Algorithm \ref{['CircuitAlgorithm']} by way of Eq. \ref{['eq:AppliedGates']} and removing phantom qubits.
  • Figure 3: Canonical phase estimation procedures. (a) RI circuit with the corresponding Bloch sphere rotations. (b) The perfectly distinguishable phases of RI with 7 unitary applications. (c) QPE circuit. (d) The perfectly distinguishable phases of QPE with 3 qubits.
  • Figure 4: The conditional probability of (a) RI with 7 unitary applications and (b) QPE with 3 qubits. The distance between phases $\theta_a$ and $\theta_b$ calculated by Eq. \ref{['eq:distance']} for (c) RI with 7 unitary applications and (d) QPE with 3 qubits.
  • Figure 5: Using an RQPE circuit with 1 and 6 unitary applications on two qubits, shown are (a) the conditional probability and (b) the distance calculated by Eq. \ref{['eq:distance']} between two phases.
  • ...and 1 more figures