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Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer

Qisheng Wang, Zhicheng Zhang

TL;DR

An improved (multi-)samplizer for pure states is used as an algorithmic tool in the construction, through which any quantum query algorithm using $Q$ queries to the reflection operator about a pure state can be converted to a $\delta$-close quantum sample algorithm using $\Theta(Q^2/\delta)$ samples of $|\psi\rangle$.

Abstract

Trace distance and infidelity (induced by square root fidelity), as basic measures of the closeness of quantum states, are commonly used in quantum state discrimination, certification, and tomography. However, the sample complexity for their estimation still remains open. In this paper, we solve this problem for pure states. We present a quantum algorithm that estimates the trace distance and square root fidelity between pure states to within additive error $\varepsilon$, given sample access to their identical copies. Our algorithm achieves the optimal sample complexity $Θ(1/\varepsilon^2)$, improving the long-standing folklore $O(1/\varepsilon^4)$. Our algorithm is composed of a samplized phase estimation of the product of two Householder reflections. Notably, an improved (multi-)samplizer for pure states is used as an algorithmic tool in our construction, through which any quantum query algorithm using $Q$ queries to the reflection operator about a pure state $|ψ\rangle$ can be converted to a $δ$-close (in the diamond norm) quantum sample algorithm using $Θ(Q^2/δ)$ samples of $|ψ\rangle$. This samplizer for pure states is shown to be optimal.

Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer

TL;DR

An improved (multi-)samplizer for pure states is used as an algorithmic tool in the construction, through which any quantum query algorithm using queries to the reflection operator about a pure state can be converted to a -close quantum sample algorithm using samples of .

Abstract

Trace distance and infidelity (induced by square root fidelity), as basic measures of the closeness of quantum states, are commonly used in quantum state discrimination, certification, and tomography. However, the sample complexity for their estimation still remains open. In this paper, we solve this problem for pure states. We present a quantum algorithm that estimates the trace distance and square root fidelity between pure states to within additive error , given sample access to their identical copies. Our algorithm achieves the optimal sample complexity , improving the long-standing folklore . Our algorithm is composed of a samplized phase estimation of the product of two Householder reflections. Notably, an improved (multi-)samplizer for pure states is used as an algorithmic tool in our construction, through which any quantum query algorithm using queries to the reflection operator about a pure state can be converted to a -close (in the diamond norm) quantum sample algorithm using samples of . This samplizer for pure states is shown to be optimal.

Paper Structure

This paper contains 26 sections, 14 theorems, 58 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Organization of this paper.
  3. Overview
  4. Properties of the product of Householder reflections
  5. Multi-samplizer for pure states
  6. Optimality
  7. Related Work
  8. Preliminaries
  9. Norms
  10. Vector norms.
  11. Matrix norms.
  12. Diamond norm.
  13. Quantum phase estimation
  14. Density matrix exponentiation
  15. Estimation with Reflections
  16. ...and 11 more sections

Key Result

Theorem 1.1

We can $\varepsilon$-estimate $\mathrm{T}\lparen\lvert\varphi\rangle, \lvert\psi\rangle\rparen$ and $\mathrm{F}\lparen\lvert\varphi\rangle, \lvert\psi\rangle\rparen$ by using $O\lparen1/\varepsilon^2\rparen$ samples of $\lvert\varphi\rangle$ and $\lvert\psi\rangle$.

Figures (3)

  • Figure 1: The SWAP test.
  • Figure 2: Phase estimation of $R_\varphi R_\psi$ on $\lvert\varphi\rangle$.
  • Figure 3: Phase estimation of $U$ on $\lvert\psi\rangle$.

Theorems & Definitions (27)

  • Theorem 1.1: Optimal sample complexity upper bound for estimating pure-state trace distance and square root fidelity, \ref{['thm:algo-analysis']} restated
  • Definition 2.1: Multi-samplizer for pure states, \ref{['def:samplizer-pure']} restated
  • Theorem 2.2: Implementation of multi-samplizer for pure states, \ref{['thm:pure-state-samplizer']} restated
  • Remark 2.1
  • Theorem 2.3: Optimality of the samplizer for pure states, \ref{['thm:lb-k-samplizer']} restated
  • Theorem 4.1: Quantum phase estimation, NC10
  • Theorem 4.2: KLL+17
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 17 more