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Error-Tolerant Quantum State Discrimination: Optimization and Quantum Circuit Synthesis

Chien-Kai Ma, Bo-Hung Chen, Tian-Fu Chen, Dah-Wei Chiou, Jie-Hong Roland Jiang

TL;DR

This work addresses robust quantum state discrimination (QSD) under moderate noise by introducing CrossQSD and FitQSD, plus a hybrid objective that blends minimum-error and unambiguous discrimination. All optimization problems are cast as convex programs (SDP/DCP) and solved efficiently, with a circuit-synthesis framework based on a modified Naimark dilation to implement POVMs using minimal ancilla. An open-source toolkit automates problem specification, POVM design, and circuit generation, enabling practical QSD on current quantum hardware. Case studies with coherent and entangled states on noisy simulators demonstrate robust performance and significant resource savings, highlighting the approach’s potential for near-term quantum devices.

Abstract

We develop error-tolerant quantum state discrimination(QSD) strategies that maintain reliable performance under moderate noise. Two complementary approaches are proposed: CrossQSD, which generalizes unambiguous discrimination with tunable confidence bounds to balance accuracy and efficiency, and FitQSD, which optimizes the measurement outcome distribution to approximate that of the ideal noiseless case. Furthermore, we provide a unified hybrid-objective QSD framework that continuously interpolates between minimum-error discrimination (MED) and FitQSD, allowing flexible trade-offs among competing objectives. The associated optimization problems are formulated as convex programs and efficiently solved via disciplined convex programming or, in many cases, semidefinite programming. Additionally, a circuit synthesis framework based on a modified Naimark dilation and isometry synthesis enables hardware-efficient implementations with substantially reduced qubit and gate resources. An open-source toolkit automates the full optimization and synthesis workflow, providing a practical route to QSD on current quantum devices.

Error-Tolerant Quantum State Discrimination: Optimization and Quantum Circuit Synthesis

TL;DR

This work addresses robust quantum state discrimination (QSD) under moderate noise by introducing CrossQSD and FitQSD, plus a hybrid objective that blends minimum-error and unambiguous discrimination. All optimization problems are cast as convex programs (SDP/DCP) and solved efficiently, with a circuit-synthesis framework based on a modified Naimark dilation to implement POVMs using minimal ancilla. An open-source toolkit automates problem specification, POVM design, and circuit generation, enabling practical QSD on current quantum hardware. Case studies with coherent and entangled states on noisy simulators demonstrate robust performance and significant resource savings, highlighting the approach’s potential for near-term quantum devices.

Abstract

We develop error-tolerant quantum state discrimination(QSD) strategies that maintain reliable performance under moderate noise. Two complementary approaches are proposed: CrossQSD, which generalizes unambiguous discrimination with tunable confidence bounds to balance accuracy and efficiency, and FitQSD, which optimizes the measurement outcome distribution to approximate that of the ideal noiseless case. Furthermore, we provide a unified hybrid-objective QSD framework that continuously interpolates between minimum-error discrimination (MED) and FitQSD, allowing flexible trade-offs among competing objectives. The associated optimization problems are formulated as convex programs and efficiently solved via disciplined convex programming or, in many cases, semidefinite programming. Additionally, a circuit synthesis framework based on a modified Naimark dilation and isometry synthesis enables hardware-efficient implementations with substantially reduced qubit and gate resources. An open-source toolkit automates the full optimization and synthesis workflow, providing a practical route to QSD on current quantum devices.
Paper Structure (17 sections, 3 theorems, 37 equations, 10 figures, 4 tables)

This paper contains 17 sections, 3 theorems, 37 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Error-Tolerant Quantum State Discrimination
  3. Modified Minimum Error Discrimination (MED${}^+$)
  4. CrossQSD
  5. FitQSD
  6. Hybrid-objective QSD
  7. Quantum Circuit Synthesis
  8. Toolkit and Demonstration
  9. Problem instance construction
  10. POVM synthesis with an optimization strategy
  11. Quantum circuit synthesis and resynthesis
  12. Case Study: UQSD for Coherent States in Noisy Quantum Circuit Simulators
  13. Proofs
  14. Proof of \ref{['thm:med_medplus']}
  15. Proof of \ref{['corollary']}
  16. ...and 2 more sections

Key Result

theorem thmcountertheorem

Let both MED and MED${}^+$ be applied under the same predefined states $\{\rho_{1}, \rho_{2}, \dots, \rho_{k}\}$ and prior probabilities $\{p_{1}, p_{2}, \dots, p_{k}\}$. Then, in the optimal MED${}^+$ solution, the inconclusive operator necessarily vanishes, i.e., $\Pi_? = 0$, rendering the optimal

Figures (10)

  • Figure 1: CrossQSD: Error-to-success ratio $P_{\mathrm{err}} / P_{\mathrm{succ}}$ vs. noise level $\lambda$. The predefined states are three coherent states truncated to three qubits; $\alpha_i = \beta_i = 0.01$; $\lambda_{\mathrm{eval}}=0.01$.
  • Figure 2: FitQSD: Success probability $P_{\mathrm{succ}}$ (left) and $L_2$ distance (right) vs. noise level $\lambda$. The predefined states are given by \ref{['eq:2-qubits states']}. The MOSEK solver precision is set to $10^{-9}$.
  • Figure 3: Hybrid-objective QSD with $\ell=1$ and the predefined states in \ref{['eq:2-qubits states']}: Success probability $P_{\mathrm{succ}}$ (left) and $L_2$ distance (right) vs. noise level $\lambda$.
  • Figure 4: Hybrid-objective QSD with $\ell=1$ and the predefined states in \ref{['eq:1-qubit states']}: Success probability $P_{\mathrm{succ}}$ (left) and $L_2$ distance (right) vs. noise level $\lambda$.
  • Figure 5: Quantum circuit synthesis workflow for QSD.
  • ...and 5 more figures

Theorems & Definitions (6)

  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • theorem thmcountertheorem
  • proof
  • proof
  • proof