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

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.

💡 Research Summary

Quantum state discrimination (QSD) is a fundamental task in quantum information science, underpinning quantum cryptography, communication, and computing. Traditional QSD strategies—minimum‑error discrimination (MED) and unambiguous discrimination (UQSD)—optimise different trade‑offs between accuracy, confidence, and efficiency, but they assume ideal, noise‑free conditions. In realistic devices, even modest depolarising noise can render these strategies ineffective: UQSD may produce only inconclusive outcomes, while MED’s performance degrades sharply.
The authors address this gap by introducing two novel error‑tolerant QSD methods that explicitly incorporate noise models and allow the user to control permissible error rates.

1. CrossQSD generalises UQSD by adding tunable upper bounds on false‑positive (α_i) and false‑negative (β_i) errors for each hypothesis. The optimisation is formulated as a semidefinite program (SDP) that directly includes the noisy channel 𝔈_λ, enabling the designer to balance accuracy, confidence, and inconclusive‑outcome rate continuously. Numerical experiments on three truncated coherent states (α = 1, e^{‑iπ/3}, e^{‑i2π/3}) show that, provided the assumed noise level λ_est is not underestimated, the error‑to‑success ratio remains low even when the actual noise λ varies over several orders of magnitude.
2. FitQSD takes a different approach: it first solves the ideal (noiseless) UQSD problem to obtain a reference POVM {Π⁰_j}. Then, under the same noisy channel, it seeks a new POVM {Π_j} whose joint outcome probabilities p(E_λ(ρ_i), Π_j) are as close as possible to the reference distribution. Closeness is measured by an L_p‑norm (ℓ = 1 or 2) or by a sum‑of‑squares objective, leading to convex optimisation problems. An alternative formulation, FitQSD‑MECO, maximises the success probability while enforcing upper and lower bounds that keep each individual outcome probability within a neighbourhood of its reference value. Experiments on three two‑qubit entangled states demonstrate that FitQSD can preserve the ideal success probability and distribution shape despite substantial depolarising noise.
   The paper further unifies MED and FitQSD in a hybrid‑objective framework. By introducing a scalar interpolation parameter λ∈