We propose a hybrid quantum-classical framework for the unambiguous discrimination of pure states by combining the noisy intermediate-scale quantum semidefinite program solver (NSS) with the reciprocal-state representation. In NSS, a given semidefinite program (SDP) is represented within a subspace spanned by chosen ansatz states, with the required quantities estimated using quantum circuits, and the resulting optimization is performed classically. This makes NSS naturally applicable to discrimination problems involving circuit-prepared states, as finding a discrimination measurement can be formulated as an SDP. By choosing the discriminated states themselves as the ansatz and exploiting the reciprocal-state representation, the general matrix-variable NSS formulation is reduced to an optimization over N scalar variables, where N is the number of discriminated states. The resulting formulation reduces the number of optimization variables and positive-semidefinite constraints compared with the original NSS formulation. We further establish robustness guarantees against Gram matrix estimation errors, including a rescaling procedure that guarantees a valid positive operator-valued measure (POVM) under bounded perturbations. The resulting bounds quantify both the deviation in average success probability and violations of the no-error condition using only the estimated Gram matrix, its SDP solution, and a characterized perturbation bound, without access to the ideal Gram matrix. Numerical simulations validate the derived robustness bounds and characterize finite-shot scaling. We also perform experiments on IBM quantum hardware to estimate pairwise overlaps, and then evaluate the performance of the resulting POVMs.
We propose a hybrid quantum-classical framework for the unambiguous discrimination of pure states by combining the noisy intermediate-scale quantum semidefinite program solver (NSS) with the reciprocal-state representation. In NSS, a given semidefinite program (SDP) is represented within a subspace spanned by chosen ansatz states, with the required quantities estimated using quantum circuits, and the resulting optimization is performed classically. This makes NSS naturally applicable to discrimination problems involving circuit-prepared states, as finding a discrimination measurement can be formulated as an SDP. By choosing the discriminated states themselves as the ansatz and exploiting the reciprocal-state representation, the general matrix-variable NSS formulation is reduced to an optimization over N scalar variables, where N is the number of discriminated states. The resulting formulation reduces the number of optimization variables and positive-semidefinite constraints compared with the original NSS formulation. We further establish robustness guarantees against Gram matrix estimation errors, including a rescaling procedure that guarantees a valid positive operator-valued measure (POVM) under bounded perturbations. The resulting bounds quantify both the deviation in average success probability and violations of the no-error condition using only the estimated Gram matrix, its SDP solution, and a characterized perturbation bound, without access to the ideal Gram matrix. Numerical simulations validate the derived robustness bounds and characterize finite-shot scaling. We also perform experiments on IBM quantum hardware to estimate pairwise overlaps, and then evaluate the performance of the resulting POVMs.
Quantum state discrimination is a fundamental primitive in quantum information processing, underpinning tasks in quantum communication, sensing, and learning. We consider the general Bayes framework, as introduced by Helstrom, for state discrimination when, instead of a classical description of the candidate states, one has access to their \emph{source code}: the quantum circuit that prepares them. We show that the semidefinite program (SDP) for the discrimination problem can be reformulated in terms of the Gram matrix of these states, reducing the SDP variable dimensions from dL to NL, where d is the Hilbert space dimension, N is the number of candidate states, and L is the number of possible guesses. Importantly, we further introduce a quantum pre-processing procedure which efficiently constructs the reduced semidefinite program from the source code, enabling our method to operate directly on quantum data. We consider two applications. First, we characterize the optimal identifications for quantum changepoint problems under several reward structures, including multiple-changepoint settings that were previously computationally inaccessible. Second, we consider a quantum error classification problem and show how our reduction makes it tractable for systems of hundreds of qubits.
Near-term quantum computation holds potential across multiple application domains. However, imperfect preparation and evolution of states due to algorithmic and experimental shortcomings, characteristic in the near-term implementation, would typically result in measurement outcomes deviating from the ideal setting. It is thus crucial for any near-term application to quantify and bound these output errors. We address this need by deriving robustness intervals which are guaranteed to contain the output in the ideal setting. The first type of interval is based on formulating robustness bounds as semidefinite programs, and uses only the first moment and the fidelity to the ideal state. Furthermore, we consider higher statistical moments of the observable and generalize bounds for pure states based on the non-negativity of Gram matrices to mixed states, thus enabling their applicability in the NISQ era where noisy scenarios are prevalent. Finally, we demonstrate our results in the context of the variational quantum eigensolver (VQE) on noisy and noiseless simulations.
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.
The task of quantum state discrimination provides an operational characterization of distinguishability and plays a central role in quantum information science. Since the achievable success probability in quantum state discrimination depends both on the states being discriminated and on the allowed measurements, it is natural to study discrimination under physically motivated measurement constraints. Here, we investigate minimum-error quantum state discrimination under measurements implementable by both fixed and adaptive stabilizer circuits. Given arbitrary stabilizer-state ancillas, we show that fixed stabilizer circuits provide no additional discrimination power, whereas adaptive circuits do. Then, focusing on single-qubit systems, we derive analytical expressions for the success probability across both circuit classes when supplied with arbitrary (non-stabilizer) ancillas, showing that the performance gap between fixed and adaptive circuits persists. We further consider discrimination with non-stabilizerness incorporated directly into the measurement operators. Here, the optimization is formulated as a semidefinite program, and analytical bounds interpolating between the stabilizer and Helstrom limits for qubits are derived. Finally, we illustrate applications in quantum random access codes and bounds on unitary synthesis fidelity with a finite number of magic states.
In this paper we consider the general problem of designing the optimal quantum detection for minimum-error quantum state discrimination. Since it is hard to get an explicit solution and we must resort to numerical methods for the general case. We show the problem can be cast into a semidefinite programming(SDP) problem, we use the theory of SDP to solve it. Based on the idea, we propose a method for minimum-error discrimination by formulating the design of optimal quantum detection as a standard SDP problem; furthermore, based on the optimality conditions for SDP, we derive a set of optimality conditions for quantum detection. It is exemplified that this method is simple to implement and can be used to design optimum detection operators for minimum-error discrimination efficiently.
Many fundamental problems in quantum information theory can be formulated as optimization tasks over quantum states and measurements. While the convex structure of quantum theory often enables powerful semidefinite programming formulations, a major challenge persists: the exponential growth of Hilbert space dimension with system size. This dissertation discusses techniques to address these and related obstacles across several core problems in quantum information. We first study optimization over separable quantum states. Such problems are typically NP-hard and standard semidefinite programming hierarchies suffer from the curse of dimensionality. We introduce a hybrid algorithm that leverages a quantum co-processor to reduce the effective dimension of the problem. While several existing approaches provide only a numerical bound on the optimal value, our method is guaranteed to return an explicit feasible solution: a valid separable state, even in instances where it is not optimal. We apply this framework to approximate the minimum separable energy of many-body Hamiltonians, define an entanglement measure for their ground spaces, and demonstrate numerical scalability to systems of up to 28 qubits. We next consider the general Bayes framework, as introduced by Helstrom, for state discrimination when, instead of a classical description of the candidate states, one has access to their source code: the quantum circuit that prepares them. We show that the typical semidefinite program for the discrimination problem can be reformulated in terms of the Gram matrix of these states, reducing the variable dimensions from dL to NL, where d is the Hilbert space dimension, N is the number of candidate states, and L is the number of possible guesses. Importantly, we further introduce a quantum pre-processing procedure which efficiently constructs the reduced problem from the source code, enabling our method to operate directly on quantum data. We consider two applications. First, we characterize the optimal identifications for quantum changepoint problems under several reward structures whose associated discrimination problems were previously computationally inaccessible. Second, we consider a quantum error classification problem and show how our reduction makes it tractable for systems of hundreds of qubits. We also analyze structural measurement strategies. As a counterpart to the well-known pretty good measurement for minimum-error state discrimination, we introduce the pretty bad measurement for the exclusion variant, in which the goal is to identify a state that was not sent, rather than to identify the one that was, and prove that both are guaranteed to perform at least as well as blind guessing in their respective objectives. Finally, we consider evaluating Boolean functions under severe memory constraints and prove that a simple local "greedy" measurement strategy is globally optimal precisely for affine Boolean functions, while more generally achieving at least the square of the optimal global success probability. Together, these results demonstrate that high-dimensional quantum optimization problems often admit hidden structure enabling dimension reduction, hybrid computation, and provable approximation guarantees.