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Techniques For Computing and Approximating Large Optimization Problems in Quantum Information

Mohan, Ankith

Published Aug 18, 2026• 0 citations
OpenAlex

Abstract

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.

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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.
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 dLdLdL to NLNLNL, where ddd is the Hilbert space dimension, NNN is the number of candidate states, and LLL 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.
This thesis focuses on the intersection of mathematical and computational optimization and quantum information. Main contributions are open-source software code: A hybrid approach mixing "traditional" nonconvex and convex methods can make difficult problems more accessible. A demonstration of how to efficiently implement such an algorithm, avoiding interfacial bottlenecks, is provided, finding optimal protocols to establish entanglement through a lossy channel. The central software package developed addresses polynomial optimization problems. Many problems naturally involve only a polynomial objective and constraint polynomials. Such problems can automatically be cast into semidefinite programs that provide a hierarchy of outer approximations. The resulting problems are often so large and scale so unfavorably with respect to the variable number and degree involved that the boundary of the doable is reached quickly. However, technical progress both in hardware and algorithms has pushed this boundary - but software frameworks for polynomial optimization have not followed in the same manner, often now making them the bottleneck that before was the solver. The package PolynomialOptimization.jl developed during this thesis aims to fill the gap and provide a very resource-efficient intermediate layer together with a wide number of algorithms to reduce the problem size, and naturally supporting complex numbers and semidefinite constraints ubiquitous in quantum information problems. Its application on an entanglement distribution problem is demonstrated, showing that even relaxations with semidefinite matrices of three- and four-digit size can be solved conveniently. Finally, a new way to calculate interior-point barriers for the cone of sums-of-squares matrices in a nearly time-optimal way is developed, whose efficient implementation has the potential of further reducing resource consumption.
Solving optimization problems is a key task for which quantum computers could possibly provide a speedup over the best known classical algorithms. Particular classes of optimization problems including semidefinite programming (SDP) and linear programming (LP) have wide applicability in many domains of computer science, engineering, mathematics, and physics. Here we focus on semidefinite and linear programs for which the dimensions of the variables involved are exponentially large, so that standard classical SDP and LP solvers are not helpful for such large-scale problems. We propose the QSlack and CSlack methods for estimating their optimal values, respectively, which work by (1) introducing slack variables to transform inequality constraints to equality constraints, (2) transforming a constrained optimization to an unconstrained one via the penalty method, and (3) replacing the optimizations over all possible nonnegative variables by optimizations over parameterized quantum states and parameterized probability distributions. Under the assumption that the SDP and LP inputs are efficiently measurable observables, it follows that all terms in the resulting objective functions are efficiently estimable by either a quantum computer in the SDP case or a quantum or probabilistic computer in the LP case. Furthermore, by making use of SDP and LP duality theory, we prove that these methods provide a theoretical guarantee that, if one could find global optima of the objective functions, then the resulting values sandwich the true optimal values from both above and below. Finally, we showcase the QSlack and CSlack methods on a variety of example optimization problems and discuss details of our implementation, as well as the resulting performance. We find that our implementations of both the primal and dual for these problems approach the ground truth, typically achieving errors of the order 10-2.
Many problems in information theory can be reduced to optimizations over matrices, where the rank of the matrices is constrained.We establish a link between rank-constrained optimization and the theory of quantum entanglement.More precisely, we prove that a large class of rank-constrained semidefinite programs can be written as a convex optimization over separable quantum states and, consequently, we construct a complete hierarchy of semidefinite programs for solving the original problem.This hierarchy not only provides a sequence of certified bounds for the rank-constrained optimization problem, but also gives pretty good and often exact values in practice when the lowest level of the hierarchy is considered.We demonstrate that our approach can be used for relevant problems in quantum information processing, such as the optimization over pure states, the characterization of mixed unitary channels and faithful entanglement, and quantum contextuality, as well as in classical information theory including the maximum cut problem, pseudo-Boolean optimization, and the orthonormal representation of graphs.Finally, we show that our ideas can be extended to rank-constrained quadratic and higher-order programming.
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.
Semidefinite programs (SDPs) are convex optimization programs with vast applications in control theory, quantum information, combinatorial optimization, and operational research. Noisy intermediate-scale quantum (NISQ) algorithms aim to make an efficient use of the current generation of quantum hardware. However, optimizing variational quantum algorithms is a challenge as it is an nondeterministic polynomial time-hard problem that in general requires an exponential time to solve and can contain many far from optimal local minima. Here, we present a current term NISQ algorithm for solving SDPs. The classical optimization program of our NISQ solver is another SDP over a lower dimensional ansatz space. We harness the SDP-based formulation of the Hamiltonian ground-state problem to design a NISQ eigensolver. Unlike variational quantum eigensolvers, the classical optimization program of our eigensolver is convex and can be solved in polynomial time with the number of ansatz parameters, and every local minimum is a global minimum. We find numeric evidence that NISQ SDP can improve the estimation of ground-state energies in a scalable manner. Further, we efficiently solve constrained problems to calculate the excited states of Hamiltonians, find the lowest energy of symmetry constrained Hamiltonians, and determine the optimal measurements for quantum state discrimination. We demonstrate the potential of our approach by finding the largest eigenvalue of up to 21000{2}^{1000}21000 dimensional matrices and solving graph problems related to quantum contextuality. We also discuss NISQ algorithms for rank-constrained SDPs. Our work extends the application of NISQ computers onto one of the most successful algorithmic frameworks of the past few decades.