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arXiv:2306.16560 · Journal of Physics A Mathematical and Theoretical

Semi-definite programming and quantum information

Piotr Mironowicz

Published Feb 21, 2024• 23 citations
arXivHTMLPDF

Abstract

Abstract This paper presents a comprehensive exploration of semi-definite programming (SDP) techniques within the context of quantum information. It examines the mathematical foundations of convex optimization, duality, and SDP formulations, providing a solid theoretical framework for addressing optimization challenges in quantum systems. By leveraging these tools, researchers and practitioners can characterize classical and quantum correlations, optimize quantum states, and design efficient quantum algorithms and protocols. The paper also discusses implementational aspects, such as solvers for SDP and modeling tools, enabling the effective employment of optimization techniques in quantum information processing. The insights and methodologies presented in this paper have proven instrumental in advancing the field of quantum information, facilitating the development of novel communication protocols, self-testing methods, and a deeper understanding of quantum entanglement.

DOI
10.1088/1751-8121/ad2b85

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Abstract This paper presents a comprehensive exploration of semi-definite programming (SDP) techniques within the context of quantum information. It examines the mathematical foundations of convex optimization, duality, and SDP formulations, providing a solid theoretical framework for addressing optimization challenges in quantum systems. By leveraging these tools, researchers and practitioners can characterize classical and quantum correlations, optimize quantum states, and design efficient quantum algorithms and protocols. The paper also discusses implementational aspects, such as solvers for SDP and modeling tools, enabling the effective employment of optimization techniques in quantum information processing. The insights and methodologies presented in this paper have proven instrumental in advancing the field of quantum information, facilitating the development of novel communication protocols, self-testing methods, and a deeper understanding of quantum entanglement.
This work is concerned with the issue of applications of the semi-definite programming (SDP) in the field of quantum information science. Our results of the analysis of certain quantum information protocols using this optimization technique are presented, and an implementation of a relevant numerical tool is introduced. The key method used is NPA discovered by Navascues et al. [Phys. Rev. Lett. 98, 010401 (2007)]. In chapter 1 a brief overview of mathematical methods used in this work is presented. In chapter 2 an introduction to quantum information science is given. Chapter 3 concerns the device-independent (DI) and semi-device-independent (SDI) approaches in quantum cryptography. In chapter 4 our results regarding quantum information protocols which we developed using SDP are described. The results include a new type of quantum key distribution protocol based on Hardy's paradox; several protocols for quantum random number expansion; a method of formulation of SDP relaxations of SDI protocols; and a DI Santha-Vazirani source of randomness amplification protocol. In chapter 5 an introduction to interior point methods is given. In chapter 6 the implementation of our numerical tool dedicated to problems occurring in chapter 4 are discussed and compared with other solutions.
As the title indicates, this chapter presents a brief, self-contained introduction to five fundamental problems in Quantum Information Science (QIS) that are especially well-suited to be formulated as Semi-definite Programs (SDP). We have in mind two audiences. The primary audience comprises of Operations Research (and Computer Science) graduate students who have familiarity with SDPs, but have found it daunting to become even minimally conversant with pre-requisites of QIS. The second audience consists of Physicists (and Electrical Engineers) already knowledgeable with modeling of QIS via SDP but interested in computational tools that are applicable more generally. For both audiences, we strive for rapid access to the unfamiliar material. For the first, we provide just enough required background material (from Quantum Mechanics, treated via matrices, and mapping them in Dirac notation) and simultaneously for the second audience we recreate, computationally in Jupyter notebooks, known closed-form solutions. We hope you will enjoy this introduction and gain understanding of the marvelous connection between SDP and QIS by self-study, or as a short seminar course. Ultimately, we hope this disciplinary outreach will fuel advances in QIS through their fruitful study via SDPs.
Sometimes a mathematical tool emerges as uniquely useful and even a defining feature within some branch of physics such as Feynman diagrams. In quantum information theory, the semidefinite program (SDP) has emerged as such a tool. SDP is an optimization task in which a linear objective function is maximized over a set of Hermitian matrices with positive eigenvalues. This review discusses the highly efficient algorithms available for the SDP, and shows how comprehensively the SDP has been deployed in problems of entanglement characterization, quantum nonlocality, quantum channel capacities, and the bounding of ground-state energies.
Semidefinite Programming (SDP) is a class of 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 NP-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 SDP. The classical optimization program of our NISQ solver is another SDP over a smaller 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, can be solved in polynomial time with the number of ansatz parameters and every local minimum is a global minimum. Further, we demonstrate the potential of our NISQ SDP solver by finding the largest eigenvalue of up to 210002^{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.
Abstract In this paper, building on the formulation of quantum Markov decision processes (q-MDPs) presented in our previous work [ N. Saldi, S. Sanjari, and S. Yüksel , Quantum Markov Decision Processes: General Theory, Approximations, and Classes of Policies , SIAM Journal on Control and Optimization, 2024], our focus shifts to the development of semi-definite programming approaches for optimal policies and value functions of both open-loop and classical-state-preserving closed-loop policies. First, by using the duality between the dynamic programming and the semi-definite programming formulations of any q-MDP with open-loop policies, we establish that the optimal value function is linear and there exists a stationary optimal policy among open-loop policies. Then, using these results, we establish a method for computing an approximately optimal value function and formulate computation of optimal stationary open-loop policy as a bi-linear program. Next, we turn our attention to classical-state-preserving closed-loop policies. Dynamic programming and semi-definite programming formulations for classical-state-preserving closed-loop policies are established, where duality of these two formulations similarly enables us to prove that the optimal policy is linear and there exists an optimal stationary classical-state-preserving closed-loop policy. Then, similar to the open-loop case, we establish a method for computing the optimal value function and pose computation of optimal stationary classical-state-preserving closed-loop policies as a bi-linear program.
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.