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arXiv:2110.03400

A Cutting-plane Method for Semidefinite Programming with Potential Applications on Noisy Quantum Devices

Jakub Mareček, Albert Akhriev

Published Oct 7, 2021• 1 citation
arXivHTMLPDF

Abstract

There is an increasing interest in quantum algorithms for optimization problems. Within convex optimization, interior-point methods and other recently proposed quantum algorithms are non-trivial to implement on noisy quantum devices. Here, we discuss how to utilize an alternative approach to convex optimization, in general, and semidefinite programming (SDP), in particular. This approach is based on a randomized variant of the cutting-plane method. We show how to leverage quantum speed-up of an eigensolver in speeding up an SDP solver utilizing the cutting-plane method. For the first time, we demonstrate a practical implementation of a randomized variant of the cutting-plane method for semidefinite programming on instances from SDPLIB, a well-known benchmark. Furthermore, we show that the RCP method is very robust to noise in the boundary oracle, which may make RCP suitable for use even on noisy quantum devices.

DOI
10.48550/arxiv.2110.03400

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There is an increasing interest in quantum algorithms for optimization problems. Within convex optimization, interior-point methods and other recently proposed quantum algorithms are non-trivial to implement on noisy quantum devices. Here, we discuss how to utilize an alternative approach to convex optimization, in general, and semidefinite programming (SDP), in particular. This approach is based on a randomized variant of the cutting-plane method. We show how to leverage quantum speed-up of an eigensolver in speeding up an SDP solver utilizing the cutting-plane method. For the first time, we demonstrate a practical implementation of a randomized variant of the cutting-plane method for semidefinite programming on instances from SDPLIB, a well-known benchmark. Furthermore, we show that the RCP method is very robust to noise in the boundary oracle, which may make RCP suitable for use even on noisy quantum devices.
This paper studies a fundamental problem in convex optimization, which is to solve semidefinite programming (SDP) with high accuracy. This paper follows from the existing robust SDP-based interior point method analysis due to [Huang, Jiang, Song, Tao and Zhang, FOCS 2022]. While, the previous work only provides an efficient implementation in the classical setting. This work provides a novel quantum implementation. We give a quantum second-order algorithm with high-accuracy in both the optimality and the feasibility of its output, and its running time depending on log⁡(1/ε)\log(1/ε)log(1/ε) on well-conditioned instances. Due to the limitation of quantum itself or first-order method, all the existing quantum SDP solvers either have polynomial error dependence or low-accuracy in the feasibility.
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.
Semidefinite programming plays a key role in optimization, with broad impact across control theory, machine learning, and combinatorial optimization. Although semidefinite programs are polynomially solvable, several commonly used algorithms rest on a linear-algebraic step whose running time grows cubically with the matrix dimension and which requires the matrix itself to be held in memory, at quadratic cost. In this study, we propose replacing it with a variational quantum eigensolver, whose qubit requirement is logarithmic in the matrix dimension, and present the first end-to-end implementation of such an approach within a cutting-plane framework, together with an operator-derived ansatz whose entanglement structure is read directly from the Pauli support of the candidate matrix. Evaluated on the control family of SDPLIB against an identical scheme driven by an exact eigendecomposition, the variational oracle produces valid cuts throughout, closing 32 to 82% of the initial optimality gap against a near-constant 75 to 82% for the exact oracle. Implementing and measuring the method end to end surfaces several effects not visible from theoretical analyses alone: where memory is actually consumed, how the padding required to fit a matrix onto a quantum register can mislead the variational optimizer, and why the candidate matrices prove dense in the Pauli basis, reducing the operator-derived ansatz to full entanglement. We report these findings and discuss their implications for near-term hybrid quantum-classical approaches.
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.
A semidefinite program (SDP) is a particular kind of convex optimization problem with applications in operations research, combinatorial optimization, quantum information science, and beyond. In this work, we propose variational quantum algorithms for approximately solving SDPs. For one class of SDPs, we provide a rigorous analysis of their convergence to approximate locally optimal solutions, under the assumption that they are weakly constrained (i.e., N ≫ M , where N is the dimension of the input matrices and M is the number of constraints). We also provide algorithms for a more general class of SDPs that requires fewer assumptions. Finally, we numerically simulate our quantum algorithms for applications such as MaxCut, and the results of these simulations provide evidence that convergence still occurs in noisy settings.
Semidefinite programming (SDP) is a fundamental convex optimization problem with wide-ranging applications. However, solving large-scale instances remains computationally challenging due to the high cost of solving linear systems and performing eigenvalue decompositions. In this paper, we present a quantum alternating direction method of multipliers (QADMM) for SDPs, building on recent advances in quantum computing. An inexact ADMM framework is developed, which tolerates errors in the iterates arising from block-encoding approximation and quantum measurement. Within this robust scheme, we design a polynomial proximal operator to address the semidefinite conic constraints and apply the quantum singular value transformation to accelerate the most costly projection updates. We prove that the scheme converges to an ϵ -optimal solution of the SDP problem under the strong duality assumption. A detailed complexity analysis shows that the QADMM algorithm achieves favorable scaling with respect to dimension compared to the classical ADMM algorithm and quantum interior point methods, highlighting its potential for solving large-scale SDPs.