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arXiv:2106.03891 · Physical Review A

Noisy intermediate-scale quantum algorithm for semidefinite programming

Kishor Bharti, Tobias Haug, Vlatko Vedral, L. C. Kwek

Published May 31, 2022• 31 citations
arXivHTMLPDF

Abstract

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 ${2}^{1000}$ 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.

DOI
10.1103/physreva.105.052445

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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.
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.
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.
This study explores the application of variational quantum algorithms to solve NP-hard combinatorial optimization problems, focusing specifically on the MaxBisection problem. The research introduces an enhanced hybrid quantum-classical framework, HTAAC-QSDP, which combines problem-informed ansatz design with semidefinite programming (SDP) relaxation and iterative optimization. Simulation results across multiple graph instances show that the proposed method achieves approximation ratios consistently above 0.95, approaching classical SDP benchmarks. Experiments also demonstrate the sensitivity of performance to key hyperparameters such as circuit depth, rotation scaling, and learning rate. Additionally, hardware implementation on IBM quantum devices reveals the model’s resilience under noise, provided that initialization and optimization strategies are carefully tuned. The findings suggest that variational quantum approaches can provide high-quality approximations for complex NP-hard problems, even in the noisy intermediate-scale quantum (NISQ) era. The results lay the groundwork for scalable quantum optimization frameworks and support the advancement of quantum advantage in real-world applications.
Solving combinatorial optimization problems using variational quantum algorithms (VQAs) might be a promise application in the NISQ era. However, the limited trainability of VQAs could hinder their scalability to large problem sizes. In this paper, we improve the trainability of variational quantum eigensolver (VQE) by utilizing convex interpolation to solve portfolio optimization. Based on convex interpolation, the location of the ground state can be evaluated by learning the property of a small subset of basis states in the Hilbert space. This enlightens naturally the proposals of the strategies of close-to-solution initialization, regular cost function landscape, and recursive ansatz equilibrium partition. The successfully implementation of a 404040-qubit experiment using only 101010 superconducting qubits demonstrates the effectiveness of our proposals. Furthermore, the quantum inspiration has also spurred the development of a prototype greedy algorithm. Extensive numerical simulations indicate that the hybridization of VQE and greedy algorithms achieves a mutual complementarity, combining the advantages of both global and local optimization methods. Our proposals can be extended to improve the trainability for solving other large-scale combinatorial optimization problems that are widely used in real applications, paving the way to unleash quantum advantages of NISQ computers in the near future.
Semidefinite programs (SDPs) are a particular class of convex optimization problems with applications in combinatorial optimization, operational research, and quantum information science. Seminal work by Brandão and Svore shows that a ``quantization'' of the matrix multiplicative-weight algorithm can provide approximate solutions to SDPs quadratically faster than the best classical algorithms by using a quantum computer as a Gibbs-state sampler. We propose a modification of this quantum algorithm and show that a similar speedup can be obtained by replacing the Gibbs-state sampler with the preparation of thermal pure quantum (TPQ) states. While our methodology incurs an additional problem-dependent error, which decreases as the problem size grows, it avoids the preparation of purified Gibbs states, potentially saving a number of ancilla qubits. In addition, we identify a spectral condition which, when met, reduces the resources further, and shifts the computational bottleneck from Gibbs state preparation to ground-state energy estimation. With classical state-vector simulations, we verify the efficiency of the algorithm for particular cases of Hamiltonian learning problems. We are able to obtain approximate solutions for two-dimensional spinless Hubbard and one-dimensional Heisenberg XXZ models for sizes of up to N=210N=2^{10}N=210 variables. For the Hubbard model, we provide an estimate of the resource requirements of our algorithm, including the number of Toffoli gates and the number of qubits.
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.