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arXiv · 2310.07774

Quantum Semidefinite Programming with Thermal Pure Quantum States

Oscar Watts·Yuta Kikuchi·Luuk Coopmans
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Abstract

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 Brand\~{a}o 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 $N=2^{10}$ 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.

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Oscar Watts, Yuta Kikuchi, Luuk Coopmans. 2023-10-11. Quantum Semidefinite Programming with Thermal Pure Quantum States. https://arxiv.org/abs/2310.07774

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