Differentiable matrix product states for simulating variational quantum computational chemistry

Chu Guo1, Yi Fan2, Zhiqian Xu3, and Honghui Shang4

1Henan Key Laboratory of Quantum Information and Cryptography, Zhengzhou, Henan 450000, China
2Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
3Institute of Computing Technology, Chinese Academy of Sciences, Beijing
4Key Laboratory of Precision and Intelligent Chemistry,University of Science and Technology of China, Hefei,Anhui 230026, China

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Quantum Computing is believed to be the ultimate solution for quantum chemistry problems. Before the advent of large-scale, fully fault-tolerant quantum computers, the variational quantum eigensolver (VQE) is a promising heuristic quantum algorithm to solve real world quantum chemistry problems on near-term noisy quantum computers. Here we propose a highly parallelizable classical simulator for VQE based on the matrix product state representation of quantum state, which significantly extend the simulation range of the existing simulators. Our simulator seamlessly integrates the quantum circuit evolution into the classical auto-differentiation framework, thus the gradients could be computed efficiently similar to the classical deep neural network, with a scaling that is independent of the number of variational parameters. As applications, we use our simulator to study commonly used small molecules such as HF, HCl, LiH and H$_2$O, as well as larger molecules CO$_2$, BeH$_2$ and H$_4$ with up to $40$ qubits. The favorable scaling of our simulator against the number of qubits and the number of parameters could make it an ideal testing ground for near-term quantum algorithms and a perfect benchmarking baseline for oncoming large scale VQE experiments on noisy quantum computers.

Quantum computation technologies have made enormous progress in recent years, and quantum chemistry combined with the variational quantum eigensolver is a promising candidate to realize practical quantum advantages. State of the art VQE simulator, the state-vector simulator is memory bounded and current simulations are limited within 28 qubits. We propose a differentiable MPS simulator which largely overcomes this barrier, by taking advantages from both the matrix product states tool from quantum many-body physics and the classical automatic differentiation framework. Differentiable calculations of real chemical systems with up to 40 qubits are demonstrated. Our work thus provides a timely and scalable test ground for researchers in both quantum computing and quantum chemistry.

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[3] Marcel Niedermeier, Jose L. Lado, and Christian Flindt, "Tensor-Network Simulations of Noisy Quantum Computers", arXiv:2304.01751, (2023).

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