Symmetry enhanced variational quantum spin eigensolver

Chufan Lyu1, Xusheng Xu2, Man-Hong Yung2,3,4, and Abolfazl Bayat1

1Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610051, China
2Central Research Institute, 2012 Labs, Huawei Technologies
3Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
4Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China

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The variational quantum-classical algorithms are the most promising approach for achieving quantum advantage on near-term quantum simulators. Among these methods, the variational quantum eigensolver has attracted a lot of attention in recent years. While it is very effective for simulating the ground state of many-body systems, its generalization to excited states becomes very resource demanding. Here, we show that this issue can significantly be improved by exploiting the symmetries of the Hamiltonian. The improvement is even more effective for higher energy eigenstates. We introduce two methods for incorporating the symmetries. In the first approach, called hardware symmetry preserving, all the symmetries are included in the design of the circuit. In the second approach, the cost function is updated to include the symmetries. The hardware symmetry preserving approach indeed outperforms the second approach. However, integrating all symmetries in the design of the circuit could be extremely challenging. Therefore, we introduce hybrid symmetry preserving method in which symmetries are divided between the circuit and the classical cost function. This allows to harness the advantage of symmetries while preventing sophisticated circuit design.

Quantum simulators are rapidly emerging in various physical platforms. However, the current noisy Intermediate-Scale Quantum (NISQ) simulators suffer from imperfect initialization, noisy operation and faulty readout. Variational quantum algorithms have been proposed as the most promising approach for achieving quantum advantage on NISQ devices. In these algorithms, the complexity is divided between a parameterized quantum simulator and a classical optimizer for optimizing the parameters of the circuit. Therefore, in variational quantum algorithms we deal with both quantum and classical resources, for both of which we have to be efficient. Here, we focus on Variational Quantum Eigensolver (VQE) algorithm, which has been designed to variationally generate the low-energy eigenstates of a many-body system on a quantum simulator. We exploit symmetries of the system to improve resource efficiency in a VQE algorithm. Two methods are investigated: (i) incorporating the symmetries in the design of the circuit that naturally generates quantum states with desired symmetry; and (ii) adding extra terms to the cost function to penalize the quantum states without the relevant symmetry. Through extensive analysis, we show that the first approach is far more resource efficient, with respect to both quantum and classical resources. In realistic scenarios, one may need to use a hybrid scheme in which some symmetries are incorporated in the hardware and some are targeted through the cost function.

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[2] Margarite L. LaBorde and Mark M. Wilde, "Quantum Algorithms for Testing Hamiltonian Symmetry", Physical Review Letters 129 16, 160503 (2022).

[3] Chufan Lyu, Xiaoyu Tang, Junning Li, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat, "Variational quantum simulation of long-range interacting systems", arXiv:2203.14281, (2022).

[4] Raphael César de Souza Pimenta and Anibal Thiago Bezerra, "Revisiting semiconductor bulk hamiltonians using quantum computers", arXiv:2208.10323, (2022).

[5] Arunava Majumder, Dylan Lewis, and Sougato Bose, "Variational Quantum Circuits for Multi-Qubit Gate Automata", arXiv:2209.00139, (2022).

[6] Qingyu Li, Yuhan Huang, Xiaokai Hou, Ying Li, Xiaoting Wang, and Abolfazl Bayat, "Ensemble-learning variational shallow-circuit quantum classifiers", arXiv:2301.12707, (2023).

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