Exploiting anticommutation in Hamiltonian simulation

Qi Zhao1 and Xiao Yuan2,3

1Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, Maryland 20742, USA
2Center on Frontiers of Computing Studies, Department of Computer Science, Peking University, Beijing 100871, China
3Stanford Institute for Theoretical Physics, Stanford University, Stanford California 94305, USA

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Quantum computing can efficiently simulate Hamiltonian dynamics of many-body quantum physics, a task that is generally intractable with classical computers. The hardness lies at the ubiquitous anti-commutative relations of quantum operators, in corresponding with the notorious negative sign problem in classical simulation. Intuitively, Hamiltonians with more commutative terms are also easier to simulate on a quantum computer, and anti-commutative relations generally cause more errors, such as in the product formula method. Here, we theoretically explore the role of anti-commutative relation in Hamiltonian simulation. We find that, contrary to our intuition, anti-commutative relations could also reduce the hardness of Hamiltonian simulation. Specifically, Hamiltonians with mutually anti-commutative terms are easy to simulate, as what happens with ones consisting of mutually commutative terms. Such a property is further utilized to reduce the algorithmic error or the gate complexity in the truncated Taylor series quantum algorithm for general problems. Moreover, we propose two modified linear combinations of unitaries methods tailored for Hamiltonians with different degrees of anti-commutation. We numerically verify that the proposed methods exploiting anti-commutative relations could significantly improve the simulation accuracy of electronic Hamiltonians. Our work sheds light on the roles of commutative and anti-commutative relations in simulating quantum systems.

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Cited by

[1] John M. Martyn, Yuan Liu, Zachary E. Chin, and Isaac L. Chuang, "Efficient fully-coherent quantum signal processing algorithms for real-time dynamics simulation", The Journal of Chemical Physics 158 2, 024106 (2023).

[2] Guang Hao Low, Yuan Su, Yu Tong, and Minh C. Tran, "Complexity of Implementing Trotter Steps", PRX Quantum 4 2, 020323 (2023).

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[4] Guang Hao Low, Yuan Su, Yu Tong, and Minh C. Tran, "On the complexity of implementing Trotter steps", arXiv:2211.09133, (2022).

[5] John M. Martyn, Yuan Liu, Zachary E. Chin, and Isaac L. Chuang, "Efficient Fully-Coherent Quantum Signal Processing Algorithms for Real-Time Dynamics Simulation", arXiv:2110.11327, (2021).

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