Abstract
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.
Highlights
It is a notoriously hard problem to classically simulate an arbitrary large many-body physics systems
We explore the anti-commutative relation in Hamiltonian simulation
We find that Anti-commutative relation plays a positive role in the truncated Taylor series method
Summary
It is a notoriously hard problem to classically simulate an arbitrary large many-body physics systems. We consider the product formula methods [32] and ones based on linearcombination-of-unitaries (LCU) [16] They have shown different asymptotical gate complexities with respect to the simulation time t, the accuracy ε, and the property of H. It is intuitively straightforward to see that the commutative relation could simplify or ease the complexity of Hamiltonian simulation whereas the anti-commutative relation could make the problem hard While this seems to be true for the product formula method where Hamiltonians consisting of mutually (anti-)commutative terms have zero (maximal) approximation errors, the intuition breaks down in the LCU based methods. We propose two modified LCU algorithms that exploit the anti-commutative relation and further reduce high-order truncation errors. Our work broadens our understanding of commutative and anti-commutative relations in Hamiltonian simulation
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