  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.
