Commutator Methods For Quantum Hamiltonians And Dynamical Systems

Qi Zhao, Xiao Yuan

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.

P A M Dirac

Николай Михайлович Небогатиков

A procedure of linearizing quantum brackets (commutator of a dynamic operator with the Hamiltonian function of the system) is considered as an algorithm of terminating an infinite hierarchical system of linked equations of motion. Quantum brackets are introduced as a commutator of a dynamic operator with the Hamiltonian function of the system. This procedure is applied to the study of a configurational model of a binary atomic system using dynamical symmetry of the system. It was revealed that dynamical variables form a closed space with respect to commutation with the Hamiltonian function. This is a necessary and sufficient condition for invariance of a physical system relative to the group of dynamical symmetry of Hamiltonian.

Andrew C. Millard

Shushen Qin, M. Cramer, Christiane P. Koch +1 more

The ability to characterise a Hamiltonian with high precision is crucial for the implementation of quantum technologies. In addition to the well-developed approaches utilising optimal probe states and optimal measurements, the method of optimal control can be used to identify time-dependent pulses applied to the system to achieve higher precision in the estimation of Hamiltonian parameters, especially in the presence of noise. Here, we extend optimally controlled estimation schemes for single qubits to non-commuting dynamics as well as two interacting qubits, demonstrating improvements in terms of maximal precision, time-stability, as well as robustness over uncontrolled protocols.

Joseph V. Lepore