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A generalized circuit for the Hamiltonian dynamics through the truncated series. (English) Zbl 1402.81085

Summary: In this paper, we present a method for Hamiltonian simulation in the context of eigenvalue estimation problems, which improves earlier results dealing with Hamiltonian simulation through the truncated Taylor series. In particular, we present a fixed-quantum circuit design for the simulation of the Hamiltonian dynamics, \(\mathcal H(t)\), through the truncated Taylor series method described by D. W. Berry et al. [Phys. Rev. Lett. 114, No. 9, Article ID 090502, 5 p. (2015; doi:10.1103/PhysRevLett.114.090502)]. The circuit is general and can be used to simulate any given matrix in the phase estimation algorithm by only changing the angle values of the quantum gates implementing the time variable \(t\) in the series. The circuit complexity depends on the number of summation terms composing the Hamiltonian and requires \(O(Ln)\) number of quantum gates for the simulation of a molecular Hamiltonian. Here, \(n\) is the number of states of a spin orbital, and \(L\) is the number of terms in the molecular Hamiltonian and generally is bounded by \(O(n^4)\). We also discuss how to use the circuit in adaptive processes and eigenvalue-related problems along with a slightly modified version of the iterative phase estimation algorithm. In addition, a simple divide-and-conquer method is presented for mapping a matrix which are not given as sums of unitary matrices into the circuit. The complexity of the circuit is directly related to the structure of the matrix and can be bounded by \(O(\text{poly}(n))\) for a matrix with \(\text{poly}(n)\)-sparsity.

MSC:

81P68 Quantum computation
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81V55 Molecular physics
35P15 Estimates of eigenvalues in context of PDEs
68U20 Simulation (MSC2010)
68Q12 Quantum algorithms and complexity in the theory of computing
94C05 Analytic circuit theory
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
47C05 Linear operators in algebras
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References:

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