Simulating Hamiltonian dynamics with a truncated Taylor series

We describe a simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. Our method can simulate the time evolution of a wide variety of physical systems. As in another recent algorithm, the cost of our method depends only logarithmically on the inverse of the desired precision, which is optimal. However, we simplify the algorithm and its analysis by using a method for implementing linear combinations of unitary operations to directly apply the truncated Taylor series.Comment: 5 page

Simulating Hamiltonian dynamics using many-qudit Hamiltonians and local unitary control

When can a quantum system of finite dimension be used to simulate another quantum system of finite dimension? What restricts the capacity of one system to simulate another? In this paper we complete the program of studying what simulations can be done with entangling many-qudit Hamiltonians and local unitary control. By entangling we mean that every qudit is coupled to every other qudit, at least indirectly. We demonstrate that the only class of finite-dimensional entangling Hamiltonians that aren't universal for simulation is the class of entangling Hamiltonians on qubits whose Pauli operator expansion contains only terms coupling an odd number of systems, as identified by Bremner et. al. [Phys. Rev. A, 69, 012313 (2004)]. We show that in all other cases entangling many-qudit Hamiltonians are universal for simulation

Hamiltonian Simulation Using Linear Combinations of Unitary Operations

We present a new approach to simulating Hamiltonian dynamics based on implementing linear combinations of unitary operations rather than products of unitary operations. The resulting algorithm has superior performance to existing simulation algorithms based on product formulas and, most notably, scales better with the simulation error than any known Hamiltonian simulation technique. Our main tool is a general method to nearly deterministically implement linear combinations of nearby unitary operations, which we show is optimal among a large class of methods.Comment: 18 pages, 3 figure

On the relationship between continuous- and discrete-time quantum walk

Quantum walk is one of the main tools for quantum algorithms. Defined by analogy to classical random walk, a quantum walk is a time-homogeneous quantum process on a graph. Both random and quantum walks can be defined either in continuous or discrete time. But whereas a continuous-time random walk can be obtained as the limit of a sequence of discrete-time random walks, the two types of quantum walk appear fundamentally different, owing to the need for extra degrees of freedom in the discrete-time case. In this article, I describe a precise correspondence between continuous- and discrete-time quantum walks on arbitrary graphs. Using this correspondence, I show that continuous-time quantum walk can be obtained as an appropriate limit of discrete-time quantum walks. The correspondence also leads to a new technique for simulating Hamiltonian dynamics, giving efficient simulations even in cases where the Hamiltonian is not sparse. The complexity of the simulation is linear in the total evolution time, an improvement over simulations based on high-order approximations of the Lie product formula. As applications, I describe a continuous-time quantum walk algorithm for element distinctness and show how to optimally simulate continuous-time query algorithms of a certain form in the conventional quantum query model. Finally, I discuss limitations of the method for simulating Hamiltonians with negative matrix elements, and present two problems that motivate attempting to circumvent these limitations.Comment: 22 pages. v2: improved presentation, new section on Hamiltonian oracles; v3: published version, with improved analysis of phase estimatio