%0 Journal Article %J Physical Review Letters %D 2015 %T Simulating Hamiltonian dynamics with a truncated Taylor series %A Dominic W. Berry %A Andrew M. Childs %A Richard Cleve %A Robin Kothari %A Rolando D. Somma %X 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. %B Physical Review Letters %V 114 %P 090502 %8 2015/03/03 %G eng %U http://arxiv.org/abs/1412.4687v1 %N 9 %! Phys. Rev. Lett. %R 10.1103/PhysRevLett.114.090502 %0 Journal Article %J Proceedings of the 46th ACM Symposium on Theory of Computing (STOC 2014) %D 2014 %T Exponential improvement in precision for simulating sparse Hamiltonians %A Dominic W. Berry %A Andrew M. Childs %A Richard Cleve %A Robin Kothari %A Rolando D. Somma %X We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a $d$-sparse Hamiltonian $H$ acting on $n$ qubits can be simulated for time $t$ with precision $\epsilon$ using $O\big(\tau \frac{\log(\tau/\epsilon)}{\log\log(\tau/\epsilon)}\big)$ queries and $O\big(\tau \frac{\log^2(\tau/\epsilon)}{\log\log(\tau/\epsilon)}n\big)$ additional 2-qubit gates, where $\tau = d^2 \|{H}\|_{\max} t$. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuous- and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also simplify the analysis of this conversion, avoiding the need for a complex fault correction procedure. Our simplification relies on a new form of "oblivious amplitude amplification" that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error. %B Proceedings of the 46th ACM Symposium on Theory of Computing (STOC 2014) %P 283-292 %8 2014/05/31 %@ 978-1-4503-2710-7 %G eng %U http://arxiv.org/abs/1312.1414v2 %! Proceedings of the 46th ACM Symposium on Theory of Computing (STOC 2014) %R 10.1145/2591796.2591854 %0 Journal Article %J Theory of Computing %D 2009 %T Discrete-query quantum algorithm for NAND trees %A Andrew M. Childs %A Richard Cleve %A Stephen P. Jordan %A David Yeung %X Recently, Farhi, Goldstone, and Gutmann gave a quantum algorithm for evaluating NAND trees that runs in time O(sqrt(N log N)) in the Hamiltonian query model. In this note, we point out that their algorithm can be converted into an algorithm using O(N^{1/2 + epsilon}) queries in the conventional quantum query model, for any fixed epsilon > 0. %B Theory of Computing %V 5 %P 119 - 123 %8 2009/07/01 %G eng %U http://arxiv.org/abs/quant-ph/0702160v1 %N 1 %! Theory of Comput. %R 10.4086/toc.2009.v005a005 %0 Journal Article %D 2002 %T Exponential algorithmic speedup by quantum walk %A Andrew M. Childs %A Richard Cleve %A Enrico Deotto %A Edward Farhi %A Sam Gutmann %A Daniel A. Spielman %X We construct an oracular (i.e., black box) problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a different technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk efficiently in our oracular setting. We then show how this quantum walk can be used to solve our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve this problem with high probability in subexponential time. %8 2002/09/24 %G eng %U http://arxiv.org/abs/quant-ph/0209131v2 %! Proc. 35th ACM Symposium on Theory of Computing (STOC 2003) %R 10.1145/780542.780552