



@article {1265,
	title = {Simulating Hamiltonian dynamics with a truncated Taylor series},
	journal = {Physical Review Letters},
	volume = {114},
	year = {2015},
	month = {2015/03/03},
	pages = {090502},
	abstract = {  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.
},
	doi = {10.1103/PhysRevLett.114.090502},
	url = {http://arxiv.org/abs/1412.4687v1},
	author = {Dominic W. Berry and Andrew M. Childs and Richard Cleve and Robin Kothari and Rolando D. Somma}
}
@article {1264,
	title = {Exponential improvement in precision for simulating sparse Hamiltonians},
	journal = {Proceedings of the 46th ACM Symposium on Theory of Computing (STOC 2014)},
	year = {2014},
	month = {2014/05/31},
	pages = {283-292},
	abstract = {  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.
},
	isbn = {978-1-4503-2710-7},
	doi = {10.1145/2591796.2591854},
	url = {http://arxiv.org/abs/1312.1414v2},
	author = {Dominic W. Berry and Andrew M. Childs and Richard Cleve and Robin Kothari and Rolando D. Somma}
}
@article {1254,
	title = {Discrete-query quantum algorithm for NAND trees},
	journal = {Theory of Computing},
	volume = {5},
	year = {2009},
	month = {2009/07/01},
	pages = {119 - 123},
	abstract = {  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.
},
	doi = {10.4086/toc.2009.v005a005},
	url = {http://arxiv.org/abs/quant-ph/0702160v1},
	author = {Andrew M. Childs and Richard Cleve and Stephen P. Jordan and David Yeung}
}
@article {1263,
	title = {Exponential algorithmic speedup by quantum walk},
	year = {2002},
	month = {2002/09/24},
	abstract = {  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.
},
	doi = {10.1145/780542.780552},
	url = {http://arxiv.org/abs/quant-ph/0209131v2},
	author = {Andrew M. Childs and Richard Cleve and Enrico Deotto and Edward Farhi and Sam Gutmann and Daniel A. Spielman}
}
