



@article {1605,
	title = {Quantum algorithm for systems of linear equations with exponentially improved dependence on precision},
	journal = {SIAM Journal on Computing},
	volume = {46},
	year = {2017},
	month = {2017/12/21},
	pages = {1920-1950},
	abstract = {<p>Harrow, Hassidim, and Lloyd showed that for a suitably specified N\&times;N matrix A and N-dimensional vector b⃗ , there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations Ax⃗ =b⃗ . If A is sparse and well-conditioned, their algorithm runs in time poly(logN,1/ϵ), where ϵ is the desired precision in the output state. We improve this to an algorithm whose running time is polynomial in log(1/ϵ), exponentially improving the dependence on precision while keeping essentially the same dependence on other parameters. Our algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation. This allows us to bypass the quantum phase estimation algorithm, whose dependence on ϵ is prohibitive.</p>},
	doi = {10.1137/16M1087072},
	url = {http://epubs.siam.org/doi/10.1137/16M1087072},
	author = {Andrew M. Childs and Robin Kothari and Rolando D. Somma}
}
@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}
}
