



01362nas a2200157   4500008004100000245010600041210006900147260001500216300001400231490000700245520083800252100002301090700001901113700002301132856004901155       2017                        eng d00aQuantum algorithm for systems of linear equations with exponentially improved dependence on precision0 aQuantum algorithm for systems of linear equations with exponenti  c2017/12/21  a1920-19500 v463 a<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>1 aChilds, Andrew, M.1 aKothari, Robin1 aSomma, Rolando, D.  uhttp://epubs.siam.org/doi/10.1137/16M108707201120nas a2200181   4500008004100000245006700041210006700108260001500175300001100190490000800201520058500209100002300794700002300817700001900840700001900859700002300878856003700901       2015                        eng d00aSimulating Hamiltonian dynamics with a truncated Taylor series0 aSimulating Hamiltonian dynamics with a truncated Taylor series  c2015/03/03  a0905020 v1143 a  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.
1 aBerry, Dominic, W.1 aChilds, Andrew, M.1 aCleve, Richard1 aKothari, Robin1 aSomma, Rolando, D.  uhttp://arxiv.org/abs/1412.4687v102001nas a2200181   4500008004100000020002200041245007600063210006900139260001500208300001200223520144000235100002301675700002301698700001901721700001901740700002301759856003701782       2014                        eng d  a978-1-4503-2710-700aExponential improvement in precision for simulating sparse Hamiltonians0 aExponential improvement in precision for simulating sparse Hamil  c2014/05/31  a283-2923 a  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.
1 aBerry, Dominic, W.1 aChilds, Andrew, M.1 aCleve, Richard1 aKothari, Robin1 aSomma, Rolando, D.  uhttp://arxiv.org/abs/1312.1414v2