Exponential improvement in precision for simulating sparse Hamiltonians

TitleExponential improvement in precision for simulating sparse Hamiltonians
Publication TypeJournal Article
Year of Publication2014
AuthorsBerry, DW, Childs, AM, Cleve, R, Kothari, R, Somma, RD
JournalProceedings of the 46th ACM Symposium on Theory of Computing (STOC 2014)
Date Published2014/05/31
ISBN Number978-1-4503-2710-7

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

Short TitleProceedings of the 46th ACM Symposium on Theory of Computing (STOC 2014)