791 research outputs found
Exponential improvement in precision for simulating sparse Hamiltonians
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
-sparse Hamiltonian acting on qubits can be simulated for time
with precision using queries and
additional 2-qubit gates, where . 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.Comment: v1: 27 pages; Subsumes and improves upon results in arXiv:1308.5424.
v2: 28 pages, minor change
Parallel Quantum Algorithm for Hamiltonian Simulation
We study how parallelism can speed up quantum simulation. A parallel quantum
algorithm is proposed for simulating the dynamics of a large class of
Hamiltonians with good sparse structures, called uniform-structured
Hamiltonians, including various Hamiltonians of practical interest like local
Hamiltonians and Pauli sums. Given the oracle access to the target sparse
Hamiltonian, in both query and gate complexity, the running time of our
parallel quantum simulation algorithm measured by the quantum circuit depth has
a doubly (poly-)logarithmic dependence
on the simulation precision . This presents an exponential
improvement over the dependence of
previous optimal sparse Hamiltonian simulation algorithm without parallelism.
To obtain this result, we introduce a novel notion of parallel quantum walk,
based on Childs' quantum walk. The target evolution unitary is approximated by
a truncated Taylor series, which is obtained by combining these quantum walks
in a parallel way. A lower bound is
established, showing that the -dependence of the gate depth achieved
in this work cannot be significantly improved.
Our algorithm is applied to simulating three physical models: the Heisenberg
model, the Sachdev-Ye-Kitaev model and a quantum chemistry model in second
quantization. By explicitly calculating the gate complexity for implementing
the oracles, we show that on all these models, the total gate depth of our
algorithm has a dependence in the
parallel setting.Comment: Minor revision. 55 pages, 6 figures, 1 tabl
Hamiltonian simulation with nearly optimal dependence on all parameters
We present an algorithm for sparse Hamiltonian simulation whose complexity is
optimal (up to log factors) as a function of all parameters of interest.
Previous algorithms had optimal or near-optimal scaling in some parameters at
the cost of poor scaling in others. Hamiltonian simulation via a quantum walk
has optimal dependence on the sparsity at the expense of poor scaling in the
allowed error. In contrast, an approach based on fractional-query simulation
provides optimal scaling in the error at the expense of poor scaling in the
sparsity. Here we combine the two approaches, achieving the best features of
both. By implementing a linear combination of quantum walk steps with
coefficients given by Bessel functions, our algorithm's complexity (as measured
by the number of queries and 2-qubit gates) is logarithmic in the inverse
error, and nearly linear in the product of the evolution time, the
sparsity, and the magnitude of the largest entry of the Hamiltonian. Our
dependence on the error is optimal, and we prove a new lower bound showing that
no algorithm can have sublinear dependence on .Comment: 21 pages, corrects minor error in Lemma 7 in FOCS versio
Simulating Quantum Dynamics On A Quantum Computer
We present efficient quantum algorithms for simulating time-dependent
Hamiltonian evolution of general input states using an oracular model of a
quantum computer. Our algorithms use either constant or adaptively chosen time
steps and are significant because they are the first to have time-complexities
that are comparable to the best known methods for simulating time-independent
Hamiltonian evolution, given appropriate smoothness criteria on the Hamiltonian
are satisfied. We provide a thorough cost analysis of these algorithms that
considers discretizion errors in both the time and the representation of the
Hamiltonian. In addition, we provide the first upper bounds for the error in
Lie-Trotter-Suzuki approximations to unitary evolution operators, that use
adaptively chosen time steps.Comment: Paper modified from previous version to enhance clarity. Comments are
welcom
Efficient quantum algorithms for simulating sparse Hamiltonians
We present an efficient quantum algorithm for simulating the evolution of a
sparse Hamiltonian H for a given time t in terms of a procedure for computing
the matrix entries of H. In particular, when H acts on n qubits, has at most a
constant number of nonzero entries in each row/column, and |H| is bounded by a
constant, we may select any positive integer such that the simulation
requires O((\log^*n)t^{1+1/2k}) accesses to matrix entries of H. We show that
the temporal scaling cannot be significantly improved beyond this, because
sublinear time scaling is not possible.Comment: 9 pages, 2 figures, substantial revision
Hamiltonian Simulation by Qubitization
We present the problem of approximating the time-evolution operator
to error , where the Hamiltonian is the
projection of a unitary oracle onto the state created by
another unitary oracle. Our algorithm solves this with a query complexity
to both oracles that is optimal
with respect to all parameters in both the asymptotic and non-asymptotic
regime, and also with low overhead, using at most two additional ancilla
qubits. This approach to Hamiltonian simulation subsumes important prior art
considering Hamiltonians which are -sparse or a linear combination of
unitaries, leading to significant improvements in space and gate complexity,
such as a quadratic speed-up for precision simulations. It also motivates
useful new instances, such as where is a density matrix. A key
technical result is `qubitization', which uses the controlled version of these
oracles to embed any in an invariant subspace. A large
class of operator functions of can then be computed with optimal
query complexity, of which is a special case.Comment: 23 pages, 1 figure; v2: updated notation; v3: accepted versio
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