53 research outputs found

### Hamiltonian Simulation by Qubitization

We present the problem of approximating the time-evolution operator
$e^{-i\hat{H}t}$ to error $\epsilon$, where the Hamiltonian $\hat{H}=(\langle
G|\otimes\hat{\mathcal{I}})\hat{U}(|G\rangle\otimes\hat{\mathcal{I}})$ is the
projection of a unitary oracle $\hat{U}$ onto the state $|G\rangle$ created by
another unitary oracle. Our algorithm solves this with a query complexity
$\mathcal{O}\big(t+\log({1/\epsilon})\big)$ 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 $d$-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 $\hat{H}$ is a density matrix. A key
technical result is `qubitization', which uses the controlled version of these
oracles to embed any $\hat{H}$ in an invariant $\text{SU}(2)$ subspace. A large
class of operator functions of $\hat{H}$ can then be computed with optimal
query complexity, of which $e^{-i\hat{H}t}$ is a special case.Comment: 23 pages, 1 figure; v2: updated notation; v3: accepted versio

### Quantum imaging by coherent enhancement

Conventional wisdom dictates that to image the position of fluorescent atoms
or molecules, one should stimulate as much emission and collect as many photons
as possible. That is, in this classical case, it has always been assumed that
the coherence time of the system should be made short, and that the statistical
scaling $\sim1/\sqrt{t}$ defines the resolution limit for imaging time $t$.
However, here we show in contrast that given the same resources, a long
coherence time permits a higher resolution image. In this quantum regime, we
give a procedure for determining the position of a single two-level system, and
demonstrate that the standard errors of our position estimates scale at the
Heisenberg limit as $\sim 1/t$, a quadratic, and notably optimal, improvement
over the classical case.Comment: 4 pages, 4 figue

### Quantum Inference on Bayesian Networks

Performing exact inference on Bayesian networks is known to be #P-hard.
Typically approximate inference techniques are used instead to sample from the
distribution on query variables given the values $e$ of evidence variables.
Classically, a single unbiased sample is obtained from a Bayesian network on
$n$ variables with at most $m$ parents per node in time
$\mathcal{O}(nmP(e)^{-1})$, depending critically on $P(e)$, the probability the
evidence might occur in the first place. By implementing a quantum version of
rejection sampling, we obtain a square-root speedup, taking
$\mathcal{O}(n2^mP(e)^{-\frac12})$ time per sample. We exploit the Bayesian
network's graph structure to efficiently construct a quantum state, a q-sample,
representing the intended classical distribution, and also to efficiently apply
amplitude amplification, the source of our speedup. Thus, our speedup is
notable as it is unrelativized -- we count primitive operations and require no
blackbox oracle queries.Comment: 8 pages, 3 figures. Submitted to PR

### Fixed-point quantum search with an optimal number of queries

Grover's quantum search and its generalization, quantum amplitude
amplification, provide quadratic advantage over classical algorithms for a
diverse set of tasks, but are tricky to use without knowing beforehand what
fraction $\lambda$ of the initial state is comprised of the target states. In
contrast, fixed-point search algorithms need only a reliable lower bound on
this fraction, but, as a consequence, lose the very quadratic advantage that
makes Grover's algorithm so appealing. Here we provide the first version of
amplitude amplification that achieves fixed-point behavior without sacrificing
the quantum speedup. Our result incorporates an adjustable bound on the failure
probability, and, for a given number of oracle queries, guarantees that this
bound is satisfied over the broadest possible range of $\lambda$.Comment: 4 pages plus references, 2 figure

### Finite geometry models of electric field noise from patch potentials in ion traps

We model electric field noise from fluctuating patch potentials on conducting
surfaces by taking into account the finite geometry of the ion trap electrodes
to gain insight into the origin of anomalous heating in ion traps. The scaling
of anomalous heating rates with surface distance, $d$, is obtained for several
generic geometries of relevance to current ion trap designs, ranging from
planar to spheroidal electrodes. The influence of patch size is studied both by
solving Laplace's equation in terms of the appropriate Green's function as well
as through an eigenfunction expansion. Scaling with surface distance is found
to be highly dependent on the choice of geometry and the relative scale between
the spatial extent of the electrode, the ion-electrode distance, and the patch
size. Our model generally supports the $d^{-4}$ dependence currently found by
most experiments and models, but also predicts geometry-driven deviations from
this trend

### Effects of electrode surface roughness on motional heating of trapped ions

Electric field noise is a major source of motional heating in trapped ion
quantum computation. While the influence of trap electrode geometries on
electric field noise has been studied in patch potential and surface adsorbate
models, only smooth surfaces are accounted for by current theory. The effects
of roughness, a ubiquitous feature of surface electrodes, are poorly
understood. We investigate its impact on electric field noise by deriving a
rough-surface Green's function and evaluating its effects on adsorbate-surface
binding energies. At cryogenic temperatures, heating rate contributions from
adsorbates are predicted to exhibit an exponential sensitivity to local surface
curvature, leading to either a large net enhancement or suppression over smooth
surfaces. For typical experimental parameters, orders-of-magnitude variations
in total heating rates can occur depending on the spatial distribution of
absorbates. Through careful engineering of electrode surface profiles, our
results suggests that heating rates can be tuned over orders of magnitudes.Comment: 12 pages, 5 figure

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