26,410 research outputs found
Black-box Hamiltonian simulation and unitary implementation
We present general methods for simulating black-box Hamiltonians using
quantum walks. These techniques have two main applications: simulating sparse
Hamiltonians and implementing black-box unitary operations. In particular, we
give the best known simulation of sparse Hamiltonians with constant precision.
Our method has complexity linear in both the sparseness D (the maximum number
of nonzero elements in a column) and the evolution time t, whereas previous
methods had complexity scaling as D^4 and were superlinear in t. We also
consider the task of implementing an arbitrary unitary operation given a
black-box description of its matrix elements. Whereas standard methods for
performing an explicitly specified N x N unitary operation use O(N^2)
elementary gates, we show that a black-box unitary can be performed with
bounded error using O(N^{2/3} (log log N)^{4/3}) queries to its matrix
elements. In fact, except for pathological cases, it appears that most
unitaries can be performed with only O(sqrt{N}) queries, which is optimal.Comment: 19 pages, 3 figures, minor correction
Statistical Properties of Many Particle Eigenfunctions
Wavefunction correlations and density matrices for few or many particles are
derived from the properties of semiclassical energy Green functions. Universal
features of fixed energy (microcanonical) random wavefunction correlation
functions appear which reflect the emergence of the canonical ensemble as the
number of particles approaches infinity. This arises through a little known
asymptotic limit of Bessel functions. Constraints due to symmetries,
boundaries, and collisions between particles can be included.Comment: 13 pages, 4 figure
Stochastic Heisenberg limit: Optimal estimation of a fluctuating phase
The ultimate limits to estimating a fluctuating phase imposed on an optical
beam can be found using the recently derived continuous quantum Cramer-Rao
bound. For Gaussian stationary statistics, and a phase spectrum scaling
asymptotically as 1/omega^p with p>1, the minimum mean-square error in any
(single-time) phase estimate scales as N^{-2(p-1)/(p+1)}, where N is the photon
flux. This gives the usual Heisenberg limit for a constant phase (as the limit
p--> infinity) and provides a stochastic Heisenberg limit for fluctuating
phases. For p=2 (Brownian motion), this limit can be attained by phase
tracking.Comment: 5+4 pages, to appear in Physical Review Letter
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
Adaptive Quantum Measurements of a Continuously Varying Phase
We analyze the problem of quantum-limited estimation of a stochastically
varying phase of a continuous beam (rather than a pulse) of the electromagnetic
field. We consider both non-adaptive and adaptive measurements, and both dyne
detection (using a local oscillator) and interferometric detection. We take the
phase variation to be \dot\phi = \sqrt{\kappa}\xi(t), where \xi(t) is
\delta-correlated Gaussian noise. For a beam of power P, the important
dimensionless parameter is N=P/\hbar\omega\kappa, the number of photons per
coherence time. For the case of dyne detection, both continuous-wave (cw)
coherent beams and cw (broadband) squeezed beams are considered. For a coherent
beam a simple feedback scheme gives good results, with a phase variance \simeq
N^{-1/2}/2. This is \sqrt{2} times smaller than that achievable by nonadaptive
(heterodyne) detection. For a squeezed beam a more accurate feedback scheme
gives a variance scaling as N^{-2/3}, compared to N^{-1/2} for heterodyne
detection. For the case of interferometry only a coherent input into one port
is considered. The locally optimal feedback scheme is identified, and it is
shown to give a variance scaling as N^{-1/2}. It offers a significant
improvement over nonadaptive interferometry only for N of order unity.Comment: 11 pages, 6 figures, journal versio
Surprises in the suddenly-expanded infinite well
I study the time-evolution of a particle prepared in the ground state of an
infinite well after the latter is suddenly expanded. It turns out that the
probability density shows up quite a surprising behaviour:
for definite times, {\it plateaux} appear for which is
constant on finite intervals for . Elements of theoretical explanation are
given by analyzing the singular component of the second derivative
. Analytical closed expressions are obtained for some
specific times, which easily allow to show that, at these times, the density
organizes itself into regular patterns provided the size of the box in large
enough; more, above some critical time-dependent size, the density patterns are
independent of the expansion parameter. It is seen how the density at these
times simply results from a construction game with definite rules acting on the
pieces of the initial density.Comment: 24 pages, 14 figure
Superconductor-proximity effect in chaotic and integrable billiards
We explore the effects of the proximity to a superconductor on the level
density of a billiard for the two extreme cases that the classical motion in
the billiard is chaotic or integrable. In zero magnetic field and for a uniform
phase in the superconductor, a chaotic billiard has an excitation gap equal to
the Thouless energy. In contrast, an integrable (rectangular or circular)
billiard has a reduced density of states near the Fermi level, but no gap. We
present numerical calculations for both cases in support of our analytical
results. For the chaotic case, we calculate how the gap closes as a function of
magnetic field or phase difference.Comment: 4 pages, RevTeX, 2 Encapsulated Postscript figures. To be published
by Physica Scripta in the proceedings of the "17th Nordic Semiconductor
Meeting", held in Trondheim, June 199
The quantum Bell-Ziv-Zakai bounds and Heisenberg limits for waveform estimation
We propose quantum versions of the Bell-Ziv-Zakai lower bounds on the error
in multiparameter estimation. As an application we consider measurement of a
time-varying optical phase signal with stationary Gaussian prior statistics and
a power law spectrum , with . With no other
assumptions, we show that the mean-square error has a lower bound scaling as
, where is the time-averaged mean photon
flux. Moreover, we show that this accuracy is achievable by sampling and
interpolation, for any . This bound is thus a rigorous generalization of
the Heisenberg limit, for measurement of a single unknown optical phase, to a
stochastically varying optical phase.Comment: 18 pages, 6 figures, comments welcom
Optimal Heisenberg-style bounds for the average performance of arbitrary phase estimates
The ultimate bound to the accuracy of phase estimates is often assumed to be
given by the Heisenberg limit. Recent work seemed to indicate that this bound
can be violated, yielding measurements with much higher accuracy than was
previously expected. The Heisenberg limit can be restored as a rigorous bound
to the accuracy provided one considers the accuracy averaged over the possible
values of the unknown phase, as we have recently shown [Phys. Rev. A 85,
041802(R) (2012)]. Here we present an expanded proof of this result together
with a number of additional results, including the proof of a previously
conjectured stronger bound in the asymptotic limit. Other measures of the
accuracy are examined, as well as other restrictions on the generator of the
phase shifts. We provide expanded numerical results for the minimum error and
asymptotic expansions. The significance of the results claiming violation of
the Heisenberg limit is assessed, followed by a detailed discussion of the
limitations of the Cramer-Rao bound.Comment: 22 pages, 4 figure
Calculation of the Aharonov-Bohm wave function
A calculation of the Aharonov-Bohm wave function is presented. The result is
a series of confluent hypergeometric functions which is finite at the forward
direction.Comment: 12 pages in LaTeX, and 3 PostScript figure
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