2,787 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
Influence of suction on shock wave-turbulent boundary layer interactions for two- dimensional and axially symmetric flows, 16 September 1967 - 30 June 1969
Influence of suction on shock wave turbulent boundary layer interactions for two dimensional and axially symmetric flow
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
Reversible simulation of bipartite product Hamiltonians
Consider two quantum systems A and B interacting according to a product
Hamiltonian H = H_A x H_B. We show that any two such Hamiltonians can be used
to simulate each other reversibly (i.e., without efficiency losses) with the
help of local unitary operations and local ancillas. Accordingly, all non-local
features of a product Hamiltonian -- including the rate at which it can be used
to produce entanglement, transmit classical or quantum information, or simulate
other Hamiltonians -- depend only upon a single parameter. We identify this
parameter and use it to obtain an explicit expression for the entanglement
capacity of all product Hamiltonians. Finally, we show how the notion of
simulation leads to a natural formulation of measures of the strength of a
nonlocal Hamiltonian.Comment: 10 page
Asymptotic entanglement capacity of the Ising and anisotropic Heisenberg interactions
We compute the asymptotic entanglement capacity of the Ising interaction ZZ,
the anisotropic Heisenberg interaction XX + YY, and more generally, any
two-qubit Hamiltonian with canonical form K = a XX + b YY. We also describe an
entanglement assisted classical communication protocol using the Hamiltonian K
with rate equal to the asymptotic entanglement capacity.Comment: 5 pages, 1 figure; minor corrections, conjecture adde
Wave Profile for Current Bearing Antiforce Waves
For fluid dynamical analysis of breakdown waves, we employ a one-dimensional, three-component (electrons, ions and neutral particles) fluid model to describe a steady-state, ionizing wave propagating counter to strong electric fields. The electron gas temperature and therefore the electron fluid pressure is assumed to be large enough to sustain the wave motion down the discharge tube. Such waves are referred to as antiforce waves. The complete set of equations describing such waves consists of the equations of conservation of mass, momentum and energy coupled with Poisson’s equation. Inclusion of current behind the wave front alters the set of electron fluid dynamical equations and also the boundary condition on electron temperature. For a range of experimentally observed current values, using the modified boundary condition on electron temperature, we have been able to integrate our modified set of electron fluid dynamical equations through the Debye layer. Our solutions meet the expected boundary conditions at the trailing edge of the wave. We present the wave profile for electric field, electron velocity, electron number density and electron temperature within the Debye layer of the wave
Wave Profile for Anti-force Waves with Maximum Possible Currents
In the theoretical investigation of the electrical breakdown of a gas, we apply a one-dimensional, steady state, constant velocity, three component fluid model and consider the electrons to be the main element in propagation of the wave. The electron gas temperature, and therefore the electron gas partial pressure, is considered to be large enough to provide the driving force. The wave is considered to have a shock front, followed by a thin dynamical transition region. Our set of electron fluid-dynamical equations consists of the equations of conservation of mass, momentum, and energy, plus the Poisson\u27s equation. The set of equations is referred to as the electron fluid dynamical equations; and a successful solution therefor must meet a set of acceptable physical conditions at the trailing edge of the wave. For breakdown waves with a significant current behind the shock front, modifications must be made to the set of electron fluid dynamical equations, as well as the shock condition on electron temperature. Considering existence of current behind the shock front, we have derived the shock condition on electron temperature, and for a set of experimentally measured wave speeds, we have been able to find maximum current values for which solutions to our set of electron velocity, electron temperature, and electron number density within the dynamical transition region of the wave
Connecting the discrete and continuous-time quantum walks
Recently, quantized versions of random walks have been explored as effective
elements for quantum algorithms. In the simplest case of one dimension, the
theory has remained divided into the discrete-time quantum walk and the
continuous-time quantum walk. Though the properties of these two walks have
shown similarities, it has remained an open problem to find the exact relation
between the two. The precise connection of these two processes, both quantally
and classically, is presented. Extension to higher dimensions is also
discussed.Comment: 5 pages, 1 figur
Quantum rejection sampling
Rejection sampling is a well-known method to sample from a target
distribution, given the ability to sample from a given distribution. The method
has been first formalized by von Neumann (1951) and has many applications in
classical computing. We define a quantum analogue of rejection sampling: given
a black box producing a coherent superposition of (possibly unknown) quantum
states with some amplitudes, the problem is to prepare a coherent superposition
of the same states, albeit with different target amplitudes. The main result of
this paper is a tight characterization of the query complexity of this quantum
state generation problem. We exhibit an algorithm, which we call quantum
rejection sampling, and analyze its cost using semidefinite programming. Our
proof of a matching lower bound is based on the automorphism principle which
allows to symmetrize any algorithm over the automorphism group of the problem.
Our main technical innovation is an extension of the automorphism principle to
continuous groups that arise for quantum state generation problems where the
oracle encodes unknown quantum states, instead of just classical data.
Furthermore, we illustrate how quantum rejection sampling may be used as a
primitive in designing quantum algorithms, by providing three different
applications. We first show that it was implicitly used in the quantum
algorithm for linear systems of equations by Harrow, Hassidim and Lloyd.
Secondly, we show that it can be used to speed up the main step in the quantum
Metropolis sampling algorithm by Temme et al.. Finally, we derive a new quantum
algorithm for the hidden shift problem of an arbitrary Boolean function and
relate its query complexity to "water-filling" of the Fourier spectrum.Comment: 19 pages, 5 figures, minor changes and a more compact style (to
appear in proceedings of ITCS 2012
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