2,387 research outputs found
The Bose-Hubbard model is QMA-complete
The Bose-Hubbard model is a system of interacting bosons that live on the
vertices of a graph. The particles can move between adjacent vertices and
experience a repulsive on-site interaction. The Hamiltonian is determined by a
choice of graph that specifies the geometry in which the particles move and
interact. We prove that approximating the ground energy of the Bose-Hubbard
model on a graph at fixed particle number is QMA-complete. In our QMA-hardness
proof, we encode the history of an n-qubit computation in the subspace with at
most one particle per site (i.e., hard-core bosons). This feature, along with
the well-known mapping between hard-core bosons and spin systems, lets us prove
a related result for a class of 2-local Hamiltonians defined by graphs that
generalizes the XY model. By avoiding the use of perturbation theory in our
analysis, we circumvent the need to multiply terms in the Hamiltonian by large
coefficients
A new construction for a QMA complete 3-local Hamiltonian
We present a new way of encoding a quantum computation into a 3-local
Hamiltonian. Our construction is novel in that it does not include any terms
that induce legal-illegal clock transitions. Therefore, the weights of the
terms in the Hamiltonian do not scale with the size of the problem as in
previous constructions. This improves the construction by Kempe and Regev, who
were the first to prove that 3-local Hamiltonian is complete for the complexity
class QMA, the quantum analogue of NP.
Quantum k-SAT, a restricted version of the local Hamiltonian problem using
only projector terms, was introduced by Bravyi as an analogue of the classical
k-SAT problem. Bravyi proved that quantum 4-SAT is complete for the class QMA
with one-sided error (QMA_1) and that quantum 2-SAT is in P. We give an
encoding of a quantum circuit into a quantum 4-SAT Hamiltonian using only
3-local terms. As an intermediate step to this 3-local construction, we show
that quantum 3-SAT for particles with dimensions 3x2x2 (a qutrit and two
qubits) is QMA_1 complete. The complexity of quantum 3-SAT with qubits remains
an open question.Comment: 11 pages, 4 figure
Combined Error Correction Techniques for Quantum Computing Architectures
Proposals for quantum computing devices are many and varied. They each have
unique noise processes that make none of them fully reliable at this time.
There are several error correction/avoidance techniques which are valuable for
reducing or eliminating errors, but not one, alone, will serve as a panacea.
One must therefore take advantage of the strength of each of these techniques
so that we may extend the coherence times of the quantum systems and create
more reliable computing devices. To this end we give a general strategy for
using dynamical decoupling operations on encoded subspaces. These encodings may
be of any form; of particular importance are decoherence-free subspaces and
quantum error correction codes. We then give means for empirically determining
an appropriate set of dynamical decoupling operations for a given experiment.
Using these techniques, we then propose a comprehensive encoding solution to
many of the problems of quantum computing proposals which use exchange-type
interactions. This uses a decoherence-free subspace and an efficient set of
dynamical decoupling operations. It also addresses the problems of
controllability in solid state quantum dot devices.Comment: Contribution to Proceedings of the 2002 Physics of Quantum
Electronics Conference", to be published in J. Mod. Optics. This paper
provides a summary and review of quant-ph/0205156 and quant-ph/0112054, and
some new result
Asymptotic entanglement in 1D quantum walks with a time-dependent coined
Discrete-time quantum walk evolve by a unitary operator which involves two
operators a conditional shift in position space and a coin operator. This
operator entangles the coin and position degrees of freedom of the walker. In
this paper, we investigate the asymptotic behavior of the coin position
entanglement (CPE) for an inhomogeneous quantum walk which determined by two
orthogonal matrices in one-dimensional lattice. Free parameters of coin
operator together provide many conditions under which a measurement perform on
the coin state yield the value of entanglement on the resulting position
quantum state. We study the problem analytically for all values that two free
parameters of coin operator can take and the conditions under which
entanglement becomes maximal are sought.Comment: 23 pages, 4 figures, accepted for publication in IJMPB. arXiv admin
note: text overlap with arXiv:1001.5326 by other author
Encoded Universality for Generalized Anisotropic Exchange Hamiltonians
We derive an encoded universality representation for a generalized
anisotropic exchange Hamiltonian that contains cross-product terms in addition
to the usual two-particle exchange terms. The recently developed algebraic
approach is used to show that the minimal universality-generating encodings of
one logical qubit are based on three physical qubits. We show how to generate
both single- and two-qubit operations on the logical qubits, using suitably
timed conjugating operations derived from analysis of the commutator algebra.
The timing of the operations is seen to be crucial in allowing simplification
of the gate sequences for the generalized Hamiltonian to forms similar to that
derived previously for the symmetric (XY) anisotropic exchange Hamiltonian. The
total number of operations needed for a controlled-Z gate up to local
transformations is five. A scalable architecture is proposed.Comment: 11 pages, 4 figure
Overview of Quantum Error Prevention and Leakage Elimination
Quantum error prevention strategies will be required to produce a scalable
quantum computing device and are of central importance in this regard. Progress
in this area has been quite rapid in the past few years. In order to provide an
overview of the achievements in this area, we discuss the three major classes
of error prevention strategies, the abilities of these methods and the
shortcomings. We then discuss the combinations of these strategies which have
recently been proposed in the literature. Finally we present recent results in
reducing errors on encoded subspaces using decoupling controls. We show how to
generally remove mixing of an encoded subspace with external states (termed
leakage errors) using decoupling controls. Such controls are known as ``leakage
elimination operations'' or ``LEOs.''Comment: 8 pages, no figures, submitted to the proceedings of the Physics of
Quantum Electronics, 200
Continuous-time quantum walks on one-dimension regular networks
In this paper, we consider continuous-time quantum walks (CTQWs) on
one-dimension ring lattice of N nodes in which every node is connected to its
2m nearest neighbors (m on either side). In the framework of the Bloch function
ansatz, we calculate the spacetime transition probabilities between two nodes
of the lattice. We find that the transport of CTQWs between two different nodes
is faster than that of the classical continuous-time random walk (CTRWs). The
transport speed, which is defined by the ratio of the shortest path length and
propagating time, increases with the connectivity parameter m for both the
CTQWs and CTRWs. For fixed parameter m, the transport of CTRWs gets slow with
the increase of the shortest distance while the transport (speed) of CTQWs
turns out to be a constant value. In the long time limit, depending on the
network size N and connectivity parameter m, the limiting probability
distributions of CTQWs show various paterns. When the network size N is an even
number, the probability of being at the original node differs from that of
being at the opposite node, which also depends on the precise value of
parameter m.Comment: Typos corrected and Phys. ReV. E comments considered in this versio
Exchange-Only Dynamical Decoupling in the 3-Qubit Decoherence Free Subsystem
The Uhrig dynamical decoupling sequence achieves high-order decoupling of a
single system qubit from its dephasing bath through the use of bang-bang Pauli
pulses at appropriately timed intervals. High-order decoupling of single and
multiple qubit systems from baths causing both dephasing and relaxation can
also be achieved through the nested application of Uhrig sequences, again using
single-qubit Pauli pulses. For the 3-qubit decoherence free subsystem (DFS) and
related subsystem encodings, Pauli pulses are not naturally available
operations; instead, exchange interactions provide all required encoded
operations. Here we demonstrate that exchange interactions alone can achieve
high-order decoupling against general noise in the 3-qubit DFS. We present
decoupling sequences for a 3-qubit DFS coupled to classical and quantum baths
and evaluate the performance of the sequences through numerical simulations
Asymptotic entanglement in a two-dimensional quantum walk
The evolution operator of a discrete-time quantum walk involves a conditional
shift in position space which entangles the coin and position degrees of
freedom of the walker. After several steps, the coin-position entanglement
(CPE) converges to a well defined value which depends on the initial state. In
this work we provide an analytical method which allows for the exact
calculation of the asymptotic reduced density operator and the corresponding
CPE for a discrete-time quantum walk on a two-dimensional lattice. We use the
von Neumann entropy of the reduced density operator as an entanglement measure.
The method is applied to the case of a Hadamard walk for which the dependence
of the resulting CPE on initial conditions is obtained. Initial states leading
to maximum or minimum CPE are identified and the relation between the coin or
position entanglement present in the initial state of the walker and the final
level of CPE is discussed. The CPE obtained from separable initial states
satisfies an additivity property in terms of CPE of the corresponding
one-dimensional cases. Non-local initial conditions are also considered and we
find that the extreme case of an initial uniform position distribution leads to
the largest CPE variation.Comment: Major revision. Improved structure. Theoretical results are now
separated from specific examples. Most figures have been replaced by new
versions. The paper is now significantly reduced in size: 11 pages, 7 figure
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