arXiv subjects

David A. Meyer

Publications and source records attributed to David A. Meyer.

At least 37 records · Page 2Linked to original sources

arXiv2015

Distinguishing symmetric quantum oracles and quantum group multiplication

Given a unitary representation of a finite group on a finite-dimensional Hilbert space, we show how to find a state whose translates under the group are distinguishable with the highest probability. We apply this to several quantum oracle problems, including the GROUP MULTIPLICATION problem, in which the product of an ordered $n$-tuple of group elements is to be determined by querying elements of the tuple. For any finite group $G$, we give an algorithm to find the product of two elements of $G$ with a single quantum query with probability $2/|G|$. This generalizes Deutsch's Algorithm from $Z_2$ to an arbitrary finite group. We further prove that this algorithm is optimal. We also introduce the HIDDEN CONJUGATING ELEMENT PROBLEM, in which the oracle acts by conjugating by an unknown element of the group. We show that for many groups, including dihedral and symmetric groups, the unknown element can be determined with probability $1$ using a single quantum query.

quant-ph↗

arXiv2015

Completeness is Unnecessary for Fast Nonlinear Quantum Search

Although strongly regular graphs and the hypercube are not complete, they are "sufficiently complete" such that a randomly walking quantum particle asymptotically searches on them in the same $Θ(\sqrt{N})$ time as on the complete graph, the latter of which is precisely Grover's algorithm. We show that physically realistic nonlinearities of the form $f(|ψ|^2)ψ$ can speed up search on sufficiently complete graphs, depending on the nonlinearity and graph. Thus nonlinear (quantum) computation can retain its power even when a degree of noncompleteness is introduced.

quant-ph↗

arXiv2015

Connectivity is a Poor Indicator of Fast Quantum Search

A randomly walking quantum particle evolving by Schrödinger's equation searches on $d$-dimensional cubic lattices in $O(\sqrt{N})$ time when $d \ge 5$, and with progressively slower runtime as $d$ decreases. This suggests that graph connectivity (including vertex, edge, algebraic, and normalized algebraic connectivities) is an indicator of fast quantum search, a belief supported by fast quantum search on complete graphs, strongly regular graphs, and hypercubes, all of which are highly connected. In this paper, we show this intuition to be false by giving two examples of graphs for which the opposite holds true: one with low connectivity but fast search, and one with high connectivity but slow search. The second example is a novel two-stage quantum walk algorithm in which the walking rate must be adjusted to yield high search probability.

quant-ph↗

arXiv2014

A Model of Consistent Node Types in Signed Directed Social Networks

Signed directed social networks, in which the relationships between users can be either positive (indicating relations such as trust) or negative (indicating relations such as distrust), are increasingly common. Thus the interplay between positive and negative relationships in such networks has become an important research topic. Most recent investigations focus upon edge sign inference using structural balance theory or social status theory. Neither of these two theories, however, can explain an observed edge sign well when the two nodes connected by this edge do not share a common neighbor (e.g., common friend). In this paper we develop a novel approach to handle this situation by applying a new model for node types. Initially, we analyze the local node structure in a fully observed signed directed network, inferring underlying node types. The sign of an edge between two nodes must be consistent with their types; this explains edge signs well even when there are no common neighbors. We show, moreover, that our approach can be extended to incorporate directed triads, when they exist, just as in models based upon structural balance or social status theory. We compute Bayesian node types within empirical studies based upon partially observed Wikipedia, Slashdot, and Epinions networks in which the largest network (Epinions) has 119K nodes and 841K edges. Our approach yields better performance than state-of-the-art approaches for these three signed directed networks.

cs.SI↗

arXiv2014

Global Symmetry is Unnecessary for Fast Quantum Search

Grover's quantum search algorithm can be formulated as a quantum particle randomly walking on the (highly symmetric) complete graph, with one vertex marked by a nonzero potential. From an initial equal superposition, the state evolves in a two-dimensional subspace. Strongly regular graphs have a local symmetry that ensures that the state evolves in a \emph{three}-dimensional subspace, but most have no \emph{global} symmetry. Using degenerate perturbation theory, we show that quantum random walk search on known families of strongly regular graphs nevertheless achieves the full quantum speedup of $Θ(\sqrt{N})$, disproving the intuition that fast quantum search requires global symmetry.

quant-ph↗

arXiv2014

History Dependent Quantum Random Walks as Quantum Lattice Gas Automata

Quantum Random Walks (QRW) were first defined as one-particle sectors of Quantum Lattice Gas Automata (QLGA). Recently, they have been generalized to include history dependence, either on previous coin (internal, i.e., spin or velocity) states or on previous position states. These models have the goal of studying the transition to classicality, or more generally, changes in the performance of quantum walks in algorithmic applications. We show that several history dependent QRW can be identified as one-particle sectors of QLGA. This provides a unifying conceptual framework for these models in which the extra degrees of freedom required to store the history information arise naturally as geometrical degrees of freedom on the lattice.

math-ph↗

arXiv2013

Quantum Search with General Nonlinearities

Evolution by the Gross-Pitaevskii equation, which describes Bose-Einstein condensates under certain conditions, solves the unstructured search problem more efficiently than does the Schrödinger equation, because it includes a cubic nonlinearity, proportional to $|ψ|^2ψ$. This is not the only nonlinearity of the form $f(|ψ|^2)ψ$ that arises in effective equations for the evolution of real quantum physical systems, however: The cubic-quintic nonlinear Schrödinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schrödinger equation describes Bose liquids under certain conditions. Analysis of computation with such systems yields some surprising results; for example, when time-measurement precision is included in the resource accounting, searching a "database" when there is a single correct answer may be easier than searching when there are multiple correct answers. In each of these cases the nonlinear equation is an effective approximation to a multi-particle Schrödinger equation, for search by which Grover's algorithm is optimal. Thus our results lead to quantum information-theoretic bounds on the physical resources required for these effective nonlinear theories to hold, asymptotically.

quant-ph↗

arXiv2013

Nonlinear Quantum Search Using the Gross-Pitaevskii Equation

We solve the unstructured search problem in constant time by computing with a physically motivated nonlinearity of the Gross-Pitaevskii type. This speedup comes, however, at the novel expense of increasing the time-measurement precision. Jointly optimizing these resource requirements results in an overall scaling of $N^{1/4}$. This is a significant, but not unreasonable, improvement over the $N^{1/2}$ scaling of Grover's algorithm. Since the Gross-Pitaevskii equation approximates the multi-particle (linear) Schrödinger equation, for which Grover's algorithm is optimal, our result leads to a quantum information-theoretic lower bound on the number of particles needed for this approximation to hold, asymptotically.

quant-ph↗

arXiv2012

Discrete Quantum Control - State Preparation

A discrete-time method for solving problems in optimal quantum control is presented. Controlling the time discretized markovian dynamics of a quantum system can be reduced to a Markov-decision process. We demonstrate this method in this with a class of simple one qubit systems, which are also discretized in space. For the task of state preparation we solve the examples both numerically and analytically with dynamic programming techniques.

quant-ph↗

arXiv2011

Multi-query quantum sums

PARITY is the problem of determining the parity of a string $f$ of $n$ bits given access to an oracle that responds to a query $x\in\{0,1,...,n-1\}$ with the $x^{\rm th}$ bit of the string, $f(x)$. Classically, $n$ queries are required to succeed with probability greater than 1/2 (assuming equal prior probabilities for all length $n$ bitstrings), but only $\lceil n/2\rceil$ quantum queries suffice to determine the parity with probability 1. We consider a generalization to strings $f$ of $n$ elements of $\Z_k$ and the problem of determining $\sum f(x)$. By constructing an explicit algorithm, we show that $n-r$ ($n\ge r\in\N$) entangled quantum queries suffice to compute the sum correctly with worst case probability $\min\{\lfloor n/r\rfloor/k,1\}$. This quantum algorithm utilizes the $n-r$ queries sequentially and adaptively, like Grover's algorithm, but in a different way that is not amplitude amplification.

quant-ph↗

arXiv2010

On the uselessness of quantum queries

Given a prior probability distribution over a set of possible oracle functions, we define a number of queries to be useless for determining some property of the function if the probability that the function has the property is unchanged after the oracle responds to the queries. A familiar example is the parity of a uniformly random Boolean-valued function over $\{1,2,...,N\}$, for which $N-1$ classical queries are useless. We prove that if $2k$ classical queries are useless for some oracle problem, then $k$ quantum queries are also useless. For such problems, which include classical threshold secret sharing schemes, our result also gives a new way to obtain a lower bound on the quantum query complexity, even in cases where neither the function nor the property to be determined is Boolean.

quant-ph↗

arXiv2010

Lattice gas simulations of dynamical geometry in two dimensions

We present a hydrodynamic lattice gas model for two-dimensional flows on curved surfaces with dynamical geometry. This model is an extension to two dimensions of the dynamical geometry lattice gas model previously studied in one-dimension. We expand upon a variation of the two-dimensional flat space FHP model created by Frisch, Hasslacher and Pomeau, and independently by Wolfram, and modified by Boghosian, Love, and Meyer. We define a hydrodynamic lattice gas model on an arbitrary triangulation, whose flat space limit is the FHP model. Rules that change the geometry are constructed using the Pachner moves, which alter the triangulation but not the topology. We present results on the growth of the number of triangles as a function of time. Simulations show that the number of triangles lattice grows with time as the cube root of the number of time steps, in agreement a mean field prediction. We also present preliminary results on the distribution of curvature over a typical triangulation for these simulations.

cond-mat.soft↗

arXiv2009

Single query learning from abelian and non-abelian Hamming distance oracles

We study the problem of identifying an n-bit string using a single quantum query to an oracle that computes the Hamming distance between the query and hidden strings. The standard action of the oracle on a response register of dimension r is by powers of the cycle (1...r), all of which, of course, commute. We introduce a new model for the action of an oracle--by general permutations in S_r--and explore how the success probability depends on r and on the map from Hamming distances to permutations. In particular, we prove that when r = 2, for even n the success probability is 1 with the right choice of the map, while for odd n the success probability cannot be 1 for any choice. Furthermore, for small odd n and r = 3, we demonstrate numerically that the image of the optimal map generates a non-abelian group of permutations.

quant-ph↗

arXiv2009

Negative Refraction Gives Rise to the Klein Paradox

Electromagnetic negative refraction in metamaterials has attracted increasingly great interest, since its first experimental verification in 2001. It potentially leads to the applications superior to conventional devices including compact antennas for mobile stations, imaging beyond the diffraction limit, and high-resolution radars, not to mention the anamolous wave propagation in fundamental optics. Here, we report how metamaterials could be used to simulate the "negative refraction of spin-zero particles interacting with a strong potential barrier", which gives rise to the Klein paradox--a counterintuitive relativistic process. We address the underlying physics of analogous wave propagation behaviours in those two entirely different domains of quantum and classical.

quant-ph↗

arXiv2009

Faster transport with a directed quantum walk

We give the first example of faster transport with a quantum walk on an inherently directed graph, on the directed line with a variable number of self-loops at each vertex. These self-loops can be thought of as adding a number of small dimensions. This is a discrete time quantum walk using the Fourier transform coin, where the walk proceeds a distance $Θ(1)$ in constant time compared to $Θ(1/n)$ classically, independent of the number of these small dimensions. The analysis proceeds by reducing this walk to a walk with a two dimensional coin.

quant-ph↗

arXiv2006

Integrated Conditional Teleportation and Readout Circuit Based on a Photonic Crystal Single Chip

We demonstrate the design of an integrated conditional quantum teleportation circuit and a readout circuit using a two-dimensional photonic crystal single chip. Fabrication and testing of the proposed quantum circuit can be accomplished with current or near future semiconductor process technology and experimental techniques. The readout part of our device, which has potential for independent use as an atomic interferometer, can also be used on its own or integrated with other compatible optical circuits to achieve atomic state detection. Further improvement of the device in terms of compactness and robustness could be achieved by integrating it with sources and detectors in the optical regime.

quant-ph↗

arXiv2006

Creation of Entanglement and Implementation of Quantum Logic Gate Operations Using a Three-Dimensional Photonic Crystal Single-Mode Cavity

We solve the Jaynes-Cummings Hamiltonian with time-dependent coupling parameters under dipole and rotating-wave approximation for a three-dimensional (3D) photonic crystal (PC) single mode cavity with a sufficiently high quality (Q) factor. We then exploit the results to show how to create a maximally entangled state of two atoms, and how to implement several quantum logic gates: a dual-rail Hadamard gate, a dual-rail NOT gate, and a SWAP gate. The atoms in all of these operations are syncronized, which is not the case in previous studies [1,2] in PCs. Our method has the potential for extension to N-atom entanglement, universal quantum logic operations, and the implementation of other useful, cavity QED based quantum information processing tasks.

quant-ph↗

arXiv2005

Periodicity and Growth in a Lattice Gas with Dynamical Geometry

We study a one-dimensional lattice gas "dynamical geometry model" in which local reversible interactions of counter-rotating groups of particles on a ring can create or destroy lattice sites. We exhibit many periodic orbits and and show that all other solutions have asymptotically growing lattice length in both directions of time. We explain why the length grows as $\sqrt{t}$ in all cases examined. We completely solve the dynamics for small numbers of particles with arbitrary initial conditions.

nlin.CG↗