Graph identification by quantum walks
This paper has been withdrawn by the authors; replaced by arXiv:0705.2531v2 [quant-ph].
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arXiv subjects
J. B. Wang
Publications and source records attributed to J. B. Wang.
Solid State Implementation of Quantum Random Walks on General Graphs
Advances in recent years have made it possible to explore quantum dots as a viable technology for scalable quantum information processing. Charge qubits for example can be realized in the lowest bound states of coupled quantum dots and the precision control of the confinement potential allows for the realization of a full set of universal qubit gates, including arbitrary single-qubit rotations and two-qubit C-NOT gates. In this work we describe a novel scheme for implementing quantum random walks on arbitrarily complex graphs by extending these elementary operations to the control of a two-dimensional quantum dot grid. As single-qubit rotations constitute the essential building blocks of our implementation scheme, we also present numerical simulations of one such mechanism by directly solving the corresponding time-dependent Schrodinger equation.
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Classical approach to the graph isomorphism problem using quantum walks
Given the extensive application of classical random walks to classical algorithms in a variety of fields, their quantum analogue in quantum walks is expected to provide a fruitful source of quantum algorithms. So far, however, such algorithms have been scarce. In this work, we enumerate some important differences between quantum and classical walks, leading to their markedly different properties. We show that for many practical purposes, the implementation of quantum walks can be efficiently achieved using a classical computer. We then develop both classical and quantum graph isomorphism algorithms based on discrete-time quantum walks. We show that they are effective in identifying isomorphism classes of large databases of graphs, in particular groups of strongly regular graphs. We consider this approach to represent a promising candidate for an efficient solution to the graph isomorphism problem, and believe that similar methods employing quantum walks, or derivatives of these walks, may prove beneficial in constructing other algorithms for a variety of purposes.
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Quantum Walks in an array of Quantum Dots
Quantum random walks are shown to have non-intuitive dynamics, which makes them an attractive area of study for devising quantum algorithms for well-known classical problems as well as those arising in the field of quantum computing. In this work we propose a novel scheme for the physical implementation of a discrete-time quantum random walk using laser excitations of the electronic states of an array of quantum dots. These dots represent the discrete nodes of the walk, while transitions between the energy levels inside each dot correspond to the required coin operation and stimulated Raman adiabatic passage (STIRAP) processes are employed to induce the steps of the walk. The quantum dot design is tailored in such a way as to enable selective coupling of the energy levels. Our simulation results show a close agreement with the ideal quantum walk distribution as well as modest robustness towards noise disturbance.
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Continuous-Time Quantum Random Walks Require Discrete Space
Quantum random walks are shown to have non-intuitive dynamics which makes them an attractive area of study for devising quantum algorithms for long-standing open problems as well as those arising in the field of quantum computing. In the case of continuous-time quantum random walks, such peculiar dynamics can arise from simple evolution operators closely resembling the quantum free-wave propagator. We investigate the divergence of quantum walk dynamics from the free-wave evolution and show that in order for continuous-time quantum walks to display their characteristic propagation, the state space must be discrete. This behavior rules out many continuous quantum systems as possible candidates for implementing continuous-time quantum random walks.
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Readout of solid-state charge qubits using a single-electron pump
A major difficulty in realizing a solid-state quantum computer is the reliable measurement of the states of the quantum registers. In this paper, we propose an efficient readout scheme making use of the resonant tunneling of a ballistic electron produced by a single electron pump. We treat the measurement interaction in detail by modeling the full spatial configuration, and show that for pumped electrons with suitably chosen energy the transmission coefficient is very sensitive to the qubit state. We further show that by using a short sequence of pumping events, coupled with a simple feedback control procedure, the qubit can be measured with high accuracy.
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Transmission spectra of two-dimensional quantum structures
To study the ballistic transport of charge carriers in nano-structured quantum devices, a highly efficient numerical technique is developed, which provides continuous transmission spectra for arbitrarily complex potential geometries in two dimensions. We apply the proposed method to single and double barrier structures and compare the results with those obtained using standard techniques for computing transmission coefficients. Excellent numerical agreement as well as considerable computational saving is demonstrated.
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Theory and Computation of the Spheroidal Wave Functions
In this paper we report on a package, written in the Mathematica computer algebra system, which has been developed to compute the spheroidal wave functions of Meixner [J. Meixner and R.W. Schaefke, Mathieusche Funktionen und Sphaeroidfunktionen, 1954] and is available online (www.physics.uwa.edu.au/~falloon/spheroidal/spheroidal.html). This package represents a substantial contribution to the existing software, since it computes the spheroidal wave functions to arbitrary precision for general complex parameters mu, nu, gamma and argument z; existing software can only handle integer mu, nu and does not give arbitrary precision. The package also incorporates various special cases and computes analytic power series and asymptotic expansions in the parameter gamma. The spheroidal wave functions of Flammer [C. Flammer, Spheroidal Wave Functions, 1957] are included as a special case of Meixner's more general functions. This paper presents a concise review of the general theory of spheroidal wave functions and a description of the formulas and algorithms used in their computation, and gives high-precision numerical examples.
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