Searcharxiv⌕ Search

arXiv subjects

Colin P. Williams

Publications and source records attributed to Colin P. Williams.

12 recordsLinked to original sources
arXiv2012

A Near-Term Quantum Computing Approach for Hard Computational Problems in Space Exploration

In this article, we show how to map a sampling of the hardest artificial intelligence problems in space exploration onto equivalent Ising models that then can be attacked using quantum annealing implemented in D-Wave machine. We overview the existing results as well as propose new Ising model implementations for quantum annealing. We review supervised and unsupervised learning algorithms for classification and clustering with applications to feature identification and anomaly detection. We introduce algorithms for data fusion and image matching for remote sensing applications. We overview planning problems for space exploration mission applications and algorithms for diagnostics and recovery with applications to deep space missions. We describe combinatorial optimization algorithms for task assignment in the context of autonomous unmanned exploration. Finally, we discuss the ways to circumvent the limitation of the Ising mapping using a "blackbox" approach based on ideas from probabilistic computing. In this article we describe the architecture of the D-Wave One machine and report its benchmarks. Results on random ensemble of problems in the range of up to 96 qubits show improved scaling for median core quantum annealing time compared with classical algorithms; whether this scaling persists for larger problem sizes is an open question. We also review previous results of D-Wave One benchmarking studies for solving binary classification problems with a quantum boosting algorithm which is shown to outperform AdaBoost. We review quantum algorithms for structured learning for multi-label classification and introduce a hybrid classical/quantum approach for learning the weights. Results of D-Wave One benchmarking studies for learning structured labels on four different data sets show a better performance compared with an independent Support Vector Machine approach with linear kernel.

quant-ph↗
arXiv2004

A Random Matrix Model of Adiabatic Quantum Computing

We present an analysis of the quantum adiabatic algorithm for solving hard instances of 3-SAT (an NP-complete problem) in terms of Random Matrix Theory (RMT). We determine the global regularity of the spectral fluctuations of the instantaneous Hamiltonians encountered during the interpolation between the starting Hamiltonians and the ones whose ground states encode the solutions to the computational problems of interest. At each interpolation point, we quantify the degree of regularity of the average spectral distribution via its Brody parameter, a measure that distinguishes regular (i.e., Poissonian) from chaotic (i.e., Wigner-type) distributions of normalized nearest-neighbor spacings. We find that for hard problem instances, i.e., those having a critical ratio of clauses to variables, the spectral fluctuations typically become irregular across a contiguous region of the interpolation parameter, while the spectrum is regular for easy instances. Within the hard region, RMT may be applied to obtain a mathematical model of the probability of avoided level crossings and concomitant failure rate of the adiabatic algorithm due to non-adiabatic Landau-Zener type transitions. Our model predicts that if the interpolation is performed at a uniform rate, the average failure rate of the quantum adiabatic algorithm, when averaged over hard problem instances, scales exponentially with increasing problem size.

quant-ph↗
arXiv2004

From Linear Optical Quantum Computing to Heisenberg-Limited Interferometry

The working principles of linear optical quantum computing are based on photodetection, namely, projective measurements. The use of photodetection can provide efficient nonlinear interactions between photons at the single-photon level, which is technically problematic otherwise. We report an application of such a technique to prepare quantum correlations as an important resource for Heisenberg-limited optical interferometry, where the sensitivity of phase measurements can be improved beyond the usual shot-noise limit. Furthermore, using such nonlinearities, optical quantum nondemolition measurements can now be carried out at the single-photon level.

quant-ph↗
arXiv2004

Realization of a General Three-Qubit Quantum Gate

We prove that a generic three-qubit quantum logic gate can be implemented using at most 98 one-qubit rotations about the $y$- and $z$-axes and 40 CNOT gates, beating an earlier bound of 64 CNOT gates.

quant-ph↗
arXiv2002

Practical quantum repeaters with linear optics and double-photon guns

We show how to create practical, efficient, quantum repeaters, employing double-photon guns, for long-distance optical quantum communication. The guns create polarization-entangled photon pairs on demand. One such source might be a semiconducter quantum dot, which has the distinct advantage over parametric down-conversion that the probability of creating a photon pair is close to one, while the probability of creating multiple pairs vanishes. The swapping and purifying components are implemented by polarizing beam splitters and probabilistic optical CNOT gates.

quant-ph↗
arXiv2000

Quantum Clock Synchronization Based on Shared Prior Entanglement

We demonstrate that two spatially separated parties (Alice and Bob) can utilize shared prior quantum entanglement, and classical communications, to establish a synchronized pair of atomic clocks. In contrast to classical synchronization schemes, the accuracy of our protocol is independent of Alice or Bob's knowledge of their relative locations or of the properties of the intervening medium.

quant-ph↗
arXiv2000

Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat The Diffraction Limit

Classical, interferometric, optical lithography is diffraction limited to writing features of a size lambda/2 or greater, where lambda is the optical wavelength. Using nonclassical photon number states, entangled N at a time, we show that it is possible to write features of minimum size lambda/(2N) in an N-photon absorbing substrate. This result surpasses the usual classical diffraction limit by a factor of N. Since the number of features that can be etched on a two-dimensional surface scales inversely as the square of the feature size, this allows one to write a factor of N^2 more elements on a semiconductor chip. A factor of N = 2 can be achieved easily with entangled photon pairs generated from optical parametric downconversion. It is shown how to write arbitrary 2D patterns by using this method.

quant-ph↗
arXiv1999

Fast quantum algorithms for numerical integrals and stochastic processes

We discuss quantum algorithms that calculate numerical integrals and descriptive statistics of stochastic processes. With either of two distinct approaches, one obtains an exponential speed increase in comparison to the fastest known classical deterministic algorithms and a quadratic speed increase in comparison to classical Monte Carlo (probabilistic) methods. We derive a simpler and slightly faster version of Grover's mean algorithm, demonstrate how to apply quantum counting to the problem, develop some variations of these algorithms, and show how both (apparently quite different) approaches can be understood from the same unified framework. Finally, we discuss how the exponential speed increase appears to (but does not) violate results obtained via the method of polynomials, from which it is known that a bounded-error quantum algorithm for computing a total function can be only polynomially more efficient than the fastest deterministic classical algorithm.

quant-ph↗
arXiv1999

Maxwell Duality, Lorentz Invariance, and Topological Phase

We discuss the Maxwell electromagnetic duality relations between the Aharonov-Bohm, Aharonov-Casher, and He-McKellar-Wilkens topological phases, which allows a unified description of all three phenomena. We also elucidate Lorentz transformations that allow these effects to be understood in an intuitive fashion in the rest frame of the moving quantum particle. Finally, we propose two experimental schemes for measuring the He-McKellar-Wilkens phase.

quant-ph↗
arXiv1999

Generalized Quantum Search with Parallelism

We generalize Grover's unstructured quantum search algorithm to enable it to use an arbitrary starting superposition and an arbitrary unitary matrix simultaneously. We derive an exact formula for the probability of the generalized Grover's algorithm succeeding after n iterations. We show that the fully generalized formula reduces to the special cases considered by previous authors. We then use the generalized formula to determine the optimal strategy for using the unstructured quantum search algorithm. On average the optimal strategy is about 12% better than the naive use of Grover's algorithm. The speedup obtained is not dramatic but it illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. We extend the analysis to the case of a society of k quantum searches acting in parallel. We derive an analytic formula that connects the degree of parallelism with the optimal strategy for k-parallel quantum search. We then derive the formula for the expected speed of k-parallel quantum search.

quant-ph↗
arXiv1998

Quantum Wavelet Transforms: Fast Algorithms and Complete Circuits

The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies $D^{(4)}$ transforms. Our approach is to factor the operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform.

quant-ph↗