The theoretical study of quantum computation has yielded efficient algorithms for some traditionally hard problems. Correspondingly, experimental work on the underlying physical implementation technology has progressed steadily. However, almost no work has yet been done which explores the architecture design space of large scale quantum computing systems. In this paper, we present a set of tools that enable the quantitative evaluation of architectures for quantum computers. The infrastructure we created comprises a complete compilation and simulation system for computers containing thousands of quantum bits. We begin by compiling complete algorithms into a quantum instruction set. This ISA enables the simple manipulation of quantum state. Another tool we developed automatically transforms quantum software into an equivalent, fault-tolerant version required to operate on real quantum devices. Next, our infrastructure transforms the ISA into a set of low-level micro architecture specific control operations. In the future, these operations can be used to directly control a quantum computer. For now, our simulation framework quickly uses them to determine the reliability of the application for the target micro architecture. Finally, we propose a simple, regular architecture for iontrap based quantum computers. Using our software infrastructure, we evaluate the design trade offs of this micro architecture. 1

This paper is an exploration of relationships between the Jones polynomial and quantum computing. We discuss the structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra, and give an example of a unitary representation of the braid group. We discuss the evaluation of the polynomial as a generalized quantum amplitude and show how the braiding part of the evaluation can be construed as a quantum computation when the braiding representation is unitary. The question of an efficient quantum algorithm for computing the whole polynomial remains open. 1

This paper is a quick introduction to key relationships between the theories of knots,links, three-manifold invariants and the structure of quantum mechanics. In section 2 we review the basic ideas and principles of quantum mechanics. Section 3 shows how the idea of a quantum amplitude is applied to the construction of invariants of knots and links. Section 4 explains how the generalisation of the Feynman integral to quantum fields leads to invariants of knots, links and three-manifolds. Section 5 is a discussion of a general categorical approach to these issues. Section 6 is a brief discussion of the relationships of quantum topology to quantum computing. This paper is intended as an introduction that can serve as a springboard for working on the interface between quantum topology and quantum computing. Section 7 summarizes the paper.

This paper initiates the study of quantum computing within the constraints of using a polylogarithmic (O(log k n),k ≥ 1) number of qubits and a polylogarithmic number of computation steps. The current research in the literature has focussed on using a polynomial number of qubits. A new mathematical model of computation called Quantum Neural Networks (QNNs) is defined, building on Deutsch’s model of quantum computational network. The model introduces a nonlinear and irreversible gate, similar to the speculative operator defined by Abrams and Lloyd. The precise dynamics of this operator are defined and while giving examples in which nonlinear Schrödinger’s equations are applied, we speculate on its possible implementation. The many practical problems associated with the current model of quantum computing are alleviated in the new model. It is shown that QNNs of logarithmic size and constant depth have the same computational power as threshold circuits, which are used for modeling neural networks. QNNs of polylogarithmic size and polylogarithmic depth can solve the problems in NC, the class of problems with theoretically fast parallel solutions. Thus, the new model may indeed provide an approach for building scalable parallel computers.

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The theory of computational complexity has some interesting links to physics, in particular to quantum computing and statistical mechanics. This article contains an informal introduction to this theory and its links to physics.

In contrast to the usual approach to simulating quantum computing algorithms as a series of operations to be performed on a set of “qubits, ” we have used the Haskell programming language to implement Grover’s fast search algorithm as a composition of functional transformations applied, as a final step, to a set of qubits (a “quantum register”). In this approach, which has been (at least implicitly) suggested in the literature, but, as far as we have been able to determine, not been realized in a working program, the composition is constructed by means of common matrix manipulations and takes advantage of the associativity of matrix operations to eliminate complicated computations usually associated with simulating quantum computations. We present the crucial code sections, along with actual program results. We discuss the implications of our approach in the areas of quantum program simulation, algorithm analysis, algorithm construction, and construction of languages for quantum programming.

A new type of seismic imaging, based on Feynman path integrals for waveform modelling, is capable of producing accurate subsurface images without any need for a reference velocity model. Instead of the usual optimization for traveltime curves with maximal signal semblance, a weighted summation over all representative curves avoids the need for velocity analysis, with its common difficulties of subjective and timeconsuming manual picking. The summation over all curves includes the stationary one that plays a preferential role in classical imaging schemes, but also multiple stationary curves when they exist. Moreover, the weighted summation over all curves also accounts for non-uniqueness and uncertainty in the stacking/migration velocities. The path-integral imaging can be applied to stacking to zero-offset and to time and depth migration. In all these cases, a properly defined weighting function plays a vital role: to emphasize contributions from traveltime curves close to the optimal one and to suppress contributions from unrealistic curves. The path-integral method is an authentic macromodel-independent technique in the sense that there is strictly no parameter optimization or estimation involved. Development is still in its initial stage, and several conceptual and implementation issues are yet to be solved. However, application to synthetic and real data examples shows that it has the potential for becoming a fully automatic imaging technique.

Reversible logic was originally proposed as a way to build classical computers with very low energy requirements. Its greatest impact so far has in fact been on the field of quantum computing, where it is used pervasively -- but for an entirely different purpose. Quantum interference effects offer the potential for speeding up certain classical calculations. But if state information is expelled from a quantum computer into the unmodeled environment -- as happens whenever an irreversible operation is performed -- then it becomes impossible to predict or control when interference will occur. Reversibility is necessary because it allows bits to be erased by uncomputing them rather than simply expelling them, preserving control over interference effects.

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