We set the ground for a theory of quantum walks on graphsthe generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.

In the past year many developments have taken place in the area of quantum error corrections. Recently Shor showed how to perform fault tolerant quantum computation when,~, the probability for a fault in one time step per qubit or per gate, is polylogarithmically small. This paper closes the gap and shows how to perform fault tolerant quantum computation when the error probability, q, is smaller than some constant threshold, q.. The cost is polylogarithmic in time and space, and no measurements are used during the quantum computation. The same result is shown also for quantum circuits which operate on nearest neighbors only.To achieve this noise resistance, we use concatenated quantum error correcting codes. The scheme presented is general, and works with any quantum code, that satisfies certain restm"ctions, namely that it is a "proper quantum code". The constant threshold r10 is a function of the parameters of the specifc proper code used. We present two explicit classes of proper quantum codes. The first class generalizes classical secret sharing with polynomials.The codes are defined over a field with p elements, which means that the elementary quantum particle is not a qubit but a "qupit". The second class uses a known class of quantum codes and converts it to a proper code.We estimate the threshold qO to be = 10-6. Hopefully, this paper motivates a search for proper quantum codes with higher thresholds, at which point quantum computation becomes practical.

Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the conventional quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor interactions. The equivalence between the models provides a new vantage point from which to tackle the central issues in quantum computation, namely designing new quantum algorithms and constructing fault tolerant quantum computers. In particular, by translating the main open questions in the area of quantum algorithms to the language of spectral gaps of sparse matrices, the result makes these questions accessible to a wider scientific audience, acquainted with mathematical physics, expander theory and rapidly mixing Markov chains.

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Abstract. Shor has showed how to perform fault tolerant quantum computation when the probability for an error in a qubit or a gate, η, decays with the size of the computation polylogarithmically, an assumption which is physically unreasonable. This paper improves this result and shows that quantum computation can be made robust against errors and inaccuracies, when the error rate, η, is smaller than a constant threshold, ηc. The cost is polylogarithmic in space and time. The result holds for a very general noise model, which includes probabilistic errors, decoherence, amplitude damping, depolarization, and systematic inaccuracies in the gates. Moreover, we allow exponentially decaying correlations between the errors both in space and in time. Fault tolerant computation can be performed with any universal set of gates. The result also holds for quantum particles with p > 2 states, namely qupits, and is also generalized to one dimensional quantum computers with only nearest neighbor interactions. No measurements, or classical operations, are required during the quantum computation.We use Calderbank-Shor Steane (CSS) quantum error correcting codes, generalized to qupits, and introduce a new class of CSS codes over Fp, called polynomial codes. It is shown how to apply a universal set of gates fault tolerantly on states encoded by general CSS codes, based on modifications of Shor's procedures, and on states encoded by polynomial codes, where the procedures for polynomial codes have a simple and systematic structure based on the algebraic properties of the code.Geometrical and group theoretical arguments are used to prove the universality of the two sets of gates. Our key theorem asserts that applying computation on encoded states recursively achieves fault tolerance against constant error rate. The generalization to general noise models is done using the framework of quantum circuits with density matrices. Finally, we calculate the threshold to be ηc ≃ 10 −6 , in the best case.The paper contains new and significantly simpler proofs for most of the known results which we use. For example, we give a simple proof that it suffices to correct bit and phase flips, we significantly simplify Calderbank and Shor's original proof of the correctness of CSS codes. We also give a simple proof of the fact that two-qubit gates are universal. The paper thus provides a self contained and complete proof for universal fault tolerant quantum computation.1. Outline. Quantum computation has recently gained a lot of attention, due to oracle results [65,10,11] and quantum algorithms [25,65,36], in particular Shor's factorization algorithm [62]. These results indicate a possibility that quantum computers are exponentially more powerful than classical computers. It is yet unclear whether and how quantum computers will be physically realizable [19,22,30,37,50,51,55,59,69] but as any physical system, they in principle will be subjected to noise, such as decoherence [76,71], and inaccuracies. Thus, the question of correcting noise cannot be sepa...