Dorit Aharonov

Quantum walks on graphs

We set the ground for a theory of quantum walks on graphs-the 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.

Fault-tolerant quantum computation with constant error

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.

Quantum circuits with mixed states

We define the model of quantum circuits with density matrices, where non-unitary gates are allowed. Measurements in the middle of the computation, noise and decoherence are implemented in a natural way in this model, which is shown to be equivalent in computational power to standard quantum circuits. nThe main result in this paper is a solution for the subroutine problem: The general function that a quantum circuit outputs is a probabilistic function, but using pure state language, such a function can not be used as a black box in other computations. We give a natural definition of using general subroutines, and analyze their computational power. nWe suggest convenient metrics for quantum computing with mixed states. For density matrices we analyze the so called ``trace metric, and using this metric, we define and discuss the ``diamond metric on superoperators. These metrics enable a formal discussion of errors in the computation. nUsing a ``causality lemma for density matrices, we also prove a simple lower bound for probabilistic functions.

Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation

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 allows stating the main open problems in quantum computation using well-studied mathematical objects such as eigenvectors and spectral gaps of sparse matrices.

Fault-Tolerant Quantum Computation with Constant Error Rate

This paper shows that quantum computation can be made fault-tolerant against errors and inaccuracies when

Adiabatic quantum state generation and statistical zero knowledge

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Adiabatic quantum computation is equivalent to standard quantum computation

, the probability for an error in a qubit or a gate, is smaller than a constant threshold

Quantum bit escrow

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Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation

. This result improves on Shors result [Proceedings of the 37th Symposium on the Foundations of Computer Science, IEEE, Los Alamitos, CA, 1996, pp. 56-65], which shows how to perform fault-tolerant quantum computation when the error rate

A lattice problem in quantum NP

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