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Faulttolerant quantum computation with local gates
 Jour. of Modern Optics
"... I discuss how to perform faulttolerant quantum computation with concatenated codes using local gates in small numbers of dimensions. I show that a threshold result still exists in three, two, or one dimensions when nexttonearestneighbor gates are available, and present explicit constructions. In ..."
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I discuss how to perform faulttolerant quantum computation with concatenated codes using local gates in small numbers of dimensions. I show that a threshold result still exists in three, two, or one dimensions when nexttonearestneighbor gates are available, and present explicit constructions. In two or three dimensions, I also show how nearestneighbor gates can give a threshold result. In all cases, I simply demonstrate that a threshold exists, and do not attempt to optimize the error correction circuit or determine the exact value of the threshold. The additional overhead due to the faulttolerance in both space and time is polylogarithmic in the error rate per logical gate. 1
On universal and faulttolerant quantum computing: a novel basis and a new constructive proof of universality for Shor’s basis
 In Proceedings of the 40th Annual Symposium on Foundations of Computer Science
, 1999
"... A novel universal and faulttolerant basis (set of gates) for quantum computation is described. Such a set is necessary to perform quantum computation in a realistic noisy environment. The new basis consists of two singlequbit gates 1 (Hadamard and σz 4), and one doublequbit gate (ControlledNOT). ..."
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Cited by 39 (3 self)
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A novel universal and faulttolerant basis (set of gates) for quantum computation is described. Such a set is necessary to perform quantum computation in a realistic noisy environment. The new basis consists of two singlequbit gates 1 (Hadamard and σz 4), and one doublequbit gate (ControlledNOT). Since the set consisting of ControlledNOT and Hadamard gates is not universal, the new basis achieves universality by including only one additional elementary (in the sense that it does not include angles that are irrational multiples of π) singlequbit gate, and hence, is potentially the simplest universal basis that one can construct. We also provide an alternative proof of universality for the only other known class of universal and faulttolerant basis proposed in [25, 17]. 1
Accuracy Threshold for Quantum Computation
, 1996
"... We have previously [11] shown that for quantum memories and quantum communication, a state can be transmitted over arbitrary distances with error ffl provided each gate has error at most cffl. We discuss a similar concatenation technique which can be used with fault tolerant networks to achieve any ..."
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We have previously [11] shown that for quantum memories and quantum communication, a state can be transmitted over arbitrary distances with error ffl provided each gate has error at most cffl. We discuss a similar concatenation technique which can be used with fault tolerant networks to achieve any desired accuracy when computing with classical initial states, provided a minimum gate accuracy can be achieved. The technique works under realistic assumptions on operational errors. These assumptions are more general than the stochastic error heuristic used in other work. Methods are proposed to account for leakage errors, a problem not previously recognized. 1 Introduction Three recent events are promising to make extensive quantum computations as practical as classical computations. The first is the discovery by Shor [13], Steane [15] and Calderbank et al. [4, 3] of quantum errorcorrecting codes email: knill@lanl.gov y laflamme@lanl.gov z whz@lanl.gov which can be used to main...
Unpaired majorana fermions in quantum wires
, 2000
"... Certain onedimensional Fermi systems have an energy gap in the bulk spectrum while boundary states are described by one Majorana operator per boundary point. A finite system of length L possesses two ground states with an energy difference proportional to exp(−L/l0) and different fermionic parities ..."
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Certain onedimensional Fermi systems have an energy gap in the bulk spectrum while boundary states are described by one Majorana operator per boundary point. A finite system of length L possesses two ground states with an energy difference proportional to exp(−L/l0) and different fermionic parities. Such systems can be used as qubits since they are intrinsically immune to decoherence. The property of a system to have boundary Majorana fermions is expressed as a condition on the bulk electron spectrum. The condition is satisfied in the presence of an arbitrary small energy gap induced by proximity of a 3dimensional pwave superconductor, provided that the normal spectrum has an odd number of Fermi points in each half of the Brillouin zone (each spin component counts separately).
The Computational Complexity of Linear Optics
 in Proceedings of STOC 2011
"... We give new evidence that quantum computers—moreover, rudimentary quantumcomputers built entirely out of linearoptical elements—cannotbeefficientlysimulatedbyclassical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linearoptical n ..."
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We give new evidence that quantum computers—moreover, rudimentary quantumcomputers built entirely out of linearoptical elements—cannotbeefficientlysimulatedbyclassical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linearoptical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomialtime classical algorithm that samples from the same probability distribution as a linearoptical network, then P #P = BPP NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the PermanentofGaussians Conjecture, which says that it is #Phard to approximate the permanent of a matrixAofindependentN (0,1)Gaussianentries, withhigh probability over A; and the Permanent AntiConcentration Conjecture, which says that Per(A)  ≥ √ n!/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. For the 96page full version, see www.scottaaronson.com/papers/optics.pdf
On optimal quantum codes
 Int. J. Quantum Inform
, 2004
"... We present families of quantum errorcorrecting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over qdimensional quantum systems, where q is an arbitrary prime power. ..."
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We present families of quantum errorcorrecting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over qdimensional quantum systems, where q is an arbitrary prime power.
Secure Multiparty Quantum Computation
 STOC'02
, 2002
"... Secure multiparty computing, also called secure function evaluation, has been extensively studied in classical cryptography. We consider the extension of this task to computation with quantum inputs and circuits. Our protocols are informationtheoretically secure, i.e. no assumptions are made on th ..."
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Secure multiparty computing, also called secure function evaluation, has been extensively studied in classical cryptography. We consider the extension of this task to computation with quantum inputs and circuits. Our protocols are informationtheoretically secure, i.e. no assumptions are made on the computational power of the adversary. For the weaker task of verifiable quantum secret sharing, we give a protocol which tolerates any t < n/4 cheating parties (out of n). This is shown to be optimal. We use this new tool to show how to perform any multiparty quantum computation as long as the number of dishonest players is less than n/6.
Faulttolerant quantum computation for local nonmarkovian noise
 Phys. Rev. A
, 2005
"... We derive a threshold result for faulttolerant quantum computation for local nonMarkovian noise models. The role of error amplitude in our analysis is played by the product of the elementary gate time t0 and the spectral width of the interaction Hamiltonian between system and bath. We discuss exte ..."
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We derive a threshold result for faulttolerant quantum computation for local nonMarkovian noise models. The role of error amplitude in our analysis is played by the product of the elementary gate time t0 and the spectral width of the interaction Hamiltonian between system and bath. We discuss extensions of our model and the applicability of our analysis for several physical decoherence processes. 1
Architectural implications of quantum computing technologies
 ACM Journal on Emerging Technologies in Computing Systems (JETC
, 2006
"... In this article we present a classification scheme for quantum computing technologies that is based on the characteristics most relevant to computer systems architecture. The engineering tradeoffs of execution speed, decoherence of the quantum states, and size of systems are described. Concurrency, ..."
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Cited by 27 (4 self)
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In this article we present a classification scheme for quantum computing technologies that is based on the characteristics most relevant to computer systems architecture. The engineering tradeoffs of execution speed, decoherence of the quantum states, and size of systems are described. Concurrency, storage capacity, and interconnection network topology influence algorithmic efficiency, while quantum error correction and necessary quantum state measurement are the ultimate drivers of logical clock speed. We discuss several proposed technologies. Finally, we use our taxonomy to explore architectural implications for common arithmetic circuits, examine the implementation of quantum error correction, and discuss clusterstate quantum computation.
Quantum computation and the localization of modular functors
"... Kevin Walker, and Zhenghan Wang. Their work has been the inspiration for this lecture. The mathematical problem of localizing modular functors to neighborhoods of points is shown to be closely related to the physical problem of engineering a local Hamiltonian for a computationally universal quantum ..."
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Kevin Walker, and Zhenghan Wang. Their work has been the inspiration for this lecture. The mathematical problem of localizing modular functors to neighborhoods of points is shown to be closely related to the physical problem of engineering a local Hamiltonian for a computationally universal quantum medium. For genus = 0 surfaces, such a local Hamiltonian is mathematically defined. Braiding defects of this medium implements a representation associated to the Jones polynomial and this representation is known to be universal for quantum computation. 1 The Picture Principle Reality has the habit of intruding on the prodigies of purest thought and encumbering them with unpleasant embellishments. So it is astonishing when the chthonian hammer of the engineer resonates precisely to the gossamer fluttering of theory. Such a moment may soon be at hand in the practice and theory of quantum computation. The most compelling theoretical question, “localization, ” is yielding an answer which points the way to a solution of Based on lectures prepared for the joint Microsoft/University of Washington celebration