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142
Faulttolerant quantum computation
 In Proc. 37th FOCS
, 1996
"... It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information i ..."
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Cited by 264 (5 self)
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It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer, making long computations impossible. A further difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering long computations unreliable. However, these obstacles may not be as formidable as originally believed. For any quantum computation with t gates, we show how to build a polynomial size quantum circuit that tolerates O(1 / log c t) amounts of inaccuracy and decoherence per gate, for some constant c; the previous bound was O(1 /t). We do this by showing that operations can be performed on quantum data encoded by quantum errorcorrecting codes without decoding this data. 1.
Quantum Walks on Graphs
, 2002
"... 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 meas ..."
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Cited by 126 (6 self)
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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.
Optimal lower bounds for quantum automata and random access codes
"... Consider the finite regular ¢¤£¦¥¨§�©�����©�� language ©������� �. In [3] it was shown that while this language is accepted by a deterministic finite automaton of ������ � size, any oneway quantum finite automaton (QFA) for it has ���¤ � £��� � ����£� � size. This was based on the fact that the e ..."
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Cited by 120 (9 self)
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Consider the finite regular ¢¤£¦¥¨§�©�����©�� language ©������� �. In [3] it was shown that while this language is accepted by a deterministic finite automaton of ������ � size, any oneway quantum finite automaton (QFA) for it has ���¤ � £��� � ����£� � size. This was based on the fact that the evolution of a QFA is required to be reversible. When arbitrary intermediate measurements are allowed, this intuition breaks down. Nonetheless, we show ���� � £�� a lower bound for such QFA ¢ £ for, thus also improving the previous bound. The improved bound is obtained from simple entropy arguments based on Holevo’s theorem [8]. This method also allows us to obtain an asymptotically op���������������� � timal bound for the dense quantum codes (random access codes) introduced in [3]. We then turn to Holevo’s theorem, and show that in typical situations, it may be replaced by a tighter and more transparent inprobability bound.
Succinct Quantum Proofs for Properties of Finite Groups
 In Proc. IEEE FOCS
, 2000
"... In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NPtype proof. Specifically, we consider quantum proofs for properties of blackbox groups, which are finite g ..."
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Cited by 83 (3 self)
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In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NPtype proof. Specifically, we consider quantum proofs for properties of blackbox groups, which are finite groups whose elements are encoded as strings of a given length and whose group operations are performed by a group oracle. We prove that for an arbitrary group oracle there exist succinct (polynomiallength) quantum proofs for the Group NonMembership problem that can be checked with small error in polynomial time on a quantum computer. Classically this is impossibleit is proved that there exists a group oracle relative to which this problem does not have succinct proofs that can be checked classically with bounded error in polynomial time (i.e., the problem is not in MA relative to the group oracle constructed). By considering a certain subproblem of the Group NonMembership problem we obtain a simple proof that there exists an oracle relative to which BQP is not contained in MA. Finally, we show that quantum proofs for nonmembership and classical proofs for various other group properties can be combined to yield succinct quantum proofs for other group properties not having succinct proofs in the classical setting, such as verifying that a number divides the order of a group and verifying that a group is not a simple group.
Adiabatic quantum computation is equivalent to standard quantum computation
 SIAM Journal on Computing
"... 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 convention ..."
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Cited by 80 (12 self)
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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 twodimensional 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. 1
Parallelization, Amplification, and Exponential Time Simulation of Quantum Interactive Proof Systems
 In Proceedings of the 32nd ACM Symposium on Theory of Computing
, 2000
"... In this paper we consider quantum interactive proof systems, which are interactive proof systems in which the prover and verier may perform quantum computations and exchange quantum information. We prove that any polynomialround quantum interactive proof system with twosided bounded error can be p ..."
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Cited by 77 (19 self)
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In this paper we consider quantum interactive proof systems, which are interactive proof systems in which the prover and verier may perform quantum computations and exchange quantum information. We prove that any polynomialround quantum interactive proof system with twosided bounded error can be parallelized to a quantum interactive proof system with exponentially small onesided error in which the prover and verier exchange only 3 messages. This yields a simplied proof that PSPACE has 3message quantum interactive proof systems. We also prove that any language having a quantum interactive proof system can be decided in deterministic exponential time, implying that singleprover quantum interactive proof systems are strictly less powerful than multipleprover classical interactive proof systems unless EXP = NEXP. 1. INTRODUCTION Interactive proof systems were introduced by Babai [3] and Goldwasser, Micali, and Racko [17] in 1985. In the same year, Deutsch [10] gave the rst for...
Twoway finite automata with quantum and classical states
"... We introduce 2way finite automata with quantum and classical states (2qcfa's). This is a variant on the 2way quantum finite automata (2qfa) model which may be simpler to implement than unrestricted 2qfa's; the internal state of a 2qcfa may include a quantum part that may be in a (mixed ..."
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Cited by 60 (0 self)
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We introduce 2way finite automata with quantum and classical states (2qcfa's). This is a variant on the 2way quantum finite automata (2qfa) model which may be simpler to implement than unrestricted 2qfa's; the internal state of a 2qcfa may include a quantum part that may be in a (mixed) quantum state, but the tape head position is required to be classical. We show two languages for which 2qcfa's are better than classical 2way automata. First, 2qcfa's can recognize palindromes, a language that cannot be recognized by 2way deterministic or probabilistic finite automata. Second, in polynomial time 2qcfa's can recognize fa n b n
The Quantum Communication Complexity of Sampling
 SIAM J. COMPUT
, 1998
"... Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f : X 1} and a probability distribution over X Y , we define the sampling complexity of (f, as the minimum number of bits Alice and Bob must communica ..."
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Cited by 55 (3 self)
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Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f : X 1} and a probability distribution over X Y , we define the sampling complexity of (f, as the minimum number of bits Alice and Bob must communicate for Alice to pick x X and Bob to pick y Y as well as a value z such that the resulting distribution of (x, y, z) is close to the distribution (D, f(D)). In this paper
Zeroknowledge against quantum attacks
 STOC
, 2006
"... This paper proves that several interactive proof systems are zeroknowledge against general quantum attacks. This includes the wellknown GoldreichMicaliWigderson classical zeroknowledge protocols for Graph Isomorphism and Graph 3Coloring (assuming the existence of quantum computationally conce ..."
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Cited by 54 (0 self)
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This paper proves that several interactive proof systems are zeroknowledge against general quantum attacks. This includes the wellknown GoldreichMicaliWigderson classical zeroknowledge protocols for Graph Isomorphism and Graph 3Coloring (assuming the existence of quantum computationally concealing commitment schemes in the second case). Also included is a quantum interactive protocol for a complete problem for the complexity class of problems having "honest verifier" quantum statistical zeroknowledge proofs, which therefore establishes that honest verifier and general quantum statistical zeroknowledge are equal: QSZK = QSZKHV. Previously no nontrivial proof systems were known to be zeroknowledge against quantum attacks, except in restricted settings such as the honestverifier and common reference string models. This paper therefore establishes for the first time that true zeroknowledge is indeed possible in the presence of quantum information and computation.