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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 46 (10 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
A lambda calculus for quantum computation with classical control
 IN PROCEEDINGS OF THE 7TH INTERNATIONAL CONFERENCE ON TYPED LAMBDA CALCULI AND APPLICATIONS (TLCA), VOLUME 3461 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2005
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The twoeigenvalue problem and density of Jones representation of braid groups
 Commun. Math. Phys.
, 2002
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Quantum and Classical Strong Direct Product Theorems and Optimal TimeSpace Tradeoffs
 SIAM Journal on Computing
, 2004
"... A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum ..."
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Cited by 42 (7 self)
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A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function. Our direct product theorems...
Quantum search on boundederror inputs
 In Proc. of 30th ICALP
, 2003
"... Abstract. Suppose we have n algorithms, quantum or classical, each computing some bitvalue with bounded error probability. We describe a quantum algorithm that uses O ( √ n) repetitions of the base algorithms and with high probability finds the index of a 1bit among these n bits (if there is such ..."
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Cited by 42 (6 self)
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Abstract. Suppose we have n algorithms, quantum or classical, each computing some bitvalue with bounded error probability. We describe a quantum algorithm that uses O ( √ n) repetitions of the base algorithms and with high probability finds the index of a 1bit among these n bits (if there is such an index). This shows that it is not necessary to first significantly reduce the error probability in the base algorithms to O(1/poly(n)) (which would require O ( √ nlog n) repetitions in total). Our technique is a recursive interleaving of amplitude amplification and errorreduction, and may be of more general interest. Essentially, it shows that quantum amplitude amplification can be made to work also with a boundederror verifier. As a corollary we obtain optimal quantum upper bounds of O ( √ N) queries for all constantdepth ANDOR trees on N variables, improving upon earlier upper bounds of O ( √ Npolylog(N)). 1
Toward An Architecture For Quantum Programming
, 2003
"... It is becoming increasingly clear that, if a useful device for quantum computation will ever be built, it will be embodied by a classical computing machine with control over a truly quantum subsystem, this apparatus performing a mixture of classical and quantum computation. This paper investigates ..."
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Cited by 41 (0 self)
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It is becoming increasingly clear that, if a useful device for quantum computation will ever be built, it will be embodied by a classical computing machine with control over a truly quantum subsystem, this apparatus performing a mixture of classical and quantum computation. This paper investigates a possible approach to the problem of programming such machines: a template high level quantum language is presented which complements a generic general purpose classical language with a set of quantum primitives.
Communicating quantum processes
 In POPL 2005
, 2005
"... We define a language CQP (Communicating Quantum Processes) for modelling systems which combine quantum and classical communication and computation. CQP combines the communication primitives of the picalculus with primitives for measurement and transformation of quantum state; in particular, quantum ..."
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Cited by 39 (10 self)
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We define a language CQP (Communicating Quantum Processes) for modelling systems which combine quantum and classical communication and computation. CQP combines the communication primitives of the picalculus with primitives for measurement and transformation of quantum state; in particular, quantum bits (qubits) can be transmitted from process to process along communication channels. CQP has a static type system which classifies channels, distinguishes between quantum and classical data, and controls the use of quantum state. We formally define the syntax, operational semantics and type system of CQP, prove that the semantics preserves typing, and prove that typing guarantees that each qubit is owned by a unique process within a system. We illustrate CQP by defining models of several quantum communication systems, and outline our plans for using CQP as the foundation for formal analysis and verification of combined quantum and classical systems. 1
Hidden translation and orbit coset in quantum computing
 In Proc. 35th ACM STOC
, 2003
"... We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently ..."
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Cited by 38 (6 self)
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We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Z n p, whenever p is a fixed prime. For the induction step, we introduce the problem Orbit Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful selfreducibility result: Orbit Coset in a finite group G is reducible to Orbit Coset in G/N and subgroups of N, for any solvable normal subgroup N of G. Our selfreducibility framework combined with Kuperberg’s subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group. 1
Quantum algorithms for solvable groups
 In Proceedings of the 33rd ACM Symposium on Theory of Computing
, 2001
"... ABSTRACT In this paper we give a polynomialtime quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, r ..."
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Cited by 38 (1 self)
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ABSTRACT In this paper we give a polynomialtime quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, reduce to computing orders of solvable groups and therefore admit polynomialtime quantum algorithms as well. Our algorithm works in the setting of blackbox groups, wherein none of these problems have polynomialtime classical algorithms. As an important byproduct, our algorithm is able to produce a pure quantum state that is uniform over the elements in any chosen subgroup of a solvable group, which yields a natural way to apply existing quantum algorithms to factor groups of solvable groups. 1.