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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...
Quantum cryptanalysis of hidden linear functions
 in Proceedings of Crypto’95, Lecture Notes in Comput. Sci. 963
, 1995
"... Abstract. Recently there has been a great deal of interest in the power of \Quantum Computers " [4, 15, 18]. The driving force is the recent beautiful result of Shor that shows that discrete log and factoring are solvable in random quantum polynomial time [15]. We use a method similar to Shor&a ..."
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Cited by 74 (0 self)
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Abstract. Recently there has been a great deal of interest in the power of \Quantum Computers " [4, 15, 18]. The driving force is the recent beautiful result of Shor that shows that discrete log and factoring are solvable in random quantum polynomial time [15]. We use a method similar to Shor's to obtain a general theorem about quantum polynomial time. We show that any cryptosystem based on what we refer to as a `hidden linear form ' can be broken in quantum polynomial time. Our results imply that the discrete log problem is doable in quantum polynomial time over any group including Galois elds and elliptic curves. Finally, we introduce the notion of `junk bits ' which are helpful when performing classical computations that are not injective. 1
Improved simulation of stabilizer circuits
 Phys. Rev. Lett
"... The GottesmanKnill theorem says that a stabilizer circuit—that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates—can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. • By removing the need for Gaussian elimination, we ..."
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Cited by 66 (6 self)
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The GottesmanKnill theorem says that a stabilizer circuit—that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates—can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. • By removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freelyavailable program called CHP (CNOTHadamardPhase), which can handle thousands of qubits easily. • We show that the problem of simulating stabilizer circuits is complete for the classical complexity class ⊕L, which means that stabilizer circuits are probably not even universal for classical computation. • We give efficient algorithms for computing the inner product between two stabilizer states, putting any nqubit stabilizer circuit into a “canonical form ” that requires at most O ( n 2 /log n) gates, and other useful tasks. • We extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of nonstabilizer gates, and circuits acting on general tensorproduct initial states but containing only a limited number of measurements. 1
Quantum automata and quantum grammars
 Theoretical Computer Science
"... Abstract. To study quantum computation, it might be helpful to generalize structures from language and automata theory to the quantum case. To that end, we propose quantum versions of finitestate and pushdown automata, and regular and contextfree grammars. We find analogs of several classical the ..."
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Cited by 52 (2 self)
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Abstract. To study quantum computation, it might be helpful to generalize structures from language and automata theory to the quantum case. To that end, we propose quantum versions of finitestate and pushdown automata, and regular and contextfree grammars. We find analogs of several classical theorems, including pumping lemmas, closure properties, rational and algebraic generating functions, and Greibach normal form. We also show that there are quantum contextfree languages that are not contextfree. 1
FaultTolerant Error Correction with Efficient Quantum Codes
, 1996
"... We exhibit a simple, systematic procedure for detecting and correcting errors using any of the recently reported quantum errorcorrecting codes. The procedure is shown explicitly for a code in which one qubit is mapped into five. The quantum networks obtained are fault tolerant, that is, they can fu ..."
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Cited by 51 (4 self)
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We exhibit a simple, systematic procedure for detecting and correcting errors using any of the recently reported quantum errorcorrecting codes. The procedure is shown explicitly for a code in which one qubit is mapped into five. The quantum networks obtained are fault tolerant, that is, they can function successfully even if errors occur during the error correction. Our construction is derived using a recently introduced grouptheoretic framework for unifying all known quantum codes.
Quantum multiprover interactive proof systems with limited prior entanglement
 Journal of Computer and System Sciences
"... This paper gives the first formal treatment of a quantum analogue of multiprover interactive proof systems. In quantum multiprover interactive proof systems there can be two natural situations: one is with prior entanglement among provers, and the other does not allow prior entanglement among prov ..."
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Cited by 46 (5 self)
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This paper gives the first formal treatment of a quantum analogue of multiprover interactive proof systems. In quantum multiprover interactive proof systems there can be two natural situations: one is with prior entanglement among provers, and the other does not allow prior entanglement among provers. This paper focuses on the latter situation and proves that, if provers do not share any prior entanglement each other, the class of languages that have quantum multiprover interactive proof systems is equal to NEXP. It implies that the quantum multiprover interactive proof systems without prior entanglement have no gain to the classical ones. This result can be extended to the following statement of the cases with prior entanglement: if a language L has a quantum multiprover interactive proof system allowing at most polynomially many prior entangled qubits among provers, L is necessarily in NEXP. Another interesting result shown in this paper is that, in the case the prover does not have his private qubits, the class of languages that have singleprover quantum interactive proof systems is also equal to NEXP. Our results are also of importance in the sense of giving exact correspondances between quantum and classical complexity classes, because there have been known only a few results giving such correspondances.
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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Cited by 44 (2 self)
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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
Quantum summation with an application to integration
, 2001
"... We study summation of sequences and integration in the quantum model of computation. We develop quantum algorithms for computing the mean of sequences which satisfy a psummability ( condition and for d integration of functions from Lebesgue spaces Lp [0, 1] ) and analyze their convergence rates. We ..."
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Cited by 43 (11 self)
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We study summation of sequences and integration in the quantum model of computation. We develop quantum algorithms for computing the mean of sequences which satisfy a psummability ( condition and for d integration of functions from Lebesgue spaces Lp [0, 1] ) and analyze their convergence rates. We also prove lower bounds which show that the proposed algorithms are, in many cases, optimal within the setting of quantum computing. This extends recent results of Brassard, Høyer, Mosca, and Tapp (2000) on computing the mean for bounded sequences and complements results of Novak (2001) on integration of functions from Hölder classes.
Quantum MerlinArthur Proof Systems: Are Multiple Merlins More Helpful to Arthur?
, 2008
"... This paper introduces quantum “multipleMerlin”Arthur proof systems in which Arthur receives multiple quantum proofs that are unentangled with each other. Although classical multiproof systems are obviously equivalent to classical singleproof systems (i.e., usual MerlinArthur proof systems), it ..."
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Cited by 41 (8 self)
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This paper introduces quantum “multipleMerlin”Arthur proof systems in which Arthur receives multiple quantum proofs that are unentangled with each other. Although classical multiproof systems are obviously equivalent to classical singleproof systems (i.e., usual MerlinArthur proof systems), it is unclear whether or not quantum multiproof systems collapse to quantum singleproof systems (i.e., usual quantum MerlinArthur proof systems). This paper presents a necessary and sufficient condition under which the number of quantum proofs is reducible to two. It is also proved that, in the case of perfect soundness, using multiple quantum proofs