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On the Power of Quantum Finite State Automata
 Proceedings of the 38th IEEE Conference on Foundations of Computer Science
, 1997
"... In this paper, we introduce 1way and 2way quantum finite state automata (1qfa's and 2qfa's), which are the quantum analogues of deterministic, nondeterministic and probabilistic 1way and 2way finite state automata. We prove the following facts regarding 2qfa's. 1. For any ffl ? 0, ..."
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Cited by 105 (5 self)
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In this paper, we introduce 1way and 2way quantum finite state automata (1qfa's and 2qfa's), which are the quantum analogues of deterministic, nondeterministic and probabilistic 1way and 2way finite state automata. We prove the following facts regarding 2qfa's. 1. For any ffl ? 0, there is a 2qfa M which recognizes the nonregular language L = fa m b m j m 1g with (onesided) error bounded by ffl, and which halts in linear time. Specifically, M accepts any string in L with probability 1 and rejects any string not in L with probability at least 1 \Gamma ffl. 2. For every regular language L, there is a reversible (and hence quantum) 2way finite state automaton which recognizes L and which runs in linear time. In fact, it is possible to define 2qfa's which recognize the noncontextfree language fa m b m c m jm 1g, based on the same technique used for 1. Consequently, the class of languages recognized by linear time, bounded error 2qfa's properly includes the regular l...
An exact quantum polynomialtime algorithm for Simon's problem
 IN PROCEEDINGS OF THE 5TH ISRAELI SYMPOSIUM ON THEORY OF COMPUTING AND SYSTEMS (ISTCS'97
, 1997
"... We investigate the power of quantum computers when they are required to return an answer that is guaranteed to be correct after a time that is upperbounded by a polynomial in the worst case. We show that a natural generalization of Simon’s problem can be solved in this way, whereas previous algorit ..."
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Cited by 96 (10 self)
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We investigate the power of quantum computers when they are required to return an answer that is guaranteed to be correct after a time that is upperbounded by a polynomial in the worst case. We show that a natural generalization of Simon’s problem can be solved in this way, whereas previous algorithms required quantum polynomial time in the expected sense only, without upper bounds on the worstcase running time. This is achieved by generalizing both Simon’s and Grover’s algorithms and combining them in a novel way. It follows that there is a decision problem that can be solved in exact quantum polynomial time, which would require expected exponential time on any classical boundederror probabilistic computer if the data is supplied as a black box.
Universality in quantum computation
 Proc. R. Soc. London A
, 1995
"... We show that in quantum computation almost every gate that operates on two or more bits is a universal gate. We discuss various physical considerations bearing on the proper definition of universality for computational components such as logic gates. ..."
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Cited by 95 (2 self)
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We show that in quantum computation almost every gate that operates on two or more bits is a universal gate. We discuss various physical considerations bearing on the proper definition of universality for computational components such as logic gates.
Complexity Limitations on Quantum Computation
 Journal of Computer and System Sciences
, 1997
"... We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP.  BQP is low for PP, i.e., PP BQP = PP.  There exists a relativized world where P = ..."
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Cited by 95 (3 self)
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We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP.  BQP is low for PP, i.e., PP BQP = PP.  There exists a relativized world where P = BQP and the polynomialtime hierarchy is infinite.  There exists a relativized world where BQP does not have complete sets.  There exists a relativized world where P = BQP but P 6= UP " coUP and oneway functions exist. This gives a relativized answer to an open question of Simon.
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.
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 Algorithms for Element Distinctness
 SIAM Journal of Computing
, 2001
"... We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N) quantum upper bound for the element distinctness problem in the comparison c ..."
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Cited by 75 (9 self)
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We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N) quantum upper bound for the element distinctness problem in the comparison complexity model. This contrasts with Θ(N log N) classical complexity. We also prove a lower bound of Ω ( √ N) comparisons for this problem and derive bounds for a number of related problems. 1
Communication complexity lower bounds by polynomials
 In Proc. of the 16th Conf. on Computational Complexity (CCC
, 2001
"... The quantum version of communication complexity allows Alice and Bob to communicate qubits and/or to make use of prior entanglement (shared EPRpairs). Some lower bound techniques are available for qubit communication [17, 11, 2], but except for the inner product function [11], no bounds are known f ..."
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Cited by 68 (13 self)
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The quantum version of communication complexity allows Alice and Bob to communicate qubits and/or to make use of prior entanglement (shared EPRpairs). Some lower bound techniques are available for qubit communication [17, 11, 2], but except for the inner product function [11], no bounds are known for the model with unlimited prior entanglement. We show that the “log rank ” lower bound extends to the strongest model (qubit communication + prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the “logrank conjecture ” and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for boundederror protocols. 1
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