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PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
 SIAM J. on Computing
, 1997
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 1278 (4 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
Algorithms for Quantum Computation: Discrete Logarithms and Factoring
, 1994
"... A computer is generally considered to be a universal computational device; i.e., it is believed able to simulate any physical computational device with a cost in computation time of at most a polynomial factol: It is not clear whether this is still true when quantum mechanics is taken into consider ..."
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Cited by 1107 (5 self)
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A computer is generally considered to be a universal computational device; i.e., it is believed able to simulate any physical computational device with a cost in computation time of at most a polynomial factol: It is not clear whether this is still true when quantum mechanics is taken into consideration. Several researchers, starting with David Deutsch, have developed models for quantum mechanical computers and have investigated their computational properties. This paper gives Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored. These two problems are generally considered hard on a classical computer and have been used as the basis of several proposed cryptosystems. (We thus give the first examples of quantum cryptanulysis.)
Elementary Gates for Quantum Computation
, 1995
"... We show that a set of gates that consists of all onebit quantum gates (U(2)) and the twobit exclusiveor gate (that maps Boolean values (x, y)to(x, x⊕y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2 n)) can be expressed as compositions of these gates. We in ..."
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Cited by 279 (11 self)
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We show that a set of gates that consists of all onebit quantum gates (U(2)) and the twobit exclusiveor gate (that maps Boolean values (x, y)to(x, x⊕y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2 n)) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized DeutschToffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two and threebit quantum gates, the asymptotic number required for nbit DeutschToffoli gates, and make some observations about the number required for arbitrary nbit unitary operations.
Quantum measurements and the Abelian stabilizer problem
"... We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor’s results [7]. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure ..."
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Cited by 195 (0 self)
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We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor’s results [7]. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group. The paper also contains a rather detailed introduction to the theory of quantum computation.
TwoBit Gates Are Universal for Quantum Computation
, 1995
"... A proof is given, which relies on the commutator algebra of the unitary Lie groups, that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit. The best previous result had shown the universality of threebit gates, by analogy to the universality of ..."
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Cited by 181 (9 self)
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A proof is given, which relies on the commutator algebra of the unitary Lie groups, that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit. The best previous result had shown the universality of threebit gates, by analogy to the universality of the Toffoli threebit gate of classical reversible computing. Twobit quantum gates may be implemented by magnetic resonance operations applied to a pair of electronic or nuclear spins. A "gearbox quantum computer" proposed here, based on the principles of atomic force microscopy, would permit the operation of such twobit gates in a physical system with very long phase breaking (i.e., quantum phase coherence) times. Simpler versions of the gearbox computer could be used to do experiments on EinsteinPodolskyRosen states and related entangled quantum states.
Oracle quantum computing
 Brassard & U.Vazirani, Strengths and weaknesses of quantum computing
, 1994
"... \Because nature isn't classical, dammit..." ..."
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Cited by 115 (8 self)
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\Because nature isn't classical, dammit..."
A functional quantum programming language
 In: Proceedings of the 20th Annual IEEE Symposium on Logic in Computer Science
, 2005
"... This thesis introduces the language QML, a functional language for quantum computations on finite types. QML exhibits quantum data and control structures, and integrates reversible and irreversible quantum computations. The design of QML is guided by the categorical semantics: QML programs are inte ..."
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Cited by 64 (10 self)
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This thesis introduces the language QML, a functional language for quantum computations on finite types. QML exhibits quantum data and control structures, and integrates reversible and irreversible quantum computations. The design of QML is guided by the categorical semantics: QML programs are interpreted by morphisms in the category FQC of finite quantum computations, which provides a constructive operational semantics of irreversible quantum computations, realisable as quantum circuits. The quantum circuit model is also given a formal categorical definition via the category FQC. QML integrates reversible and irreversible quantum computations in one language, using first order strict linear logic to make weakenings, which may lead to the collapse of the quantum wavefunction, explicit. Strict programs are free from measurement, and hence preserve superpositions and entanglement. A denotational semantics of QML programs is presented, which maps QML terms
Quantum computation in brain microtubules? The Penrose–Hameroff “Orch OR” model of consciousness.
 Philos Trans R Soc Lond Ser A, Math Phys Sci
, 1998
"... Potential features of quantum computation could explain enigmatic aspects of consciousness. The PenroseHameroff model (orchestrated objective reduction: 'Orch OR') suggests that quantum superposition and a form of quantum computation occur in microtubulescylindrical protein lattices of ..."
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Cited by 58 (13 self)
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Potential features of quantum computation could explain enigmatic aspects of consciousness. The PenroseHameroff model (orchestrated objective reduction: 'Orch OR') suggests that quantum superposition and a form of quantum computation occur in microtubulescylindrical protein lattices of the cell cytoskeleton within the brain's neurons. Microtubules couple to and regulate neurallevel synaptic functions, and they may be ideal quantum computers because of dynamical lattice structure, quantumlevel subunit states and intermittent isolation from environmental interactions. In addition to its biological setting, the Orch OR proposal differs in an essential way from technologically envisioned quantum computers in which collapse, or reduction to classical output states, is caused by environmental decoherence (hence introducing randomness). In the Orch OR proposal, reduction of microtubule quantum superposition to classical output states occurs by an objective factorRoger Penrose's quantum gravity threshold stemming from instability in Planckscale separations (superpositions) in spacetime geometry. Output states following Penrose's objective reduction are neither totally deterministic nor random, but influenced by a noncomputable factor ingrained in fundamental spacetime. Taking a modern panpsychist view in which protoconscious experience and Platonic values are embedded in Planckscale spin networks, the Orch OR model portrays consciousness as brain activities linked to fundamental ripples in spacetime geometry. Keywords: consciousness; quantum computation; objective reduction; orchestrated objective reduction (Orch OR); microtubules; brain Quantum computation and consciousness Proposals for quantum computation rely on superposed states implementing multiple computations simultaneously, in parallel, according to quantum linear superposition (see, for example,
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