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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.
A Quick Glance at Quantum Cryptography
, 1998
"... The recent application of the principles of quantum mechanics to cryptography has led to a remarkable new dimension in secret communication. As a result of these new developments, it is now possible to construct cryptographic communication systems which detect unauthorized eavesdropping should it oc ..."
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Cited by 11 (2 self)
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The recent application of the principles of quantum mechanics to cryptography has led to a remarkable new dimension in secret communication. As a result of these new developments, it is now possible to construct cryptographic communication systems which detect unauthorized eavesdropping should it occur, and which give a guarantee of no eavesdropping should it not occur. Contents 1 Cryptographic systems before quantum cryptography 3 2 Preamble to quantum cryptography 7 Partially supported by ARL Contract #DAAL0195P1884, ARO Grant #P38804PH QC, and the LOOP Fund. 3 The BB84 quantum cryptographic protocol without noise 10 3.1 Stage 1. Communication over a quantum channel . . . . . . . 12 3.2 Stage 2. Communication in two phases over a public channel . 14 3.2.1 Phase 1 of Stage 2. Extraction of raw key . . . . . . . 14 3.2.2 Phase 2 of Stage 2. Detection of Eve's intrusion via error detection . . . . . . . . . . . . . . . . . . . . . . 15 4 The BB84 quantum cryptographic pr...
Quantum coherent nonlinear feedback with applications to quantum optics on chip
 IEEE TRANS. AUTOM. CONTROL
, 2012
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Quantum computation with Kerrnonlinear photonic crystals
, 2008
"... In this paper, we consider a method for implementing a quantum logic gate with photons whose wave function propagates in a onedimensional Kerrnonlinear photonic crystal. The photonic crystal causes the incident photons to undergo Bragg reflection by its periodic structure of dielectric materials a ..."
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Cited by 6 (0 self)
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In this paper, we consider a method for implementing a quantum logic gate with photons whose wave function propagates in a onedimensional Kerrnonlinear photonic crystal. The photonic crystal causes the incident photons to undergo Bragg reflection by its periodic structure of dielectric materials and forms the photonic band structure, namely, the light dispersion relation. This dispersion relation reduces the group velocity of the wave function of the photons, so that it enhances nonlinear interaction of the photons. (Because variation of the group velocity against the wave vector is very steep, we have to tune up the wavelength of injected photons precisely, however.) If the photonic crystal includes layers of a Kerr medium, we can rotate the phase of the wave function of the incident photons by a large angle efficiently. We show that we can construct the nonlinear signshift (NS) gate proposed by Knill, Laflamme, and Milburn (KLM) by this method. Thus, we can construct the conditional signflip gate for two qubits, which is crucial for quantum computation. Our NS gate works with probability unity in principle while KLM’s original one is a nondeterministic gate conditioned on the detection of an auxiliary photon. 1
Quantum computation with linear optics
 Online preprint quantph/9806048), QCQC 98
, 1998
"... Abstract. We present a constructive method to translate small quantum circuits into their optical analogues, using linear components of presentday quantum optics technology only. These optical circuits perform precisely the computation that the quantum circuits are designed for, and can thus be use ..."
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Abstract. We present a constructive method to translate small quantum circuits into their optical analogues, using linear components of presentday quantum optics technology only. These optical circuits perform precisely the computation that the quantum circuits are designed for, and can thus be used to test the performance of quantum algorithms. The method relies on the representation of several quantum bits by a single photon, and on the implementation of universal quantum gates using simple optical components (beam splitters, phase shifters, etc.). The optical implementation of Brassard et al.’s teleportation circuit, a nontrivial 3bit quantum computation, is presented as an illustration. 1
Reversible Arithmetic Coding for Quantum Data Compression
 IEEE Transactions on Information Theory
"... We study the problem of compressing a block of symbols (a block quantum state) emitted by a memoryless quantum Bernoulli source. We present a simpletoimplement quantum algorithm for projecting, with high probability, the block quantum state onto the typical subspace spanned by the leading eigensta ..."
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We study the problem of compressing a block of symbols (a block quantum state) emitted by a memoryless quantum Bernoulli source. We present a simpletoimplement quantum algorithm for projecting, with high probability, the block quantum state onto the typical subspace spanned by the leading eigenstates of its density matrix. We propose a fixedrate quantum ShannonFano code to compress the projected block quantum state using a per symbol code rate that is slightly higher than the von Neumann entropy limit. Finally, we propose quantum arithmetic codes to efficiently implement quantum ShannonFano codes. Our arithmetic encoder/decoder have a cubic circuit and a cubic computational complexity in the block size. The encoder and decoder are quantummechanical inverses of each other, and constitute an elegant example of reversible quantum computation. Keywords: quantum computation, quantum information theory, quantum measurement, noiseless coding, reversible computation, Schumacher coding, a...
Generalized Quantum Search with Parallelism
, 1999
"... Abstract. We generalize Grover’s unstructured quantum search algorithm to enable it to use an arbitrary starting superposition and an arbitrary unitary matrix simultaneously. We derive an exact formula for the probability of the generalized Grover’s algorithm succeeding after n iterations. We show t ..."
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Abstract. We generalize Grover’s unstructured quantum search algorithm to enable it to use an arbitrary starting superposition and an arbitrary unitary matrix simultaneously. We derive an exact formula for the probability of the generalized Grover’s algorithm succeeding after n iterations. We show that the fully generalized formula reduces to the special cases considered by previous authors. We then use the generalized formula to determine the optimal strategy for using the unstructured quantum search algorithm. On average the optimal strategy is about 12 % better than the naive use of Grover’s algorithm. The speedup obtained is not dramatic but it illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. We extend the analysis to the case of a society of k quantum searches acting in parallel. We derive an analytic formula that connects the degree of parallelism with the optimal strategy for kparallel quantum search. We then derive the formula for the expected speed of kparallel quantum search. 1
Efficient and Exact Quantum Compression and Molecular Scale O° K Heat Engines
"... This paper is concerned with the quantum computation of a certain unitary transformation, costing a quasilinear number of elementary quantum operations (in particular, O(n(log 4 n) log log n)), which can be used for two applications: optimal quantum compression /decompression and also for the ..."
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This paper is concerned with the quantum computation of a certain unitary transformation, costing a quasilinear number of elementary quantum operations (in particular, O(n(log 4 n) log log n)), which can be used for two applications: optimal quantum compression /decompression and also for the initiation of bulk quantum computers to a pure state. The first application is to compression and decompression of a noiseless source of n quantum bits (qubits), each sampled independently from a given mixed state quantum ensemble. For such a quantum source, the compression factor obtainable by classical information theory is limited by the Shannon entropy, which in general (except in the case where the quantum ensemble has only orthogonal states) is less than the quantum compression factor given by the von Neumann entropy. The quantum encoding due to Schumacher [Sch95] is known to provide quantum compression with asymptotically optimal in size and fidelity up to that provided by th...
unknown title
, 2008
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: