## Introduction to Quantum Algorithms (2001)

Citations: | 20 - 0 self |

### BibTeX

@TECHREPORT{Shor01introductionto,

author = {Peter W. Shor},

title = {Introduction to Quantum Algorithms},

institution = {},

year = {2001}

}

### OpenURL

### Abstract

Abstract. These notes discuss the quantum algorithms we know of that can solve problems significantly faster than the corresponding classical algorithms.

### Citations

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Citation Context ... convincing argument that BQP is larger than BPP, as nobody really knows the complexity of factoring. However, as factoring is a widely studied problem that is fundamental for public key cryptography =-=[35]-=-, the quantum factoring algorithm brought widespread attention to the field of quantum computing. 5. The Factoring Algorithm For factoring an L-bit number N, the best classical algorithm known is the ... |

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Citation Context ...uter science appeared before the discipline of computer science existed; in fact, even before electronic computers existed. Shortly after Gödel proved his famous incompleteness result, several papers =-=[13, 27, 32, 41]-=- were published that drew a distinction between computable and non-computable functions. These papers showed that there are some mathematically defined functions which are impossible to compute algori... |

941 | Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
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Citation Context ...f this space permits us to take the Fourier transform of an exponential length sequence. How this works will be made clearer by the following sketch of the algorithm, the full details of which are in =-=[36]-=-, along with a quantum algorithm for finding discrete logarithms. The idea behind all the fast factoring algorithms (classical or quantum) is fairly simple. To factor N, find two residues mod N such t... |

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Citation Context ...r”. Benioff [5] had already showed how quantum mechanical processes could be used as the basis of a classical Turing machine. Feynman [20] refined these ideas in a later paper. In 1985, David Deutsch =-=[15]-=- gave an abstract model of quantum computation, and also raised the question of whether quantum computers might actually be useful for classical problems. Subsequently, he and a number of other people... |

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Citation Context ...gave an abstract model of quantum computation, and also raised the question of whether quantum computers might actually be useful for classical problems. Subsequently, he and a number of other people =-=[16, 8, 39]-=- came up with rather contrived-appearing problems for which quantum computers seemed to work better than classical computers. It was by studying these algorithms, especially Dan Simon’s [39], that I f... |

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Citation Context ...s, if you can make a Toffoli gate, you can perform any reversible classical computation. Further, as long as the input is not erased, any classical computation can be efficiently performed reversibly =-=[6]-=-, and thus implemented efficiently by Toffoli gates. The matrix corresponding to a Toffoli gate is (2.2) ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0... |

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Citation Context ...gave an abstract model of quantum computation, and also raised the question of whether quantum computers might actually be useful for classical problems. Subsequently, he and a number of other people =-=[16, 8, 39]-=- came up with rather contrived-appearing problems for which quantum computers seemed to work better than classical computers. It was by studying these algorithms, especially Dan Simon’s [39], that I f... |

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Citation Context ...gave an abstract model of quantum computation, and also raised the question of whether quantum computers might actually be useful for classical problems. Subsequently, he and a number of other people =-=[16, 8, 39]-=- came up with rather contrived-appearing problems for which quantum computers seemed to work better than classical computers. It was by studying these algorithms, especially Dan Simon’s [39], that I f... |

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Citation Context ...e of an efficiently computable function) for a specific item in time O( √ N), an improvement on the optimal classical algorithm, which must look at N/2 items on average before finding a specific item =-=[25]-=-. The technique used in this algorithm can be applied to a number of other problems to also obtain a square root speed-up [26]. If you are searching an unordered database, this square root speed-up is... |

321 | Strengths and weaknesses of quantum computing
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Citation Context ...to also obtain a square root speed-up [26]. If you are searching an unordered database, this square root speed-up is as good as a quantum computer can do; this is proved using techniques developed in =-=[7]-=-. Finally, a generalization of both Grover’s search algorithm and the lower bound above gives tight bounds on how much a quantum computer can amplify a quantum procedure that has a given probability o... |

294 |
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Citation Context ...uter science appeared before the discipline of computer science existed; in fact, even before electronic computers existed. Shortly after Gödel proved his famous incompleteness result, several papers =-=[13, 27, 32, 41]-=- were published that drew a distinction between computable and non-computable functions. These papers showed that there are some mathematically defined functions which are impossible to compute algori... |

286 | Quantum circuit complexity
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Citation Context ... by studying these algorithms, especially Dan Simon’s [39], that I figured out how to design the factoring algorithm. 2. The Quantum Circuit Model In this section we discuss the quantum circuit model =-=[44]-=- for quantum computation. This is a rigorous mathematical model for a quantum computer. It is not the only mathematical model that has been proposed for quantum computation; there are also the quantum... |

277 | Good quantum error-correcting codes exist,” Phys - Calderbank, Shor - 1996 |

238 | Quantum Error Correction Via Codes Over GF(4 - Calderbank, Shor, et al. - 1998 |

231 |
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Citation Context ...trolled NOT, which negates the 3rd bit if and only if the first two are both 1. By itself the Toffoli gate is universal for reversible classical computation, as it can simulate both AND and NOT gates =-=[21]-=-. Thus, if you can make a Toffoli gate, you can perform any reversible classical computation. Further, as long as the input is not erased, any classical computation can be efficiently performed revers... |

212 | Fault-tolerant quantum computation - Shor - 1996 |

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Citation Context ...on called decoherence, which makes superpositions of states decay, and makes large-scale superpositions of states decay very quickly. A thorough, elementary, discussion of decoherence can be found in =-=[47]-=-; one reason it occurs is that we are dealing with open systems rather than closed ones. Although closed systems quantum mechanically undergo unitary evolution, open systems need not. They are subsyst... |

152 | Class of quantum error correcting codes saturating the quantum Hamming bound - Gottesman - 1996 |

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Citation Context ...he set of all one-bit gates and a specific two-bit gate called the Controlled NOT (CNOT). These gates can efficiently simulate any quantum circuits whose gates act on only a constant number of qubits =-=[4]-=-. On basis vectors, the CNOT gate negates the second (target) qubit if and only if the first (control) qubit is 1. In other words, it takes VXY to VXZ where Z = X + Y (mod 2). This corresponds to the ... |

137 | Multiple Particle Interference and Quantum Error Correction - Steane - 1996 |

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Citation Context ...amentally quantum mechanical phenomena might be used to simulate quantum mechanics much more efficiently. In much the same spirit, you could think of a wind tunnel as a “turbulence computer”. Benioff =-=[5]-=- had already showed how quantum mechanical processes could be used as the basis of a classical Turing machine. Feynman [20] refined these ideas in a later paper. In 1985, David Deutsch [15] gave an ab... |

115 | Quantum circuits with mixed states
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Citation Context ...es that do not conform to the above assumptions. However, there do not seem to be any physically realistic models which have more computational power than the ones listed above. Neither non-unitarity =-=[3]-=- nor fermions [9] add significant power to the mathematical model. Of these models, the quantum circuit model is possibly the simplest to describe. It is also easier to connect with possible physical ... |

90 | A Framework For Fast Quantum Mechanical Algorithms
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Citation Context ...m, which must look at N/2 items on average before finding a specific item [25]. The technique used in this algorithm can be applied to a number of other problems to also obtain a square root speed-up =-=[26]-=-. If you are searching an unordered database, this square root speed-up is as good as a quantum computer can do; this is proved using techniques developed in [7]. Finally, a generalization of both Gro... |

79 |
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Citation Context ...uter science appeared before the discipline of computer science existed; in fact, even before electronic computers existed. Shortly after Gödel proved his famous incompleteness result, several papers =-=[13, 27, 32, 41]-=- were published that drew a distinction between computable and non-computable functions. These papers showed that there are some mathematically defined functions which are impossible to compute algori... |

60 | Efficient simulation of quantum systems by quantum computers
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Citation Context .... This could be accomplished by showing that systems with such Hamiltonians can be efficiently simulated by a quantum computer. Some work has been done on simulating Hamiltonians on quantum computers =-=[1, 29, 45]-=-, but I do not believe this question has been completely addressed yet. 4. Simon’s Algorithm In this section, we give Dan Simon’s algorithm [39] for a problem that takes exponential time on a classica... |

54 | Parallel quantum computation
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Citation Context ...quantum computer. It is not the only mathematical model that has been proposed for quantum computation; there are also the quantum Turing machine model [8, 44] and the quantum cellular automata model =-=[31, 42]-=-. All these models result in the same class of polynomial-time quantum computable functions. These are, of course, not the only potential models for quantum computation, and some of the assumptions ma... |

42 | Simulation of many-body Fermi systems on a universal quantum computer, Physical Review Letters 79
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Citation Context .... This could be accomplished by showing that systems with such Hamiltonians can be efficiently simulated by a quantum computer. Some work has been done on simulating Hamiltonians on quantum computers =-=[1, 29, 45]-=-, but I do not believe this question has been completely addressed yet. 4. Simon’s Algorithm In this section, we give Dan Simon’s algorithm [39] for a problem that takes exponential time on a classica... |

41 |
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Citation Context ...es that occur when performing experiments that are not reflected in the quantum circuit model. This section contains a brief discussion of these issues, some of which are discussed more thoroughly in =-=[8, 18]-=-. In everyday life, objects behave very classically, and on large scales we do not see any quantum mechanical behavior. This is due to a phenomenon called decoherence, which makes superpositions of st... |

35 |
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Citation Context ...should also be classified as gates. These duplicating “gates” are not possible in the domain of quantum computing, because of the theorem that an arbitrary quantum state cannot be cloned (duplicated) =-=[17, 43]-=-. A quantum circuit is similarly built out of logical quantum wires carrying qubits, and quantum gates acting on these qubits. Each wire corresponds to one of the n qubits. We assume each gate acts on... |

27 | Bounds for small-error and zero-error quantum algorithms
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Citation Context ... a generalization of both Grover’s search algorithm and the lower bound above gives tight bounds on how much a quantum computer can amplify a quantum procedure that has a given probability of success =-=[10]-=-. The quantum search algorithm can be thought of in these terms; the procedure is just that of choosing a random element of the N-element list, so the probability of success is 1/N. A quantum computer... |

26 | Fault-tolerant quantum computation, in Introduction to quantum computation and information, edited by - Preskill - 1998 |

25 |
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Citation Context ...e gates and obtain an approximate Fourier transform which is close enough to the actual Fourier transform that it barely changes the probability that the factoring algorithm14 PETER W. SHOR succeeds =-=[14]-=-. This reduces the number of gates required for the quantum Fourier transform from O(k 2 ) to O(k log k). 6. Grover’s Algorithm Another very important algorithm in quantum computing is L. K. Grover’s ... |

25 | Fast version of Shor’s quantum factoring algorithm
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Citation Context ...e of the first register in order to make d/r likely to be the closest fraction to c/2 2L with numerator and denominator at most N. More details of this algorithm can be found in [36]. Recently, Zalka =-=[46]-=- has analyzed the resources required by this algorithm much more thoroughly, improving upon their original values in many respects. For example, he shows that you can use only 3L + o(L) qubits, wherea... |

18 |
Universal quantum simulators, Science 273
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Citation Context .... This could be accomplished by showing that systems with such Hamiltonians can be efficiently simulated by a quantum computer. Some work has been done on simulating Hamiltonians on quantum computers =-=[1, 29, 45]-=-, but I do not believe this question has been completely addressed yet. 4. Simon’s Algorithm In this section, we give Dan Simon’s algorithm [39] for a problem that takes exponential time on a classica... |

17 |
Simulating physics with computers, Internat
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Citation Context ...uter time, even while trying to solve relatively simple problems. One of these is turbulence, about which I unfortunately have nothing further to say. The other is quantum mechanics. In 1982, Feynman =-=[19]-=- argued that simulating quantum mechanics inherently required an exponential amount of overhead, so that it must take enormous amounts of computer time no matter how clever you are. This realization w... |

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Citation Context ...quantum computer. It is not the only mathematical model that has been proposed for quantum computation; there are also the quantum Turing machine model [8, 44] and the quantum cellular automata model =-=[31, 42]-=-. All these models result in the same class of polynomial-time quantum computable functions. These are, of course, not the only potential models for quantum computation, and some of the assumptions ma... |

10 | Quantum computing, Documenta Mathematica Extra Vol - Shor - 1998 |

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Citation Context ...head, so that it must take enormous amounts of computer time no matter how clever you are. This realization was come to independently, and somewhat earlier, in 1980, in the Soviet Union by Yuri Manin =-=[30]-=-. It is not true that all quantum mechanical systems are difficult to simulate; some of them have exact solutions and others have very clever computational shortcuts, but it does appear to be true whe... |

2 | Fermionic quantum computation, lanl e-print quant-ph/0003137 - Bravyi |

2 |
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Citation Context ... spirit, you could think of a wind tunnel as a “turbulence computer”. Benioff [5] had already showed how quantum mechanical processes could be used as the basis of a classical Turing machine. Feynman =-=[20]-=- refined these ideas in a later paper. In 1985, David Deutsch [15] gave an abstract model of quantum computation, and also raised the question of whether quantum computers might actually be useful for... |

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2 | Fault-tolerant quantum computation, to appear in Introduction to Quantum Computation - Preskill - 1998 |

1 |
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Citation Context ...e revised for dealing with noisy gates, an area not covered in this paper. In fact, it can be shown that with these requirements, noisy unitary gates make it impossible to carry out long computations =-=[2]-=-; some means of eliminating noise by resetting qubits to values near 0 is required. Quantum gates acting on one or two qubits (C2 or C4 ) naturally induce a transformation on the state space of the en... |

1 |
An introduction to quantum error correction, in this volume
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Citation Context ...re the quantum state becomes so noisy as to usually give the wrong answer [8]. Active error correction can improve this situation substantially; this is discussed in Gottesman’s notes for this course =-=[24]-=-. In some proposed physical architectures for quantum computers, there are restrictions that are more severe than the quantum circuit model given in the preceding section. Many of these restrictions d... |

1 |
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Citation Context ...should also be classified as gates. These duplicating “gates” are not possible in the domain of quantum computing, because of the theorem that an arbitrary quantum state cannot be cloned (duplicated) =-=[17, 43]-=-. A quantum circuit is similarly built out of logical quantum wires carrying qubits, and quantum gates acting on these qubits. Each wire corresponds to one of the n qubits. We assume each gate acts on... |

1 |
Limitations of noisy reversible computation, Also LANL e-print quant-ph/9611028 available online at http://xxx.lanl.gov
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Citation Context ...e revised for dealing with noisy gates, an area not covered in this paper. In fact, it can be shown that with these requirements, noisy unitary gates make it impossible to carry out long computations =-=[2]-=-; some means of eliminating noise by resetting qubits to values near 0 is required. Quantum gates acting on one or two qubits (C2 or C4 ) naturally induce a transformation on the state space of the en... |

1 | Fermionic quantum computation, Also LANL e-print quant-ph/0003137 available online at http://xxx.lanl.gov/. 20 - Bravyi, Kitaev - 1999 |