## Quantum Computation (1999)

Venue: | In Annual Review of Computational Physics VI, D. Stauffer, Ed., World Scientific |

Citations: | 17 - 0 self |

### BibTeX

@INPROCEEDINGS{Aharonov99quantumcomputation,

author = {Dorit Aharonov},

title = {Quantum Computation},

booktitle = {In Annual Review of Computational Physics VI, D. Stauffer, Ed., World Scientific},

year = {1999},

pages = {259--346},

publisher = {Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool, capable of performing tasks which seem intractable for classical computers. This review is about to tell the story of theoretical quantum computation. I left out the developing topic of experimental realizations of the model, and neglected other closely related topics which are quantum information and quantum communication. As a result of narrowing the scope of this paper, I hope it has gained the benefit of being an almost self contained introduction to the exciting field of quantum computation. The review begins with background on theoretical computer science, Turing machines and Boolean circuits. In light of these models, I define quantum computers, and discuss the issue of universal quantum gates. Quantum algorithms, including Shor’s factorization algorithm and Grover’s algorithm for searching databases, are explained. I will devote much attention to understanding what the origins of the quantum computational power are, and what the limits of this power are. Finally, I describe the recent theoretical results which show that quantum computers maintain their complexity power even in the presence of noise, inaccuracies and finite precision. This question cannot be separated from that of quantum complexity, because any realistic model will inevitably be subject to such inaccuracies. I tried to put all results in their context, asking what the implications to other issues in computer science and physics are. In the end of this review I make these connections explicit, discussing the possible implications of quantum computation on fundamental physical questions, such as the transition from quantum to classical physics. 1

### Citations

1950 | Sloane The Theory of Error-Correcting Codes - MacWilliams, A - 1977 |

878 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
- Shor
- 1997
(Show Context)
Citation Context ... the most important quantum algorithm known today: Shor's algorithm. Shor's algorithm (1994) is a polynomial quantum algorithm for factoring integers, and for finding the logarithm over a finite field=-=[172]-=-. For both problems, the best known classical algorithms are exponential. However, there is no proof that classical efficient algorithms do not exist. Shor's result is regarded as extremely important ... |

689 |
Quantum Theory: Concepts and Methods
- Peres
- 1993
(Show Context)
Citation Context ...um mechanics such as Hilbert spaces, Schrodinger equation and measurements I recommend to consult the books by Sakurai[167], and by Cohen-Tanoudji[71]. As for more advanced material, the book by Peres=-=[161]-=- would be a good reference. However, I will give here all the necessary definitions. A quantum circuit is a system built of two state quantum particles, called qubits. We will work with n qubits, the ... |

650 | Quantum theory, the Church-Turing principle, and the universal quantum computer
- Deutsch
- 1985
(Show Context)
Citation Context ...wn that P ` QP and BPP ` BQP , as we will see very soon. 16 3 The Model of Quantum Computation Deutsch was the first to define a rigorous model of quantum computation, first of quantum Turing machines=-=[78]-=- and then of quantum circuits[79]. I will describe first the model of quantum circuits, which is much simpler. At the end of the chapter, I present the model of quantum Turing machines, for completene... |

614 | Molecular computation of solutions to combinatorial problems
- Adleman
- 1994
(Show Context)
Citation Context ...ht seem to contradict this thesis at first sight. One such model is the DNA computer which enables a solution of NP - complete problems (these are hard problems to be defined later) in polynomial time=-=[4, 140]-=-. However, the cost of the solution is exponential because the number of molecules in the system grows exponentially with the size of the computation. Vergis et al[194] suggested a machine which seems... |

506 |
On the Einstein-Podolsky-Rosen paradox
- Bell
- 1964
(Show Context)
Citation Context ...ut is some superposition which describes quantum correlations between these particles. Such particles are said to be quantumly entangled. The Einstein Podolski Rosen paradox[89], and Bell inequalities=-=[25, 26, 68, 108]-=-, correspond to this puzzling quantum feature by which a quantum particle does not have a state of its own. Because of the entanglement or quantum correlations between the n quantum particles, the sta... |

479 | Quantum complexity theory
- Bernstein, Vazirani
- 1997
(Show Context)
Citation Context ...f excitement all over the world. It is important that the quantum computation power does not rely on unreasonable precision but a polynomial amount of precision in the computational elements is enough=-=[38]-=-. This means that the new model requires physically reasonable resources, in terms of time, space, and precision, but yet it is (possibly) exponentially stronger than the ordinary model of probabilist... |

457 |
Logical reversibility of computation
- Bennett
(Show Context)
Citation Context ...which is inevitable in irreversible operations) and as a solution to the "maxwell demon" paradox[136, 30, 31, 121]. It turns out that any classical function can be represented as a reversibl=-=e function[137, 29]-=- on a few more bits, and the computation of f can be made reversible without losing much in efficiency. Moreover, if f can be computed classically by polynomially many elementary reversible steps, the... |

426 |
A mathematical theory of communication, Bell Syst
- Shannon
- 1948
(Show Context)
Citation Context ...he original state of the k qubits can be recovered. k=n tends asymptotically to a constant transmission rate which is non zero. This is analogous to Shannon's result from noisy classical communication=-=[171]-=-. Many different examples of quantum error correcting codes followed[181, 134, 59, 131, 165, 138], and a group theoretical framework for most quantum codes was established[55, 54, 106]. Resilient quan... |

422 | Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Physical Review Letters 70
- Bennett, Brassard, et al.
- 1993
(Show Context)
Citation Context ...blishes the existence of correlations between quantum particles, which are stronger than any classical model can explain. Another "quantum pearl" which builds on quantum entanglement, is tel=-=eportation[34]-=-. This is an amazing quantum recipe which enables two parties (Alice and Bob) which are far apart, to transfer an unknown quantum state of a particle in Alice's hands onto a particle in Bob's hand, wi... |

360 |
Probabilistic logics and the synthesis of reliable organisms from unreliable components
- Neumann
- 1956
(Show Context)
Citation Context ...evitably suffer from inaccuracies. Will physical realizations of the model of quantum computation still be as powerful as the ideal model? In classical computation, it was already shown by von-Neumann=-=[153]-=- how to compute when the elements of the computation are faulty, using redundant information. Indeed, nowadays error corrections are seldom used in computers because of extremely high reliability of t... |

356 | On the power of quantum computation
- Simon
- 1994
(Show Context)
Citation Context ...e restriction of exactness. Deutsch and Jozsa made use of the most powerful tool in quantum algorithms, the Fourier transform, which indeed manifests interference and exponentiality. Simon's algorithm=-=[177]-=- uses similar techniques, and was the seed for the most important quantum algorithm known today: Shor's algorithm. Shor's algorithm (1994) is a polynomial quantum algorithm for factoring integers, and... |

340 |
Rapid solution of problems by quantum computation
- Deutsch, Jozsa
- 1992
(Show Context)
Citation Context ...the system yields the correct output. The first quantum algorithm which combines interference and exponentiality to solve a problem faster than classical computers, was discovered by Deutsch and Jozsa=-=[80]. This algorithm add-=-resses the problem we have encountered before in connection with probabilistic algorithms: Distinguish between "constant" and "balanced" databases. The quantum algorithm solves thi... |

313 | Strengths and weaknesses of quantum computing
- Bennett, Bernstein, et al.
- 1997
(Show Context)
Citation Context ...tum complexity power and classical complexity power when the two models are allowed to have access to an oracle, i.e. a black box which can compute a certain (possibly difficult) function in one step =-=[38, 36, 40, 41]-=-. In fact, the result of Bernstein and Vazirani[38] from 1993 demonstrating a superpolynomial gap between quantum and classical computational comlexity with an access to a certain oracle initialized t... |

297 | Quantum Mechanics helps in searching for a needle in a haystack. Phys
- Grover
- 1997
(Show Context)
Citation Context ...uring machine. As such, it is the only model which really threatens the modern Church thesis. There are a few major developing directions of research in the area of quantum computation. In 1995 Grover=-=[110]-=- discovered an algorithm which searches an unsorted database of N items and finds a specific item in p N time steps. This result is surprising, because intuitively, one cannot search the database with... |

278 | Quantum circuit complexity
- Yao
- 1997
(Show Context)
Citation Context ...s, has a "uniform" and "non-uniform" versions. Again, we will restrict ourselves to the uniform model, i.e. quantum circuits which can be designed in polynomial time on a classical=-= Turing Machine. Yao[202]-=- showed that uniform quantum circuits are polynomially equivalent to quantum Turing machines, by a proof which is surprisingly complicated. This proof enables us the freedom of choosing whichever mode... |

263 |
An unsolvable problem of elementary number theory
- Church
(Show Context)
Citation Context ...o which direction it will move and what will be the new machine 's state at time t + 1. The Turing machine model seems to capture the entire concept of computability, according to the following thesis=-=[62]: 2 Church-=- Turing Thesis: A Turing machine can compute any function computable by a reasonable physical device What does "reasonable physical device" mean? This thesis is a physical statement, and as ... |

256 |
Introduction to Coding Theory
- Lint
- 1998
(Show Context)
Citation Context ...plies that the distance between two possible code words is at least 2d+ 1, so that each word is corrected uniquely. For an introduction to the subject of classical error correcting codes, see van Lint=-=[139]-=-. 57 We define a quantum code in a similar way. The state of k qubits is mapped into the state of m qubits. The term logical state is used for the original state of the k qubits. We say that such a co... |

245 |
Quantum computational networks
- Deutsch
- 1989
(Show Context)
Citation Context ... we will see very soon. 16 3 The Model of Quantum Computation Deutsch was the first to define a rigorous model of quantum computation, first of quantum Turing machines[78] and then of quantum circuits=-=[79]-=-. I will describe first the model of quantum circuits, which is much simpler. At the end of the chapter, I present the model of quantum Turing machines, for completeness. For background on basic quant... |

232 | Quantum error-correction via codes over GF(4
- Calderbank, Rains, et al.
- 1998
(Show Context)
Citation Context ... classical communication[171]. Many different examples of quantum error correcting codes followed[181, 134, 59, 131, 165, 138], and a group theoretical framework for most quantum codes was established=-=[55, 54, 106]-=-. Resilient quantum computation is more complicated than simply protecting quantum information which is sent through a noisy quantum channel. Naturally, to protect the information we would compute on ... |

222 |
Quantum Mechanical Computers
- Feynman
- 1985
(Show Context)
Citation Context ...modern Church thesis, since they require exponential physical resources. However, note that all the suggestions mentioned above rely on classical physics. In the early 80's Benioff[27, 28] and Feynman=-=[94]-=- started to discuss the question of whether computation can be done in the scale of quantum physics. In classical computers, the elementary information unit is a bit, i.e. a value which is either 0 or... |

219 |
Scheme for reducing decoherence in quantum computer memory
- Shor
- 1995
(Show Context)
Citation Context ...g classical error correction codes techniques in the quantum setting. Shor was the first to present a scheme that reduces the affect of noise and inaccuracies, building on the discretization of errors=-=[173]-=-. As in classical error correcting codes, quantum states of k qubits are encoded on states of more qubits. Spreading the state of a few qubits on more qubits, allows correction of the information, if ... |

213 |
NP is as easy as detecting unique solutions
- Valiant, Vazirani
- 1986
(Show Context)
Citation Context ...e expect this 47 probability to be almost one! There are several ways to generalize this algorithm to the general case where the number of "good" items is not known. One is a known classical=-= reduction[192]. Another generaliza-=-tion was suggested in [44]. This suggestion not only finds a "good" item regardless of what the number, t, of "good" items is, but also gives a good estimation of t. The idea is th... |

206 |
Quantum Noise
- Gardiner, Zoller
- 1999
(Show Context)
Citation Context ...ation. Errors, that might occur, will behave, presumably, according to the same law of constant probability for error per element per time step. Perhaps the most severe problem was that of decoherence=-=[151, 184, 205, 156, 100]-=-. Decoherence is the physical process, in which quantum system lose some of their quantum characteristics due to interactions with environment. Such interactions are inevitable because no system can b... |

204 | Fault-tolerant quantum computation
- Shor
- 1996
(Show Context)
Citation Context ...n in a distributed manner, such that each qubit can effect only a small number of other qubits. Kitaev[124] showed how to perform the computation of error correction with faulty gates. Shor discovered=-=[174]-=- how to perform a general computation in the presence of noise, under the unphysical assumption that the noise decreases (slowly) with the size of the computation. A more physically reasonable assumpt... |

192 | Fault-Tolerant Quantum Computation With Constant Error Rate,” ArXiv e-prints
- Aharonov, Ben-Or
- 1999
(Show Context)
Citation Context ...ise, we apply a concatenation of Shor's scheme. We encode the state once, and then encode the encoded state, and so on for for several levels. This technique enabled the proof of the threshold theorem=-=[127, 128, 107, 5, 125, 162]-=-, which asserts that it is possible to perform resilient quantum computation for as long as we wish, if the noise is smaller than a certain threshold. Decoherence and imprecision are therefore no long... |

163 |
Decoherence and the transition from quantum to classical. Phys. Today
- Zurek
- 1991
(Show Context)
Citation Context ...mation is extremely fragile, due to inevitable interactions between the system and its environment. These interactions cause the system to lose part of its quantum nature, a process called decoherence=-=[184, 205]-=-. In addition, quantum elementary operations (called gates) will inevitably suffer from inaccuracies. Will physical realizations of the model of quantum computation still be as powerful as the ideal m... |

162 |
Experimental test of Bell’s inequalities using time varying analyzers
- Aspect, Dalibard, et al.
- 1982
(Show Context)
Citation Context ... is difficult to create and preserve[65]. 10 So far, entangled pairs of photons were created successfully[133, 185], and entanglement features such as violation of Bell inequalities were demonstrated =-=[10, 11]-=-. Even entangled pairs of atoms were created[114]. However quantum computation is advantageous only when macroscopically many particles are entangled[118, 6], a task which seems impossible as of now. ... |

153 |
A class of quantum error-correcting codes saturating the quantum hamming bound
- Gottesman
- 1996
(Show Context)
Citation Context ...t flips or only phase flips occur than one qubit can be encoded on less than 5 qubits. The theory of quantum error correcting codes has further developed. A group theoretical structure was discovered =-=[54, 55, 105, 106, 129, 175]-=-, which most of the known quantum error correcting codes obey. Codes that obey this structure are called stabilizer codes[105, 106], and their group theoretical structure gives a recipe for constructi... |

148 |
On the degree of boolean functions as real polynomials
- Nisan, Szegedy
- 1994
(Show Context)
Citation Context ...cept X i = 1. Since the function which the algorithm computes is OR, this is a contradiction. Beals et. al.[23] recently generalized the above result building on classical results by Nisan and Szegedi=-=[154]-=-. Beals et.al. compare the minimal number of queries to the oracle which are needed in a quantum algorithm, with the minimal number of queries which are needed in a classical algorithm. Let us denote ... |

146 | Two-bit gates are universal for quantum computation
- DiVincenzo
- 1995
(Show Context)
Citation Context ... elementary gate, which operates on three qubits[79]. Bernstein and Vazirani[38] gave another proof of universality in terms of QTM . It was then shown by DiVincenzo that two-qubit gates are universal=-=[84]-=-. This is an important result, since it seems impossible to control interactions between three particles, whereas two particle interactions are likely to be much easier to implement. It was a surprisi... |

142 |
Perfect quantum error correction code
- Laflamme, Miquel, et al.
- 1996
(Show Context)
Citation Context ...lly to a constant transmission rate which is non zero. This is analogous to Shannon’s result from noisy classical communication[171]. Many different examples of quantum error correcting codes followed=-=[181, 134, 59, 131, 165, 138]-=-, and a group theoretical framework for most quantum codes was established[55, 54, 106]. Resilient quantum computation is more complicated than simply protecting quantum information which is sent thro... |

142 |
A single quantum cannot be cloned. Nature 299, 802. Discussion P. Marcer (BCS Cybernetic Machine Group
- Wootters, Zurek
- 1982
(Show Context)
Citation Context ...t of quantum noise is necessary. Physicists were pessimistic about the question of whether such a correction method exists[135, 189]. The reason is that quantum information in general cannot be cloned=-=[83, 200, 20]-=-, and so the information cannot be simply protected by redundancy, as is done classically. Another problem is that in contrast to the discreteness of digital computers, a quantum system can be in a su... |

136 | Multiple particle interference and quantum error correction
- Steane
- 1996
(Show Context)
Citation Context ...k qubits are encoded on states of more qubits. Spreading the state of a few qubits on more qubits, allows correction of the information, if part of it has been contaminated. These ideas were extended =-=[53, 180]-=- to show that a quantum state of k qubits can be encoded on n qubits, such that if the n qubits are sent through a noisy channel, the original state of the k qubits can be recovered. k=n tends asympto... |

130 |
Elementary gates for quantum computation”, Phys
- Barenco, Bennett, et al.
- 1995
(Show Context)
Citation Context ...ll operations. Barenco[13] and Deutsch et.al[81] showed that almost any two-bit gate is universal (See also Lloyd [141, 143]). An improvement of DiVincenzo's result was achieved later by Barenco et.al=-=[16]-=-, where it was shown that the classical controlled not gate, together with all one-qubit gates construct a universal set as well! In fact, one 1-qubit gate and the controlled not gate will do. This is... |

121 |
The Computer as a Physical System: A Microscopic Quantum Mechanical Hamiltonian Model of Computers as Represented by Turing Machines
- Benioff
- 1980
(Show Context)
Citation Context ...terexamples for the modern Church thesis, since they require exponential physical resources. However, note that all the suggestions mentioned above rely on classical physics. In the early 80's Benioff=-=[27, 28]-=- and Feynman[94] started to discuss the question of whether computation can be done in the scale of quantum physics. In classical computers, the elementary information unit is a bit, i.e. a value whic... |

114 | Oracle quantum computing
- Berthiaume, Brassard
- 1994
(Show Context)
Citation Context ...tum complexity power and classical complexity power when the two models are allowed to have access to an oracle, i.e. a black box which can compute a certain (possibly difficult) function in one step =-=[38, 36, 40, 41]-=-. In fact, the result of Bernstein and Vazirani[38] from 1993 demonstrating a superpolynomial gap between quantum and classical computational comlexity with an access to a certain oracle initialized t... |

114 |
Tight bounds on quantum searching. Fortschritte der Physik
- Boyer, Brassard, et al.
- 1998
(Show Context)
Citation Context ...nnot search the database without going through all the items. Grover's solution is quadratically better than any possible classical algorithms, and was followed by numerous extensions and applications=-=[44, 111, 112, 87, 47, 48]-=-, all achieving polynomial advantage over classical algorithms. A promising new branch in quantum complexity theory is the study of a class of problems which is the quantum analog of the complexity cl... |

114 |
Reversible Computing
- Toffoli
- 1980
(Show Context)
Citation Context ...ts to m+ n bits: f : i 7\Gamma! f(i) + f r : (i; j) 7\Gamma! (i; f(i) \Phi j): (15) Applying this method, for example, to the logical AND gate, (a; b) 7\Gamma! ab it will become the known Toffoli gate=-=[186]-=- (a; b; c) 7\Gamma! (a; b; c \Phi ab); which is described by the unitary matrix on three qubits: T = 0 B B B B B B B B B B B B @ 1 1 1 1 1 1 0 1 1 0 1 C C C C C C C C C C C C A (16) The Toffoli gate a... |

96 | Complexity limitations on quantum computations
- Fortnow, Rogers
- 1990
(Show Context)
Citation Context ...centive is analyzing the complexity power of quantum computers. For this the set suggested by Solovay and by Adleman et. al. seems more appropriate. (Fortnow recently reported on bounds using this set=-=[95]-=-). We will see that for error correction purposes, we will need a completely different universal set of gates. An important question should arise here. If our computer is built using one set, how can ... |

92 |
Time/space trade-offs for reversible computation
- Bennett
- 1989
(Show Context)
Citation Context ...rocess of conversion to reversible operations, each gate is replaced by a gate operating on more qubits. This means that making circuits reversible costs in adding a linear number of extra qubits. In =-=[32]-=-, Bennett used a nice pebbling argument, to show that the space cost can be decreased to a logarithmic factor with only a minor cost in time: T 7\Gamma! T 1+ffl . Thus the above conversion to quantum ... |

92 | Grover’s quantum searching algorithm is optimal
- Zalka
- 1999
(Show Context)
Citation Context ...assical 2 n , which simply goes over all the 2 n items in the database. Grover's algorithm provides a quadratic advantage over any possible classical algorithm, which is optimal, due to Bennett et.al.=-=[36, 44, 204]-=-, a result which I will discuss when dealing with quantum lower bounds in section 10. Let me now describe several variants on Grover's algorithm, all using Grover's iteration as the basic step. (These... |

85 | Bulk spin resonance quantum computation
- Gershenfeld, Chuang
- 1997
(Show Context)
Citation Context ...ication was successfully tested[116, 147]. Bouwmeester et. al. have recently reported on experimental realization of quantum teleportation[43] . Suggestions for implementations of quantum computation =-=[63, 74, 104, 142, 85, 117, 37, 145, 117, 160, 163, 182]-=- include quantum dots, cold trapped ions and nuclear magnetic resonance, and some of these suggestions were already implemented [150, 187, 147, 104, 75]. Unfortunately, these implementations were so f... |

84 | A framework for fast quantum mechanical algorithms. To appear
- Grover
- 1998
(Show Context)
Citation Context ...nnot search the database without going through all the items. Grover's solution is quadratically better than any possible classical algorithms, and was followed by numerous extensions and applications=-=[44, 111, 112, 87, 47, 48]-=-, all achieving polynomial advantage over classical algorithms. A promising new branch in quantum complexity theory is the study of a class of problems which is the quantum analog of the complexity cl... |

80 |
Universal quantum simulator
- Lloyd
- 1996
(Show Context)
Citation Context ...es which is universal. Note that one qubit gate is certainly not enough to construct all operations. Barenco[13] and Deutsch et.al[81] showed that almost any two-bit gate is universal (See also Lloyd =-=[141, 143]-=-). An improvement of DiVincenzo's result was achieved later by Barenco et.al[16], where it was shown that the classical controlled not gate, together with all one-qubit gates construct a universal set... |

78 |
Quantum mechanical Hamiltonian models of Turing machines
- Benioff
- 1982
(Show Context)
Citation Context ...terexamples for the modern Church thesis, since they require exponential physical resources. However, note that all the suggestions mentioned above rely on classical physics. In the early 80's Benioff=-=[27, 28]-=- and Feynman[94] started to discuss the question of whether computation can be done in the scale of quantum physics. In classical computers, the elementary information unit is a bit, i.e. a value whic... |

74 |
Thermodynamics of computation- a review
- Bennett
(Show Context)
Citation Context ...letely different problems, e.g. the problem of whether computation can be done without generating heat (which is inevitable in irreversible operations) and as a solution to the "maxwell demon&quo=-=t; paradox[136, 30, 31, 121]-=-. It turns out that any classical function can be represented as a reversible function[137, 29] on a few more bits, and the computation of f can be made reversible without losing much in efficiency. M... |

69 |
Leiserson C, Rivest R, “Introduction to Algorithms
- Cormen
- 1998
(Show Context)
Citation Context ... error. This gave rise to much more rapid solutions to different problems, which make use of random coin flips, such as the Miller-Rabin randomized algorithm to test whether an integer is prime or not=-=[73]. Here is -=-a simple example of the advantage of probabilistic algorithms: we have access to a database of N bits, and we are told that they are either all equal, ("constant") or half are 0 and half are... |

68 | Quantum Computation
- DiVincenzo
- 1995
(Show Context)
Citation Context ...ication was successfully tested[116, 147]. Bouwmeester et. al. have recently reported on experimental realization of quantum teleportation[43] . Suggestions for implementations of quantum computation =-=[63, 74, 104, 142, 85, 117, 37, 145, 117, 160, 163, 182]-=- include quantum dots, cold trapped ions and nuclear magnetic resonance, and some of these suggestions were already implemented [150, 187, 147, 104, 75]. Unfortunately, these implementations were so f... |

66 |
Quantum computation of Fourier transforms over symmetric groups
- BEALS
- 1997
(Show Context)
Citation Context ...Fourier transforms over non-Abelian groups would be helpful tools, however they are much more complicated operations since the Fourier coefficients are complex matrices, and not complex numbers! Beals=-=[22]-=- made the first (and only) step in this direction by discovering an efficient quantum Fourier transform algorithm for the non-Abelian permutations group, S n , building on the classical FFT over S n [... |