## Quantum complexity theory (1993)

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Venue: | in Proc. 25th Annual ACM Symposium on Theory of Computing, ACM |

Citations: | 483 - 5 self |

### BibTeX

@INPROCEEDINGS{Bernstein93quantumcomplexity,

author = {Ethan Bernstein and Umesh Vazirani},

title = {Quantum complexity theory},

booktitle = {in Proc. 25th Annual ACM Symposium on Theory of Computing, ACM},

year = {1993},

pages = {11--20}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. In this paper we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing machine in Deutsch’s model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97–117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specified. We prove that O(log T) bits of precision suffice to support a T step computation. This justifies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church–Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class BPP. The class BQP of languages that are efficiently decidable (with small error-probability) on a quantum Turing machine satisfies BPP ⊆ BQP ⊆ P ♯P. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.

### Citations

850 | A fast quantum mechanical algorithm for database search
- Grover
- 1996
(Show Context)
Citation Context ...lative to a permutation oracle chosen uniformly at random, with probability 1, the class NP # co--NP cannot be solved on a QTM in time o(2 n/3 ). The former bound is tight since recent work of Grover =-=[28]-=- shows how to accept the class NP relative to any oracle on a quantum computer in time O(2 n/2 ). Several designs have been proposed for realizing quantum computers [17, 23, 31]. A number of authors h... |

809 | Algorithms for quantum computation: discrete logarithms and factoring
- Shor
- 1994
(Show Context)
Citation Context ...s not address the nondeterminism issue). More importantly, Simon’s paper also introduced an important new technique which was one of the ingredients in a remarkable result proved subsequently by Shor =-=[37]-=-. Shor gave polynomial time quantum algorithms for the factoring and discrete log problems. These two problems have been well studied, and their presumed intractability forms the basis of much of mode... |

658 | Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer
- Deutsch
- 1985
(Show Context)
Citation Context ...r we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing Machine in Deutsch's model of a quantum Turing Machine =-=[20]-=-. This construction is substantially more complicated than the corresponding construction for classical Turing Machines - in fact, even simple primitives such as looping, branching and composition are... |

462 |
Logical Reversability of Computation
- Bennett
- 1973
(Show Context)
Citation Context ...hey concluded that as far as deterministic computation is concerned, the only additional constraint imposed by quantum mechanics is that the computation must be reversible, and therefore by Bennett's =-=[7]-=- work it follows that quantum computers are at least as powerful as classical computers. The issue of the extra computational power of quantum mechanics over probabilistic computers was first raised b... |

397 | Simulating physics with computers
- Feynman, Shor
- 1982
(Show Context)
Citation Context ...ows that quantum computers are at least as powerful as classical computers. The issue of the extra computational power of quantum mechanics over probabilistic computers was first raised by by Feynman =-=[25]-=- in 1982. In that paper, Feynman pointed out a very curious problem: the natural simulation of a quantum physical system on a probabilistic Turing Machine requires an exponential slowdown. Moreover, i... |

353 | On the power of quantum computation
- Simon
- 1997
(Show Context)
Citation Context ...in n o(log n) time on a probabilistic Turing Machine with any fixed error probability ! 1=2. A detailed discussion about the implications of these oracle results is in the introduction to x8.4. Simon =-=[39]-=- subsequently strengthened our result in the time parameter by proving the existence of an oracle relative to which a certain problem can be solved in polynomial time on a quantum Turing Machine, but ... |

342 |
Rapid solution of problems by quantum computation
- Deutsch, Jozsa
- 1992
(Show Context)
Citation Context ...the special case of linearly bounded quantum cellular automata [24, 45]. Finally, several researchers have explored the computational power of quantum Turing Machines. Early work by Deutsch and Jozsa =-=[22]-=- showed how to exploit some inherently quantum mechanical features of QTMs. Their results, in conjunction with subsequent results by Berthiaume and Brassard [12, 13], established the existence of orac... |

315 | Vaizirani U., Strengths and weaknesses of quantum computing
- Bennett, Bernstein, et al.
- 1997
(Show Context)
Citation Context ...ss there is a major breakthrough in complexity theory. It is natural to ask whether quantum Turing Machines can solve every problem in NP in polynomial time. Bennett, Bernstein, Brassard and Vazirani =-=[9]-=- give evidence showing the limitations of quantum Turing Machines. They show that relative to an oracle chosen uniformly at random, with probability 1, the class NP cannot be solved on a quantum Turin... |

309 |
Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes
- Babai, Moran
- 1988
(Show Context)
Citation Context ...eplaced by a symbol from \Sigma i other than $ and #. The first phase of the simulation will be handled by the machine M 1 with state set given by the union of sets Q, Q \Theta Q \Theta \Sigma \Theta =-=[1; 4]-=-, Q \Theta [5; 7], and fq a ; q b g. Its start state will be q a and the final state q f . 58 The transitions of M 1 are defined as follows. First, we mark the position of the tape head of M in the st... |

276 | Quantum circuit complexity, in
- Yao
- 1993
(Show Context)
Citation Context ...rk [11] left open the question of whether standard variants of a quantum Turing Machine, such as machines with multiple tapes or with modified tape access, are more powerful than the basic model. Yao =-=[46]-=- showed that these models are polynomially equivalent to the basic model, as are quantum circuits (which were introduced in [21]. The efficiency of Yao's simulation has been improved in [10] to show t... |

262 | Good quantum error-correcting codes exist
- Calderbank, Shor
- 1996
(Show Context)
Citation Context ... have been a sequence of important results showing how to implement quantum errorcorrecting codes and also use these codes to make quantum algorithms (quite) robust against the effects of decoherence =-=[16]-=- and [38]. Quantum computation touches upon the foundations of both computer science and quantum physics. The nature of quantum physics was clarified by the Einstein-Podolsky-Rosen paradox and Bell's ... |

246 |
Quantum computational networks
- Deutsch
- 1989
(Show Context)
Citation Context ...stic except for a single kind of quantum coin-flip. The decomposition of an arbitrary unitary transformation into a product of simple unitary transformations is similar to work carried out by Deutsch =-=[21]-=-. Deutsch's work, although phrased in terms of quantum computation networks, can be viewed as showing that a d dimensional unitary transformation can be decomposed into a product of transformations wh... |

222 |
Quantum mechanical computers
- Feynman
- 1985
(Show Context)
Citation Context ...x n ; y n ) = (x n ; U y n ). Taking limits, we get that ( Ux; y) = (x; U y), as desired. As an aside, we should briefly mention that another resolution of this issue is achieved by following Feynman =-=[26]-=-, who suggested that if we use a quantum mechanical system with Hamiltonian U +U , then the resulting system has a local, time invariant Hamiltonian. It is easy to probabilistically recover the comput... |

219 | Conservative Logic - Fredkin, Toffoli - 1982 |

216 |
Various techniques used in connection with random digits
- Neumann
- 1951
(Show Context)
Citation Context ... work showing that probabilistic computation can be carried out in a such a way that it is so insensitive to the transition probabilities that they can be allowed to vary arbitrarily in a large range =-=[34, 44, 47]-=-. In this paper, we show in a similar sense, that quantum Turing Machines are discrete devices: the transition amplitudes need only be accurate to O(log T ) bits of precision to support T steps of com... |

207 |
Proof verification and intractability of approximation problems
- Arora, Lund, et al.
- 1998
(Show Context)
Citation Context ...sical probabilistic TMs with bounded error probability [11]. One could ask what the relevance of these oracle results is, in view of the non-relativizing results on probabilistically checkable proofs =-=[36, 3]. Moreover-=-, Arora et al. [2] make the case that the fact that the P versus NP question relativizes does not imply that the question "cannot be resolved by current techniques in complexity theory". On ... |

204 | Quantum computing
- Shor
- 1998
(Show Context)
Citation Context ...n a sequence of important results showing how to implement quantum errorcorrecting codes and also use these codes to make quantum algorithms (quite) robust against the effects of decoherence [16] and =-=[38]-=-. Quantum computation touches upon the foundations of both computer science and quantum physics. The nature of quantum physics was clarified by the Einstein-Podolsky-Rosen paradox and Bell's inequalit... |

145 | Two bit gates are universal for quantum computation
- DiVincenzo
- 1995
(Show Context)
Citation Context ...ce recent work of Grover [28] shows how to accept the class NP relative to any oracle on a quantum computer in time O(2 n=2 ). Several designs have been proposed for realizing quantum computers [17], =-=[23]-=-. A number of authors have argued that there are fundamental problems in building quantum computers, most notably the effects of the decoherence of quantum superpositions, or the entanglement of the s... |

136 | Quantum Mechanics - Cohen-Tannoudji, Diu, et al. - 1977 |

132 |
Elementary gates for quantum computation, Phys. Rev. A 52
- Barenco, Bennett, et al.
- 1995
(Show Context)
Citation Context ...ion can itself be e#ciently computed given the desired unitary transformation as input. For more recent work on the e#cient simulation of a unitary transformation by a quantum computation network see =-=[5]-=- and the references therein. 6.1. Measuring errors in approximated transformations. In this section, we will deal with operators (linear transformations) on finite-dimensional Hilbert spaces. It is of... |

113 | Oracle quantum computing
- Berthiaume, Brassard
- 1994
(Show Context)
Citation Context ...hines. Early work by Deutsch and Jozsa [22] showed how to exploit some inherently quantum mechanical features of QTMs. Their results, in conjunction with subsequent results by Berthiaume and Brassard =-=[12, 13]-=-, established the existence of oracles under which there are computational problems that QTMs can solve in polynomial time with certainty, whereas if we require a classical probabilistic Turing machin... |

109 | Quantum computability
- Adleman, Demarrais, et al.
- 1997
(Show Context)
Citation Context ...is su#cient to physically realize a simple quantum operation on a single bit (in addition to maintaining coherence and carrying out deterministic operations, of course). Adleman, DeMarrais, and Huang =-=[1]-=- and Solovay and Yao [40] have further clarified this point by showing that quantum coin flips with amplitudes 3 5 and 4 5 are su#cient for universal quantum computation. Quantum computation is necess... |

93 | Time/space tradeoffs for reversible computation - Bennett - 1989 |

74 |
A potentially realizable quantum computer
- Lloyd
- 1993
(Show Context)
Citation Context ...ght since recent work of Grover [28] shows how to accept the class NP relative to any oracle on a quantum computer in time O(2n/2 ). Several designs have been proposed for realizing quantum computers =-=[17, 23, 31]-=-. A number of authors have argued that there are fundamental problems in building quantum computers, most notably the effects of the decoherence of quantum superpositions or the entanglement of the sy... |

59 |
General weak random sources
- Zuckerman
- 1990
(Show Context)
Citation Context ... work showing that probabilistic computation can be carried out in a such a way that it is so insensitive to the transition probabilities that they can be allowed to vary arbitrarily in a large range =-=[34, 44, 47]-=-. In this paper, we show in a similar sense, that quantum Turing Machines are discrete devices: the transition amplitudes need only be accurate to O(log T ) bits of precision to support T steps of com... |

53 | The Quantum Challenge to Structural Complexity Theory
- Berthiaume, Brassard
(Show Context)
Citation Context ...hines. Early work by Deutsch and Jozsa [22] showed how to exploit some inherently quantum mechanical features of QTMs. Their results, in conjunction with subsequent results by Berthiaume and Brassard =-=[12, 13]-=-, established the existence of oracles under which there are computational problems that QTMs can solve in polynomial time with certainty, whereas if we require a classical probabilistic Turing machin... |

46 |
Is quantum mechanics useful
- Landauer
- 1995
(Show Context)
Citation Context ... argued that there are fundamental problems in building quantum computers, most notably the effects of the decoherence of quantum superpositions or the entanglement of the system with the environment =-=[14, 18, 29, 35, 42]-=-. Very recently, there has been a sequence of important results showing how to implement quantum error-correcting codes and also how to use these codes to make quantum algorithms (quite) robust agains... |

34 |
Maintaining coherence in quantum computers, Phys
- Unruh
- 1995
(Show Context)
Citation Context ... argued that there are fundamental problems in building quantum computers, most notably the effects of the decoherence of quantum superpositions or the entanglement of the system with the environment =-=[14, 18, 29, 35, 42]-=-. Very recently, there has been a sequence of important results showing how to implement quantum error-correcting codes and also how to use these codes to make quantum algorithms (quite) robust agains... |

32 | Quantum computation and dissipation
- Palma, Suominen, et al.
- 1996
(Show Context)
Citation Context ...e argued that there are fundamental problems in building quantum computers, most notably the e#ects of the decoherence of quantum superpositions or the entanglement of the system with the environment =-=[14, 18, 29, 35, 42]-=-. Very recently, there has been a sequence of important results showing how to implement quantum error-correcting codes and also how to use these codes to make quantum algorithms (quite) robust agains... |

30 | Learning DNF over the uniform distribution using a quantum example oracle
- Bshouty, Jackson
- 1999
(Show Context)
Citation Context ...gests a broader definition, which may be useful in other contexts. For example, oracles that perform more general, non-Boolean unitary operations have been considered in computational learning theory =-=[15]-=-, and used to obtain large separations [32] between quantum and classical relativized complexity classes. See [9] for a discussion of more general definitions of oracle quantum computing. 8.4 Fourier ... |

21 | Quantum computers, factoring, and decoherence
- Chuang, Laflamme, et al.
- 1995
(Show Context)
Citation Context ...e argued that there are fundamental problems in building quantum computers, most notably the e#ects of the decoherence of quantum superpositions or the entanglement of the system with the environment =-=[14, 18, 29, 35, 42]-=-. Very recently, there has been a sequence of important results showing how to implement quantum error-correcting codes and also how to use these codes to make quantum algorithms (quite) robust agains... |

19 | Bicontinuous Extensions of Invertible Combinatorial Functions - Toffoli |

19 |
Random polynomial time is equal to semi-random polynomial time
- Vazirani, Vazirani
- 1988
(Show Context)
Citation Context ... work showing that probabilistic computation can be carried out in a such a way that it is so insensitive to the transition probabilities that they can be allowed to vary arbitrarily in a large range =-=[34, 44, 47]-=-. In this paper, we show in a similar sense, that quantum Turing Machines are discrete devices: the transition amplitudes need only be accurate to O(log T ) bits of precision to support T steps of com... |

16 |
Simulating physics with computers, Internat
- Feynman
- 1982
(Show Context)
Citation Context ...ollows that quantum computers are at least as powerful as classical computers. The issue of the extra computational power of quantum mechanics over probabilistic computers was first raised by Feynman =-=[25]-=- in 1982. In that paper, Feynman pointed out a very curious problem: the natural simulation of a quantum physical system on a probabilistic TM requires an exponential slowdown. Moreover, it is unclear... |

14 | A decision procedure for unitary linear quantum cellular automata
- Dürr, Santha
- 1996
(Show Context)
Citation Context ...t represents many overlapping sites of activity into a product of simple, local unitary transformations. This problem has been solved in the special case of linearly bounded quantum cellular automata =-=[24, 45]-=-. Finally, several researchers have explored the computational power of QTMs. Early work by Deutsch and Jozsa [22] showed how to exploit some inherently quan1414 ETHAN BERNSTEIN AND UMESH VAZIRANI tum... |

13 |
Quantum mechanical computers, Found
- Feynman
- 1986
(Show Context)
Citation Context ..., y n ) = (x n , U # y n ). Taking limits, we get that (sUx, y) = (x,sU # y), as desired. As an aside, we should briefly mention that another resolution of this issue is achieved by following Feynman =-=[26]-=-, who suggested that if we use a quantum mechanical system with Hamiltonian U +U # , then the resulting system has a local, time invariant Hamiltonian. It is easy to probabilistically recover the comp... |

13 | The stabilisation of quantum computations
- Berthiaume, Deutsch, et al.
- 1994
(Show Context)
Citation Context ... argued that there are fundamental problems in building quantum computers, most notably the effects of the decoherence of quantum superpositions or the entanglement of the system with the environment =-=[14, 18, 29, 35, 42]-=-. Very recently, there has been a sequence of important results showing how to implement quantum error-correcting codes and also how to use these codes to make quantum algorithms (quite) robust agains... |

11 | Time/Space Trade-Os for Reversible Computation - Bennett - 1989 |

10 |
A 1-tape 2-symbol reversible Turing machine
- Morita, Shirasaki, et al.
- 1989
(Show Context)
Citation Context ...versible TM that computes the same function. For both constructions, he used a multi-tape TM and also suggested how the simulation could be carried out using only a single-tape machine. Morita et.al. =-=[33]-=- use Bennett's ideas, and some further techniques, to show that any deterministic TM can be simulated by a generalized reversible TM with a two symbol alphabet. We will give a slightly different simul... |

9 |
On one dimensional quantum cellular automata
- Watrous
- 1995
(Show Context)
Citation Context ...t represents many overlapping sites of activity into a product of simple, local unitary transformations. This problem has been solved in the special case of linearly bounded quantum cellular automata =-=[24, 45]-=-. Finally, several researchers have explored the computational power of quantum Turing Machines. Early work by Deutsch and Jozsa [22] showed how to exploit some inherently quantum mechanical features ... |

4 | Phase information in quantum oracle computing
- Machta
- 1996
(Show Context)
Citation Context ...eful in other contexts. For example, oracles that perform more general, non-Boolean unitary operations have been considered in computational learning theory [15], and used to obtain large separations =-=[32]-=- between quantum and classical relativized complexity classes. See [9] for a discussion of more general definitions of oracle quantum computing. 8.4 Fourier Sampling and the power of QTMs In this sect... |

3 | On the role of the Cook–Levin theorem in complexity theory. (Escript available at http//www.cs.berkeley.edu/~vazirani - Arora, Impagliazzo, et al. - 1994 |

2 |
A decision procedure for well-formed linear quantum cellular automata
- urr, eThanh, et al.
- 1997
(Show Context)
Citation Context ...t represents many overlapping sites of activity into a product of simple, local unitary transformations. This problem has been solved in the special case of linearly bounded quantum cellular automata =-=[24, 45]-=-. Finally, several researchers have explored the computational power of quantum Turing Machines. Early work by Deutsch and Jozsa [22] showed how to exploit some inherently quantum mechanical features ... |

1 |
Quantum computability, manuscript
- Adleman, DeMarrais, et al.
- 1994
(Show Context)
Citation Context ...es that it is sufficient to physically realize a simple quantum operation on a single bit (in addition to maintaining coherence and carrying out deterministic operations, of course). Adleman, et. al. =-=[1]-=- and Solovay and Yao [40] have further clarified this point by showing that quantum coin flips with amplitudes 3=5 and 4=5 are sufficient for universal quantum computation. Quantum computation is nece... |

1 |
Elementary gates for quantum computation, manuscript
- Barenco, Bennett, et al.
- 1995
(Show Context)
Citation Context ...can itself be efficiently computed given the desired unitary transformation as input. For more recent work on the efficient simulation of a unitary transformation by a quantum computation network see =-=[5]-=- and the references therein. 6.1 Measuring errors in approximated transformations In this section, we will deal with operators (linear transformations) on finite dimensional Hilbert spaces. It is ofte... |

1 |
Quantum Hamiltonian models of Turing machines
- Benioff
- 1982
(Show Context)
Citation Context ...serts that the Universe is quantum physical. Can we get inherently new kinds of (discrete) computing devices based on quantum physics? Early work on the computational possibilities of quantum physics =-=[6]-=- asked the opposite question: does quantum mechanic's insistence on unitary evolution restrict the class of efficiently computable problems? They concluded that as far as deterministic computation is ... |

1 |
Quantum computation using trapped cold ions, manuscript
- Chirac, Zoller
- 1995
(Show Context)
Citation Context ...ht since recent work of Grover [28] shows how to accept the class NP relative to any oracle on a quantum computer in time O(2 n=2 ). Several designs have been proposed for realizing quantum computers =-=[17]-=-, [23]. A number of authors have argued that there are fundamental problems in building quantum computers, most notably the effects of the decoherence of quantum superpositions, or the entanglement of... |

1 | Quantum computers, factoring and decoherence, manuscript - Chuang, LaFlamme, et al. - 1995 |

1 |
Searching for a needle in a haystack -- a fast quantum mechanical algorithm, manuscript
- Grover
- 1995
(Show Context)
Citation Context ...tion oracle chosen uniformly at random, with probability 1, the class NP " co--NP cannot be solved on a quantum Turing Machine in time o(2 n=3 ). The former bound is tight since recent work of Gr=-=over [28]-=- shows how to accept the class NP relative to any oracle on a quantum computer in time O(2 n=2 ). Several designs have been proposed for realizing quantum computers [17], [23]. A number of authors hav... |

1 | Quantum computers and dissipation, manuscript - Palma, Suominen, et al. - 1995 |