## On the power of quantum computation (1998)

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Venue: | Philosophical Transactions of the Royal Society of London, Series A |

Citations: | 17 - 3 self |

### BibTeX

@ARTICLE{Vazirani98onthe,

author = {Umesh Vazirani},

title = {On the power of quantum computation},

journal = {Philosophical Transactions of the Royal Society of London, Series A},

year = {1998},

pages = {1759--1768}

}

### Years of Citing Articles

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### Abstract

This paper surveys the use of the ‘hybrid argument ’ to prove that quantum corrections are insensitive to small perturbations. This property of quantum computations is used to establish that quantum circuits are robust against inaccuracy in the implementation of its elementary gates. The insensitivity to small perturbations is also used to establish lower-bounds, including showing that relative to an oracle, the class NP requires exponential time on a quantum computer; and that quantum algorithms are polynomially related to deterministic algorithms in the black-box model.

### Citations

892 | A fast quantum-mechanical algorithm for database search
- Grover
- 1996
(Show Context)
Citation Context ...y theory, by showing that relative to a random oracle, NP 6` BQT IME(o(2 n=2 )). This result is best possible, up to constant factors, since there is a matching upper bound that follows from Grover's =-=[Gr]-=- quantum search algorithm. Boyer, et. al. [BBHT] obtain the exact constants in the upper and lower bounds, thus exhibiting a tight bound on quantum search. [BBBV] also showed that relative to a random... |

854 | Algorithms for quantum computation: Discrete logarithms and factoring
- Shor
- 1994
(Show Context)
Citation Context ...n and Vazirani [BV], building upon earlier work of Deutsch and Jozsa [DJ]. This was followed by an important group theoretic quantum algorithm by Simon [Si], and then by the remarkable result of Shor =-=[Sh]-=- showing that factoring and computing discrete logs are computable in polynomial time on a quantum computer. In view of these results it is natural to ask whether quantum computers can solve all probl... |

823 | The complexity of theorem-proving procedures
- Cook
- 1971
(Show Context)
Citation Context ...ulation and diagonalization arguments holds relative to all oracles. However, they also pointed out that there is one non-relativizing technique in complexity theory --- namely the Cook-Levin theorem =-=[Co, Le]-=-. The non-relativizing form of the Cook-Levin theorem says that any language with a polynomial time checkable proof of membership also has a checkable proof of membership. [AIV] also argue that all th... |

694 | Quantum theory, the Church-Turing principle and the universal quantum computer - Deutsch - 1985 |

491 | Quantum complexity theory
- Bernstein, Vazirani
- 1997
(Show Context)
Citation Context ...an be simulated by a probabilistic Turing Machine with at most polynomial factor simulation overhead. The first evidence that quantum computers violate this thesis was given by Bernstein and Vazirani =-=[BV]-=-, building upon earlier work of Deutsch and Jozsa [DJ]. This was followed by an important group theoretic quantum algorithm by Simon [Si], and then by the remarkable result of Shor [Sh] showing that f... |

426 | Simulating physics with computers - Feynman - 1982 |

362 | On the power of quantum computation
- Simon
- 1994
(Show Context)
Citation Context ...puters violate this thesis was given by Bernstein and Vazirani [BV], building upon earlier work of Deutsch and Jozsa [DJ]. This was followed by an important group theoretic quantum algorithm by Simon =-=[Si]-=-, and then by the remarkable result of Shor [Sh] showing that factoring and computing discrete logs are computable in polynomial time on a quantum computer. In view of these results it is natural to a... |

351 |
Rapid solution of problems by quantum computation
- Deutsch, Jozsa
- 1992
(Show Context)
Citation Context ... at most polynomial factor simulation overhead. The first evidence that quantum computers violate this thesis was given by Bernstein and Vazirani [BV], building upon earlier work of Deutsch and Jozsa =-=[DJ]-=-. This was followed by an important group theoretic quantum algorithm by Simon [Si], and then by the remarkable result of Shor [Sh] showing that factoring and computing discrete logs are computable in... |

321 | Strengths and weaknesses of quantum computing
- Bennett, Bernstein, et al.
- 1997
(Show Context)
Citation Context ...time on a quantum computer. In view of these results it is natural to ask whether quantum computers can solve all problems in the class NP (or NP " co \Gamma NP ) in polynomial time. Bennett, et.=-= al. [BBBV]-=- gave evidence that this question is unlikely to be resolved without a major breakthrough in complexity theory, by showing that relative to a random oracle, NP 6` BQT IME(o(2 n=2 )). This result is be... |

282 | Quantum circuit complexity - Yao - 1993 |

252 | Quantum computational networks - Deutsch - 1989 |

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

205 |
Proof verification and intractability of approximation problems
- Arora, Lund, et al.
- 1992
(Show Context)
Citation Context ... " co \Gamma NP 6` BQTIME(o(2 n=3 )). How significant are these oracle results in view of the recent non-relativizing results in complexity theory, such as IP = PSPACE [Sha2], and NP = PCP (log n=-=; 1) [ALMSS]-=-? Arora, et. al. [AIV] formalize the folk notion that any result in complexity theory that is proved using only simulation and diagonalization arguments holds relative to all oracles. However, they al... |

193 | Fault-Tolerant Quantum Computation With Constant Error Rate
- Ben-Or
- 1998
(Show Context)
Citation Context ...quantum error-correction techniques, fault-tolerance quantum circuits can be created that are resilient to constant error in the rotation gates of the circuit, independent of the size of the circuit (=-=Aharanov & Ben-Or 1996-=-; Gottesman 1997). How does one explain the power of quantum computation? The dimension of the Hilbert space associated with an n-qubit system is 2 n . Therefore, just describing the state of this sys... |

116 | Tight bounds on quantum searching
- Boyer, Brassard, et al.
- 1996
(Show Context)
Citation Context ... oracle, NP 6` BQT IME(o(2 n=2 )). This result is best possible, up to constant factors, since there is a matching upper bound that follows from Grover's [Gr] quantum search algorithm. Boyer, et. al. =-=[BBHT] obta-=-in the exact constants in the upper and lower bounds, thus exhibiting a tight bound on quantum search. [BBBV] also showed that relative to a random permutation oracle NP " co \Gamma NP 6` BQT IME... |

85 |
CREWPRAMs and decision trees
- Nisan
- 1991
(Show Context)
Citation Context ...ity of f, bs(f) is defined to be the maximum number t such that there exists an input x ∈{0,1} n and t disjoint subsets S1,...St ⊆{1,2,...,n}such that for all 1 � i � t, f(x) ̸= f(x (Si)). Lemma 4.2 (=-=Nisan 1989-=-). √ C(f) � bs(f) � C(f). Corollary 4.3. bs(f) � C(f) � D(f) � C(f) 2 � bs(f) 4 . Beals et al. (1998) improve this bound by showing: Lemma 4.4. D(f) � C(f)bs(f) � bs(f) 3 . We are now ready to prove t... |

75 |
Generic oracles and oracle classes
- Blum, Impagliazzo
- 1987
(Show Context)
Citation Context ...ber of bits of the input that must be revealed (by someone who knows all the input bits) to convince a deterministic algorithm about the value of f(x). A key result, that was first discovered by Blum =-=[BI]-=-, shows that in the black box model the deterministic and nondeterministic (certificate) complexity of a function are polynomially related. Recall that in the black box model we only count the number ... |

64 | Resilient Quantum Computation: Error Models and Thresholds - Knill, Laflamme, et al. - 1997 |

52 |
Theory of fault-tolerant quantum computation
- Gottesman
- 1997
(Show Context)
Citation Context ... techniques, fault-tolerance quantum circuits can be created that are resilient to constant error in the rotation gates of the circuit, independent of the size of the circuit (Aharanov & Ben-Or 1996; =-=Gottesman 1997-=-). How does one explain the power of quantum computation? The dimension of the Hilbert space associated with an n-qubit system is 2 n . Therefore, just describing the state of this system requires 2 n... |

39 | The complexity of analog computation - Vergis, Steiglitz, et al. - 1986 |

37 | Universal’nyĭe perebornyĭe zadachi (Universal search problems - Levin - 1973 |

15 |
Factoring Numbers in O(log n) Arithmetic Steps
- Shamir
- 1979
(Show Context)
Citation Context ...s the quantum computation model? There are several "unreasonable" classical models of computation, in which it is possible to factor numbers or even solve NP-complete problems in polynomial =-=time (see [Sha1]-=- for a particularly delightful example) . In each case, these models rely on being able to carry out operations on numbers with exponentially many bits in a single computational step. This is unrealis... |

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
(Show Context)
Citation Context ...E(o(2 n=3 )). How significant are these oracle results in view of the recent non-relativizing results in complexity theory, such as IP = PSPACE [Sha2], and NP = PCP (log n; 1) [ALMSS]? Arora, et. al. =-=[AIV]-=- formalize the folk notion that any result in complexity theory that is proved using only simulation and diagonalization arguments holds relative to all oracles. However, they also pointed out that th... |

3 |
Universal’nyĭe perebornyĭe zadachi
- Levin
- 1973
(Show Context)
Citation Context ...nd diagonalization arguments holds relative to all oracles. However, they also pointed out that there is one non-relativizing technique in complexity theory, namely the Cook–Levin theorem (Cook 1971; =-=Levin 1973-=-). The non-relativizing form of the Cook–Levin theorem says that any language with a polynomial-time checkable proof of membership also has a log space checkable proof of membership. Arora et al. (199... |

1 |
de Wolf "Quantum Lower Bounds by Polynomials", quantph /9802049
- Beals, Buhrman, et al.
(Show Context)
Citation Context ...m to compute a boolean function f : f0; 1g ! f0; 1g, and let Q(f) denote the number of queries made by a quantum algorithm that computes f with error probability less by 1=3. Recently, Beals, et. al. =-=[BBCMW]-=- introduced a new framework, involving polynomials, for establishing lower bounds in the black box model; they proved a general result showing that D(f) = O(Q(f) 6 ). In this paper we prove the weaker... |

1 |
CREW PRAM's and decision trees", STOC
- Nisan
- 1989
(Show Context)
Citation Context ... the C(f) 2 upper bound on the deterministic complexity is established by giving an algorithm that requires 2 C(f) steps of computation, but only C(f) 2 queries. Lemma 3 (BI) C(f)sD(f)sC(f) 2 . Nisan =-=[Ni]-=- established another fundamental result, that shows that the black box complexity of a boolean function f is closely related to a structural property of f called its block sensitivity. To define this ... |

1 | Generic oracles and oracle classes", 28th FOCS - Blum, Impagliazzo - 1987 |

1 | Generic oracles and oracle classes - Bernstein, Vazirani - 1997 |

1 |
Complex Systems Institute. (Available at InterJournal at http://interjournal.org
- Cook
- 1971
(Show Context)
Citation Context ...imulation and diagonalization arguments holds relative to all oracles. However, they also pointed out that there is one non-relativizing technique in complexity theory, namely the Cook–Levin theorem (=-=Cook 1971-=-; Levin 1973). The non-relativizing form of the Cook–Levin theorem says that any language with a polynomial-time checkable proof of membership also has a log space checkable proof of membership. Arora... |

1 | The complexity of analog computation. Math. Computers Simul. 28, 91–113. Discussion P. Marcer (BCS Cybernetic Machine Group, Keynsham, UK ). If quantum theory is regarded as a Lie transformational system (as it is in relation to quantum holography), then - Simon - 1997 |