## An introduction to quantum complexity theory (2000)

Venue: | Collected Papers on Quantum Computation and Quantum Information Theory |

Citations: | 22 - 0 self |

### BibTeX

@INPROCEEDINGS{Cleve00anintroduction,

author = {Richard Cleve},

title = {An introduction to quantum complexity theory},

booktitle = {Collected Papers on Quantum Computation and Quantum Information Theory},

year = {2000},

pages = {103--127},

publisher = {World Scientific}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

10903 |
Computers and Intractability: A Guide to the Theory of NP-Completeness
- Johnson
- 1977
(Show Context)
Citation Context ...uantity is 2 n . It is not known whether or not there is a polynomially-bounded circuit family for circuit satisfiability. In fact, circuit satisfiability is one of the so-called NP-complete problems =-=[19, 26]-=-, for which a polynomially-bounded circuit family would yield polynomially-bounded circuits for all problems in NP. 1.2 Quantum circuits To develop a theory of computational complexity for quantum inf... |

876 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
- Shor
- 1997
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Citation Context ...ard technicality (at least with the quantum algorithms discovered so far). Perhaps the most remarkable quantum algorithm that has been discovered to date is the factoring algorithm, due to Peter Shor =-=[44]-=-. Theorem 2 ([44]) There exists a quantum circuit family of size O(n 2 log d (n/ε)) that solves the factoring problem within accuracy ε (for some constant d). Note that this circuit size is essentiall... |

841 | A fast quantum mechanical algorithm for database search
- Grover
- 1996
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Citation Context ...n is nevertheless considerably smaller than 2 n for large values of n. The quantum algorithm is a consequence of a remarkable algorithm in the query complexity model that was discovered by Lov Grover =-=[27]-=- (explained in the next section). 2 Query complexity This is an abstract scenario which can be thought of as a game, like “twenty questions”. The goal is to determine some information by asking as few... |

650 | Quantum theory, the Church-Turing principle and the universal quantum computer
- Deutsch
- 1985
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Citation Context ...n is a transformation that is confined to a small number of bits or qubits (such as two or three). The above property is satisfied by Turing machines and circuits, and also by quantum Turing machines =-=[7, 21]-=- and quantum circuits [22, 52] (see also [39]). We shall find it most convenient to work with circuit models here. 1.1 Classical circuits For classical circuits, the basic operations can be taken as t... |

477 | Quantum complexity theory
- Bernstein, Vazirani
- 1997
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Citation Context ...n is a transformation that is confined to a small number of bits or qubits (such as two or three). The above property is satisfied by Turing machines and circuits, and also by quantum Turing machines =-=[7, 21]-=- and quantum circuits [22, 52] (see also [39]). We shall find it most convenient to work with circuit models here. 1.1 Classical circuits For classical circuits, the basic operations can be taken as t... |

455 |
Logical reversibility of computation
- Bennett
- 1973
(Show Context)
Citation Context ...ciently simulate an fa,N-query gate. Moreover, this simulation can be implemented in 16terms of quantum gates, such as not, c-not, and c 2 -not (using techniques for reversible classical computation =-=[5]-=-). By doing this simulation for each fa,N-query gate in the quantum circuit for the order-finding problem, one obtains a quantum circuit of size O(n 2 log d (n/ε)) that takes a and N as input and prod... |

383 |
Some complexity questions related to distributive computing
- Yao
- 1979
(Show Context)
Citation Context ...a protocol, where the parties send information to each other (possibly in both directions and over several rounds) until one of them (say, Bob) obtains the answer. This 19model was introduced by Yao =-=[51]-=- and has been widely studied in the classical context (see [35] for a survey). An interesting example is the equality problem, where the function is EQ, defined as EQ(x,y) = { 1 if x = y 0 if x ̸= y. ... |

355 | On the power of quantum computation
- Simon
- 1997
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Citation Context ...h the above advantage is small, there are generalizations of Deutsch’s problem for which the discrepancy between classical and quantum query complexity is much larger. One of these is Simon’s problem =-=[46]-=-, which is defined as follows. For a function f : {0,1} n → {0,1} n , define s ∈ {0,1} n to be an XOR-mask of f if: f(x) = f(y) if and only if x ⊕ y ∈ {0 n ,s} (where ⊕ is defined over {0,1} n × {0,1}... |

313 | Strengths and weaknesses of quantum computing
- Bennett, Bernstein, et al.
- 1997
(Show Context)
Citation Context ...hm that solves the search problem for f : {0,1} n → {0,1} with O( √ 2 n log(1/ε)) queries to f, and errs with probability at most ε. The efficiency of the above algorithm has been shown to be optimal =-=[6, 8, 14, 53]-=-. Clearly, Grover’s algorithm can also be used to solve the existential search problem, where the goal is just to determine whether or not there 17exists an x ∈ {0,1} n such that f(x) = 1 (a problem ... |

278 | Quantum circuit complexity
- Yao
- 1993
(Show Context)
Citation Context ... confined to a small number of bits or qubits (such as two or three). The above property is satisfied by Turing machines and circuits, and also by quantum Turing machines [7, 21] and quantum circuits =-=[22, 52]-=- (see also [39]). We shall find it most convenient to work with circuit models here. 1.1 Classical circuits For classical circuits, the basic operations can be taken as the binary ∧ (and) gate, the bi... |

245 |
Quantum computational networks
- Deutsch
- 1989
(Show Context)
Citation Context ... confined to a small number of bits or qubits (such as two or three). The above property is satisfied by Turing machines and circuits, and also by quantum Turing machines [7, 21] and quantum circuits =-=[22, 52]-=- (see also [39]). We shall find it most convenient to work with circuit models here. 1.1 Classical circuits For classical circuits, the basic operations can be taken as the binary ∧ (and) gate, the bi... |

204 | Fault-tolerant quantum computation
- Shor
- 1996
(Show Context)
Citation Context ...rorcorrecting codes and fault-tolerant techniques then even gates with constant inaccuracies (and that are subject to “decoherence”) can in principle be employed in arbitrarily large quantum circuits =-=[1, 31, 45]-=- (see [42] for an extensive review). For quantum circuit families, we must also consider the issue of uniformity: a legitimate quantum circuit family should be finitely specifiable in a compu11tation... |

192 | Fault-Tolerant Quantum Computation With Constant Error Rate,” ArXiv e-prints
- Aharonov, Ben-Or
- 1999
(Show Context)
Citation Context ...rorcorrecting codes and fault-tolerant techniques then even gates with constant inaccuracies (and that are subject to “decoherence”) can in principle be employed in arbitrarily large quantum circuits =-=[1, 31, 45]-=- (see [42] for an extensive review). For quantum circuit families, we must also consider the issue of uniformity: a legitimate quantum circuit family should be finitely specifiable in a compu11tation... |

183 | Unbiased bits from sources of weak randomness and probabilistic communication complexity
- Chor, Goldreich
- 1988
(Show Context)
Citation Context ...th of these problems are more difficult than EQ when probabilistic protocols are considered: any probabilistic protocol with error probability up to (say) 1 3 requires Ω(n) bits of communication (see =-=[15]-=- for IP, and [29] for IN ; also [35]). It is natural to ask whether any reduction in communication can be obtained by somehow using quantum information. Define a quantum communication protocol as one ... |

154 |
Quantum computations: algorithms and error correction
- Kitaev
- 1997
(Show Context)
Citation Context ...ementioned one-qubit Hadamard gate H (Eq. 4) and the two-qubit controlled-V gate (where V is defined in Eq. 8) are an example of such a set. The precise result is best stated as a theorem. Theorem 1 (=-=[33, 48]-=-) Let B be any two-qubit gate and ε > 0. Then there exists a quantum circuit of size O(log d (1/ε)) (where d is a constant) consisting of only H and controlled-V gates which computes a unitary operati... |

146 | Quantum measurements and the Abelian stabilizer problem
- Kitaev
- 1995
(Show Context)
Citation Context ... probability 1−ε using only O(log(1/ε)) queries to fa,N and O(n 2 log d (n/ε)) auxiliary gates (for some constant d). Detailed explanations of the algorithm can be found in several sources, including =-=[18, 32, 44]-=-. A significant property of the function fa,N is that there exists a classical circuit of size O(n 2 log n log log n) that takes N (an n-bit number), a ∈ {1,... ,N −1} (such that gcd(a,N) = 1), and x,... |

145 | Quantum algorithms revisited
- Cleve, Ekert, et al.
- 1998
(Show Context)
Citation Context ...o be shown that no classical algorithm exists that computes c1 with a single f-query (since it is impossible to determine f(0) ⊕ f(1) from just f(0) or f(1) alone). But the quantum circuit in Fig. 10 =-=[18, 21]-=- computes c1 with a single f-query gate. |0〉 H f H |1〉 H ♠ H Figure 10: Quantum circuit for Deutsch’s problem using one query. Here the initial state of the two-qubit system is |0〉 |1〉 and its final s... |

130 |
Elementary gates for quantum computation”, Phys
- Barenco, Bennett, et al.
- 1995
(Show Context)
Citation Context ...cted from this quantum state by a measurement). The three-qubit operation that is simulated in Fig. 6 is a so-called Toffoli gate (also called a controlled-controlled-not, or c 2 -not for short). See =-=[3, 23, 47]-=- for some similar constructions. For classical circuits, there are finite sets of gates which are universal in the sense that they can be used to simulate any other set of gates. For quantum circuits,... |

127 | Quantum vs. classical communication and computation
- Buhrman, Cleve, et al.
- 1998
(Show Context)
Citation Context ...n2 → {0,1}, and n1,n2 are implicit parameters satisfying n1+n2 = n. By a suitable recursive application of Grover’s algorithm for OR, this problem can be solved with O( √ 2 n n log(1/ε)) queries to f =-=[13]-=- (the extra factor of √ n is to amplify the accuracy of the bottom level algorithm for AND). In fact, one can extend the above to k alternations of quantifiers: OR-AND- · · ·-Q(f) = (∃x1)(∀x2) · · · (... |

114 |
Tight bounds on quantum searching. Fortschritte der Physik
- Boyer, Brassard, et al.
- 1998
(Show Context)
Citation Context ...d to err with probability (say) 1 3 . Lov Grover [27] discovered a remarkable quantum algorithm that accomplishes this with O( √ 2n ) queries (some detailed explanations of the algorithm are found in =-=[8, 27, 37]-=-). Grover’s result, with some later refinements [8, 10, 14, 37, 54] incorporated into it, is summarized as follows. Theorem 6 ([27]) There is a quantum algorithm that solves the search problem for f :... |

107 |
On distinguishing prime numbers from composite numbers
- Adleman, Pomerance, et al.
- 1983
(Show Context)
Citation Context ...or of x be produced. It turns out that the smallest currently-known uniform circuit family for this problem has size O(n dlog log n ) (for some constant d), which is shy of being polynomially-bounded =-=[2]-=-. There exist probabilistic circuit families that solve primality testing more efficiently. A probabilistic circuit is one that can flip coins during its execution, and the evolution of the computatio... |

101 | Quantum counting
- Brassard, Høyer, et al.
(Show Context)
Citation Context ...vered a remarkable quantum algorithm that accomplishes this with O( √ 2n ) queries (some detailed explanations of the algorithm are found in [8, 27, 37]). Grover’s result, with some later refinements =-=[8, 10, 14, 37, 54]-=- incorporated into it, is summarized as follows. Theorem 6 ([27]) There is a quantum algorithm that solves the search problem for f : {0,1} n → {0,1} with O( √ 2 n log(1/ε)) queries to f, and errs wit... |

94 | Complexity limitations on quantum computation
- Fortnow, Rogers
- 1998
(Show Context)
Citation Context ...pace that is polynomial in S(n) and T(n) (but still with an exponential number of operations), and there are also more esoteric computational models that subsume the power of quantum circuit families =-=[25]-=-. Regarding the circuit satisfiability problem, it is currently unknown whether or not there exists a polynomially-bounded quantum circuit family that solves it. What is known is that quantum algorith... |

92 | Grover’s quantum searching algorithm is optimal
- Zalka
- 1999
(Show Context)
Citation Context ...hm that solves the search problem for f : {0,1} n → {0,1} with O( √ 2 n log(1/ε)) queries to f, and errs with probability at most ε. The efficiency of the above algorithm has been shown to be optimal =-=[6, 8, 14, 53]-=-. Clearly, Grover’s algorithm can also be used to solve the existential search problem, where the goal is just to determine whether or not there 17exists an x ∈ {0,1} n such that f(x) = 1 (a problem ... |

84 | An exact quantum polynomial-time algorithm for Simon's problem
- Brassard, Hyer
- 1997
(Show Context)
Citation Context ... with only O(n log(1/ε)) queries to f (see [46] for the details). There is also a refinement to Simon’s original algorithm that makes a polynomial number of queries and solves Simon’s problem exactly =-=[11]-=-. Although the primary resource under consideration is the number of queries, the number of auxiliary operations (i.e. the non-query gates) is also of interest, and it is desirable to bound both quant... |

79 | Exponential separation of quantum and classical communication complexity
- Raz
- 1999
(Show Context)
Citation Context ...l can solve with exponentially less communication than the best classical probabilistic protocol. The description of the problem is more complicated than EQ, IN, and IP, and the reader is referred to =-=[43]-=- for the details. The methodology used to establish Theorem 7 involved the conversion of an algorithm in the query model (for OR) to a communication protocol (for IN (x,y) = OR(fx ∧ fy)). This convers... |

58 | The hidden subgroup problem and eigenvalue estimation on a quantum computer - Mosca, Ekert - 1998 |

51 | Strong communication complexity or generating quasirandom sequences form two communicating semi-random sources - Vazirani - 1987 |

48 | A limit on the speed of quantum computation in determining parity. Quantum Physics e-Print archive
- Farhi, Goldstone, et al.
(Show Context)
Citation Context ...ity model (explained in the next section) is to show that this is not possible: at least Ω(2n /n) queries must be made by any quantum algorithm. In fact, a stronger lower bound of 1 22n is also known =-=[4, 24]-=- (using different methods). It is important to note that, although upper bounds in the query model translate into upper bounds in the computational model, the converse of this need not be true. For ex... |

44 |
Some estimates of the information transmitted by quantum communication channels. Problems of Information Transmission
- Holevo
- 1973
(Show Context)
Citation Context ...ntum information occurred in [12, 16, 20, 34, 52]. There are fundamental results in quantum information theory which imply that classical information cannot be “compressed” within quantum information =-=[28]-=-. For example, Alice cannot convey more than r classical bits of information to Bob by sending him an r-qubit message. Based on this, one might mistakenly think that there is no advantage to using qua... |

43 |
A rigorous time bound for factoring integers
- Pomerance
- 1992
(Show Context)
Citation Context ... of x. This is apparently much harder than primality testing, since the smallest currently-known circuit family for this problem is probabilistic and has size O(2 d√ n log n ) (where d is a constant) =-=[36, 41]-=-, which is far from being polynomially-bounded. One of the reasons why quantum algorithms are of interest is that there exists a quantum circuit family of polynomial-size that solves the factoring pro... |

43 |
Substituting quantum entanglement for communication
- Cleve, Buhrman
- 1997
(Show Context)
Citation Context ... the ownership of qubits). The output is then taken as the outcome of some measurement of Bob’s qubits. Various preliminary results about communication complexity with quantum information occurred in =-=[12, 16, 20, 34, 52]-=-. There are fundamental results in quantum information theory whichimply that classical information cannot be “compressed” within quantum information [28]. For example, Alice cannot convey more than r... |

32 |
A fast Monte Carlo test for primality
- Solovay, Strassen
- 1977
(Show Context)
Citation Context ...ut and is understood to emit one uniformlydistributed random bit when executed during a computation. If m random bits are required then m /c-gates can be inserted into a circuit. Solovay and Strassen =-=[49]-=- discovered a remarkable probabilistic algorithm for primality testing that can be expressed in terms of probabilistic circuits. For any ε > 0, there is a probabilistic circuit of size O(n 3 log(1/ε))... |

27 | Multiparty Quantum Communication Complexity”, Los Alamos preprint archive
- Dam, Hoyer, et al.
(Show Context)
Citation Context ... the ownership of qubits). The output is then taken as the outcome of some measurement of Bob’s qubits. Various preliminary results about communication complexity with quantum information occurred in =-=[12, 16, 20, 34, 52]-=-. There are fundamental results in quantum information theory which imply that classical information cannot be “compressed” within quantum information [28]. For example, Alice cannot convey more than ... |

26 |
Threshold Accuracy for Quantum Computation,” ArXiv e-prints
- Knill, Laflamme, et al.
- 1996
(Show Context)
Citation Context ...rorcorrecting codes and fault-tolerant techniques then even gates with constant inaccuracies (and that are subject to “decoherence”) can in principle be employed in arbitrarily large quantum circuits =-=[1, 31, 45]-=- (see [42] for an extensive review). For quantum circuit families, we must also consider the issue of uniformity: a legitimate quantum circuit family should be finitely specifiable in a compu11tation... |

25 | On Universal and Fault-Tolerant Quantum Computing: A Novel Basis and a New Constructive Proof of Universality for Shor’s Basis
- Boykin, Mor, et al.
- 1999
(Show Context)
Citation Context ...tes that is universal in the approximate sense is: H, W, and c-not, where ( 1 0 W = 0 eiπ/4 ) . (9) In fact, with W and c-not gates, one can simulate a controlled-V gate, as shown in Fig. 7 (see also =-=[9]-=-). W ✈ ✈ ✈ W ❧ W † ❧ ≡ V Figure 7: Simulation of a controlled-V gate (note: W † = W 7 ). As in the classical case, the measure of computational complexity for quantum circuits is most interesting when... |

19 |
Quantum searching, counting and amplitude amplification by eigenvector analysis
- Mosca
- 1998
(Show Context)
Citation Context ...d to err with probability (say) 1 3 . Lov Grover [27] discovered a remarkable quantum algorithm that accomplishes this with O( √ 2n ) queries (some detailed explanations of the algorithm are found in =-=[8, 27, 37]-=-). Grover’s result, with some later refinements [8, 10, 14, 37, 54] incorporated into it, is summarized as follows. Theorem 6 ([27]) There is a quantum algorithm that solves the search problem for f :... |

19 |
rigorous factorization and discrete logarithm algorithms
- Pomerance, Fast
- 1987
(Show Context)
Citation Context ... of x. This is apparently much harder than primality testing, since the smallest currently-known circuit family for this problem is probabilistic and has size O(2 d√ n log n ) (where d is a constant) =-=[36, 41]-=-, which is far from being polynomially-bounded. One of the reasons why quantum algorithms are of interest is that there exists a quantum circuit family of polynomial-size that solves the factoring pro... |

18 |
Substituting quantum entanglement for communication”, Phys. Rev. A 56
- Cleve, Buhrman
- 1997
(Show Context)
Citation Context ... the ownership of qubits). The output is then taken as the outcome of some measurement of Bob’s qubits. Various preliminary results about communication complexity with quantum information occurred in =-=[12, 16, 20, 34, 52]-=-. There are fundamental results in quantum information theory which imply that classical information cannot be “compressed” within quantum information [28]. For example, Alice cannot convey more than ... |

17 | Computational complexity of uniform quantum circuit families and quantum Turing machines”, preprint quantph/9906095
- Nishimura, Ozawa
- 1999
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Citation Context ...l number of bits or qubits (such as two or three). The above property is satisfied by Turing machines and circuits, and also by quantum Turing machines [7, 21] and quantum circuits [22, 52] (see also =-=[39]-=-). We shall find it most convenient to work with circuit models here. 1.1 Classical circuits For classical circuits, the basic operations can be taken as the binary ∧ (and) gate, the binary ∨ (or) gat... |

15 | Quantum gates and circuits
- DiVincenzo
(Show Context)
Citation Context ...cted from this quantum state by a measurement). The three-qubit operation that is simulated in Fig. 6 is a so-called Toffoli gate (also called a controlled-controlled-not, or c 2 -not for short). See =-=[3, 23, 47]-=- for some similar constructions. For classical circuits, there are finite sets of gates which are universal in the sense that they can be used to simulate any other set of gates. For quantum circuits,... |

6 | Lower bounds for quantum search and derandomization”, preprint quant-ph/9811046
- Buhrman, Wolf
- 1998
(Show Context)
Citation Context ...vered a remarkable quantum algorithm that accomplishes this with O( √ 2n ) queries (some detailed explanations of the algorithm are found in [8, 27, 37]). Grover’s result, with some later refinements =-=[8, 10, 14, 37, 54]-=- incorporated into it, is summarized as follows. Theorem 6 ([27]) There is a quantum algorithm that solves the search problem for f : {0,1} n → {0,1} with O( √ 2 n log(1/ε)) queries to f, and errs wit... |

6 |
Realizable universal quantum logic gates”, preprint
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(Show Context)
Citation Context ...cted from this quantum state by a measurement). The three-qubit operation that is simulated in Fig. 6 is a so-called Toffoli gate (also called a controlled-controlled-not, or c 2 -not for short). See =-=[3, 23, 47]-=- for some similar constructions. For classical circuits, there are finite sets of gates which are universal in the sense that they can be used to simulate any other set of gates. For quantum circuits,... |

5 |
Quantum entanglement and communication complexity”, preprint quant-ph/9705033
- Buhrman, Cleve, et al.
- 1997
(Show Context)
Citation Context ... the ownership of qubits). The output is then taken as the outcome of some measurement of Bob’s qubits. Various preliminary results about communication complexity with quantum information occurred in =-=[12, 16, 20, 34, 52]-=-. There are fundamental results in quantum information theory which imply that classical information cannot be “compressed” within quantum information [28]. For example, Alice cannot convey more than ... |

5 |
The complexity of theorem proving procedures,” Proc
- Cook
- 1971
(Show Context)
Citation Context ...uantity is 2 n . It is not known whether or not there is a polynomially-bounded circuit family for circuit satisfiability. In fact, circuit satisfiability is one of the so-called NP-complete problems =-=[19, 26]-=-, for which a polynomially-bounded circuit family would yield polynomially-bounded circuits for all problems in NP. 1.2 Quantum circuits To develop a theory of computational complexity for quantum inf... |

5 |
Realizable universal quantum logic
- Sleator, Weinfurter
(Show Context)
Citation Context ...acted from this quantum state by a measurement). The three-qubit operation that is simulated in Fig. 6 is a so-called Toffoli gate (also called a controlled-controlled-not, orc 2 -not for short). See =-=[3, 23, 47]-=- for some similar constructions. For classical circuits, there are finite sets of gates which are universal in the sense that they can be used to simulate any other set of gates. For quantum circuits,... |

4 |
Quantum Communication, MSc
- Kremer
- 1995
(Show Context)
Citation Context |

4 |
Lie groups and quantum circuits
- Solovay
- 1999
(Show Context)
Citation Context ...ementioned one-qubit Hadamard gate H (Eq. 4) and the two-qubit controlled-V gate (where V is defined in Eq. 8) are an example of such a set. The precise result is best stated as a theorem. Theorem 1 (=-=[33, 48]-=-) Let B be any two-qubit gate and ε > 0. Then there exists a quantum circuit of size O(log d (1/ε)) (where d is a constant) consisting of only H and controlled-V gates which computes a unitary operati... |

3 |
Fast quantum verification for the formulas of predicate calculus”, preprint quant-ph/9809015
- Ozhigov
- 1998
(Show Context)
Citation Context ...ending on whether k is even or odd, and f : {0,1} n1 ×· · ·×{0,1} nk → {0,1} with n1+· · ·nk = n. In this case, √ the recursive application of Grover’s technique makes O( queries to f (see [13]; also =-=[40]-=- for a related result). 2 n n k−1 log(1/ε)) 18For all of these variations of OR and AND, it can be shown that any classical algorithm for one of these problems must make Ω(2 n ) queries, and the quan... |

2 | A Grover-based quantum search of optimal order for an unknown number of marked elements”, preprint quant-ph/9902049
- Zalka
- 1999
(Show Context)
Citation Context ...vered a remarkable quantum algorithm that accomplishes this with O( √ 2n ) queries (some detailed explanations of the algorithm are found in [8, 27, 37]). Grover’s result, with some later refinements =-=[8, 10, 14, 37, 54]-=- incorporated into it, is summarized as follows. Theorem 6 ([27]) There is a quantum algorithm that solves the search problem for f : {0,1} n → {0,1} with O( √ 2 n log(1/ε)) queries to f, and errs wit... |