is called the \u03c0 8or T gate CNOT 1 1 1 1 in the two qubit 10 11 204

# Is called the π 8or t gate cnot 1 1 1 1 in the two

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is called the π 8 or T-gate , CNOT = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 in the two-qubit basis | 00 , | 01 , | 10 , | 11 .
204 ERIC C. ROWELL AND ZHENGHAN WANG The CNOT gate is called controlled-NOT because the first qubit is the control bit, so that when it is | 0 , nothing is done to the second qubit, but when it is | 1 , the NOT gate is applied to the second qubit. Definition 3.3. (1) An n -qubit quantum circuit over a gate set S is a map U L : ( C 2 ) n ( C 2 ) n composed of finitely many matrices of the form Id p g Id q , where g ∈ S and p, q can be 0. (2) A gate set is universal if the collection of all n -qubit circuits forms a dense subset of SU(2 n ) for any n . The gate set S = { H, σ 1 / 4 z , CNOT } will be called the standard gate set, which we will use unless stated otherwise. Theorem 3.2. (1) The standard gate set is universal. (2) Every matrix in U (2 n ) can be eﬃciently approximated up to an overall phase by a circuit over S . Note that (2) means that eﬃcient approximations of unitary matrices follows from approximations for free due to the Solovay–Kitaev theorem; see [105]. 3.4. Complexity class BQP. Let C T be the class of computable functions C T , and let P be the subclass of functions eﬃciently computable by a Turing machine. 20 Defining a computing model X is the same as selecting a class of computable func- tions from C T , denoted X P . The class X P of eﬃciently computable functions codifies the computational power of computing machines in X . Quantum com- puting selects a new class BQP–bounded-error quantum polynomial time. The class BQP consists of those problems that can be solved eﬃciently by a quantum computer. In theoretical computer science, separation of complexity classes is ex- tremely hard, as the millennium problem P vs. NP problem shows. It is generally believed that the class BQP does not contain NP-complete problems. Therefore, good target problems will be those NP problems which are not known to be NP complete. Three candidates are factoring integers, graph isomorphism, and finding the shortest vector in lattices. Definition 3.4. Let S be any finite universal gate set with eﬃciently computable matrix entries. A problem f : { 0 , 1 } → { 0 , 1 } (represented by { f n } : Z n 2 Z m ( n ) 2 ) is in BQP (i.e., can be solved eﬃciently by a quantum computer) if there exist polynomials a ( n ) , g ( n ) : N N satisfying n + a ( n ) = m ( n ) + g ( n ) and a classical eﬃcient algorithm to output a map δ ( n ) : N → { 0 , 1 } describing a quantum circuit U δ ( n ) over S of size O (poly( n )) such that U δ ( n ) | x, 0 a ( n ) = I a I | I , | I = | f n ( x ) ,z | a I | 2 3 4 , where z Z g ( n ) 2 . 20 More precisely, this should be denoted as FP since P usually denotes the subset of decision problems.
MATHEMATICS OF TOPOLOGICAL QUANTUM COMPUTING 205 The a ( n ) qubits are an ancillary working space, so we initialize an input | x by appending a ( n ) zeros and identify the resulting bit string as a basis vector in ( C 2 ) ( n + a ( n )) . The g ( n ) qubits are garbage. The classical algorithm takes as input the length n and returns a description of the quantum circuit U δ ( n ) . For a given | x , the probability that the first m ( n ) bits of the output equal f n ( x ) is 3 4 .

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