nt - p h / 99 09 02 7 v 2 9 S ep 1 99 9 Non - Resonant Effects in Implementation of Quantum Shor Algorithm

We simulate Shor's algorithm on an Ising spin quantum computer. The influence of non-resonant effects is analyzed in detail. It is shown that our " 2πk "-method successfully suppresses non-resonant effects even for relatively large values of the Rabi frequency. The quantum Shor algorithm [1] provides an exciting opportunity for prime-factorization of large integers – a problem beyond the capabilities of today's powerful digital computers. Shor's algorithm utilizes two quantum registers (the x-and y-registers), which contain quantum bits two-level quantum systems called qubits [1]-[3]. First, the quantum computer produces the uniform superposition of all states in the x-register – all possible values of x. Second, the quantum computer computes the periodic function: y(x) = q x (mod N), where N is the number to factorize, and q is any number which is coprime to N. Third, the quantum computer creates a discrete Fourier transform of the x-register. The measurement of the state of the x-register yields the period, T , of the function y(x), which is used to produce a factor of the number N. In Dirac notation, the wave function of the quantum computer can be represented as a superposition of digital states, |a L−1 a L−2 ...a 1 a 0 , b M −1 b M −2 ...b 1 b 0 , (1) where a k (0 ≤ k ≤ L − 1) denotes the state of the k-th qubit in the x-register, and b n (0 ≤ n ≤ M − 1) denotes the state of the n-th qubit in the y-register. For example, if the k-th qubit of the x-register is in the ground state, then: a k = 0, and if it is in the excited state, a k = 1. In decimal notation, the digital state can be represented as |x, y, where x = L−1 k=0 a k 2 k , y = M −1 n=0 b n 2 n. (2) In this notation, the initial wave function of a quantum system is: |0, 0. The uniform superposition of the states created in the x-register can be written as, Ψ = 1 √ D x |x, 0, (3) where D = 2 L is the number of states in the x-register. After computation of the function y(x), we have, Ψ = 1 √ D x |x, y(x). (4) After the discrete Fourier transform, one measures the state of the x-register. The probability of …