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. 1 I. The Quantum Shor AlgorithmThe 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,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 , whereIn 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,where D = 2 L is the number of states in the x-register. After computation of the function y(x), we have,After the discrete Fourier transform, one measures the state of the x-register. The probability of the measurement, P (x), must be a peaked distribution with peak separation, ∆x, equal to a multiple of 1/T . In particular, if the number of states, D, in the x-register is divisible by the period, T , then: ∆x = D/T . From the value ∆x one can find the period, T . A factor of the number N can be found by computing the greatest common divisor of (q T /2 + 1) and N, or (q T /2 − 1) and N (for even T ). 2It was shown in [4], that the simplest demonstration of the quantum Shor's algorithm can be done with only four qubits. (Two qubits represent the x-register and two qubits represent the y-register.) This primitive quantum computer is able to find a factor of the number 4. For N = 4, the only coprime number is q = 3. The function, y(x) = 3x (mod 4), has only two values: y = 1 and y = 3, and a period is, T = 2. In Dirac notation, the wave function (4) has the form, Ψ = 1 2 (|00, 01 + |01, 11 + |10, 01 + |11, 11 ).After the Discrete Fourier transform, the wave function of the quantum computer is, Ψ = 1 2 (|00, 01 + |00, 11 + |10, 01 + |10, 11 ).Measuring x can produce two ...

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Abstract. The perturbation theory is developed based on small parameters which naturally appear in solid state quantum computation. We report the simulations of the dynamics of quantum logic operations with a large number of qubits (up to 1000). A nuclear spin chain is considered in which selective excitations of spins are provided by having a uniform gradient of the external magnetic field. Quantum logic operations are utilized by applying resonant electromagnetic pulses. The spins interact with their nearest neighbors. We simulate the creation of the long-distance entanglement between remote qubits in the spin chain. Our method enables us to minimize unwanted non-resonant effects in a controlled way. The method we use cannot simulate complicated quantum logic (a quantum computer is required to do this), but it can be useful to test the experimental performance of simple quantum logic operations. We show that: (a) the probability distribution of unwanted states has a "band" structure, (b) the directions of spins in typical unwanted states are highly correlated, and (c) many of the unwanted states are high-energy states of a quantum computer (a spin chain). Our approach can be applied to simple quantum logic gates and fragments of quantum algorithms involving a large number of qubits.

It is shown that the statistical distribution of an ensemble of one-channel S-matrices is uniquely determined by requiring that: I) S has poles only in the lower half of the energy plane and 2) the function S(E) is ergodic in a sense to be defined. A Monte Carlo calculation was performed to illustrate numerically the above statement.

We present the study of a quantum Controlled-Controlled-Not gate, implemented in a chain of three nuclear spins weakly Ising interacting between all of them, that is, taking into account first and second neighbor spin interactions. This implementation is done using a single resonant π-pulse on the initial state of the system (digital and superposition). The fidelity parameter is used to determine the behavior of the CCN quantum gate as a function of the ratio of the second neighbor interaction coupling constant to the first neighbor interaction coupling constant (J ′ /J). We found that for J ′ /J ≥ 0.02 we can have a well defined CCN quantum gate.

This paper presents a powerfull method to integrate general monomials on the classical groups with respect to their invariant (Haar) measure. The method has first been applied to the orthogonal group in [J. Math. Phys. 43, 3342 (2002)], and is here used to obtain similar integration formulas for the unitary and the unitary symplectic group. The integration formulas turn out to be of similar form. They are all recursive, where the recursion parameter is the number of column (row) vectors from which the elements in the monomial are taken. This is an important difference to other integration methods. The integration formulas are easily implemented in a computer algebra environment, which allows to obtain analytical expressions very efficiently. Those expressions contain the matrix dimension as a free parameter.Comment: 16 page

We report the first simulations of the dynamics of quantum logic operations with a large number of qubits (up to 1000). A nuclear spin chain in which selective excitations of spins is provided by the gradient of the external magnetic field is considered. The spins interact with their nearest neighbors. We simulate the quantum CONTROL-NOT (CN) gate implementation for remote qubits which provides the long-distance entanglement. Our approach can be applied to any implementation of quantum logic gates involving a large number of qubits.

Given a constant of motion for the one-dimensional harmonic oscillator with linear dissipation in the velocity, the problem to get the Hamiltonian for this system is pointed out, and the quantization up to second order in the perturbation approach is used to determine the modification on the eigenvalues when dissipation is taken into consideration. This quantization is realized using the constant of motion instead of the Hamiltonian.