Nonresonant effects in the implementation of the quantum Shor algorithm

Abstract

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 ...

Avoiding quantum chaos in quantum computation

“…The physically important regime of quantum computation is a selective excitation which corresponds to the range of parameters: Ω p ≪ J k,n ≪ δω k ≪ ω k , where δω k = |ω k+1 − ω k | [11]. However, in this Letter, we consider a regime of non-selective excitation which is defined by the conditions, Ω p ≫ δω k ≫ J, see details in [10].…”

Modeling Full Adder in Ising Spin Quantum Computer With 1000 Qubits Using Quantum Maps

“…We assume that the Larmor frequency of kth spin is ω k = w 0 + kδω, so that the Larmor frequency difference δω = ω k+1 −ω k between the neighboring spins is independent of the spin number, k. Below we omit the index n which indicates the pulse number. The long-range dipole-dipole interaction is suppressed by choosing the angle between the chain and the external permanent magnetic field to be equal to the magic angle [10]. Note, that the Hamiltonian (1) allows the transitions associated with flip of only a single spin.…”

“…In order to flip the kth qubit in the first two states in Eq. (2) and to suppress the transitions in the third and fourth states, the value of the Rabi frequency Ω should satisfy the 2πK-condition [10] Ω…”

Method for Implementation of Universal Quantum Logic Gates in a Scalable Ising Spin Quantum Computer

“…The same effect takes place for the heteronuclear dipole-dipole interaction in solids. We should note, that the long-range dipole-dipole interaction in a spin chain can be effectively suppressed if the angle between the chain and the external magnetic field equals to the magic angle [10,12].…”

“…The third approach is based on the application of the selective electromagnetic pulses with the Rabi frequency Ω < J (see, for example, [12,15,16]). These pulses drive a resonant spin depending on the states of its neighbors.…”