Perturbation theory for quantum computation with a large number of qubits

Abstract

We describe a new and consistent perturbation theory for solid-state quantum computation with many qubits. The errors in the implementation of simple quantum logic operations caused by non-resonant transitions are estimated. We verify our perturbation approach using exact numerical solution for relatively small (L = 10) number of qubits. A preferred range of parameters is found in which the errors in processing quantum information are small. Our results are needed for experimental testing of scalable solid-state quantum computers.Pacs numbers: 32.80. Pj, 42.50.Vk, 05.45.Mt Several proposals for scalable solid-state quantum computers have been recently published [1][2][3][4][5][6][7][8][9][10][11]. For the most effective quantum information processing all of these proposals require operations with large number of qubits. In Ref.[12] quantum logic operations between remote qubits were simulated in a chain of L = 1000 nuclear spins. Such a quantum problem can be solved only using approximate methods since the dimensionality of the Hilbert space increases as 2 L . The approximate procedure used in [12] was based on the selection of states generated as a result of resonant or near-resonant transitions, while the other transitions were neglected in a controlled way. In this Letter we develop a consistent approach to this problem based on perturbation theory. We use our procedure to analytically estimate the probability of generation of unwanted states caused by non-resonant transitions and verify that these agree with the exact numerical solution for the same problem with relatively small number of qubits (L = 10), for which the dimensionality of the Hilbert space is not very large (N = 2 10 = 1024).

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

“…First, we simulate the full adder protocol for a small number of qubits using an exact numerical solution [2,4] in order to calculate the phase errors and to test the quantum map approach. For the relation between the numerical and physical parameters see Refs.…”

“…(18) are characterized by the large detuning |D| ≈ |k − k ′ |δω [1,2,4,9], which is approximately equal to the distance between the kth and k ′ th spins measured in the frequency units.…”

“…(ii) Due to the nonlocality of interaction of the rf pulses with the qubits, the direct simulation of the dynamics requires either the solution of large system of coupled differential equations for a long period of time or the diagonalization of large matrices. In order to overcome these difficulties a perturbation theory was developed in our previous papers [1][2][3][4][5][6][7][8][9]. In this paper we apply our perturbation approach to simulate the dynamics of a full adder with 1000 qubits.…”

Influence of qubit displacements on quantum logic operations in a silicon-based quantum computer with constant interaction

“…The error in the probability amplitude due to the nonresonant transitions is proportional to the ratio [23] …”

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

“…VIII of this paper or, for example, Ref. [17].) Moreover, any quantum logic operation between remote qubits can be divided into elementary logic gates between neighboring qubits.…”