We present protocols for implementation of universal quantum gates on an arbitrary superposition of quantum states in a scalable solid-state Ising spin quantum computer. The spin chain is composed of identical spins 1/2 with the Ising interaction between the neighboring spins. The selective excitations of the spins are provided by the gradient of the external magnetic field. The protocols are built of rectangular radio-frequency pulses. The phase and probability errors caused by unwanted transitions are minimized and computed numerically.
We simulated the quantum dynamics for magnetic resonance force microscopy (MRFM) in the oscillating cantilever-driven adiabatic reversals (OSCAR) technique.We estimated the frequency shift of the cantilever vibrations and demonstrated that this shift causes the formation of a Schrödinger cat state which has some similarities and differences from the conventional MRFM technique which uses cyclic adiabatic reversals of spins. The interaction of the cantilever with the environment is shown to quickly destroy the coherence between the two possible cantilever trajectories.We have shown that using partial adiabatic reversals, one can produce a significant increase in the OSCAR signal.
In this paper we analyze the dynamics in a spin-model of quantum computer. Main attention is paid to the dynamical fidelity (associated with dynamical errors) of an algorithm that allows to create an entangled state for remote qubits. We show that in the regime of selective resonant excitations of qubits there is no any danger of quantum chaos. Moreover, in this regime a modified perturbation theory gives an adequate description of the dynamics of the system. Our approach allows to explicitly describe all peculiarities of the evolution of the system under time-dependent pulses corresponding to a quantum protocol. Specifically, we analyze, both analytically and numerically, how the fidelity decreases in dependence on the model parameters.
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).
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.
A lattice-gas algorithm for the one-dimensional diffusion equation is realized using radio frequency pulses in a one-dimensional spin system. The model is a large array of quantum two-qubit nodes interconnected by the nearest-neighbor classical communication channels. We present a quantum protocol for implementation of the quantum collision operator and a method for initialization and reinitialization of quantum states. Numerical simulations of the quantum-classical dynamics are in good agreement with the analytic solution for the diffusion equation.
The dynamics of the nuclear-spin quantum computer with large number (L = 1000) of qubits is considered using a perturbation approach, based on approximate diagonalization of exponentially large sparse matrices. Small parameters are introduced and used to compute the error in implementation of entanglement between remote qubits, by applying a sequence of resonant radio-frequency pulses. The results of the perturbation theory are tested using exact numerical solutions for small number of qubits.
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