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arXiv:quant-ph/0110069

Perturbation Approach for a Solid-State Quantum Computation

G. P. Berman, D. I. Kamenev, V. I. Tsifrinovich

Published Oct 10, 2001• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0110069

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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.
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.
We develop a dynamical perturbation theory for solid-state quantum computation with many qubits. The errors in the implementation of simple quantum logic operations caused by nonresonant transitions are estimated. We verify our perturbation approach using exact numerical solution for relatively small (N=10)(N=10)(N=10) number of qubits. A preferred range of parameters is found in which the errors in processing quantum information are reasonably small. Our results can be useful for understanding the mechanisms of errors and for experimental testing of scalable solid-state quantum computers.
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 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 us 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.
Creation of entanglement is considered theoretically and numerically in an ensemble of spin chains with dipole-dipole interaction between the spins. The unwanted effect of the long-range dipole interaction is compensated by the optimal choice of the parameters of radio-frequency pulses implementing the protocol. The errors caused by (i) the influence of the environment, (ii) nonselective excitations, (iii) influence of different spin chains on each other, (iv) displacements of qubits from their perfect locations, and (v) fluctuations of the external magnetic field are estimated analytically and calculated numerically. For the perfectly entangled state the zzz component MMM of the magnetization of the whole system is equal to zero. The errors lead to a finite value of MMM. If the number of qubits in the system is large, MMM can be detected experimentally. Using the fact that MMM depends differently on the parameters of the system for each kind of error, varying these parameters would allow one to experimentally determine the most significant source of errors and to optimize correspondingly the quantum computer design in order to decrease the errors and $\ensuremath{\mid}M\ensuremath{\mid}$. Using our approach one can benchmark the quantum computer, decrease the errors, and prepare the quantum computer for implementation of more complex quantum algorithms.