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arXiv:0801.3136 · Journal of Physics A Mathematical and Theoretical

Manifold algorithmic errors in quantum computers with static internal imperfections

Murat Çetinbaş, Joshua Wilkie

Published Jan 29, 2008• 5 citations
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Abstract

The inevitable existence of static internal imperfections and residual interactions in some quantum computer architectures results in internal decoherence, dissipation and destructive unitary shifts of active algorithms. By exact numerical simulations, we determine the relative importance and origin of these errors for a Josephson charge-qubit quantum computer. In particular we determine that the dynamics of a CNOT gate interacting with its idle neighboring qubits via native residual coupling behaves much like a perturbed kicked top in the exponential decay regime, where fidelity decay is only weakly dependent on perturbation strength. This means that retroactive removal of gate errors (whether unitary or non-unitary) may not be possible, and that effective error correction schemes must operate concurrently with the implementation of subcomponents of the gate.

DOI
10.1088/1751-8113/41/6/065302

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We study effects of static inter-qubit interactions and random errors in quantum gates on the stability of various quantum algorithms including the Grover quantum search algorithm and the quantum chaos maps. For the Grover algorithm our numerical and analytical results show existence of regular and chaotic phases depending on the static imperfection strength εεε. The critical border εcε_cεc​ between two phases drops polynomially with the number of qubits nqn_qnq​ as εc∼nq−3/2ε_c \sim n_q^{-3/2}εc​∼nq−3/2​. In the regular phase (ε<εc)(ε< ε_c)(ε<εc​) the algorithm remains robust against imperfections showing the efficiency gain εc/εε_c / εεc​/ε for ε>2−nq/2ε> 2^{-n_q/2}ε>2−nq​/2. In the chaotic phase (ε>εc)(ε> ε_c)(ε>εc​) the algorithm is completely destroyed. The results for the Grover algorithm are compared with the imperfection effects for quantum algorithms of quantum chaos maps where the universal law for the fidelity decay is given by the Random Matrix Theory (RMT). We also discuss a new gyroscopic quantum error correction method which allows to reduce the effect of static imperfections. In spite of this decay GYQEC allows to obtain a significant gain in the accuracy of quantum computations.
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The stability of quantum computating of the kicked Harper model with various perturbations is investigated. Above a certain threshold of the imperfections, quantum chaos sets in. The effects of the noise errors and the static imperfections on the quantum computation are analyzed by comparing the statistical ergodic properties and the Husimi functions of the perturbed eigenstates with the ideal eigenstates of the Floquet operator. It is shown that the fidelity decay with static imperfections is exponential while it is Gaussian with noise errors. The time scales of reliable computation with these perturbations are obtained through numerical simulations. Due to the errors in classical computation the distance of two initial states increases exponentially, while the stability of quantum computation is independent of the integrable or chaotic nature of the underlying dynamics.
We analyze the stability of a quantum algorithm simulating the quantum dynamics of a system with different regimes, ranging from global chaos to integrability. We compare, in these different regimes, the behavior of the fidelity of quantum motion when the system's parameters are perturbed or when there are unitary errors in the quantum gates implementing the quantum algorithm. While the first kind of errors has a classical limit, the second one has no classical analog. It is shown that, whereas in the first case ("classical errors") the decay of fidelity is very sensitive to the dynamical regime, in the second case ("quantum errors") it is almost independent of the dynamical behavior of the simulated system. Therefore, the rich variety of behaviors found in the study of the stability of quantum motion under "classical" perturbations has no correspondence in the fidelity of quantum computation under its natural perturbations. In particular, in this latter case it is not possible to recover the semiclassical regime in which the fidelity decays with a rate given by the classical Lyapunov exponent.