arXiv:quant-ph/0303043 · Physical Review Letters

Imperfection Effects for Multiple Applications of the Quantum Wavelet Transform

M. Terraneo, Dima L. Shepelyansky

Published Jun 26, 2003• 29 citations

Abstract

We study analytically and numerically the effects of various imperfections in a quantum computation of a simple dynamical model based on the quantum wavelet transform. The results for fidelity time scales, obtained for a large range of error amplitudes and number of qubits, imply that for static imperfections the threshold for fault-tolerant quantum computation is decreased by a few orders of magnitude compared to the case of random errors.

DOI: 10.1103/physrevlett.90.257902

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arXiv:quant-ph/0303043 · Physical Review Letters

Imperfection Effects for Multiple Applications of the Quantum Wavelet Transform

M. Terraneo, Dima L. Shepelyansky

Published Jun 26, 2003• 29 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.90.257902

Related papers

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ε

. The critical border

ε_c

between two phases drops polynomially with the number of qubits

n_q

ε_c \sim n_q^{-3/2}

. In the regular phase

(ε< ε_c)

the algorithm remains robust against imperfections showing the efficiency gain

ε_c / ε

for

ε> 2^{-n_q/2}

. In the chaotic phase

(ε> ε_c)

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.

We analyze and study the effects of locality on the fault-tolerance threshold for quantum computation. We analytically estimate how the threshold will depend on a scale parameter

r

which characterizes the scale-up in the size of the circuit due to encoding. We carry out a detailed seminumerical threshold analysis for concatenated coding using the seven-qubit CSS code in the local and the `nonlocal' setting. First, we find that the threshold in the local model for the ⟦7,1,3⟧ code has a

1∕r

dependence, which is in correspondence with our analytical estimate. Second, the threshold, beyond the

1∕r

dependence, does not depend too strongly on the noise levels for transporting qubits. Beyond these results, we find that it is important to look at more than one level of concatenation in order to estimate the threshold and that it may be beneficial in certain places, like in the transportation of qubits, to do error correction only infrequently.