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arXiv:quant-ph/0310163

Stability of quantum Fourier transformation on Ising quantum computer

G. L. Celardo, Carlos Pineda, Marko Žnidarič

Published Oct 27, 2003• 3 citations
arXivHTMLPDF

Abstract

We analyze the influence of errors on the implementation of the quantum Fourier transformation (QFT) on the Ising quantum computer (IQC). Two kinds of errors are studied: (i) due to spurious transitions caused by pulses and (ii) due to external perturbation. The scaling of errors with system parameters and number of qubits is explained. We use two different procedures to fight each of them. To suppress spurious transitions we use correcting pulses (generalized $2πk$ method) while to suppress errors due to external perturbation we use an improved QFT algorithm. As a result, the fidelity of quantum computation is increased by several orders of magnitude and is thus stable in a much wider range of physical parameters.

DOI
10.48550/arxiv.quant-ph/0310163

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The performance of today's quantum computers are affected by noise. This effect can be analyzed in the result of simple quantum algorithms in real quantum computers. The noise can be characterized as a decoherence error or a systematic error, the last could be corrected by a unitary rotation. In this article we propose two methods to model a systematic error, in the Quantum Fourier Transform algorithm (QFT). The first method uses the isotropic index presented in `` [1] and needs to reconstruct the density matrix of the experimental state, while the second method, although less general, only needs to reconstruct the reduced density matrices for each qubit. In both methods, a unitary transformation is proposed, which approximates the experimental result to the expected theoretical state. As an example, the QFT algorithm is analyzed for two qubit states, in quantum IBM Q computer ibmq\_santiago.
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.
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Abstract The effectiveness of the implementation of the quantum Fourier transform (QFT) and the Grover algorithm on qudit systems is investigated. A comparative analysis of computing resources for quantum states of N ≥ 1024 dimension is carried out. It is shown that using qudits can significantly reduce the number of quantum operations in Grover’s algorithm compared to its qubit counterparts, while for QFT the main advantage is to reduce the number of physical carriers. The effect of noise using the Monte Carlo method on the accuracy of algorithms for qubit systems of various dimensions is analyzed.
The quantum Fourier transform (QFT) is a key primitive for quantum computing that is typically used as a subroutine within a larger computation, for instance for phase estimation. As such, we may have little control over the state that is input to the QFT. Thus, in implementing a good QFT, we may imagine that it needs to perform well on arbitrary input states. Verifying this worst-case correct behaviour of a QFT-implementation would be exponentially hard (in the number of qubits) in general, raising the concern that this verification would be impossible in practice on any useful-sized system. In this paper we show that, in fact, we only need to have good average-case performance of the QFT to achieve good worst-case performance for key tasks – phase estimation, period finding and amplitude estimation. Further we give a very efficient procedure to verify this required average-case behaviour of the QFT.