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International Journal of Quantum Information

STABILITY OF THE QUANTUM FOURIER TRANSFORMATION ON THE ISING QUANTUM COMPUTER

G. L. Celardo, Carlos Pineda, Marko Žnidarič

Published Sep 1, 2005• 7 citations
Publisher

Abstract

We analyze the influence of errors on the implementation of the quantum Fourier transformation. Two kinds of errors are studied: (i) systematic errors due to off-resonant transitions and (ii) errors due to an external perturbation. The scaling of errors with system parameters and the number of qubits is analyzed. To suppress off-resonant transitions, we use correcting pulses while in order to suppress errors due to an external perturbation, we use an improved quantum Fourier transformation 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
10.1142/s0219749905001122

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We analyze the influence of errors on the implementation of the quantum Fourier transformation. Two kinds of errors are studied: (i) systematic errors due to off-resonant transitions and (ii) errors due to an external perturbation. The scaling of errors with system parameters and the number of qubits is analyzed. To suppress off-resonant transitions, we use correcting pulses while in order to suppress errors due to an external perturbation, we use an improved quantum Fourier transformation 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.
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 2πk2πk2πk 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.
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.
Coherent control errors, for which ideal Hamiltonians are perturbed by unknown multiplicative noise terms, are a major obstacle for reliable quantum computing. In this paper we present a framework for analyzing the robustness of quantum algorithms against coherent control errors using Lipschitz bounds. We derive worst-case fidelity bounds which show that the resilience against coherent control errors is mainly influenced by the norms of the Hamiltonians generating the individual gates. These bounds are explicitly computable even for large circuits and they can be used to guarantee fault tolerance via threshold theorems. Moreover, we apply our theoretical framework to derive a guideline for robust quantum algorithm design and transpilation, which amounts to reducing the norms of the Hamiltonians. Using the three-qubit quantum Fourier transform as an example application, we demonstrate that this guideline targets robustness more effectively than existing ones based on circuit depth or gate count. Furthermore, we apply our framework to study the effect of parameter regularization in variational quantum algorithms. The practicality of the theoretical results is demonstrated via implementations in simulation and on a quantum computer.
Quantum computers will allow calculations beyond existing classical computers. However, current technology is still too noisy and imperfect to construct a universal digital quantum computer with quantum error correction. Inspired by the evolution of classical computation, an alternative paradigm merging the flexibility of digital quantum computation with the robustness of analog quantum simulation has emerged. This universal paradigm is known as digital-analog quantum computing. Here, we introduce an efficient digital-analog quantum algorithm to compute the quantum Fourier transform, a subroutine widely employed in several relevant quantum algorithms. We show that, under reasonable assumptions about noise models, the fidelity of the quantum Fourier transformation improves considerably using this approach when the number of qubits involved grows. This suggests that, in the noisy intermediate-scale quantum era, hybrid protocols combining digital and analog quantum computing could be a sensible approach to reach useful quantum supremacy.