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arXiv:2112.04411

Quantum phase estimation in presence of glassy disorder

Soubhadra Maiti, Kornikar Sen, Ujjwal Sen

Published Dec 8, 2021• 1 citation
arXivHTMLPDF

Abstract

We investigate the response to noise, in the form of glassy disorder present in circuit elements, in the success probability of the quantum phase estimation algorithm, a subroutine used to determine the eigenvalue - a phase - corresponding to an eigenvector of a unitary gate. We prove that when a large number of auxiliary qubits are involved in the circuit, the probability does not depend on the actual type of disorder but only on the mean and strength of the disorder. For further analysis, we consider three types of disorder distributions: Haar-uniform with a circular cut-off, Haar-uniform with an elliptical or squeezed cut-off, and spherical normal. There is generally a depreciation of the disorder-averaged success probability in response to the disorder incorporation. Even in the presence of the disorder, increasing the number of auxiliary qubits helps to get a better precision of the phase, albeit to a lesser extent (probability) than that in the clean case. We find a concave to convex transition in the dependence of probability on the strength of disorder, and a log-log dependence is witnessed between the point of inflection and the number of auxiliary qubits used.

DOI
10.48550/arxiv.2112.04411

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We investigate the response to noise, in the form of glassy disorder present in circuit elements, in the success probability of the quantum phase estimation algorithm, a subroutine used to determine the eigenvalue - a phase - corresponding to an eigenvector of a unitary gate. We prove that when a large number of auxiliary qubits are involved in the circuit, the probability does not depend on the actual type of disorder but only on the mean and strength of the disorder. For further analysis, we consider three types of disorder distributions: Haar-uniform with a circular cut-off, Haar-uniform with an elliptical or squeezed cut-off, and spherical normal. There is generally a depreciation of the disorder-averaged success probability in response to the disorder incorporation. Even in the presence of the disorder, increasing the number of auxiliary qubits helps to get a better precision of the phase, albeit to a lesser extent (probability) than that in the clean case. We find a concave to convex transition in the dependence of probability on the strength of disorder, and a log-log dependence is witnessed between the point of inflection and the number of auxiliary qubits used.
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The quantum singular value transformation (QSVT) algorithm is a general framework to implement most of the known algorithms and provides a way forward for designing new algorithms. In the present work, the impact of noise on the QSVT algorithm is examined for bit flip, phase flip, bit-phase flip, and depolarizing noise models for a small number of qubits. The small number of noisy qubits approximates the currently available noisy quantum computers. For simulation results, the QSVT implementation of the Grover search and quantum phase estimation (QPE) algorithms is considered. These algorithms are among the basic quantum algorithms and form the building blocks of various applications of quantum algorithms. The results showed that the QSVT implementation of the Grover search and QPE algorithms has a consistently worse dependence upon noise than the original implementation for all four noise models. The probability of success of the Grover algorithm and phase measured by the QPE algorithm were found to exponentially depend upon the error probability in the noisy channels but only linearly dependent on the number of qubits.
Abstract We consider the average quantum coherences of typical redits and qudits—vectors of real and complex Hilbert spaces—with the analytical forms stemming from the symmetry of Haar-uniformly distributed random pure states. We subsequently study the response to disorder in spread of the typical quantum coherence in response to glassy disorder. The disorder is inserted in the state parameters. Even in the absence of disorder, the quantum coherence distributions of redits and qudits are not uniform over the range of quantum coherence, and the spreads are relatively lower for higher dimensions. On insertion of disorder, the spreads decrease. This decrease in the spread of quantum coherence distribution in response to disorder is seen to be a generic feature of typical pure states: we observe the feature for different strengths of disorder and for various types of disorder distributions, viz. Gaussian, uniform, and Cauchy–Lorentz. We also find that the quantum coherence distributions become less asymmetric with increase in dimension and with infusion of glassy disorder.