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arXiv:2212.11814

A quantum algorithm for counting zero-crossings

Shukla, Alok

Published Dec 21, 2022• 0 citations
arXivHTMLPDF

Abstract

We present a zero-crossings counting problem that is a generalization of the Bernstein-Vazirani problem. The goal of this problem is to count the number of zero-crossings (or sign changes) in a special type of sequence S, whose definition depends upon a secret string. A quantum algorithm is presented to solve this problem. The proposed quantum algorithm requires only one oracle query to solve the problem, whereas a classical algorithm would need at least n oracle queries, where $2^n$ is the size of the sequence S. In addition to solving the zero-crossings counting problem, we also give a quantum circuit for performing the Walsh-Hadamard transforms in sequency ordering. The Walsh-Hadamard transform in sequency ordering is used in a wide range of scientific and engineering applications, including in digital signal and image processing. Therefore, the proposed quantum circuit for computing the Walsh-Hadamard transforms in sequency ordering may be helpful in quantum computing algorithms for applications for which the computation of the Walsh-Hadamard transform in sequency ordering is required.

DOI
10.48550/arxiv.2212.11814

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This presentation gives a comprehensive overview of quantum query algorithms, beginning with an introduction to the query model of computation where input is accessed via an oracle function. It then details several foundational quantum algorithms that demonstrate a quantum advantage over classical approaches: Deutsch's problem is solved with a single query using phase kickback to determine if a function is constant or balanced; the Deutsch-Jozsa algorithm extends this to n-bit inputs, efficiently distinguishing constant from balanced functions with one query versus an exponential need classically; the Bernstein-Vazirani algorithm identifies a hidden string with one query, outperforming classical linear requirements; and Simon's algorithm, which finds a hidden period in a two-to-one function with a linear number of queries, showcases an exponential speedup over the best classical probabilistic algorithms.
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A quantum algorithm to solve the parity problem is better than its most efficient classical counter- part with a separation that is polynomial in the number of queries. This was shown by E. Bernstein and U. Vazirani and was one of the earliest indications that the quantum information processing can outperform the classical one by a significant margin. The problem and its solution both is usually stated for a 2-level system since we generally work with bits/qubits. However, many works have been done generalizing known quantum computing techniques to higher level systems. Following this, we look at a generalization of the Bernstein-Vazirani algorithm implemented on a general qudit system.
We present a quantum algorithm for estimating the amplitude content of user-specified sequency bands in quantum-encoded signals. The method employs a sequency-ordered Quantum Walsh-Hadamard Transform (QWHT), a comparator-based oracle that coherently marks basis states within an arbitrary sequency range, and Quantum Amplitude Estimation (QAE) to estimate the total probability mass in the selected band. This enables the detection of structured signal components, including both high- and low-sequency features, as well as the identification of rapid sign-change behavior associated with noise or anomalies. The proposed method can be embedded as a module within a larger quantum algorithm; in this setting, both the input and output remain fully quantum, enabling seamless integration with upstream and downstream quantum operations. We show that the sequency-ordered QWHT can be implemented with circuit depth O(log⁡2N)O(\log_2 N)O(log2​N) (equivalently O(n)O(n)O(n) for N=2nN=2^nN=2n) when acting on an amplitude-encoded quantum state, whereas computing the full Walsh-Hadamard spectrum of an explicit length-NNN classical signal requires O(Nlog⁡2N)O(N\log_2 N)O(Nlog2​N) operations via the fast Walsh-Hadamard transform. This results in an exponential quantum advantage when the QWHT is used as a modular block within a larger quantum algorithm, relative to classical fast Walsh-Hadamard transform-based approaches operating on explicit data. From an application perspective, the proposed sequency band-energy estimation may be interpreted as a structure-based anomaly indicator, enabling the detection of unexpected high-sequency components relative to a nominal low-sequency signal class. The algorithm is applicable to quantum-enhanced signal processing tasks such as zero-crossing analysis, band-limited noise estimation, and feature extraction in the Walsh basis.
The complexity of certain classical computer algorithms can be reduced by a quantum algorithm completing the same task. We examine the implementation of the Bernstein–Vazirani algorithm. It finds the hidden n-bit string s from the function fs(x)=sx that takes an n-bit input string x and outputs one bit, the dot product. This algorithm reduces the complexity from n queries for the best classical algorithm to just one. Though the quantum algorithm theoretically always gives the correct string, in practice, factors such as decoherence of the physical qubits due to external noise reduce the algorithm’s accuracy. By implementing the algorithm on the IBM 5-qubit and 14-qubit quantum computers, we are able to analyze correlations between the accuracy of the results and the properties of the physical qubits used. Not all qubits are pairwise connected, nor do they have the same coherence times. Using Python to build the quantum circuits, we create several implementations of the circuit with different qubits. We establish that 1/T1, 1/T2, and gate errors are positively correlated with calculation errors. Consequently, we observe that the 14-qubit quantum computer, on average, is more prone to errors than the 5-qubit quantum computer.