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arXiv:0809.0932 · IEEE International Symposium on Multiple-Valued Logic (ISMVL)

A Generalization of the Deutsch-Jozsa Algorithm to Multi-Valued Quantum Logic

Yale Fan

Published May 1, 2007• 26 citations
arXivHTMLPDF

Abstract

We generalize the binary Deutsch-Jozsa algorithm to n- valued logic using the quantum Fourier transform. Our algorithm is not only able to distinguish between constant and balanced Boolean functions in a single query, but can also find closed expressions for classes of affine functions in quantum oracles, accurate to a constant term.
DOI
10.1109/ismvl.2007.3

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We present a generalized Deutsch-Jozsa (DJ) quantum algorithm that not only determines both the global type of an unknown Boolean function (constant or balanced) but also determines explicit output values of the function in a single oracle query. Unlike the original DJ algorithm, which identifies only whether a function is constant or balanced, our generalization retrieves actual function output values at the same time with using a Bell state as ancilla. This makes a richer function characterization with minimal queries to have practical quantum advantages, e.g. data classification, logistic regression, and quantum cryptography.
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Boolean functions are important building blocks in cryptography for their wide application in both stream and block cipher systems. For cryptanalysis of such systems one tries to find out linear functions that are correlated to the Boolean functions used in the crypto system. Let fff be an nnn-variable Boolean function and its Walsh spectra is denoted by Wf(ω)W_f(ω)Wf​(ω) at the point ω∈{0,1}nω\in \{0, 1\}^nω∈{0,1}n. The Boolean function is available in the form of an oracle. We like to find an ωωω such that Wf(ω)≠0W_f(ω) \neq 0Wf​(ω)=0 as this will provide one of the linear functions which are correlated to fff. We show that the quantum algorithm proposed by Deutsch and Jozsa (1992) solves the above mentioned problem in constant time. However, the best known classical algorithm to solve this problem requires exponential time in nnn. We also analyse certain classes of cryptographically significant Boolean functions and highlight how the basic Deutsch-Jozsa algorithm performs on them.
This mathematically intensive chapter takes us through our first steps in the domain of quantum computation (QC) algorithms . The simplest of them is the Deutsch algorithm , which makes it possible to determine whether or not a Boolean function is constant for any input. The key result is that this QC algorithm provides the answer at once, whereas in the classical case it would take two independent calculations. I describe next the generalization of the former algorithm to n qubits, referred to as the Deutsch–Jozsa algorithm . Although they have no specific or useful applications in quantum computing, both algorithms represent a most elegant means of introducing the concept of quantum computation parallelism . I then describe two most important QC algorithms, which nicely exploit quantum parallelism. The first is the quantum Fourier transform (QFT), for which a detailed analysis of QFT circuits and quantum-gate requirements is also provided. As will be shown in the next chapter, a key application of QFT concerns the famous Shor's algorithm , which makes it possible to factor numbers into primes in terms of polynomials. The second algorithm, no less famous than Shor's, is referred to as the Grover quantum database search , whose application is the identification of database items with a quadratic gain in speed. Deutsch algorithm Our exploration of quantum algorithms shall begin with the solution of a very basic problem: finding whether or not a Boolean function f ( x ) is a constant.
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.