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Quantum Information and Computation

Quantum guessing via Deutsch-Jozsa

Michael Nathanson

Published Sep 1, 2010• 1 citation
Publisher

Abstract

We examine the "Guessing Secrets" problem arising in internet routing, in which the goal is to discover the identity of two objects from a known finite set $\Omega$ by asking yes/no questions. The best known classical algorithm requires $O(\log N)$ questions and $O(\log^2 N)$ steps to process the answers, where $N = \vert \Omega \vert$. We apply the Deutsch-Jozsa algorithm and show that the number of necessary calls to the oracle is independent of the size of the domain and that the output from each run of the algorithm has immediate meaning. In doing so, we extend the types of questions that the quantum algorithms can be used to solve.

DOI
10.26421/qic10.9-10-9

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We examine the "Guessing Secrets" problem arising in internet routing, in which the goal is to discover the identity of two objects from a known finite set Ω\OmegaΩ by asking yes/no questions. The best known classical algorithm requires O(log⁡N)O(\log N)O(logN) questions and O(log⁡2N)O(\log^2 N)O(log2N) steps to process the answers, where N=∣Ω∣N = \vert \Omega \vertN=∣Ω∣. We apply the Deutsch-Jozsa algorithm and show that the number of necessary calls to the oracle is independent of the size of the domain and that the output from each run of the algorithm has immediate meaning. In doing so, we extend the types of questions that the quantum algorithms can be used to solve.
We examine the "Guessing Secrets" problem arising in internet routing, in which the goal is to discover two or more objects from a known finite set. We propose a quantum algorithm using O(1) calls to an O(logN) oracle. This improves upon the best known classical result, which uses O(logN) questions and requires an additional O(logN^3) steps to produce the answer. In showing the possibilities of this algorithm, we extend the types of questions and function oracles that the Deutsch-Jozsa algorithm can be used to solve.
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 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.
A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough. Two examples of quantum speed-up are the Deutsch–Jozsa and Simon’s problem, both efficiently solvable on a quantum Turing machine, and both believed to lack efficient classical solutions. Here we present a framework that can simulate both quantum algorithms efficiently, solving the Deutsch–Jozsa problem with probability 1 using only one oracle query, and Simon’s problem using linearly many oracle queries, just as expected of an ideal quantum computer. The presented simulation framework is in turn efficiently simulatable in a classical probabilistic Turing machine. This shows that the Deutsch–Jozsa and Simon’s problem do not require any genuinely quantum resources, and that the quantum algorithms show no speed-up when compared with their corresponding classical simulation. Finally, this gives insight into what properties are needed in the two algorithms and calls for further study of oracle separation between quantum and classical computation.
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.