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arXiv:quant-ph/0610220

De-Quantising the Solution of Deutsch's Problem

Cristian S. Calude

Published Oct 26, 2006• 1 citation
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Abstract

Probably the simplest and most frequently used way to illustrate the power of quantum computing is to solve the so-called {\it Deutsch's problem}. Consider a Boolean function $f: \{0,1\} \to \{0,1\}$ and suppose that we have a (classical) black box to compute it. The problem asks whether $f$ is constant (that is, $f(0) = f(1)$) or balanced ($f(0) \not= f(1)$). Classically, to solve the problem seems to require the computation of $f(0)$ and $ f(1)$, and then the comparison of results. Is it possible to solve the problem with {\em only one} query on $f$? In a famous paper published in 1985, Deutsch posed the problem and obtained a ``quantum'' {\em partial affirmative answer}. In 1998 a complete, probability-one solution was presented by Cleve, Ekert, Macchiavello, and Mosca. Here we will show that the quantum solution can be {\it de-quantised} to a deterministic simpler solution which is as efficient as the quantum one. The use of ``superposition'', a key ingredient of quantum algorithm, is--in this specific case--classically available.

DOI
10.48550/arxiv.quant-ph/0610220

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Probably the simplest and most frequently used way to illustrate the power of quantum computing is to solve the so-called {\it Deutsch's problem}. Consider a Boolean function f:{0,1}→{0,1}f: \{0,1\} \to \{0,1\}f:{0,1}→{0,1} and suppose that we have a (classical) black box to compute it. The problem asks whether fff is constant (that is, f(0)=f(1)f(0) = f(1)f(0)=f(1)) or balanced (f(0)≠f(1)f(0) \not= f(1)f(0)=f(1)). Classically, to solve the problem seems to require the computation of f(0)f(0)f(0) and f(1) f(1)f(1), and then the comparison of results. Is it possible to solve the problem with {\em only one} query on fff? In a famous paper published in 1985, Deutsch posed the problem and obtained a quantum'' {\em partial affirmative answer}. In 1998 a complete, probability-one solution was presented by Cleve, Ekert, Macchiavello, and Mosca. Here we will show that the quantum solution can be {\it de-quantised} to a deterministic simpler solution which is as efficient as the quantum one. The use of superposition'', a key ingredient of quantum algorithm, is--in this specific case--classically available.
Probably the simplest and most frequently used way to illustrate the power of quantum computing is to solve the so-called Deutsch's problem. Consider a Boolean function f: {0,1} → {0,1} and suppose that we have a (classical) black box to compute it. The problem asks whether f is constant [that is, f(0) = f(1)] or balanced [f(0) ≠ f(1)]. Classically, to solve the problem seems to require the computation of f(0) and f(1), and then the comparison of results. Is it possible to solve the problem with only one query on f? In a famous paper published in 1985, Deutsch posed the problem and obtained a "quantum" partial affirmative answer. In 1998, a complete, probability-one solution was presented by Cleve, Ekert, Macchiavello and Mosca. Here we will show that the quantum solution can be de-quantized to a deterministic simpler solution which is as efficient as the quantum one. The use of "superposition," a key ingredient of quantum algorithm, is — in this specific case — classically available.
Deutsch's problem is the simplest and most frequently examined example of computational problem used to demonstrate the superiority of quantum computing over classical computing. Deutsch's quantum algorithm has been claimed to be faster than any classical ones solving the same problem, only to be discovered later that this was not the case. Various de-quantised solutions for Deutsch's quantum algorithm—classical solutions which are as efficient as the quantum one—have been proposed in the literature. These solutions are based on the possibility of classically simulating "superpositions", a key ingredient of quantum algorithms, in particular, Deutsch's algorithm. The de-quantisation proposed in this note is based on a classical simulation of the quantum measurement achieved with a model of observed system. As in some previous de-quantisations of Deutsch's quantum algorithm, the resulting de-quantised algorithm is deterministic. Finally, we classify observers (as finite state automata) that can solve the problem in terms of their "observational power".
The Deutsch-Jozsa algorithm is one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm. It was proposed by Deutsch and Jozsa in 1992 with improvements by Cleve, Ekert, Macchiavello, and Mosca in 1998. The Deutsch-Jozsa problem is a promise problem and we can equivalently describe it as a partial function ${\text{DJ}}_{n}^{0}:{{0,1}}^{n}\ensuremath{\rightarrow}{0,1}$ defined as DJn0(x)=1{\text{DJ}}_{n}^{0}(x)=1DJn0​(x)=1 for ∣x∣=n/2,DJn0(x)=0|x|=n/2, {\text{DJ}}_{n}^{0}(x)=0∣x∣=n/2,DJn0​(x)=0 for ∣x∣=0,n|x|=0,n∣x∣=0,n, and it is undefined for the rest of the cases, where nnn is even, and ∣x∣|x|∣x∣ is the Hamming weight of xxx. The optimal quantum algorithm needs only one query to compute DJn0{\text{DJ}}_{n}^{0}DJn0​ but the classical deterministic algorithm requires ${2}^{n\ensuremath{-}1}+1$ queries to compute it in the worse case. In this article, we generalize the Deutsch-Jozsa problem as DJnk(x)=1{\text{DJ}}_{n}^{k}(x)=1DJnk​(x)=1 for ∣x∣=n/2,DJnk(x)=0|x|=n/2, {\text{DJ}}_{n}^{k}(x)=0∣x∣=n/2,DJnk​(x)=0 for ∣x∣|x|∣x∣ in the set ${0,1,...,k,n\ensuremath{-}k,n\ensuremath{-}k+1,...,n}$, and it is undefined for the rest of the cases, where $0\ensuremath{\le}k<n/2$. In particular, we give and prove an optimal exact quantum query algorithm with complexity k+1k+1k+1 for computing the generalized Deutsch-Jozsa problem DJnk{\text{DJ}}_{n}^{k}DJnk​. It is clear that the case of k=0k=0k=0 is in accordance with the Deutsch-Jozsa problem. Also, we give a method for finding the approximate and exact degrees of symmetric partial Boolean functions.
Quantum computation has shown much promise at providing, at least in some cases, a significant advantage over classical computation. However, the nature of quantum computation is still far from being well understood. In order to develop quantum algorithms effectively, it is important to understand the true nature of the differences between classical and quantum computation. We investigate these differences more closely by looking at de-quantising quantum algorithms into classical counterparts which retain the benefit provided by, and thought to be intrinsic to, the quantum algorithms. We extend a previous de-quantisation of Deutsch’s problem to show that in some situations the quantum algorithm solving the Deutsch-Jozsa problem can be de-quantised into an equivalent classical one. We quantify the entanglement in this problem and show that the inability to easily extend the de-quantisation to the general case is a result of the entanglement destroying the concise classical state description needed to de-quantise such a black-box algorithm. We further show that the quantum Fourier transform, an important process in many quantum algorithms, in its standard form is de-quantisable. This highlights key misconceptions about both the quantum Fourier transform and quantum computation itself. In such cases it is the linearity of quantum mechanics which allows constructing quantum algorithms which offer advantage over classical algorithms. The careful investigation of de-quantisation in relation to these problems allows a deeper insight into the nature of quantum computation.
Although the basic ideas of quantum computation are fairly straightforward, the detailed manipulations required to extract a useful result from an entangled superposition of answers can be rather complicated. There are, however, a small number of algorithms which are simple enough to explain using only elementary methods. The best example is Deutsch's algorithm: as this only requires two qubits its properties are amenable to brute-force matrix calculations, and these calculations can be simplified using a variety of short cuts, which also give some insight into how and why the algorithm works. We will also explore the simplest case of Grover's quantum search algorithm, which is once again simple enough to be tacked by brute force. Finally we will look briefly at two methods used to stabilize quantum computers against errors. Deutsch's algorithm The invention of Deutsch's algorithm can be taken as defining the start of modern quantum computation, and it remains a key example, exhibiting many of the key properties of quantum algorithms in a particularly simple form. (Note, however, that the version of Deutsch's algorithm described here is not in fact the original but a later modification which is both more powerful and easier to understand.) Consider a binary function f from one bit to one bit, that is a function which takes in either 0 or 1 as its input and returns either 0 or 1 as its output. There are four such functions which may be conveniently labeled by their outputs as shown in Table 8.1.
Quantum-state filtering is a variant of the unambiguous state discrimination problem: the states are grouped in sets and we want to determine to which particular set a given input state belongs. The simplest case, when the NNN given states are divided into two subsets and the first set consists of one state only while the second consists of all of the remaining states, is termed quantum-state filtering. We derived previously the optimal strategy for the case of NNN nonorthogonal states, ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$, for distinguishing $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. In a previous paper [Phys. Rev. Lett. 90, 257901 (2003)], we sketched an appplication of the results to probabilistic quantum algorithms. Here we fill the gaps and give the complete derivation of the probabilstic quantum algorithm that can optimally distinguish between two classes of Boolean functions, that of the balanced functions and that of the biased functions. The algorithm is probabilistic, it fails sometimes but when it does it lets us know that it did. Our approach can be considered as a generalization of the Deutsch-Jozsa algorithm that was developed for the discrimination of balanced and constant Boolean functions.