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De-quantisation in Quantum Computation

Alastair A. Abbott

Published Jan 1, 2009• 1 citation
OpenAlex

Abstract

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.

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AIP conference proceedings•Jan 1, 2012•0 citations
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.
While it seems possible that quantum computers may allow for algorithms offering a computational speed-up over classical algorithms for some problems, the issue is poorly understood. We explore this computational speed-up by investigating the ability to de-quantise quantum algorithms into classical simulations of the algorithms which are as efficient in both time and space as the original quantum algorithms. The process of de-quantisation helps formulate conditions to determine if a quantum algorithm provides a real speed-up over classical algorithms. These conditions can be used to develop new quantum algorithms more effectively (by avoiding features that could allow the algorithm to be efficiently classically simulated), as well as providing the potential to create new classical algorithms (by using features which have proved valuable for quantum algorithms). Results on many different methods of de-quantisations are presented, as well as a general formal definition of de-quantisation. De-quantisations employing higher-dimensional classical bits, as well as those using matrix-simulations, put emphasis on entanglement in quantum algorithms; a key result is that any algorithm in which the entanglement is bounded is de-quantisable. These methods are contrasted with the stabiliser formalism de-quantisations due to the Gottesman-Knill Theorem, as well as those which take advantage of the topology of the circuit for a quantum algorithm. The benefits of the different methods are contrasted, and the importance of a range of techniques is emphasised. We further discuss some features of quantum algorithms which current de-quantisation methods do not cover.
While it seems possible that quantum computers may allow for algorithms offering a computational speed-up over classical algorithms for some problems, the issue is poorly understood. We explore this computational speed-up by investigating the ability to de-quantise quantum algorithms into classical simulations of the algorithms which are as efficient in both time and space as the original quantum algorithms. The process of de-quantisation helps formulate conditions to determine if a quantum algorithm provides a real speed-up over classical algorithms. These conditions can be used to develop new quantum algorithms more effectively (by avoiding features that could allow the algorithm to be efficiently classically simulated) and to create new classical algorithms (by using features which have proved valuable for quantum algorithms). Results on many different methods of de-quantisations are presented, as well as a general formal definition of de-quantisation. De-quantisations employing higher-dimensional classical bits, as well as those using matrix-simulations, put emphasis on entanglement in quantum algorithms; a key result is that any algorithm in which the entanglement is bounded is de-quantisable. These methods are contrasted with the stabiliser formalism de-quantisations due to the Gottesman-Knill Theorem, as well as those which take advantage of the topology of the circuit for a quantum algorithm. The benefits and limits of the different methods are discussed, and the importance of utilising a range of techniques is emphasised. We further discuss some features of quantum algorithms which current de-quantisation methods do not cover and highlight several important open questions in the area.
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".
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)$, 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.
One of the most basic tasks in quantum information processing, communication and security (QIPCC) research, theoretically deep and practically important, is to find bounds on how really important are inherently quantum resources for speeding up computations. This area of research is bringing a variety of results that imply, often in a very unexpected and counter-intuitive way, that: (a) surprisingly large classes of quantum circuits and algorithms can be efficiently simulated on classical computers; (b) the border line between quantum processes that can and cannot be efficiently simulated on classical computers is often surprisingly thin; (c) the addition of a seemingly very simple resource or a tool often enormously increases the power of available quantum tools. These discoveries have put also a new light on our understanding of quantum phenomena and quantum physics and on the potential of its inherently quantum and often mysteriously looking phenomena. The paper motivates and surveys research and its outcomes in the area of de-quantisation, especially presents various approaches and their outcomes concerning efficient classical simulations of various families of quantum circuits and algorithms. To motivate this area of research some outcomes in the area of de-randomization of classical randomized computations.