Many worlds, the cluster-state quantum computer, and the problem of the preferred basis

Abstract

I argue that the many worlds explanation of quantum computation is not licensed by, and in fact is conceptually inferior to, the many worlds interpretation of quantum mechanics from which it is derived. I argue that the many worlds explanation of quantum computation is incompatible with the recently developed cluster state model of quantum computation. Based on these considerations I conclude that we should reject the many worlds explanation of quantum computation.

Introduction

The source of quantum computational ‘speedup’—the ability of a quantum computer to achieve, for some problem domains,¹ a dramatic reduction in processing time over any known classical algorithm—is still a matter of debate. On one popular view (the ‘quantum parallelism thesis’²), the speedup is due to a quantum computer's ability to simultaneously evaluate (using a single circuit) a function for many different values of its input. Thus one finds, in textbooks on quantum computation, pronouncements such as the following:

[a] qubit can exist in a superposition of states, giving a quantum computer a hidden realm where exponential computations are possible ... This feature allows a quantum computer to do parallel computations using a single circuit—providing a dramatic speedup in many cases (McMahon, 2008, p. 197).

Unlike classical parallelism, where multiple circuits each built to compute f(x) are executed simultaneously, here a single f(x) circuit is employed to evaluate the function for multiple values of x simultaneously, by exploiting the ability of a quantum computer to be in superpositions of different states (Nielsen & Chuang, 2000, p. 31).

Among textbook writers, Mermin is, perhaps, the most cautious with respect to the significance of this ‘quantum parallelism’:

One cannot say that the result of the calculation is $2^n$ evaluations of f, though some practitioners of quantum computation are rather careless about making such a claim. All one can say is that those evaluations characterize the form of the state that describes the output of the computation. One knows what the state is only if one already knows the numerical values of all those 2ⁿ evaluations of f. Before drawing extravagant practical, or even only metaphysical, conclusions from quantum parallelism, it is essential to remember that when you have a collection of Qbits in a definite but unknown state, there is no way to find out what that state is (2007, p. 38).

Mermin's reservations notwithstanding, the quantum parallelism thesis is frequently associated with (and held to provide evidence for) the many worlds explanation of quantum computation, which draws its inspiration from the Everettian interpretation of quantum mechanics. According to the many worlds explanation of quantum computing, when a quantum computer effects a transition such as

$$\sum _{x=0}^{2^n-1}|x〉|0〉\to \sum _{x=0}^{2^n-1}|x〉|f(x)〉,$$

it literally performs, simultaneously and in different physical worlds, local function evaluations on all of the possible values of x.

The many worlds explanation of quantum computing is a very attractive explanation of quantum speedup if one accepts the quantum parallelism thesis, for, since the many worlds explanation of quantum computing directly answers the question of where this parallel processing occurs (i.e., in distinct physical universes) in a way in which other explanations do not, it is, arguably, the most intuitive explanation of quantum speedup. Indeed, for some, the many worlds explanation of quantum computing is the only possible explanation of quantum speedup. Deutsch (2010, p. 542), for instance, writes: “no single-universe theory can explain even the Einstein–Podolsky–Rosen experiment, let alone, say, quantum computation. That is because any process (hidden variables, or whatever) that accounts for such phenomena ... contains many autonomous streams of information, each of which describes something resembling the universe as described by classical physics”. Deutsch (1997, p. 217) issues a challenge to those who would explain quantum speedup without many worlds: “[t]o those who still cling to a single-universe world-view, I issue this challenge: Explain how Shor's algorithm works”.

Recently, the development of an alternative model of quantum computation—the cluster state model—has cast some doubt on these claims. The standard network model (which I will also refer to as the ‘circuit’ model) and the cluster state model are computationally equivalent in the sense that one can be used to efficiently simulate the other; however, while an explanation of the network model in terms of many worlds seems intuitive and plausible, it has been pointed out by Steane (2003, pp. 474–475), among others, that it is by no means natural to describe cluster state computation in this way.

While Steane is correct, I will argue that the problem that the cluster state model presents to the many worlds explanation of quantum computation runs deeper than this. I will argue that the many worlds explanation of quantum computing is not only unnatural as an explanation of cluster state quantum computing, but that it is, in fact, incompatible with it.³ I will show how this incompatibility is brought to light through a consideration of the familiar preferred basis problem, for a preferred basis with which to distinguish the worlds inhabited by the cluster state neither emerges naturally as the result of a dynamical process, nor can be chosen a priori in any principled way. In the process I will provide a much needed exposition of cluster state computation to the philosophical community.⁴

In addition, I will argue that the many worlds explanation of quantum computing is inadequate as an explanation of even the standard network model of quantum computation. This is because, first, unlike its close cousin, the neo-Everettian many worlds interpretation of quantum mechanics,⁵ where the decoherence criterion is able to fulfil the role assigned to it, of determining the preferred basis for world decomposition with respect to macro-experience,⁶ the corresponding criterion for world decomposition in the context of quantum computing cannot fulfil this role except in an ad hoc way. Second: alternative explanations of quantum computation exist which, unlike the many worlds explanation, are compatible with both the network and cluster state model.

The quantum parallelism thesis, and the many worlds explanation of quantum computation that is so often associated with it, is undoubtedly of great heuristic value for the purposes of algorithm analysis and design, at least with regard to the network model. This is a fact which I should not be misunderstood as disputing. What I am disputing is that we should therefore be committed to the claim that these computational worlds are, in fact, ontologically real, or that they are indispensable for any explanation of quantum speedup.

My essay will proceed as follows. In order to exhibit the motivations and intuitions for adopting a many worlds view of quantum computation, I will begin, in Section 2, with an example of a simple quantum algorithm. In Section 3, I will argue that, despite its intuitive appeal, the many worlds view of quantum computation is not licensed by, and in fact is conceptually inferior to, the neo-Everettian version of the many worlds interpretation of quantum mechanics from which it receives its inspiration. In Section 4, I will describe the cluster state model of quantum computation and show how the cluster state model and the many worlds explanation are incompatible. In Section 5 I will argue, based on the conclusions of 3 Neo-Everett and quantum computing, 4 Cluster state quantum computing, that we should reject the many worlds explanation of quantum computation tout court.