## Abstract

From the philosopher’s perspective, the interest in quantum computation stems primarily from the way that it combines fundamental concepts from two distinct sciences: Physics, in particular Quantum Mechanics, and Computer Science, each long a subject of philosophical speculation and analysis in its own right. Quantum computing combines both of these more traditional areas of inquiry into one wholly new, if not quite independent, science. Over the course of this chapter we will be discussing some of the most important philosophical questions that arise from this merger and philosophical lessons to be learned.

My work on this chapter benefited significantly from my interactions with students and other audience members during and after a series of lectures I gave at the University of Urbino’s twenty-third international summer school in philosophy of physics, held online in June 2020, in the midst of the first wave of the COVID-19 pandemic, as well as a further lecture I gave for Michel Janssen’s “The Age of Entanglement” honors seminar at the University of Minnesota in December 2020, as the second wave of the pandemic began in earnest. Thanks to Ari Duwell, Eduardo Reck Miranda, Philip Papayannopoulos, and Lev Vaidman for comments on a previous draft of this chapter. I am also grateful for informal discussions, over the years, with Guido Bacciagaluppi, Jim Baggot, Michel Janssen, Christoph Lehner, Lev Vaidman, and David Wallace; my presentation of the Everett interpretation in Sect. 3, in particular, is significantly informed by what I take myself to have learned from these discussions, though I hold only myself responsible for any mistakes or misunderstandings in my presentation of the Everettian view. Section 2, on “Fundamental concepts” is heavily informed by my recent work on related topics with Stephan Hartmann, Michael Janas, Michel Janssen, and Markus Müller, as well as by informal correspondence with Jeffrey Bub and (the late) Bill Demopoulos; though here again, I take sole responsibility for any mistakes. Finally, I gratefully acknowledge the generous financial support of the *Alexander von Humboldt Stiftung*.

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## Notes

- 1.
Space does not permit me to exhaustively survey all of the philosophical issues brought up by quantum computing. The interested reader can find a summary of other important issues in Hagar and Cuffaro (2019).

- 2.
By “quantum mechanics” I mean the fundamental theoretical framework shared in common by every specific quantum-mechanical theory (quantum field theories, for instance) of a particular class of systems; see Aaronson (2013b, pp. 110–111), Janas et al. (2022, Chap. 1 and §6.3), Nielsen and Chuang (2000, p. 2), and Wallace (2019).

- 3.
These include Newtonian, Lagrangian, Hamiltonian, relativistic, and classical statistical mechanics. For a recent comparison and philosophical discussion of Lagrangian and Hamiltonian mechanics, see Curiel (2014).

- 4.
I say “logico-mathematical” because logical operations on bits can be thought of as modulo-2 arithmetical operations (see Boole, 1847).

- 5.
- 6.
The role of human computers in the United States of America’s space program, for instance, has been documented in Shetterly (2016).

- 7.
- 8.
A Turing machine’s tape does not need to be actually infinite in length. What is required is only that the tape be

*indefinitely*long, so that, for a given (finite) computation, the machine can be supplied with enough tape to carry it out. To put it another way: What constitutes ‘enough tape’ to carry out a computation is not part of the general definition of a Turing machine. It is, rather, assumed, for the purposes of that definition, that enough tape to carry out a given computation can be supplied. That said, certain variations on the Turing machine model restrict the ways in which tape can be read by the control unit in various ways. For instance, some variants employ separate tape(s) for the machine to write “rough work” on in addition to an output tape, some variants only allow the read-write head to move in one direction along the tape, and so on. - 9.
Although the von Neumann architecture, or ‘von Neumann machine’ is only one of a number of various types of stored-program computer, the terms have today (inappropriately, from a historical point of view) come to be understood synonymously (Copeland, 2017).

- 10.
“On the order of” is a technical term, usually symbolized in “big-oh notation” as

*O*(*T*(*n*)). An algorithm is*O*(*T*(*n*)) for some function*T*(*n*) if for every sufficiently large*n*, its actual running time \(t(n) \le c \cdot T(n)\) for some constant*c*. For instance, an algorithm that never takes more than \(5n^3\) steps is \(O(n^3)\). - 11.
A famous example of a problem for which only exponential-time solutions are known is the Traveling Salesman Problem (Cook, 2012).

- 12.
In some literature it is referred to as the

*Cobham-Edmonds thesis*. Kurt Gödel anticipated the principle, to some extent, in a private letter he wrote to John von Neumann in 1956. For further discussion, see Cuffaro (2018b). - 13.
This correspondence is not perfect, but the usefulness of the polynomial principle is such that we appeal to it despite this (Cuffaro, 2018b, §11.6).

- 14.
It is easy to see this: Consider a program that consists of \(n^k\) calls of a subroutine that takes \(n^l\) steps, where

*n*is the number of bits used to represent the input, and*k*and*l*are finite constants. The total number of steps needed to carry out this program will be \(n^{k+l}\). If*k*and*l*are finite constants then so is \(k+l\). In other words, \(n^{k+l}\) is still a polynomial. - 15.
Note that it makes sense to talk about solving a given problem just as easily on \(\mathfrak {M}_1\) as on \(\mathfrak {M}_2\) even when the problem under consideration is actually intractable for both. For instance, if some problem requires \(2^n\) steps to solve on \(\mathfrak {M}_1\) and \(2^n + n^3\) steps to solve on \(\mathfrak {M}_2\) then it is no harder, from the point of view of the polynomial principle, to solve it on \(\mathfrak {M}_2\) than on \(\mathfrak {M}_1\).

- 16.
For more on probabilistic and nondeterministic Turing machines and how they compare to their deterministic counterparts, see Cuffaro (2018b, §11.3).

- 17.
- 18.
We could have also expressed the thesis in terms of

**P**rather than**BPP**. Although it was thought, for many years, that there are more problems efficiently solvable on a probabilistic Turing machine than on a standard Turing machine, a number of recent results have pointed in the opposite direction (e.g., Agrawal et al., 2004), and it is now generally believed that classical probabilistic computation does not offer any performance advantage over classical deterministic computation (Arora , Barak, 2009, Chap. 20). In other words, it is now widely believed that \(\mathbf {P} = \mathbf {BPP},\) or that it is just as easy to solve a given problem on a deterministic Turing machine as it is on a probabilistic one. We have nevertheless stated the universality thesis in terms of**BPP**because this will prove convenient when it comes time to discuss the differences between classical and quantum computation. A quantum computer is, from one point of view, just another kind of probabilistic computer (that calculates probabilities differently), and it has the same success criterion as a classical probabilistic computer, i.e., we only demand that a given “solution” be correct with “high enough” probability. - 19.
See also: Goldreich (2008, p. 33), who names it differently.

- 20.
- 21.
- 22.
See note 2 above.

- 23.
This is analogous to the way we interpret probabilities in classical

*statistical*mechanics. - 24.
- 25.
- 26.
There are also problems for which a quantum computer, despite being unable to solve them easily, can nevertheless solve them significantly

*more*easily than a classical computer can. An example is the problem to search an unstructured database, for which a quantum (“Grover’s”) algorithm can reduce the number of steps required by a quadratic factor over any known classical algorithm. See: Bennett et al. (1997), Grover (1996), and for further discussion see Cuffaro (2018b, p. 269). - 27.
For the meaning of ‘on the order of’ see fn. 10.

- 28.
The interpretation of quantum mechanics that we will be discussing in this section is one of a number of related interpretations of quantum mechanics that are collectively referred to as the “Everett interpretation”. These include, but are not limited to Hugh Everett III’s original formulation (Everett III, 1956), the “Berlin Everettianism” of Lehner (1997), Lev Vaidman’s version of Everett (Vaidman, 1998), so-called “many minds” variants (Albert & Loewer, 1988), and finally the “many-worlds” variants that are the direct inspiration for the many-worlds explanation of quantum computing. Belonging to the last group are DeWitt’s (1973 [1971]) view, as well as the “Oxford Everett” interpretation (Deutsch, 1997; Saunders, 1995; Wallace, 2003, 2012) with which we will be mostly concerned here.

- 29.
In addition to Deutsch’s 1997 book, see Deutsch (2010), and see also Vaidman (2018 [2002], §7) and Wallace (2012, Chap. 10). The strongest and most in-depth defence of the many-worlds explanation of quantum computing that I am aware of is the one given by Hewitt-Horsman (2009).

- 30.
At the very least, the gravitational effects of other distant systems will not be able to be neglected.

- 31.
Some philosophers have questioned whether we should think of even the universe as a whole as a closed system (see, for instance, Cuffaro & Hartmann, 2021).

- 32.
The preferred basis problem is not the only challenge that needs to be met by an advocate of the Everett interpretation of quantum mechanics. Another issue that has been much discussed in recent literature is the problem of how to account for probabilities on the Everettian view. For more on this issue, see Adlam (2014), Dawid and Thébault (2015), Greaves and Myrvold (2010), Vaidman (1998, 2012), and Wallace (2007).

- 33.
- 34.
If the outcomes were completely uncorrelated, the probability distribution would be

- 35.
This state is identical to the one given in Eq. (2.11) but we repeat it here for convenience.

- 36.
For further discussion, see Myrvold et al. (2020).

- 37.
For a review of some of the uses envisioned for quantum computers in music, see Miranda (2021).

- 38.
- 39.
The

*n*-party case, for \(n \ge 3\), introduces subtleties which we will discuss in the next section. - 40.
- 41.
Note that I am taking ‘tractable’ here in a relative sense. That is, the resources required by a classical computer to reproduce a particular effect should differ tractably from those required by a quantum computer. Or in other words: it should

*not be essentially harder*for the classical system to produce the same effect as the quantum system. - 42.
- 43.
This statement is not meant to express any sort of theological opinion. It is merely a statement about how science operates, at least in this century.

- 44.
A similar point is made in Janas et al. (2022, Sect. 6.3, note 22).

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Cuffaro, M.E. (2022). The Philosophy of Quantum Computing. In: Miranda, E.R. (eds) Quantum Computing in the Arts and Humanities. Springer, Cham. https://doi.org/10.1007/978-3-030-95538-0_3

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