Chapter 12: The Principle of Computational Equivalence

Section 8: Undecidability and Intractability

Quantum computers

In an ordinary classical setup one typically describes the state of something like a 2-color cellular automaton with n cells just by giving a list of n color values. But the standard formalism of quantum theory (see page 1058) implies that for an analogous quantum system—like a line of n quantum spins each either up or down—one instead has to give a whole vector of probability amplitudes for each of the 2n possible complete underlying spin configurations. And these amplitudes ai are assumed to be complex numbers with a continuous range of possible values, subject only to the conventional constraint of unit total probability Sum[Abs[ai]2,{i,2n}]==1. The evolution of such a quantum system can then formally be represented by successive multiplication of the vector of amplitudes by appropriate 2n× 2n unitary matrices.

In a classical system like a cellular automaton with n cells a probabilistic ensemble of states can similarly be described by a vector of 2n probabilities pi—now satisfying Sum[pi,{i,2n}]==1, and evolving by multiplication with 2n× 2n matrices having a single 1 in each row. (If the system is reversible—as in the quantum case—then the matrices are invertible.) But even if one assumes that all 2n states in the ensemble somehow manage to evolve in parallel, it is still fairly clear that to do reliable computations takes essentially as much effort as evolving single instances of the underlying system. For even though the vector of probabilities can formally give outcomes for 2n different initial conditions, any specific individual outcome could have probability as small as 2-n—and so would take 2n trials on average to detect.

The idea of setting up quantum analogs of systems like Turing machines and cellular automata began to be pursued in the early 1980s by a number of people, including myself. At first it was not clear what idealizations to make, but by the late 1980s—especially through the work of David Deutsch—the concept had emerged that a quantum computer should be described in terms of a network of basic quantum gates. The idea was to have say n quantum spins (each representing a so-called qubit), then to do computations much like in the reversible logic systems of page 1097 or the sorting networks of page 1142 by applying some appropriate sequence of elementary operations. It was found to be sufficient to do operations on just one and two spins at a time, and in fact it was shown that any 2n× 2n unitary matrix can be approximated arbitrarily closely by a suitable sequence of for example underlying 2-spin {x, y} -> {x, Mod[x+y, 2]} operations (assuming values 0 and 1), together with 1-spin arbitrary phase change operations. Such phase changes can be produced by repeatedly applying a single irrational rotation, and using the fact that Mod[h s, 2 Pi] will eventually for some s come close to any given phase (see page 903). From the involvement of continuous numbers, one might at first imagine that it should be possible to do fundamentally more computations than can be done say in ordinary discrete cellular automata. But all the evidence is that—just as discussed on page 1128—this will not in fact be possible if one makes the assumption that at some level discrete operations must be used to set up the initial values of probability amplitudes.

From the fact that the basic evolution of an n-spin quantum system in effect involves superpositions of 2n spin configurations one might however still imagine that in finite computations exponential speedups should be possible. And as a potential example, consider setting up a quantum computer that evaluates a given Boolean function—with its initial configurations of spins encoding possible inputs to the function, and the final configuration of a particular spin representing the output from the function. One might imagine that with such a computer it would be easy to solve the NP-complete problem of satisfiability from page 768: one would just start off with a superposition in which all 2n possible inputs have equal amplitude, then look at whether the spin representing the output from the function has any amplitude to be in a particular configuration. But in an actual physical system one does not expect to be able to find values of amplitudes directly. For according to the standard formalism of quantum theory all amplitudes do is to determine probabilities for particular outcomes of measurements. And with the setup described, even if a particular function is ultimately satisfiable the probability for a single output spin to be measured say as up can be as little as 2-n—requiring on average 2n trials to distinguish from 0, just as in the classical probabilistic case.

With a more elaborate setup, however, it appears sometimes to be possible to spread out quantum amplitudes so as to make different outcomes correspond to much larger probability differences. And indeed in 1994 Peter Shor found a way to do this so as to get quantum computers at least formally to factor integers of size n using resources only polynomial in n. As mentioned in the note above, it becomes straightforward to factor m if one can get the values of MultiplicativeOrder[a, m]. But these correspond to periodicities in the list Mod[a^Range[m], m]. Given n spins one can imagine using their 2n possible configurations to represent each element of Range[m]. But now if one sets up a superposition of all these configurations, one can compute Mod[a^#, m]&, then essentially use Fourier to find periodicities—all with a polynomial number of quantum gates. And depending on FactorInteger[m] the resulting amplitudes show fairly large differences which can then be detected in the probabilities for different outcomes of measurements.

In the mid-1990s it was thought that quantum computers might perhaps give polynomial solutions to all NP problems. But in fact only a very few other examples were found—all ultimately based on very much the same ideas as factoring. And indeed it now seems decreasingly likely that quantum computers will give polynomial solutions to NP-complete problems. (Factoring is not known to be NP-complete.)

And even in the case of factoring there are questions about the idealizations used. It does appear that only modest precision is needed for the initial amplitudes. And it seems that perturbations from the environment can be overcome using versions of error-correcting codes. But it remains unclear just what might be needed actually to perform for example the final measurements required.

Simple physical versions of individual quantum gates have been built using particles localized for example in ion traps. But even modestly larger setups have been possible only in NMR and optical systems—which show formal similarities to quantum systems (and for example exhibit interference) but presumably do not have any unique quantum advantage. (There are other approaches to quantum computation that involve for example topology of 4D quantum fields. But it is difficult to see just what idealizations are realistic for these.)

From Stephen Wolfram: A New Kind of Science [citation]