Chapter 1. Impact 7.1

Impact…ON TECHNOLOGY: I7.1 Quantum computing

The material in the Chapter lies at the heart of what might prove one day to be a revolution in the way in which certain calculations are carried out as in a new generation of computers. Quantum computing is potentially capable of solving in seconds problems that conventional computing might be unable to solve in the lifetime of the universe. It might also undermine one of the most cherished aspects of commerce, war, and government: secrecy.

Conventional digital computing is based on the manipulation of strings of 0s and 1s, or bits (‘binary digits’). Each bit might be realized in practice by the physical state of an object, such as a potential difference across a semiconductor junction or the state of polarization (‘left-circularly polarized’ or ‘right-circularly polarized’) of a photon. We could denote these physical states 0 and 1, described respectively by the wavefunctions ψ₀ and ψ₁. In essence, a conventional digital computer carries out operations on |ψ₀|² and|ψ₁|². We have seen, however, that a state in quantum mechanics might be described by a linear combination of these wavefunctions such as c₀ψ₀ + c₁ψ₁, with arbitrary (or, at least, specified) values of the two coefficients. This superposition of states is called a qubit (for ‘quantum bit’). The core idea of quantum computing is starting to emerge: instead of carrying out operations on certain states of the system individually, computations can be carried out on several, and perhaps many, states simultaneously.

The core of conventional computing are various logical circuits such as AND and NOT. Thus, an AND gate gives an output of 1 if both its inputs are 1; a NOT gate gives an output of 0 if the input is 1 and an output of 1 if the input is 0. Conventional computers string together these and other so-called ‘Boolean operations’ and realize them in terms of potential differences at semiconductor junctions. To envision quantum computing, we need to take into account the Born interpretation of the wavefunction (Topic 7B) that only its square has a direct physical significance. You might suspect that because a quantum computer carries out operations on the wavefunction rather than the probability itself that the logical circuits will in some sense be acting as the square-root of the usual Boolean operations. That is, instead of NOT acting on the classical bits, the square-root of NOT (denoted \(\sqrt{\mathrm {NOT}} \)) will be acting on the quantum qubits. Something like that turns out to be the case.

It would take us too far astray to go into the details of quantum logic and its practical implementation (which is still at a very primitive stage), but it is worth taking some of the mystery out of \(\sqrt{\mathrm {NOT}} \) to see that it has a real, physically realizable meaning. To do so, consider the arrangement in Fig. I7.1a, which shows a photon incident on a half-silvered mirror. If the photon is reflected it enters what we could call the state 0 and if it is transmitted it enters what we shall call the state 1: the states are physically distinct as the photon occupies different parts of space. However, because we have not made an observation of the location of the photon, there are equal probabilities that it is in either state: according to the postulates of quantum theory we have prepared the qubit described by the linear combination ψ₀ + ψ₁ from the state 0 (the incident photon was above the mirror). If we were to detect the photon, half the time we would find it above the mirror and half the time below.

Now suppose we join a second similar arrangement to the first, as in Fig. I7.1b. According to quantum mechanics, the linear combination ψ₀ + ψ₁ recombines in the second arrangement to result in a photon that emerges below the second half-silvered mirror; that is, in the state 1. In other words, the combination of the two arrangements acts as a NOT gate. The implication is that each individual component contributes \(\sqrt{\mathrm {NOT}} \), because the succession \(\sqrt{\mathrm {NOT}} \)\(\sqrt{\mathrm {NOT}} \) is equivalent to NOT.

Quantum computation is obviously a highly subtle technology, but it is just starting to emerge into practical implementations. It builds conceptually on the postulates and will make use of the quantized real systems that we encounter in the text.