INTRODUCTORY NOTES ON QUANTUM COMPUTATION Keith Hannabuss Balliol College, Oxford Hilary Term 2009 Notation. In these notes we shall often use the physicists bra-ket notation, writing ψ for a vector ψ in a complex inner product space H, and ψ for the dual vector in H which sends φ H to the inner product ψ φ. Definition. The linear operator sending a general vector ψ to ξ η ψ = η ψ ξ, is denoted by ξ η. The tensor product of vectors φ K and ψ H is denoted by φ ψ or φψ. In this notation the projection P j onto a normalised vector φ j, given by can be written as P j = φ j φ j. P j : ψ φ j ψ φ j = φ j φ j ψ, 1
1 Quantum Measurement 1.1. Introduction Over the last 50 years the processing power of computers has steadily increased following the 1965 observation enunciated by Gordon Moore (cofounder of Intel) that the number of transistors per device doubles every 2 years. This power has mainly be achieved by miniaturising the components, and they are now so small (32nm transistors) that they are the size of large molecules, and so reach the quantum scale. Of course quantum effects are already used in the semiconductor chips at the heart of the modern machines and in the lasers which read and write the CDs and DVDs, but the actual processing is done with classical currents. Twenty-five years ago various people suggested that it should be possible to build a quantum computer in which the actual processing was done quantum mechanically. The key idea is that ordinary computers use binary arithmetic or logic, reducing everything to strings of 0s and 1s. The quantum mechanical analogue would be a collection of systems each described by a two-dimensional space V = C 2, which one could imagine as having basis e 0 and e 1 or 0 and 1, corresponding to the classical 0 and 1. Just as a choice of 0 or 1 represents 1 bit (binary digit) of information, a choice of state vector in V would represent one qubit (quantum bit) of quantum information. (Similarly combinations of two such systems represented by states in the space V V = C 4 carry two qubits of information, states in V V V = C 8 carry three qubits, and so on.) Definition 1.1.1. A qubit is represented by a non-zero vector in a two-dimensional inner product space V = C 2, spanned by orthonormal vectors 0 and 1 ; k qubits are represented by a non-zero vector in k V, (defined inductively by 1 V = V, k V = V ( k 1 V ). Remark: The two qubit space is spanned by the four vectors 0 0, 0 1, 1 0, 1 1. The power of the idea lies in the fact that where the classical bit is either a 0 or a 1, the quantum state can be in a superposition, or linear combination, of 0 and 1, and there are uncountably infinitely many such superpositions, vastly increasing the power of information processing machines. In effect, when a computer works with a non-trivial linear combination of 0 and 1 it is simultaneously working with 0 and 1, so that it can be vastly faster than ordinary computers. (One could similarly start with a three dimensional space V getting a qutrit, but we shall concentrate on the more usual case of qubits.) In the 1990s quantum algorithms were discovered which would run significantly faster than their classical analogues. Grover s algorithm will sort or search a list of N items in time of order N, rather than the classical N. The quantum Fourier Transform calculates Fourier transforms much faster than the classical methods. Shor s algorithm exploits this to factorise numbers faster than the classical methods, indeed fast enough to threaten the current public key encryption method which relies on our inability to factorise very large numbers fast. Fortunately quantum mechanics can also help with the solution, because observations in quantum theory change the state of the system, and this can be used as the basis of quantum communication channels where any attempt to eavesdrop is detectable. We shall discuss the quantum Fourier transform and Shor s algorithm in detail towards the end of the course. A quantum computer will need a method of processing information, some algorithms which exploit the quantum nature of the machine (otherwise one might as well use a classical computer, and a method of reading and writing data. One of the big problems is that interactions between the computer and the outside world tend to get rid of the details of superpositions on which it relies. This decoherence currently places strong limits on what can be done, so it is not yet known whether practical quantum 2
Quantum Measurement 3 computers can be built, but their superiority for certain kinds of calculation have encouraged a huge effort to develop one. There are many different lines of attack on the problem, but we shall concentrate on the main theoretical ideas. The main tools needed for quantum computing are 1. Methods of reading the output. 2. Algorithms which exploit the quantum nature of the machine, such as Shor s and Gorver s algorithms. 3. Processing units or logic gates which handle the qubits. We shall start by discussing how to read the output, because that links in to questions of measurement in quantum systems which have been discussed over a long period for other reasons. We shall then discuss the processing units, and finally give examples of some of the most efficient quantum algorithms. 1.2. Quantum Measurement Ideally the measurement of an observable A will give a precise result, with no dispersion. This is quite a strong requirement, as we recall from the following simple result: Theorem 1.2.1. The dispersion of A vanishes in the state ψ if and only if Aψ = E ψ [A]ψ, that is ψ is an eigenvector of A with eigenvalue E ψ [A]. Proof. We simply note that (for normalised ψ) ψ [A] 2 = E ψ [(A E ψ [A]) 2 ] = ψ (A E ψ [A]) 2 ψ = (A E ψ [A])ψ (A E ψ [A])ψ = (A E ψ [A])ψ 2, from which the result immediately follows. This means that after the measurement one must have an eigenstate of A. Suppose that A has an orthonormal set of eigenvectors φ j, satisfying Aφ j = α j φ j with non-degenerate eigenvalues α j, and one starts with a state ψ = j c j φ j. (Taking the inner product with φ k gives φ k ψ = j c j φ k φ j = c k, and so determines the coefficients.) If a measurement of A yields the value α j, then the state afterwards must be φ j, and this happens with probability c j 2. It is convenient to break the process into two parts: first the measurement ψ c j φ j = φ j ψ φ j, and then normalisation taking one to φ j. The first step just gives the projection P j ψ = c j ψ j = φ j ψ φ j of ψ onto φ j. We also note that the probability of this is c j 2 = P j ψ 2. Postulate (von Neumann Lüders). After the measurement of an observable A the state ψ becomes P ψ, where P is the orthogonal projection onto the subspace of states compatible with the outcome of the measurement. The probability of that outcome is P ψ 2. This postulate encapsulates the above discussion for non-degenerate eigenvalues, and extends it to degenerate ones.
4 Quantum Measurement 1.3. Polarised light There is actually one classical system familiar from everyday life, which displays the same kind of projection: the polarisation of light. Light can be polarised vertically or horizontally, a fact exploited by polaroid sunglasses, and we can describe its polarisation states in a vector space spanned by the two vectors v, for vertically polarised, and h, for horizontally polarised light. In general the polarisation state of a photon is a superposition α v + β h. In particular there are the states associated with light polarised at an angle θ to the vertical θ = cos θ v + sin θ h. Malus discovered in 1809 that for two filters are at an angle θ only a proportion cos 2 θ of light transmitted by the first filter passes through the second filter If, for example, the first filter measures v and the second θ then a state v emeges form the vertical filter, and after passing through the second filter at an angle θ it is projected to θ v θ = cos θ θ, with coeficient squared cos 2 θ. In particular, when θ = π/2 a proportion cos 2 (π/2) = 0 is transmitted. However, by inserting a middle filter at an angle θ to the vertical, a proportion cos 2 θ passed through the second filter and a proportion sin 2 θ of that through the third, giving a total proportion of cos 2 θ sin 2 θ = sin 2 (θ)/4, which is non-zero for all θ (0, π/2). Polarised light provides a good introduction to some of the paradoxes of quantum mechanics. Polarisation is not just a macroscopic effect, but each individual photon in the light beam is polarised, and the quantum space describing the polarisation of an individual photon is also spanned by the two vectors v and h. In the photon picture the idea of the proportion of the light which is transmitted is the probability that a photon will pass through the filter, and it turns out that there this quantum probability behaves differently from classical probability. The usual explanation of a polaroid filter is that it cuts out light polarised in the wrong direction, but then inserting an extra filter should cut out even more light not enhance it. After the development of quantum mechanics, and, in particular, Born s probabilistic interpretation of the wave function, some physicists got on with exploring the new quantum world, but others including Einstein and Schrödinger, struggled with the question of whether the new theory was already complete or whether the puzzles were an indication that something was missing, some hidden variables which could explain the statistical aspects, much as happened in statistical mechanics. At that time it was impossible to test many of their ideas experimentally, but over the last 30 years that has changed, settling many of the questions with which they struggled. 1.4. The Fréchet Bell Inequalities Theorem (Fréchet 1919, Bell 1961) 1.4.1. Suppose that A, B, and C are events in (that is subsets of) a space with probability measure P, so that P[A \ B] denotes the probability that A happens but B does not. Then one has the inequality P(A \ C) P(A \ B) + P(B \ C). Proof. One can show directly, or by consideration of a Venn diagram, that P(A \ B) + P(B \ C) P(A \ C) = P((A C) \ B) + P(B \ (A C)), from which the non-negativity of probabilities gives P(A \ C) P(A \ B) + P(B \ C).
The Fréchet Bell Inequalities 5 This is essentially what physicists know as Bell s inequality, after John Bell who rediscovered it in 1961. Bell s important contribution was to realise that this inequality, valid for all classical systems, breaks down for quantum systems. Let us consider three polaroid filters F A, F B, and F C, with angles γ between F A and F B, β between F A and F C and α between F B and F C. Let A be the event that a photon passes through the first filter, B that it passes through the second and C that it passes through the third. Then A \ B is the event that it passes through the first but not the second, which, according to our previous analysis is 1 cos 2 γ = sin 2 γ and similarly P(B \ C) = sin 2 α and P(A \ C) = sin 2 β. Now, the triangle with sides parallel to the filter directions has sides of length sin α, sin β and sin γ. If we have a right-angled triangle with β = π/2, then Pythagoras Theorem ensures that whilst for larger angles β so that sin 2 β = sin 2 α + sin 2 γ, sin 2 β > sin 2 α + sin 2 γ, P(A \ C) > P(A \ B) + P(B \ C), violating Bell s Inequality. Alternatively, we can use the cosine rule to see that sin 2 β = sin 2 α + sin 2 γ 2 sin α sin γ cos β, which violates Bell s Inequality when β > π/2 and cos β is negative. Before quantum theory there was no problem here. Fréchet s Inequality holds when we are measuring the probability of the same state or element of the probability space, but once the photon emerges from the second filter its state has been altered compared with the state in the first filter. More precisely, the photon enters the first filter in the sate ω Ω, and leaves it in the state ω A which then proceeds to the second filter, we are measuring P[ω A \ B], and P[ω A B \ C] rather than P[ω A \ B], and P[ω B \ C]. It was Einstein, Podolsky, and Rosen who put their finger on the underlying problem of quantum theory. (We give a modified version of their idea, based on a later suggestion of Bohm.) An atom can be induced to emit two photons in opposite directions and with identical polarisations, although we don t know what that polarisation may be. It is equally likely that both left and right moving photons are vertically polarised v v, or both horizontally polarised h h, so the overall state is ψ = 1 2 ( v v + h h ). Suppose we now let them travel some distance and then measure the polarisation of one of them with filter F A, and afterwards measure the polarisation of the other with filter F B. As soon as the first photon has gone through its filter measuring its vertical polarisation the state ψ must project to the 1 part consistent with that, 2 v v. But this produces exactly the same effect as if the other photon had been measured, and it will pass through F B with probability cos 2 θ, even if the filters were far enough apart that not even light could have travelled between them in time to let the second photon know the fate of the first. The answer is therefore exactly the same as for our discussion of a single photon going through successive filters. By repeating the experiment with various filters, we can find all the values needed in the Bell Fréchet inequality, and it is violated. Einstein, Podolsky and Rosen thought that the paradoxical elements of their thought experiment revealed the incompleteness of quantum theory, but in the 1980s Alain Aspect and his team did the experiments with polarised light and found that the Fréchet Inequality was indeed violated for polarised photons, even when the orientation of the polarisers was only determined whilst the photons were in flight. These results have subsequently been confirmed by numerous other experiments. It seems that
6 Quantum Measurement either there are no hidden variables or, if they do exist, they must themselves have many paradoxical features. We conclude by noting that the feature of the Einstein Podolsky Rosen paradox which made things work was the special state 1 2 ( v v + h h ), linking properties of the two photons. Schrödinger gave this the name entanglement.