The Quantum in Quantum Computing

I am sure, by now, all of you have seen Canada's prime minister  "explain" quantum computers at Perimeter. It's really great that politicians care about these things and he managed to say what is the standard explanation for the speed up of quantum computers compared to their classical cousins: It is because you can have superpositions of initial states and therefore "perform many operations in parallel".

Except of course, that this is bullshit. This is not the reason for the speed up, you can do the same with a classical computer, at least with a  probabilistic one: You can also as step one perform a random process (throw a coin, turn a Roulette wheel, whatever) to determine the initial state you start your computer with. Then looking at it from the outside, the state of the classical computer is mixed and the further time evolution also "does all the computations in parallel". That just follows from the formalism of (classical) statistical mechanics.

Of course, that does not help much since the outcome is likely also probabilistic. But it has the same parallelism. And as the state space of a qubit is all of a Bloch sphere, the state space of a classical bit (allowing mixed states) is also an interval allowing a continuum of intermediate states.

The difference between quantum and classical is elsewhere. And it has to do with non-commuting operators (as those are essential for quantum properties) and those allow for entanglement.

To be more specific, let us consider one of the most famous quantum algorithms, Grover's database lookup, There the problem (at least in its original form) is to figure out which of $N$ possible "boxes" contains the hidden coin. Classically, you cannot do better than opening one after the other (or possibly in a random pattern), which takes $O(N)$ steps (on average).

For the quantum version, you first have to say how to encode the problem. The lore is, that you start with an $N$-dimensional Hilbert space with a basis $|1\rangle\cdots|N\rangle$. The secret is that one of these basis vectors is picked. Let's call it $|\omega\rangle$ and it is given to you in terms of a projection operator $P=|\omega\rangle\langle\omega|$.

Furthermore, you have at your disposal a way to create the flat superposition $|s\rangle = \frac1{\sqrt N}\sum_{i=1}^N |i\rangle$ and a number operator $K$ that act like $K|k\rangle= k|k\rangle$, i.e. is diagonal in the above basis and is able to distinguish the basis elements in terms of its eigenvalues.

Then, what you are supposed to do is the following: You form two unitary operators $U_\omega = 1 - 2P$  (this multiplies $|\omega\rangle$ by -1 while being the identity on the orthogonal subspace, i.e. is a reflection on the plane orthogonal to $|\omega\rangle$) and $U_s = 2|s\rangle\langle s| - 1$ which reflects the vectors orthogonal to $|s\rangle$.

It is not hard to see that both $U_s$ and $U_\omega$ map the two dimensional place spanned by $|s\rangle$ and $|\omega\rangle$ into itself. They are both reflections and thus their product is a rotation by twice the angle between the two planes which is given in terms of the scalar product $\langle s|\omega\rangle =1/\sqrt{N}$ as $\phi =\sin^{-1}\langle s|\omega\rangle$.

But obviously, using a rotation by $\cos^{-1}\langle s|\omega\rangle$, one can rotate $|s\rangle$ onto $\omega$. So all we have to do is to apply the product $(U_sU\omega)^k$ where $k$ is the ratio between these two angles which is $O(\sqrt{N})$. (No need to worry that this is not an integer, the error is $O(1/N)$ and has no influence). Then you have turned your initial state $|s\rangle$ into $|omega\rangle$ and by measuring the observable $K$ above you know which box contained the coin.

Since this took only $O(\sqrt{N})$ steps this is a quadratic speed up compared to the classical case.

So how did we get this? As I said, it's not the superposition. Classically we could prepare the probabilistic state that opens each box with probability $1/N$. But we have to expect that we have to do that $O(N)$ times, so this is essential as fast as systematically opening one box after the other.

To have a better unified classical-quantum language, let us say that we have a state space spanned by $N$ pure states $1,\ldots,N$. What we can do in the quantum case is to turn an initial state which had probability $1/N$ to be in each of these pure states into one that is deterministically in the sought after state.

Classically, this is impossible since no time evolution can turn a mixed state into a pure state. One way to see this is that the entropy of the probabilistic state is $\log(N)$ while it is 0 for the sought after state. If you like classically, we only have the observables given by C*-algebra generated by $K$, i.e. we can only observe which box we are dealing with. Both $P$ and $U_\omega$ are also in this classical algebra (they are diagonal in the special basis) and the strict classical analogue would be that we are given a rank one projector in that algebra and we have to figure out which one.

But quantum mechanically, we have more, we also have $U_s$ which does not commute with $K$ and is thus not in the classical algebra. The trick really is that in this bigger quantum algebra generated by both $K$ and $U_s$, we can form a pure state that becomes the probabilistic state when restricted to the classical algebra. And as a pure state, we can come up with a time evolution that turns it into the pure state $|\omega\rangle$.

So, this is really where the non-commutativity and thus the quantumness comes in. And we shouldn't really expect Trudeau to be able to explain this in a two sentence statement.

PS: The actual speed up in the end comes of course from the fact that probabilities are amplitudes squared and the normalization in $|s\rangle$ is $1/\sqrt{N}$ which makes the angle to be rotated by proportional to $1/\sqrt{N}$.