CyberZombie

scientiststhesis:

Finally, after a whole lot of posts about quantum mechanics, I’m ready to talk about what I wanted to talk about: quantum computing.

I have explained quite a bit about computability theory, or its classical version anyway. As of the writing of this post, I’m not done - there are things I still haven’t mentioned, like the Halting Problem. Still, this is sufficiently distinct from classical computation that I don’t think anyone will be confused.

On to definitions. Quantum computation is… computation on quantum information! And quantum information is… information that’s quantum! And information is…

…

Uh…

I dunno, man, don’t ask me complicated questions.

But anyway. In the same way we have bits, 0 and 1, that form strings that encode messages and information, and on which we operate, and that’s classical computation, we have qubits, |0⟩ and |1⟩, that form strings etc etc.

But, as we all know by now (one hopes), quantum mechanical phenomena are very different from classical. And we can’t really be straightforward in converting classical information to quantum, and vice-versa. Qubits get entangled, and into superposition, and interfere with each other in all those cute ways we’ve seen, which ends up creating a whole new and exciting field of knowledge!

Ok, so, let’s start with the basics.

Qubits

So, as I said, the basic units of quantum information are the quantum versions of the classical bits, the so-called qubits. For each qubit, we have two orthonormal states, |0⟩ and |1⟩.

How do we implement a qubit? Don’t know, don’t care. I’ve shown in my examples ways in which we can do it: the polarisation of a photon and the direction in which it’s going are two possibilities. In practice, this isn’t as easy as I’m making it sound, and building physical quantum computers is in fact a very delicate task that needs to strike a precise balance between isolation to guarantee quantum evolution and connection with the world to guarantee that we can actually perform computation.

But in the end, it makes no difference. All we need to know in order to perform quantum computation is that we have basic quantum mechanical systems that possess a pair of states as orthonormal basis.

And as you know by now, quantum mechanical systems exist in what are complex superpositions of states. Therefore, a given qubit will normally be in state |ψ⟩ = α|0⟩ + β|1⟩, and to make our lives easier, we will henceforth suppose that |α|² + |β|² = 1 so that our qubit will always be in a normalised state.

Now, you may recall that any given complex number z can be expressed as |z|e^iφ where φ is a real number; this follows from Euler’s Formula. And in particular, for any real φ, |e^iφ| = 1. So our arbitrary state |ψ⟩ can be expressed as

for some combination of real γ, θ and φ.

Why is this true? Since we have stipulated that |α|² + |β|² = 1, we might as well say that |α| = cos(θ/2) and |β| = sin(θ/2). Once again, we can express the complex numbers α and β themselves as the multiplication of their respective moduli and e to an imaginary power, from which the above follows.

But it turns out that the e^iγ factor is completely irrelevant. It has no observable consequences whatsoever and can be completely ignored. I will show that in a bit. This means, therefore, that we can simplify even further our representation:

The number γ is frequently called a global phase while φ is frequently called a relative phase. From this we conclude that the global phase is irrelevant, while the relative phase is meaningful (which will also be shown). Furthermore, the above reduction to θ and φ with an imaginary element only in the |1⟩ basis leads to a useful visual representation of one qubit:

That is called a Bloch sphere, and we can represent a single qubit as a point on its surface. Unfortunately, there is no simple generalisation of the Bloch sphere to more qubits (as they would understandably need more dimensions to be visualised and humans aren’t notoriously good at visualising more than 3 dimensions).

As I’ve already discussed, it’s impossible to know the exact state of an arbitrary qubit, which means that even though there’s an infinity of possible values for |ψ⟩ to take, we can only extract finite information from it (in the form of repeated collapse by measurement).

And that is something to bear in mind, too: you will only observe states |0⟩ and |1⟩, and your results will be probabilistic in nature, depending on your observations. That’s another fundamental difference between quantum computation and classical computation.

And then, what happens when we add more qubits? The hint was given in the two-photon experiment: we then have the basis states |00⟩, |01⟩, |10⟩, and |11⟩. And the generalisation follows for n qubits which will normally be in a superposition of 2ⁿ basis states.

A few noteworthy states are:

And the famous Bell states:

Keep reading

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