Making sense of quantum annealing

One of the tougher things about writing and reading about quantum mechanics is keeping up with how the meaning of some words change as they graduate from being used in the realm of classical mechanics – where things are what they look like – to that of the quantum – where we have no idea what the things even are. If we don’t keep up but remain fixated on what a word means in one specific context, then we’re likely to experience a cognitive drag that limits our ability to relearn, and reacquire, some knowledge.

For example, teleportation in the classical sense is the complete disintegration of an individual or object in one location in space and its reappearance in another almost instanetaneously. In quantum mechanics, teleportation is almost always used to mean the simultaneous realisation of information at two points in space, not necessarily their transportation.

Another way to look at this: to a so-called classicist, teleportation means to take object A, subject it to process B and so achieve C. But when a quantumist enters the picture, claiming to take object A, subjecting it to a different process B* and so achieving C – and still calling it teleportation, we’re forced to jettison the involvement of process B or B* from our definition of teleportation. Effectively, teleportation to us goes from being A –> B –> C to being just A –> C.

Alfonso de la Fuente Ruiz, an engineering student at the Universidad de Burgos, Spain, in 2011, wrote in an article,

In some way, all methods for annealing, alloying, tempering or crystallisation are metaphors of nature that try to imitate the way in which the molecules of a metal order themselves when magnetisation occurs, or of a crystal during the phase transition that happens for instance when water freezes or silicon dioxide crystallises after having been previously heated up enough to break its chemical bonds.

So put another way, going from A –> B –> C to A –> C would be us re-understanding a metaphor of nature, and maybe even nature itself.

The thing called annealing has a similar curse upon it. In metallurgy, annealing is the process by which a metal is forced to recrystallise by heating it above its recrystallisation temperature and then letting it cool down. This way, the metal’s internal stresses are removed and the material becomes the stronger for it. Quantum annealing, however, is referred by Wikipedia as a “metaheuristic”. A heuristic is any technique that lets people learn something by themselves. A metaheuristic then is any technique that produces a heuristic. It is commonly found in the context of computing. What could it have to do with the quantum nature of matter?

To understand whatever is happening first requires us to acknowledge that a lot of what happens in quantum mechanics is simply mathematics. This isn’t always because physicists are dealing with unphysical entities; sometimes it’s because they’re dealing with objects that exist in ways that we can’t even comprehend (such as in extra dimensions) outside the language of mathematics.

So, quantum annealing is a metaheuristic technique that helps physicists, for example, look for one specific kind of solution to a problem that has multiple independent variables and a very large number of ways in which they can influence the state of the system. This is a very broad definition. A specific instance where it could be used is to find the ground state of a system of multiple particles. Each particle’s ground state comes to be when that particle has the lowest energy it can have and still exist. When it is supplied a little more energy, such as by heating, it starts to vibrate and move around. When it is cooled, it loses the extra energy and returns to its ground state.

But in a larger system consisting of more than a few particles, a sense of the system’s ground state doesn’t arise simply by knowing what each particle’s ground state is. It also requires analysing how the particles’ interactions with each other modifies their individual and cumulative energies. These calculations are performed using matrices with 2^N rows if there are N particles. It’s easy to see that the calculations can become quickly mind-boggling: if there are 10 particles, then the matrix is a giant grid with 1,048,576 cells. To avoid this, physicists take recourse through quantum annealing.

In the classical metallurgical definition of annealing, a crystal (object A) is heated beyond its recrystallisation temperature (process B) and then cooled (outcome C). Another way to understand this is by saying that for A to transform into C, it must undergo B, and then that B would have to be a process of heating. However, in the quantum realm, there can be more than one way for A to transform into C. A visualisation of the metallurgical annealing process shows how:

The x-axis marks time, the y-axis marks heat, or energy. The journey of the system from A to C means that, as it moves through time, its energy rises and then falls in a certain way. This is because of the system’s constitution as well as the techniques we’re using to manipulate it. However, say the system included a set of other particles (that don’t change its constitution), and that for those particles to go from A to C didn’t require conventional energising but a different kind of process (B) and that B is easier to compute when we’re trying to find C.

These processes actually exist in the quantum realm. One of them is called quantum tunneling. When the system – or let’s say a particle in the system – is going downhill from the peak of the energy mountain (in the graph), sometimes it gets stuck in a valley on the way, akin to the system being mostly in its ground state except in one patch, where a particle or some particles have knotted themselves up in a configuration such that they don’t have the lowest energy possible. This happens when the particle finds an energy level on the way down where it goes, “I’m quite comfortable here. If I’m to keep going down, I will need an energy-kick.” Such states are also called metastable states.

In a classical system, the particle will have to be given some extra energy to move up the energy barrier, and then roll on down to its global ground state. In a quantum system, the particle might be able to tunnel through the energy barrier and emerge on the other side. This is thanks to Heisenberg’s uncertainty principle, which states that a particle’s position and momentum (or velocity) can’t both be known simultaneously with the same accuracy. One consequence of this is that, if we know the particle’s velocity with great certainty, then we can only suspect that the particle will pop up in a given point in spacetime with fractional surety. E.g., “I’m 50% sure that the particle will be in the metastable part of the energy mountain.”

What this also means is that there is a very small, but non-zero, chance that the particle will pop up on the other side of the mountain after having borrowed some energy from its surroundings to tunnel through the barrier.

In most cases, quantum tunneling is understood to be a problem of statistical mechanics. What this means is that it’s not understood at a per-particle level but at the population level. If there are 10 million particles stuck in the metastable valley, and if there is a 1% chance for each particle to tunnel through the valley and come out the other side, then we might be able to say 1% of the 10 million particles will tunnel; the remaining 90% will be reflected back. There is also a strange energy conservation mechanism at work: the tunnelers will borrow energy from their surroundings and go through while the ones bouncing back will do so at a higher energy than they had when they came in.

This means that in a computer that is solving problems by transforming A to C in the quickest way possible, using quantum annealing to make that journey will be orders of magnitude more effective than using metallurgical annealing because more particles will be delivered to their ground state, fewer will be left behind in metastable valleys. The annealing itself is a metaphor: if a piece of metal recalibrates itself during annealing, then a problematic quantum system resolves itself through quantum annealing.

To be a little more technical: quantum annealing is a set of algorithms that introduces new variables into the system (A) so that, with their help, the algorithms can find a shortcut for A to turn into C.

The world’s most famous quantum annealer is the D-Wave system. Ars Technica wrote this about their 2000Q model in January 2017:

Annealing involves a series of magnets that are arranged on a grid. The magnetic field of each magnet influences all the other magnets—together, they flip orientation to arrange themselves to minimize the amount of energy stored in the overall magnetic field. You can use the orientation of the magnets to solve problems by controlling how strongly the magnetic field from each magnet affects all the other magnets.

To obtain a solution, you start with lots of energy so the magnets can flip back and forth easily. As you slowly cool, the flipping magnets settle as the overall field reaches lower and lower energetic states, until you freeze the magnets into the lowest energy state. After that, you read the orientation of each magnet, and that is the solution to the problem. You may not believe me, but this works really well—so well that it’s modeled using ordinary computers (where it is called simulated annealing) to solve a wide variety of problems.

As the excerpt makes clear, an annealer can be used as a computer if system A is chosen such that it can evolve into different Cs. The more kinds of C there are possible, the more problems that A can be used to solve. For example, D-Wave can find better solutions than classical computers can for problems in aerodynamic modelling using quantum annealing – but it still can’t crack Shor’s algorithm, used widely in data encryption technologies. So the scientists and engineers working on D-Wave will be trying to augment their A such that Shor’s algorithm is also within reach.

Moreover, because of how 2000Q works, the same solution can be the result of different magnetic configurations – perhaps even millions of them. So apart from zeroing in on a solution, the computer must also figure out the different ways in which the solution can be achieved. But because there are so many possibilities, D-Wave must be ‘taught’ to identify some of them, all of them or a sample of them in an unbiased manner.

Thus, such are the problems that people working on the edge of quantum computing have to deal with these days.

(To be clear: the ‘A’ in the 2000Q is not a system of simple particles as much as it is an array of qubits, which I’ll save for a different post.)

Featured image credit: Engin_Akyurt/pixabay.