Hands up, who remembers D-Wave? No? Well, don't worry; it's still around. D-Wave is a company that is pursuing the development of a scalable quantum computer. And, just in case you forgot about them, they wheeled out a black box that apparently calculates stuff. It does this slower than your average iPad, but, you know, it's just a prototype.

Cynicism aside, the company's approach, called adiabatic quantum computing, actually has some interesting properties, and early studies indicated that it might be able to do exciting mathematical things. Unfortunately, a couple of snags have gotten in the way of further progress: adiabatic quantum computing has been shown to be identical to normal quantum computing, and there are no verifiable experimental results. Now, just to really put the screws on any optimists out there, a new paper has shown that adiabatic computers are actually quite bad at hard math problems.

Why NP-complete problems are a big deal

This result isn't too hard to understand once a bit of background scenery has been set up. So, let us first wallpaper in the pristine mountains of pure mathematics. Mathematicians seem to love categorizing stuff and, at some point in the past, someone noticed that all mathematical problems could be classified by how hard they were to solve.

This is perhaps best explained by example: the traveling salesman problem. The idea is that you have a bunch of locations on a map and you have to find the shortest route that visits every location. With two locations, this is easy; with three it is a bit harder, etc. The point is that the difficulty in finding the solution—or, more relevantly, the time taken to find the solution—increases as we add locations.

Now, if adding a location only added a fixed amount of time (or a slowly increasing amount of time) to the effort involved in finding the solution, this would be an easy problem. However, if the time taken increases exponentially with added locations, then the problem is difficult. These problems are bad, because a very small increase in the number of (in this example) locations can result in a problem that would take us on the order of the age of the universe to solve. For the record, the traveling salesman problem is considered to be fall into this class.

Mathematicians call this set of problems NP-complete and, so far, all our solutions to them take an exponentially increasing amount of time to solve. The impressive thing is that any problem in the NP-complete set can be reformulated as any of the other problems, so if we can find a way to solve any NP-complete problem efficiently, we can solve them all efficiently.

Quantum computing and decoherence

At the base of our pristine peaks are a bunch of jaw-jutting, adventurous physicists, mainly of the variety that use the word "quantum" and actually mean it. They began contemplating the dizzy heights of mathematics when it was discovered that the correlations, coherences, and superpositions of quantum systems could enable certain mathematical problems to be solved much faster.

What does this actually mean? First, let's deal with superposition. Essentially, superposition says that as long as you don't know what the value of something is, you must consider all possible values. Say I have a quantum bit (qubit) that has two possible values—until I measure it, I must consider that it can have both values at the same time. Only when I measure the qubit does it assume a single value, and the probability of obtaining a particular value is given by how the superposition behaves.

Coherence relates to this, but in subtle way. It describes how things change: imagine that I have two qubits and both are in a superposition state. As time progresses, each state is going to change, and that change would be reflected in the probabilities assigned to different measurement results. Now, if these two qubits are coherent, then their states will change in identical ways. They don't have to have the same probabilities of getting particular measurement results, but the changes to these probabilities should be related to each other.

But if I very gently shake one of the qubits, it begins to diverge in its behavior from that of its partner, so the two are no longer coherent. And therein lies public enemy number one for quantum physicists: decoherence.

Finally, we come to correlations, referred to as entanglement. This takes qubits and couples them together so that we can't consider them as qubit one and qubit two, but rather as qubit one-two. If we now gently shake qubit one, it will not diverge in its behavior from its partner. Effectively, this means that if we perform an operation on one qubit—as part of a database search, for instance—this affects the superposition states of all other entangled qubits and (hopefully) doesn't destroy the coherence of the system. If decoherence is public enemy number one, sudden death of entanglement is the misunderstood kid who sets fire to cats—we just hope it never happens to us.

To sum up: qubits can take on an indeterminate value, have that value change in synchrony with other qubits, and playing with one qubit effects the behavior of any entangled qubits. Given these properties, certain algorithms become a bit easier. For instance, factoring should, in principle, be easy on a quantum computer. Also, and with no hint of irony, quantum systems are best simulated with a quantum computer. But the jury's still out on whether NP-complete problems are made easier by quantum computers.

Actual quantum computers and NP-complete problems

Now, there are a couple of ways to put a quantum computer together. You can take the traditional processor approach, where each qubit is carefully and specifically manipulated until you perform all the logic operations on the qubits to complete the desired instruction set. At the end of this process, you read out the qubits to get an answer. All verified quantum computers are of this sort.

The alternative is to use an optimization approach. The idea is that many mathematical problems can be expressed in terms of an energy landscape, where the solution is the state that has the lowest energy. Now, formulating the energy landscape is one thing, but determining the state is usually impossible. So, optimization starts in an energy landscape for which the lowest energy state is known, and the qubits are put into that state. Then the landscape is gently and slowly prodded, bent, and twisted until it corresponds to the landscape of the problem.

If everything has gone well, the qubits still occupy the lowest energy state, and reading them out provides the answer to the problem. The speed of the calculation is simply a question of how easy it is for the qubits to jump out of the lowest energy state. If the energy gap is large, then you can transform the landscape very quickly. However, the energy gap narrows as you prod the system, and, if it narrows exponentially fast, then you have to optimize slowly.

But, and this is a key point, no one has ever shown that an NP-complete problem can be solved more quickly on a quantum computer than on a classical computer. Furthermore, all types of quantum computers have been shown to be mathematically identical, so if one won't work, none of them will.

Mind the gap: why adiabatic computing fails

This brings us, finally, to a current PNAS paper by researchers at Columbia University and NEC Laboratories who have investigated the problem-solving abilities of adiabatic quantum computers. What they have shown is that, when adiabatic quantum computers are used to solve NP-complete problems, the energy gap between the lowest energy state and the next state up is not well behaved. Instead, it narrows faster than exponentially, meaning the adiabatic quantum computing cannot, even in principle, solve NP-complete problems faster than a classical computer.

What makes this work special is that it is more general than the papers that precede it. Rather than focusing on a specific problem, the researchers looked at the statistics of how the energy gap behaves. They found that adiabatic quantum systems exhibit properties very similar to disordered quantum systems.

In particular, they found that quantum states become what's termed highly localized. The quantum state of one qubit is only influenced by a local patch of nearby neighbors, which are all quite separate from other groups. This looks rather like a phenomena called Anderson localization. Anderson localization has been studied for 50-odd years, and the researchers applied that knowledge to show how the probability of exiting the ground state increases as the optimization proceeds.

In the end, they conclude that NP-complete problems are just as hard on an adiabatic quantum computer as on a classical computer. And, since earlier work showed the equivalence between different variants of quantum computers, that pretty much shuts down the possibility of any quantum computer helping with NP-complete problems.

I don't think anyone in the field will be particularly surprised by this. The failure of earlier work to show that quantum computers offered a speed-up on any NP-complete problem indicated that it was likely that it simply was not possible.

PNAS, 2010, DOI: 10.1073/pnas.1002116107 (About DOIs).