It’s not easy to be a physicist. Richard Feynman (basically the Jimi Hendrix of physicists) once said: “The first principle is that you must not fool yourself – and you are the easiest person to fool.” This maxim motivates us to be critical of our research, even if we’re more critical when it comes to the research of others. From time to time, we even look through journals and technical magazines for arguments against the things we’re trying to do.

Last month, while I was looking for some nice criticism of quantum computing, I had the opportunity to read an article called “The Case Against Quantum Computing,” written by Mikhail Dyakonov, in IEEE Spectrum. While I was reading, I noticed two things that seemed out of the ordinary. First, all of the physics-based criticism of quantum computing was wrong, or had been addressed twenty years ago when the field was starting. The second, and perhaps more important thing, is that I could see the appeal of the article, despite its technical deficiencies.

I noticed that this article had been reviewed on the 27th of November by John Russell, here in HPCwire, so I thought that this would also be a good forum for a rebuttal (many thanks to Tiffany Trader for giving me the opportunity to write one). In the following sections, I’m going to go over two of the main technical points that Dyakonov makes, and try to give people a better idea about where we’re at in quantum computing. I’ll conclude with a comment on the article’s appeal.

Precision in Computing

Dyakonov: “A useful quantum computer needs to process a set of continuous parameters that is larger than the number of subatomic particles in the observable universe.”

No computer, classical or quantum, ever has to process even a single continuous parameter. In classical computers, we can use floating-point arithmetic to approximate continuous parameters using a finite number of bits. Most of the time, we can even manage to do it to within the desired relative precision, in order to avoid catastrophic error propagation. This is because the number of numbers which we can express using a floating-point type scales exponentially with the number of bits.

Normally, I wouldn’t belabour this point so heavily, but I’m going to do the “quantum” version of this in a minute, so let’s take a look at an animation of floating-point representations in action:

Here, I’m writing out all numbers of the form (−1)^{base sign}×significand× 10^{((−1)exp sign∗exponent)}, when the variables significand and exponent are each n-bit integers. Now, I can’t plot the whole real line (my monitor isn’t wide enough), so I’ve used a Riemann projection, drawing a ray from the center of the semi-circle shown above to the point on the real line that I’d like to show, and instead showing where that ray intersects the semi-circle, like so:

If we begin with 0 bits in the significand and exponent, we can assign any value we like to the sign bits, and the only number we can represent is 0. There are four independent ways, therefore, to represent 0, so there’s a little inefficiency in the representation. However, by the time I get up to 9 bits each in the significand and exponent, all of the points plotted are overlapping, and it’s clear that I have enough precision for the task at hand, for any real number I care to approximate.

A similar result holds in quantum computing, though the ‘data type’ we’ll consider here is a single qubit’s state, rather than a real number. The continuous complex parameters α and β mentioned by Professor Dyakonov go in a length two vector:

These parameters can also be mapped to angles θ and φ on the Bloch sphere, like so:

α = cos(^θ/₂)        β = e^iφ sin(^θ/₂)

(exercise for the reader: show that the state |0>, with α = 1 and β = 0, sits at the North Pole).

The operations we can apply in quantum computing are unitary matrices, equivalent to rotations of the Bloch sphere. For a single qubit, these matrices have two rows and two columns. Now, in fault-tolerant quantum computing, the operations which we can implement with arbitrarily low (but not exactly zero) error rates are limited to a discrete set. Let’s suppose for the sake of example, that there are two, and that they’re called H and T. Furthermore, let’s suppose that we only know how to initialise a single fixed state of our fault-tolerant qubit, the |0> state. How many qubit states can we reach with a string of Hs and Ts of fixed length n? Again, just as in floating-point arithmetic, the number of sequences I can generate scales exponentially with respect to the length of the sequence, despite a few collisions at low n (for example, HH |0> = |0>):

This animation doesn’t look quite as nice as the last one. There’s a lot more space to cover on the sphere than there is on the semi-circle that we used for floating-point arithmetic. From this, we can conclude that quantum computing is harder than classical computing, though I suspect that this does not come as a surprise.

Now, this isn’t the only thing fundamentally wrong with quantum computing, according to Professor Dyakonov. According to him, the entire discussion above is irrelevant, since imprecision and error will inevitably ruin any large-scale quantum computation before we can even think about stringing our Hs and Ts together. This is probably also not a surprise, but this was one of the first big problems that was ever solved in quantum computing, and I’ll talk about it a bit in the following section.

The Threshold Theorem

Dyakonov: “Indeed, [scientists studying quantum computing] claim that something called the threshold theorem proves it can be done. They point out that once the error per qubit per quantum gate is below a certain value, indefinitely long quantum computation becomes possible, at a cost of substantially increasing the number of qubits needed. With those extra qubits, they argue, you can handle errors by forming logical qubits using multiple physical qubits.”

The threshold theorem, initally proven by Aharonov and Ben-Or, has been around for about twenty years. The proof itself is in a 63-page paper, but the basic qualitative argument is relatively easy to grasp in a few paragraphs. At the cost of oversimplifying things, I’ll try to summarise that argument here.

Let’s define a logical gate as a small quantum computation that uses a number of physical gates acting on encoded states to simulate the effect of a single physical quantum logic gate acting on an unencoded state. Some logical gates can be made fault-tolerant by adding quantum error correction subroutines. The function of these subroutines is to correct the failure of a small number (typically one) of the physical quantum logic gates included in either the logical gate, or the error correction subroutines themselves. Each of these gadgets (that’s the technical term) contains a certain number of physical gates, let’s call it G. Also, let’s assume that, if any pair of these gates does something unanticipated, that the whole thing fails. When, then, does such a circuit have a low error probability? Let’s suppose, for the sake of simplicity, that each physical gate fails with probability p. The probability of error for the fault-tolerant gadget is , and whenever that’s less than p, we’re in business.

Now,  may not be a low enough probability of error for a given computation. In that case, we take advantage of something called concatenation, which is where you replace every physical gate in a fault-tolerant logical gate with yet another fault-tolerant logical gate, as depicted below:

If we use l levels of this concatenation, the number of gates we need to execute scales exponentially in l, but (very importantly) the final probability of error is p^2l [ed. note: p^2^l] so it’s doubly-exponentially suppressed.

If this sounds clunky and inefficient to you, you’d be more or less right. The important thing for this initial proof of concept was not that the scheme be particularly efficient, but that it use simple ideas which could be widely understood. Over the past twenty years, a small community of quantum computing researchers have been concerned with finding more efficient schemes, with fewer gates, and the ability to tolerate higher error rates, and the results have been fairly positive. They’ve also been hard at work proving that quantum computing can still be made fault-tolerant if the errors are correlated, rather than independent, as I’ve assumed above (though Aharonov and Ben-Or consider weak correlations in their original work).

During this time, people like Mikhail Dyakonov (and Gil Kalai, and other noted skeptics of quantum computing) have been career researchers. If the theorem were false, we’d expect one of these skeptics, or someone they’ve inspired, to have proven that it was false, or to show that physically-reasonable correlated noise precludes quantum computing. They have not done this. Instead, Dyakonov has loosely suggested that the theorem is false, without a direct statement, or evidence. I, for one, think that the theorem is more or less correct, and that quantum computing is possible.

These are the official fact-based rebuttals that we physicists rely on when confronted with critiques from Dyakonov and the other scientists and engineers who believe that quantum computing is doomed for some reason or another. They’ve been used before, and I suspect that they’ll be used again. In one sense, they’re perfectly sufficient, but I don’t think they’ve addressed the core problem. Dyakonov’s critiques are unfounded, and yet they endure. Why?

The Important Question

So, why was Dyakonov’s article written? Why was it published? I hope I’ve argued adequately that there’s not a lot of science behind it, so why is it so appealing?

I think this article was published because, in a sense we don’t often talk about, it’s correct. People who study quantum computing don’t view it as our responsibility to oppose the unjustified hype building up in the popular press. Times are tough for scientists in every field, as the budgets for those funding agencies Dyakonov mentions dwindle. There’s a temptation not to rock the boat, especially when the critics we do have don’t do a great job of challenging us on technical grounds.

We lament the lack of well-founded criticism, but how often, and how loudly, do we lament the abundance of unfounded optimism? Are these two things not equally dangerous to the progress of science? We’re the people best able to criticise quantum computing, is it then our responsibility to do so?

So far, we’ve left editors with little selection when they look for something to stem the tide of breathless proclamations about how quantum computing is going to solve everything. We often lament the lack of good critiques of quantum computing, but in the end, the only chance we have to elevate the level of criticism is to do it ourselves.

About the Author

Ben is a post-doctoral researcher at QuTech, part of the TU Delft in the Netherlands. His research is focused on near-term implementations of fault-tolerant quantum computing. He can be reached via Twitter (@BenCriger) and GitHub (github.com/bcriger). Scripts producing the animations in this article can be found at github.com/bcriger/examples/tree/master/articles/2019_01_HPCWire.