QUANTUM COMPUTING THE SOFT WAY

In natural science, Nature has given us a world and we’re just to discover its laws. In computers, we can stuff laws into it and create a world.

— Alan Kay

Let’s return to the question we began with: is there a single computing device that can efficiently simulate any other physical system? At the moment, the best candidate humanity has for such a computing device is a quantum computer. As you’ve probably guessed, you make such a device by combining all the elements we’ve been discussing. In this chapter we’ll discuss what a quantum computer is, why they’re useful, and whether they can be used to efficiently simulate any other physical system.

The theory of computation has traditionally been studied almost entirely in the abstract, as a topic in pure mathematics. This is to miss the point of it. Computers are physical objects, and computations are physical processes. What computers can or cannot compute is determined by the laws of physics alone, and not by pure mathematics.

— David Deutsch

In a general quantum computation, you start out with many qubits – I’ll draw four here, but in general it might be many more (or less). You apply quantum gates of various kinds, in particular, single-qubit gates and CNOT gates. And at the end of the circuit you read out the result by measuring in the computational basis. Here’s what it all looks like:

I haven’t labeled the single-qubit gates explicitly, but they might be various things – Hadamard gates, NOT gates, rotations, and perhaps others. Note also that while the computation starts in the computational basis state $|0000\rangle$, you can also start in some other computational basis state. I’ve just chosen $|0000\rangle$ for definiteness.

We can summarize the three steps in a quantum computation as follows:

1. Start in a computational basis state.

2. Apply a sequence of CNOT and single-qubit gates.

3. To obtain the result, measure in the computational basis. The probability of any result, say $00\ldots 000…0$, is just the square of the absolute value of the corresponding amplitude.

That’s all a general quantum computation is! If you understand this model, you know what a quantum computer is. It's pretty simple, really – enough so that I've sometimes heard people say “Is that all there is to it?” But while the model is simple, it contains remarkable depths, and exploring it could occupy many lifetimes.

In practice, people sometimes introduce other ideas into the way they describe quantum computations. If you’re a programmer, you can think of this as like the way programming language designers introduce higher-level abstractions to help people design different kinds of programs. In principle, those abstractions can always be reduced down to the level of AND and NOT gates. And if you understand that level – AND and NOT, or the type of quantum circuit shown above – then you have a foundation for building an understanding of the other ideas.

That may leave you wondering: does that mean you also need to master all the higher-level abstractions? The answer is no!

There are several reasons for this. One reason is that, as we discussed earlier, humanity doesn’t yet know what the higher-level abstractions are. We’re still trying to figure them out. A second reason is that it seems likely that the list of higher-level abstractions is inexhaustible. My guess – and it’s just a guess – is that we will continue to discover beautiful new abstractions forever, in both classical and quantum computing.

A related idea is that there are models of quantum computation different to the quantum circuit model. Some are merely small variations on the model I’ve described. For instance, instead of only using CNOT gates, we might allow any two-qubit unitary gate to be used in the circuit. Or perhaps instead of using qubits, we might use some other type of basic quantum system – say, the qutrit, which has three computational basis states, $|0\rangle, |1\rangle$, and $|2\rangle$. It probably won’t surprise you that the resulting models of computation are essentially equivalent to the quantum circuit model I’ve described. By this, I mean they can simulate the quantum circuit model (and vice versa) using roughly comparable numbers of gates and other physical resources.

There are also much more exotic variations, ideas such as measurement-based quantum computation, topological quantum computation, and others. I won’t describe these in any detail here, but suffice to say that they appear superficially very different to the circuit model. Nonetheless, they’re all mathematically equivalent to one another, including to the quantum circuit model. Thus a quantum computation in any of those models can be translated into an equivalent in the quantum circuit model, with only a small overhead in the cost of computation. And vice versa.

You may wonder why people bother thinking about other models, if they’re mathematically equivalent to the quantum circuit model. The reason is that just because two models are mathematically equivalent doesn’t mean they’re psychologically equivalent. Different models of computation stimulate different ways of thinking, and give rise to different ideas. And so it’s valuable to have other equivalent models.

Exercise: We saw earlier that composition of quantum gates corresponds to matrix multiplication (in reverse order). Show that the product of two unitary matrices U and V is also unitary. As a consequence, the net effect of any quantum circuit (before measurement) is to effect a unitary operation on the state space of the system.

Let me conclude this section with a brief comment about a particular class of quantum gates. These are gates which are multiples of the identity matrix, I:

$$\begin{aligned} \left[ \begin{array}{cc} e^{i\theta} & 0 \\ 0 & e^{i\theta} \end{array} \right] = e^{i\theta} \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right] = e^{i \theta} I. \end{aligned}.$$

When $\theta$ is a real number this is a unitary matrix, and so a valid quantum gate. The effect of the gate is simply to multiply the state of the quantum computer by $e^{i\theta}$. The number $e^{i\theta}$ is called a global phase factor. But even though the gate is valid, we rarely explicitly put such gates into quantum circuits. The reason is that such global phase factors have no impact on the results of the computation. To see why, imagine a quantum circuit which includes many such quantum gates, possibly for different values, $\theta_1, \theta_2, \ldots$. No matter where in the circuit the gates occur, the net effect is to multiply the state output from the circuit by an overall factor of $e^{i \theta_1 + i\theta_2 + \ldots}$ This doesn’t change the squared amplitudes of the computational basis states, and so has no impact on the measurement probabilities at the end of the computation. We could have left the gates out and it would make no difference.

Quantum computing people tend to be rather blase about such global phase factors. They’ll do things like not bother to distinguish between unitary gates such as X and -X, saying these gates are “the same up to a global phase factor”. They simply mean the -X gate is the same as doing the X gate, followed by the -I gate. Since the latter gate makes no difference to the output from the computation, it can safely be omitted. We’ll see examples in the next essay, about the quantum search algorithm, where at a couple of places it makes sense to multiply the quantum state by a global phase factor of -1.

Now we know what a quantum computer is, what are they good for? Earlier, I suggested quantum computers expand the range of operations available when computing. This is similar to the way boats expand the range of ways we can traverse space, so instead of having to get from point A to point B by land,

we can take a shortcut greatly cutting down the time required:

In order for the analogous story to hold for quantum computers, they need to be at least as capable as classical computers. Fortunately, it’s possible to convert any classical circuit into a quantum circuit. This requires some care. The obvious thing to do is to imagine that your classical circuit is expressed in terms of some standard universal set – say, the AND and NOT gates – and then to convert those gates into equivalent quantum gates.

This is easy to do with the NOT gate – we just turn it into an X gate. But what’s not so easy is the AND gate. You might wonder if there’s some quantum gate which takes two bits x and y as input, in the computational basis, $|x,y\rangle$, and then outputs a single qubit $|x \wedge y\rangle$, where $x \wedge y$ is just the logical AND of the bits x and y.

Unfortunately, that “quantum gate” makes no sense at all! Not only is it not unitary, it’s not even close: a unitary gate with two qubits as input necessarily has two qubits as output. The “gate” I described has just a single qubit as output, so there’s no way it can be unitary.

You might wonder if instead there’s some way we can find a two-qubit quantum gate which has $x \wedge y$ as one output, and something else as the other output. I won’t prove it, but it turns out that this is impossible. The proof actually isn’t all that hard – it’s a fun exercise to think through, if you want a challenge – but is more of a digression than I want to get into here.

That’s all rather disappointing. But there is a solution. It’s to use a three-qubit quantum gate called the Toffoli gate. The Toffoli gate is much like the CNOT gate, but instead of having a single control qubit, it has two control qubits, x and y, and a single target qubit, z. If both controls qubits are set, then the target is flipped. Otherwise, the target is left alone:

If the target starts out as z = 0, then you can see that the target output is just $x \wedge y$, and so the Toffoli gate can be used to simulate a classical AND gate. So if we have any classical circuit of AND and NOT gates then there’s a corresponding quantum circuit involving the same number of X and Toffoli gates which computes the same function.

Exercise: What’s a quantum circuit that can compute the NAND gate? Recall that the NAND of two bits x and y is just the NOT of $x \wedge y$.

Exercise: Can you find a way of implementing a NAND gate using just a single Toffoli gate and no other quantum gates? Note that your answer here may be the same as your answer to the previous exercise, if you answered that exercise using just a single Toffoli gate and no other quantum gates.

A wrinkle in all this is that the Toffoli gate isn’t in our standard set of basic quantum gates. However, it’s possible to build the Toffoli gate up out of CNOT and single-qubit unitary gates. One way of doing the breakdown is shown below:

There are various slick ways of “explaining” why this circuit works, but I’ll let you in on a secret: much of the earliest work on this was done by pure brute force, people simply trying lots and lots of different ways of implementing the Toffoli gate (sometimes, using a computer to assist in doing the search). Frankly, I wouldn’t worry too much about why this circuit works, just take it for granted that it does. You can look it up if you ever need to, or dig down into why. Needing to know the circuit details actually isn’t all that common, so I wouldn’t suggest memorizing it – not until you have a good reason.

Exercise: Show that the inverse of the Toffoli gate is just the Toffoli gate.

It’s comforting that we can always simulate a classical circuit – it means quantum computers aren’t slower than classical computers – but doesn’t answer the question of the last section: what problems are quantum computers good for? Can we find shortcuts that make them systematically faster than classical computers? It turns out there’s no general way known to do that. But there are some interesting classes of computation where quantum computers outperform classical.

Over the long term, I believe the most important use of quantum computers will be simulating other quantum systems. That may sound esoteric – why would anyone apart from a quantum physicist care about simulating quantum systems? But everybody in the future will (or, at least, will care about the consequences). The world is made up of quantum systems. Pharmaceutical companies employ thousands of chemists who synthesize molecules and characterize their properties. This is currently a very slow and painstaking process. In an ideal world they’d get the same information thousands or millions of times faster, by doing highly accurate computer simulations. And they’d get much more useful information, answering questions chemists can’t possibly hope to answer today. Unfortunately, classical computers are terrible at simulating quantum systems.

The reason classical computers are bad at simulating quantum systems isn’t difficult to understand. Suppose we have a molecule containing nn atoms – for a small molecule, n may be 1-10, for a complex molecule it may be hundreds or thousands or even more. And suppose we think of each atom as a qubit (not true, but go with it): to describe the system we’d need $2^n$ different amplitudes, one amplitude for each nn-bit computational basis state, e.g., $|010011\ldots\rangle$.

Of course, atoms aren’t qubits. They’re more complicated, and we need more amplitudes to describe them. Without getting into details, the rough scaling for an nn-atom molecule is that we need $k^n$ amplitudes, where $k \geq 2$. The value of k depends upon context – which aspects of the atom’s behavior are important. For generic quantum simulations k may be in the hundreds or more.

That’s a lot of amplitudes! Even for comparatively simple atoms and small values of nn, it means the number of amplitudes will be in the trillions. And it rises very rapidly, doubling or more for each extra atom. If k=100, then even n = 10 atoms will require 100 million trillion amplitudes. That’s a lot of amplitudes for a pretty simple molecule.

The result is that simulating such systems is incredibly hard. Just storing the amplitudes requires mindboggling amounts of computer memory. Simulating how they change in time is even more challenging, involving immensely complicated updates to all the amplitudes.

Physicists and chemists have found some clever tricks for simplifying the situation. But even with those tricks simulating quantum systems on classical computers seems to be impractical, except for tiny molecules, or in special situations. The reason most educated people today don’t know simulating quantum systems is important is because classical computers are so bad at it that it’s never been practical to do. We’ve been living too early in history to understand how incredibly important quantum simulation really is.

That’s going to change over the coming century. Many of these problems will become vastly easier when we have scalable quantum computers, since quantum computers turn out to be fantastically well suited to simulating quantum systems. Instead of each extra simulated atom requiring a doubling (or more) in classical computer memory, a quantum computer will need just a small (and constant) number of extra qubits. One way of thinking of this is as a loose quantum corollary to Moore’s law:

**The quantum corollary to Moore’s law**: Assuming both quantum and classical computers double in capacity every few years, the size of the quantum system we can simulate scales linearly with time on the best available classical computers, and exponentially with time on the best available quantum computers.

In the long run, quantum computers will win, and win easily.

The punchline is that it’s reasonable to suspect that if we could simulate quantum systems easily, we could greatly speed up drug discovery, and the discovery of other new types of materials.

I will risk the ire of my (understandably) hype-averse colleagues and say bluntly what I believe the likely impact of quantum simulation will be: there’s at least a 50 percent chance quantum simulation will result in one or more multi-trillion dollar industries. And there’s at least a 30 percent chance it will completely change human civilization. The catch: I don’t mean in 5 years, or 10 years, or even 20 years. I’m talking more over 100 years. And I could be wrong.

What makes me suspect this may be so important?

For most of history we humans understood almost nothing about what matter is. That’s changed over the past century or so, as we’ve built an amazingly detailed understanding of matter. But while that understanding has grown, our ability to control matter has lagged. Essentially, we’ve relied on what nature accidentally provided for us. We’ve gotten somewhat better at doing things like synthesizing new chemical elements and new molecules, but our control is still very primitive.

We’re now in the early days of a transition where we go from having almost no control of matter to having almost complete control of matter. Matter will become programmable; it will be designable. This will be as big a transition in our understanding of matter as the move from mechanical computing devices to modern computers was for computing. What qualitatively new forms of matter will we create? I don’t know, but the ability to use quantum computers to simulate quantum systems will be an essential part of this burgeoning design science.

Alright, enough speculation.

Let me also briefly mention the sober-minded conventional answer given to the question “what are quantum computers good for?” That answer is to list various algorithmic problems that we have some evidence can be solved faster on a quantum computer than on a classical computer.

The most famous example is Peter Shor’s beautiful quantum factoring algorithm. To find the prime factors of an n-bit integer seems to be a very difficult problem on a classical computer. The best existing algorithms are incredibly computationally expensive, with a cost that rises exponentially with nn. Even numbers with just a few hundred digits aren’t currently feasible to factor on classical computers. By contrast, Shor’s quantum factoring algorithm would make factoring into a comparatively easy task, if large-scale quantum computers can be built.

Factoring perhaps doesn’t seem like a very interesting application. But it turns out that the ability to factor lets you break some of the most widely-used encryption schemes, used by services such as Gmail and Amazon to keep your communications private. This ability to break encryption has made the world’s intelligence agencies very interested in factoring, and they’ve poured enormous sums of money into quantum computing research since the mid-1990s. Indeed, there’s a good (as yet unwritten) history book to be written about how the rise of quantum computing was caused by the interest of the world’s intelligence agencies in accessing humanity’s private thoughts.

There’s been surprisingly little public reflection about this on the part of the quantum computing community. One exception to this lack of public reflection is a brief discussion in Ronald de Wolf’s thoughtful essay The Potential Impact of Quantum Computers on Society (2017).

Brilliant quantum experts Talia Gershon and Scott Aaronson tell us in two short talks what quantum compuers are and are not, respectively:

© QPlayLearn 2020