# 2003 - Quantum Computation Lecture 2 - Interference

[YouTube](https://www.youtube.com/watch?v=ajgr58WN7A4)

Duration: 00:41:52

## Transcript
### David Deutsch

00:01:00 - 00:03:13

In the last lecture I gave an overview of how physical systems in general are described in quantum theory via entities called observables which can be represented by matrices and via expectation value functions that map observables to real numbers. I also told you about the simplest kind of observables, Boolean observables, which are those whose spectrum contains exactly two elements. And I told you about the simplest kind of quantum system, the qubit, a physical system all of whose non-trivial observables are Boolean. But I didn't describe any phenomena. In this lecture I'll use that theory of observables and expectation values to analyze the archetypal quantum phenomenon that of quantum interference. It can be defined as a phenomenon in which an easily measurable observable is first sharp and then becomes unsharp and then becomes sharp again. To put that in parallel universes terms, an interference phenomenon is one in which the observed outcome depends on what has been happening in more than one universe. Quantum interference can be demonstrated in a classic set of experiments involving a single qubit. In the case I'll describe, the qubit is a subsystem of a single photon or particle of light. What I mean by a subsystem of a single photon is that only some of the observables of the photon are involved and these observables evolve independently of all the others during these experiments. So it's permissible to regard them as constituting a quantum physical system in their own right.

00:03:13 - 00:05:30

Like all experiments, this one can be regarded as a computation on a single qubit. Now a classical computer with only a single bit of memory couldn't do very much. In fact there are exactly four possible computations that one can do with a single bit. Set it to one value, set it to its other possible value, flip the value, that's the not computation, or leave it alone, that's the null computation, or there's any sequence of those, though the net effect of any sequence would still be the same as one of those four. That's classical computations, but a single qubit already allows for quite a variety of quantum computations. Some of them are useful as computations, some of them are interesting in the theory of quantum computation, and some of them are interesting as physical processes. The single qubit experiment I'm going to describe has a bit of all three. Now I'll specify a qubit within an individual photon using the method I introduced in the previous lecture, namely I'll pick a Boolean observable of the photon and then I'll define our qubit as the minimal physical system containing that Boolean observable. The observable in question is which of two particular directions the photon is traveling in. We'll make sure that in this experiment it never does travel in any direction other than very close to one or other of these two directions. So measuring which one it is, we're measuring a Boolean observable, and I'll call that observable $\hat{Z}$ with eigenvalues plus and minus one, and standing for the two possible directions of travel.

00:05:30 - 00:07:49

Here's a source of photons, a laser. Here are some of them streaming out of it, about three times $10^{15}$ of them per second in fact. If we made this beam a meter long say, well the speed of light is about three times ten to the eight meters per second, so at any one instant there'd be about ten to the seven or ten million photons present in the beam. If we want to experiment on one photon at a time, we put a dark filter in front of the laser which absorbs most of the photons. If the filter reduced the intensity of light by a factor of ten million, then there'd be only one photon at a time on average on this meter long path. In the actual experiment I'm going to show you in a moment, we're going to make the intensity even lower than that, so the effective distance between successive photons entering the apparatus will be not one meter but about thirty-five meters. This is the quantum optics lab where the experiment is going to be done. This is Manuel and this is Eran. Hi. This is a source of photons, it's a laser and it's emitting about five milliwatts in the form of photons here. Now we want to do this experiment on one photon at a time, so we pass the beam through dark filters which consist of glass with embedded metal in it. There's one of them and then the light passes into this fiber optic cable, around here and into this, past this dark filter whose effect you can see here, making the beam far too weak for the naked eye to see.

00:07:49 - 00:09:57

The net effect of these filters is to attenuate the beam by a factor of $10^{10}$ so that here there are only $10^6$ photons entering the apparatus per second, one per microsecond. I said that a photon is a particle. What I mean is that in a single universe it has many of the properties that particles would have in classical physics. For instance, the photons here are moving through space at a constant speed in a straight line, just as Newton's laws would require, or actually a very slightly curved line because of gravity. And here they are bouncing off a barrier with the neat specular reflection that the conservation laws of classical mechanics would predict. Crossing two beams shows that to the extent that they resemble classical particles, photons must be really tiny, point particles to a good approximation because the beams just don't notice each other. Even with $10^{15}$ odd photons passing by each other every second here, we don't notice any absorption or deflection out of the beam through collisions. But we'll see in a moment that in other ways they don't resemble classical particles and they're not located at just one point. We could verify that none of them is deflected by putting a sensitive photon detector like this one somewhere outside the two beams. This is a solid state photon detector. If a photon goes in here it creates a small electric current which is then amplified and can be displayed on the oscilloscope. So in an experiment involving only one photon, this device measures a Boolean observable. Does the photon strike a given location where we put the detector or not?

00:09:57 - 00:12:07

Incidentally this is a destructive sort of measurement. The photon, including the qubit we're interested in, is always destroyed in the course of detecting its presence with this device. That's a limitation that won't matter in these experiments but in future lectures when I discuss more sophisticated quantum computations we'll need qubits that remain in existence after each computational step so they can participate in subsequent steps. So we can only use this device as the last step of a computation reading the output. It wouldn't do as an internal component of a quantum computer but this device would. Here's the key component in the little one qubit quantum computer that we're going to build. It's called a beam splitter. When a photon with a sharp direction of motion strikes it, its direction of motion becomes unsharp. What happens is that in half the universes it just carries on straight through and in the other half it bounces off as if from a mirror. It turns out that it can do the reverse of that too. If we were to put ordinary mirrors here and here to reflect the photon back the way it came, whichever way that was in each universe, then after it strikes the beam splitter again its direction of motion is sharp again. Let me just remind you that this isn't a case of two photons merging or becoming alike. This is a case of a single photon whose direction of motion is not sharp striking the beam splitter and its direction of motion becoming sharp again. So the beam splitter is also a beam joiner. That phenomenon of an unsharp direction of motion becoming sharp is our quantum interference phenomenon.

00:12:07 - 00:14:24

Both parts of the photon that have traveled on different paths in different universes participate in the final interference process. How can we tell? Well if we were to remove one of the mirrors so that whenever the photon takes this path it never returns then the photon never would become sharp. In fact it would end up with a three-fold unsharpness going in that direction in some universes and then the beam splitter would have nothing to join so in other universes it would end up going in this direction and in others in that direction. If we want to experiment on a single qubit we have to keep things simpler than that. A qubit has only Boolean observables so we have to make sure that none of the observables of the system or subsystem that we're investigating ever take on more than two values at a time. So the basic form of all the single qubit experiments that I'm going to describe is this. We start with a photon, we pass it through a beam splitter after which it's traveling on two paths, we define a Boolean observable of the photon as being which of the two paths it's traveling on. Then using the method I introduced in the previous lecture we'll define our qubit as the minimal physical system containing that Boolean observable. I'll call that observable $\hat{Z}$ with spectrum $\{+1,-1\}$, the eigenvalue minus 1 standing for one path and plus 1 for the other. Since $\hat{Z}$ has spectrum $\{+1,-1\}$ it'll be represented by a Hermitian matrix which may change with time but always in such a way that its eigenvalues are plus or minus 1. To give a full description of the qubit and what happens to it in these experiments we need to specify as always three things.

00:14:24 - 00:16:28

Its static constitution which is what its observables are and what the algebra is at any one time. The dynamics is how the observables change with time, remember they change in such a way as to keep their algebra at any given time constant. And the state which means how the system is programmed, how it's prepared on a particular occasion and that's all summed up in the expectation value function. So static constitution first. As I indicated last time the set of all observables of a qubit at any one instant has the same algebra as the set of all two by two Hermitian matrices. Now that algebra is most straightforwardly represented by two by two Hermitian matrices but there are higher dimensional matrix representations of the same algebra and it turns out that in most experiments we'd have to use one of those. There's a simple reason for that which I'll get to in the next lecture. Anyway for our present purposes it's sufficient to represent the observables of our qubit as two by two matrices. There's a nice way of working with two by two Hermitian matrices and more generally $2^n$ by $2^n$ ones as well where we don't have to bother getting our hands dirty with the actual components, the complex numbers. Instead we use four fixed standard Hermitian matrices namely the unit matrix I and three other matrices known as the Pauli matrices $\sigma_X$ $\sigma_Y$ and $\sigma_Z$. Every two by two Hermitian matrix can be expressed as a linear combination with real coefficients of these four matrices.

00:16:28 - 00:18:48

Here are some more properties of Pauli matrices. First you can easily verify that the eigenvalues of a Pauli matrix are plus and minus one and that the square of each Pauli matrix therefore is the unit matrix. Another property $\sigma_X\sigma_Y=i\sigma_Z$ and the other two cyclic permutations of that and also the Hermitian conjugates of those three $\sigma_Y\sigma_X=-i\sigma_Z$ and so on. These formulae completely define the algebra of two by two Hermitian matrices in the sense that if we express matrices as linear combinations of these we don't need to know the rules for adding or multiplying matrices or finding their inverses. That's all encoded in these formulae. So I'll express the observables of our qubit in terms of Pauli matrices. By the way, Pauli matrices are often called the Pauli spin matrices because they're useful in the quantum mechanics of spin and rotation but nothing in today's experiment has anything to do with spin or rotation. Nowadays when I think of Pauli matrices I don't think of spin I think of qubits. Well now I can tell you the matrix representation of the observable $\hat{Z}$ that I've defined. $\hat{Z}$ is represented at time zero by the matrix $\sigma_Z$ and the unit observable $\hat{1}$ is as always represented by the two by two unit matrix I. These equal signs should really be is represented by not equals but we save ourselves a lot of useless mathematical baggage if we gloss over that difference when we're doing calculations.

00:18:48 - 00:21:22

Now all the other two by two Hermitian matrices also represent observables of the qubit at time zero or at any other time but in particular at time zero. So there is an observable whose representation at time zero is $\sigma_X$. I'll call that observable $\hat{X}$ and I'll call the observable represented by $\sigma_Y$ at time zero $\hat{Y}$. What are $\hat{X}$ and $\hat{Y}$ physically? It'll become clear in the course of this analysis what they are how one would measure them but from this definition we already know what their algebra is at any time. Since $\hat{X}$, $\hat{Y}$, and $\hat{Z}$ have the same algebra as the Pauli matrices at time zero and since the algebra of observables is an invariant feature of any quantum system they must have the same algebra at any other time $t$ as well. So for instance $\hat{X}(t)^2$ must just equal $\hat{1}$, and the same for $\hat{Y}$ and $\hat{Z}$, and $\hat{X}(t)\hat{Y}(t)=i\hat{Z}(t)$ and so on. All other observables must be linear combinations of $\hat{X}(t)$, $\hat{Y}(t)$, and $\hat{Z}(t)$ and the unit observable with real coefficients and the coefficients must not change with time. You can check the worked examples to verify that. Well next I'll specify the state of our qubit. As I said we're going to start each of these experiments by shining the laser at the beam splitter in the $Z=+1$ direction. So at time zero Z is sharp with the value $+1$ and so its expectation value must be one. In terms of matrices that just says the expectation value of $\sigma_Z$ is one and in that way we define the expectation value function for matrices as well as observables.

00:21:22 - 00:24:08

You'll see in the worked examples that it follows from this that the expectation value of $\sigma_X$ and the expectation value of $\sigma_Y$ are both zero and as always we have that the expectation value of the unit observable is one so the expectation value of the unit matrix I is one as well. Since we can express any observable of the qubit as a linear combination of Pauli matrices and since the expectation value function is a linear function we can use these formulae to find the expectation value of any observable at any time. So I've completely specified the state of the qubit. Note that these formulae refer to one particular state the one with $Z$ initially sharp with the value $+1$. They're not inherent properties of Pauli matrices in a different state these expectation values would be different. Okay now the dynamics. The dynamics are the laws of motion of all the qubit's observables. If we think of the apparatus as a quantum computer with our qubit as its sole working register the laws of motion are the rules defining what the computer does step by step to an arbitrary single qubit input. Well the first thing that happens is that the photon passes through the beam splitter. As in classical computation it's convenient to analyze computations as proceeding in steps each of which is a relatively simple computation. For present purposes we're not interested in what happens during the operation only in its net effect on observables. So here's the law of motion in that sense for the photon or for the qubit passing through a beam splitter. $\hat{Z}(t+1)=\hat{X}(t)$, $\hat{Y}(t+1)=\hat{Y}(t)$, and $\hat{X}(t+1)=-\hat{Z}(t)$.

00:24:08 - 00:26:50

The unit observable of course never changes. By taking linear combinations of these we can determine the law of motion for any observable of the qubit. So $\hat{Z}(1)$ we said was equal to $\hat{X}(0)$ which equals $\sigma_X$ and the expectation value of $\hat{Z}(1)$ equals the expectation value of $\sigma_X$ which I said earlier is zero. So Z at time 1 is no longer sharp. Its expectation value goes to zero which means physically that the photon goes straight through on path plus 1 in half the universes and bounces back on path minus 1 in the other half. Okay next the photon bounces off an ordinary mirror on either path and you can see what that does. The operation of bouncing off either of these mirrors is just the quantum generalization of the simplest single bit classical operation namely the NOT operation converting minus 1 to plus 1 and vice versa. The law of motion for the NOT operation is, just as for the beam splitter: $\hat{Z}(t+1)=-\hat{Z}(t)$, $\hat{Y}(t+1)=\hat{Y}(t)$, and $\hat{X}(t+1)=-\hat{X}(t)$. In general I'd have to describe the dynamics of one further elementary operation before I could describe the experiment as a whole. It's simply the operation of the photon traveling unimpeded for a certain distance in a straight line in between bits of apparatus. Well the observables change periodically with distance so to make the calculation easy I'm going to pretend that all the distances traveled are a whole number of periods a whole number of wavelengths so that our whole qubit will remain unchanged in between mirrors.

00:26:50 - 00:29:36

Now here's the experiment it starts at time 0 with the photon entering the apparatus traveling in the plus 1 direction. It strikes a beam splitter and we'll call some instant after that time 1. So at time 1 it's traveling in these two different directions plus 1 and minus 1 in different universes. Then regardless of which of those directions it's traveling in it strikes an ordinary mirror and that brings us to time 2. And then again regardless of its direction of motion it again strikes the beam splitter from which there are two possible directions of exit again plus 1 and minus 1. Finally at time 3 the photon enters one or other of two photon detectors which together constitute an instrument for measuring the Boolean observable Z. Let's use quantum theory and the quantum mechanical description I've given of the qubit to predict the outcome of this measurement. Now at time t equals 0 the observable $\hat{X}$ is represented by $\sigma_X$, the observable $\hat{Y}(0)$ by $\sigma_Y$, and the observable $\hat{Z}$ by $\sigma_Z$. Now the first thing that happens is that our photon strikes the beam splitter and so what happens to $\hat{X}$, $\hat{Y}$, and $\hat{Z}$? Well by time 1 it will have hit the beam splitter and so the observables of the qubit will have changed according to the formulae that I've given for the effect of a beam splitter. So we can read off $\hat{X}(1)=-\hat{Z}(0)$ which equals minus $\sigma_Z$ and similarly for $\hat{Y}(1)$ and $\hat{Z}(1)$.

00:29:36 - 00:32:00

Well then the photon strikes the mirrors and the effect is the not operation according to these formulae that I gave from which we read off the observables at time 2. Finally the photon hits the second beam splitter and again we can read off $\hat{X}(3)=-\hat{Z}(2)$ which makes it $\sigma_X$ and so on. It turns out that all the observables of the qubit at time 3 are the same as they were initially at time 0. In particular $\hat{Z}(3)=\hat{Z}(0)$ and so $\hat{Z}(3)$ is also sharp. So finally the photon is traveling only in the plus one direction and there's our prediction. Remember there's only one photon participating in this experiment. Consider the moment just before it strikes the second beam splitter. In different universes it's coming from different directions but just consider the universes in which it's coming along here say in the northwards direction. It's approaching the beam splitter. Usually when a photon approaches a beam splitter well it goes through in half the universes and bounces off in the other half. So usually if we observe what happens half the time we detect that it's passed straight through. Not in this experiment. In this experiment the photon necessarily takes a sharp right turn and is never observed to pass straight through and similarly in universes where the photon is approaching from the eastward direction it ignores the possibility of being reflected and just passes straight through. And the net effect is that in all the universes in which the experiment is done the photon is observed at the east detector and never at the north detector.

00:32:00 - 00:34:15

Now I'll show you this experiment in action. Before I do I'd better explain a detail that might be confusing otherwise. The experimentalists like to use the same physical beam splitter for both ends of the experiment. That way they don't have to bother with finding two of them with precisely matched properties and making them exactly parallel to each other and so on. It's hard enough aligning everything as it is. They eliminate the need for one of the beam splitters by slightly changing the geometry like this. Now this is still the plus one direction and this is the minus one direction for Z and there are the two corresponding detectors and our prediction again is that all the photons will come out here into this detector and none into this one. And when we do the experiment that's exactly what we see. Each spike here on the oscilloscope represents one photon. The upper trace, the yellow one, is from the detector at the plus one position and the lower trace, the green one, from the minus one position. The few stray spikes on the minus one trace are due to various imperfections in the apparatus, especially the detector. What is it that has made the photon in the universes where it approaches the beam splitter northwards turn right? It is the presence of its counterparts in the other universes striking the same point at the same time going east and we can verify that because if we put an opaque barrier on the eastward path and let some photons come through suddenly both detectors will be firing.

00:34:15 - 00:36:59

Open up this path again, the north detector goes quiet. Close it off again and the photons again start reaching the north detector. Now let's consider this whole process as a computation. The input was one. The computation as a whole consisted of three elementary computations. The middle one was a not operation. The outer two were the same as each other. Let me call it B for beam splitter and the output was again one. And you can see from here that the whole computation is an elaborate way of performing the unit operation, otherwise known as doing nothing at all. Is that an interesting computation? Yes, because just look what's happened here. We've performed B followed by not followed by B and the result is the unit operation. So $B\,\mathrm{NOT}\,B=I$. Dot here just means performed after. It's easy to prove that there is no classical computation using just one bit with this property. No computation such that if you perform it and then flip the value of the bit, then perform it again, the value of the bit will be unchanged. So B is an example of an elementary computation, an elementary quantum computation that has no classical analog. B also has an inverse which formally you can see from here is the square root of NOT. I'll show you in later lectures that this isn't just a manner of speaking. In quantum computation not really does have a square root. Many of the new modes of computation that quantum computers are capable of make use of the roots of not. In this experiment we only ever measured Z.

00:36:59 - 00:39:14

How would we measure X or Y or the qubit's other observables? We've already seen the answer there. Look at the effect of beam splitter. $\hat{Z}(1)$ equals $\hat{X}(0)$. So if we measure $\hat{Z}$ at time 1, that's the same as measuring $\hat{X}$ at time 0. So the act of passing a photon through a beam splitter and then measuring $\hat{Z}$ amounts to a measurement of $\hat{X}$. So the beam splitter followed by the two detectors constitute a measuring instrument for measuring $\hat{X}$. Here again I've been making the simplifying assumption that our qubit's observables don't change spontaneously with time, but they do according to this. Don't worry about the details, but it follows from this that if we slightly change the path lengths in one of these experiments, and in the lab the fine adjustments always involve doing that anyway, we can easily convert an instrument for measuring $\hat{X}$ into one for measuring Y. That's how to measure $\hat{Y}$, but you may still be wondering what is $\hat{Y}$ in the sense that $\hat{Z}$ is the Boolean observable for whether the particular photon is traveling in a particular direction. What is the Boolean observable $Y$? Well I could answer Y is the Boolean observable for which of the two directions the photon would travel in if it first passed through a certain beam splitter. Here we're coming up against the fact that since quantum systems are far richer than those that everyday language and intuition are adapted to describing, there isn't always a satisfactory terminology for describing quantum systems in everyday language.

00:39:14 - 00:41:48

There's a lot more to be said on this issue, but it would take us far beyond the scope of this lecture. You'll see in the worked examples that similar methods using phase shifts as well as mirrors and beam splitters allow us to measure any of the other observables of the qubit in a similar way. Note that in the general case where the observable being measured is not sharp, we have to make sure that all the instances of the photon in all universes strike a beam splitter say at the same moment at the same spot from their two possible directions. Otherwise the photon will be on four possible paths and the observable for which path it's on is no longer Boolean and so we're no longer talking about a qubit. I described this as a single qubit experiment but to perform it we did need other physical systems aside from the qubit itself. For a start the photon has other subsystems with observables such as its polarization or components of its velocity in directions that we made sure it never traveled in. Another physical system we used was the laser to generate the photon and then the filter to make sure that only one photon at a time was in the apparatus and then there were the mirrors and the beam splitters. These are all physical systems involving vast numbers of atoms and even larger numbers of observables all engaged in intricate interactions. This experiment was carefully arranged so that the net effect of all those interactions on the qubit during the computation could be expressed as a law of motion for the qubit alone. For reasons that I'll explain next time this would not have been possible for instance with the detectors. As I've said that didn't matter in this experiment because we're not interested in what happens to the photon after its Z observable has been measured at the end of the experiment. But to understand experiments involving repeated measurements of the same quantum system within a quantum computation we need a quantum analysis of the measurement process and that's what the next lecture will be about.