What *is* a qubit?

0:00 Qubits sound really complicated but they really don't have to be.

0:03 Like this laser is a qubit and an electron by itself is a qubit,

0:08 a trapped ion can be a qubit, there can be superconducting qubits,

0:12 there are so many different things that can be qubits.

0:15 Fundamentally the only thing that matters is if it is a two level quantum

0:20 system and all two level quantum systems

0:22 are are qubits and they're basically the same, at least mathematically.

0:27 But when you're making them in the lab,

0:28 you might decide to use one type of qubit over the other

0:31 because it's more stable or it's easier to manipulate, whatever it is.

0:36 Fundamentally, at the mathematics level, they're all the same.

0:40 So let's talk about what a qubit is.

0:42 So this laser pointer here has a little polarizer

0:45 in front of it and that polarizer makes sure

0:49 that all of the light is polarized in this direction

0:52 when it comes out of this laser pointer.

0:55 So this is polarized this way but I could rotate it now it's polarized that way.

1:00 Etc.

1:00 And so there are many many different states that this light can

1:05 be in but if I was to measure the Polarization then there will

1:09 only be two outcomes that this measurement can resolve So here's an example

1:15 this block of calcite measures polarization So if I put this in front

1:21 of my laser It splits it up into two dots and those two

1:24 dots have opposite polarizations to each other So what I mean

1:27 by opposite is that they're 90 degrees away from each other So one

1:32 of them is at 45 degrees and the other is at 130 degrees.

1:40 So you could think of this calcite as measuring the polarization,

1:43 but it only allows two options.

1:45 It can allow light to go through that now

1:49 has 45 degree polarization or 130 degree polarization.

1:52 But what if I rotated this calcite by a bit?

1:56 So if I rotated it like this, now

2:00 this calcite measures a different set of polarizations.

2:04 In fact, if it was measuring 45 degree and 135 degree light.

2:08 and I rotated it like this, then

2:11 now it's measuring horizontal and vertical light.

2:14 This is why, even though this light has so many different options for what state

2:20 it's in, we call this light a two level system because when you measure it,

2:25 there's always two outcomes, no matter which way you measure it.

2:29 there's always going to be just two.

2:32 And so fundamentally this system is two dimensional.

2:35 So here's what I mean by that.

2:37 Let's suppose I start off with my light pointing

2:41 in this direction so its state is represented by this.

2:45 By the way I'm going to put that inside of a thing called a ket.

2:48 That's just a way in quantum mechanics of marking

2:51 that this thing represents the state of this thing.

2:54 So this kind of means a state.

2:56 What will happen when I measure this light

2:59 with this calcite that's oriented like this?

3:02 This calcite breaks up light into two options, horizontal or vertical.

3:07 So that's light that's pointing this way or light that's pointing that way.

3:12 Now one of the key concepts in quantum mechanics is this idea of superposition.

3:17 We say that this light is really a mixture

3:20 of both of these types of light at the same time.

3:24 Like this light is clearly pushing somewhat to this direction,

3:28 but it's also pushing upwards, and so really it's a combination of these two.

3:32 To decide how much of each of those two options we have,

3:36 we need to put coefficients in front of these vectors.

3:39 In this example, I've put one on square root two in front of both of them,

3:43 and this just means that there's an equal amount of both.

3:47 But what if we chose to measure this light in some other direction?

3:51 So instead of horizontal and vertical,

3:53 we're Let's say we rotated it kind of like this.

3:56 So now we have to write that same light in terms of two other options.

4:01 So now we have to figure out what these coefficients are going to be.

4:04 And you can see that this coefficient should be pretty

4:07 big because this is pretty much the same as that.

4:10 But this one should be quite small.

4:12 This is called a change of basis.

4:14 So we've just changed how we look at this same light.

4:18 So this light is equal to being a bit horizontal and a bit vertical,

4:23 but it's also a bit equal to this and this at the same time.

4:27 And in fact there's never anything special about

4:29 which direction you choose to do the measurements in.

4:32 For quantum computing,

4:33 we choose one direction as our sort of like fundamental direction,

4:38 not for any actually good reasons, just we choose one, and then we go, okay,

4:44 so in this basis, option one, we're going to call that zero.

4:48 And in this basis, option two is going to be called one.

4:52 So let's say that I chose horizontal and vertical to be the 0 and 1.

4:57 I've just relabelled these, I haven't actually changed

5:00 the physics of that light in any way.

5:02 But now you can kind of see why you might call this a qubit, right?

5:08 Because there's a 0 and a 1 and we're saying that this light is

5:12 in some sort of combination of both 0 and 1 at the same time,

5:17 which is what you might have heard people say about qubits.

5:20 But an important thing to note though is that this state

5:23 really is both zero and one at the same time, right?

5:26 Like if it was just zero, it would be fully horizontal.

5:29 If it was just one, it would just be vertical.

5:32 But we know it's something else.

5:33 It's something kind of in between.

5:35 And so that's all people mean when they say

5:38 that qubits are doing two things at the same time.

5:40 Okay, sorry to interrupt, but I have an announcement.

5:43 There's going to be another cohort

5:45 of the live version of this course in January.

5:48 So it's a four week course and once a week we meet to go

5:52 over the homework problems and the students

5:54 get to like ask any questions they want.

5:57 So I just finished a cohort and it was so much fun.

6:02 And a lot of the students emailed me

6:04 to say that they thought they understood quantum mechanics,

6:06 at least somewhat before doing the problems

6:09 and realizing that there's actually more to it.

6:12 I really believe that doing problems is the best way to learn physics.

6:16 So that's why I came up with this course.

6:19 So if you're interested in doing that in January,

6:21 then there's going to be a link in the description.

6:24 Light isn't the only example of a qubit though.

6:28 Electrons can also be qubits, and that's by using their property called spin.

6:34 So an electron's spin points in a particular direction.

6:37 And just like with the calcite and light,

6:40 we can measure the spin of this electron in many different directions.

6:44 So the way that you measure spin is using a Stern Gerlach machine,

6:48 and those machines can be oriented in different ways.

6:51 As it's oriented here, It's going to measure whether the spin is up.

6:56 or down.

6:57 So again, there's only two options for this particle,

7:00 even though actually the spin could be pointing in any direction at all,

7:05 when you do a measurement it collapses it to one of two options.

7:09 And that's again the sort of two dimensionality of this state.

7:13 So when I'm measuring this electron in terms of up and down,

7:17 then I can think of its state as some combination of up.

7:21 and down.

7:22 But I could also orient the Skrngalak machine in the left right

7:26 direction and write our state as a combination of left and right.

7:30 Both of these two options for writing this state are equally valid,

7:35 both of them are true, but I might decide that really I am more interested in up

7:40 and down and I might want to make them the sort of canonical directions.

7:44 And so just like with the light, I might re label up and down to zero and one.

7:50 But light and electrons are not your only

7:53 choices when it comes to making a qubit.

7:56 And actually, if you're making a quantum computer,

7:58 there are much better choices.

8:00 And that's because you want your qubits to be very stable,

8:03 but also very controllable.

8:05 And I don't think that the light or spin are, um, really either of those.

8:10 And so instead people use things like trapped ions.

8:13 So for example, a trapped ion can be made into a qubit.

8:18 Okay so I just looked this up this morning, ytterbium,

8:21 apparently that is a atom that you can make into a trapped ion.

8:26 So you take the positive version of this and you

8:29 trap it in like a magnetic trap.

8:31 And then it has two ground states.

8:34 So a ground state is sort of the lowest energy state.

8:37 And when I say that there's two ground states,

8:39 what I really mean is that there are two that are very, very close in energy.

8:43 It's like a hyperfine splitting of the ground state.

8:47 And so those two energies can kind of act as your two options for the qubit,

8:52 so one of them is zero and one of them is one.

8:55 The reason why that's apparently a good choice is because you can

9:00 use microwave radiation to jump between those two or go into superpositions,

9:04 and so all of the gates in the quantum computer are superpositioned.

9:08 are made using uh, yeah, microwaves to manipulate the state.

9:11 So yeah, that's a really nice example

9:14 because it technically isn't a two level system.

9:17 Like, yes, there are these two energies that are very close to each other,

9:20 but there's loads of other energies for this atom as well, right?

9:24 Um, like the electron could always jump to higher energies,

9:27 but they can safely ignore those higher

9:30 energies because all of the manipulations

9:32 they're going to do are only going to be between those two lowest energies.

9:36 So yeah, that's a really cool way to make

9:38 a qubit because most things aren't naturally two level systems.

9:41 So if you saw a state like this written down with a combination of zero and one,

9:47 you wouldn't be able to tell whether we're talking about

9:49 light here or we're talking about electrons or something else.

9:53 And that's kind of by design.

9:55 The purpose of writing zero and one is to abstract away all of the messy

10:00 physics and just bring it down

10:02 to what's fundamentally important about this state,

10:05 that it is some combination of these two directions.

10:08 This also helps us draw the state as a two dimensional vector,

10:12 exactly like we did with light, no matter what the actual qubit is made of.

10:18 And that's really useful.

10:19 If I have the state a times zero plus b times one,

10:23 then all I need to do to draw it is to draw zero and one here and then

10:29 I want to take a steps in the zero direction and b steps in the one direction.

10:38 So maybe two here.

10:41 And then I link them up.

10:46 This is my way of drawing the state,

10:49 which completely abstracts away the physics.

10:51 And so, no matter what type of quantum computer you're talking about,

10:56 everyone can use the same language.

10:58 This also leads to a bunch of common shorthands

11:01 in the quantum computing community that I really like.

11:04 Like for example, the plus state is an equal superposition of 0 and 1.

11:11 What do you think the minus state is?

11:16 It's zero minus one.

11:18 You might be thinking how can this be negative?

11:22 Well actually it makes perfect sense if we think about it as a vector.

11:25 If we have positive zero but minus one

11:28 then it would be pointing in this direction.

11:31 And so a combination of zero and minus one looks like this.

11:38 whereas plus looks like this.

11:41 And again, if we go back to the light example, that makes perfect sense.

11:45 This is plus and this is minus.

11:47 But you can have even weirder things.

11:50 So these numbers can actually be complex numbers.

11:53 So this is the imaginary number.

11:55 I.

11:56 Now you might freak out and think like, what are complex numbers doing here?

12:01 But it actually can make a lot of sense.

12:04 Let's go back to thinking about the spin of this electron.

12:07 We said that the spin of the electron

12:09 could be oriented all kinds of different directions.

12:13 But what about this direction?

12:15 Or something like this?

12:17 Why aren't those allowed?

12:19 Well, they are.

12:21 And the way to represent them is through complex numbers.

12:27 So for example, this state is represented by zero plus i times one.

12:35 And that kind of makes sense if you think

12:37 of the i direction as like the third dimension.

12:40 So you have your one pointing this way and your negative one pointing this way.

12:45 In between you have i.

12:46 And that's pointing upwards.

12:48 So that's how it works out.

12:50 With light it's a little bit more complicated.

12:53 Like, what would this state look like for light?

12:56 Because we said that we're allowed to rotate this light around in any direction,

13:01 and that's all fine, but those are all just like the real ones.

13:04 What would this state look like?

13:05 do.

13:05 Unlike with the electron, we can't just sort of rotate the laser,

13:08 because that doesn't really work.

13:10 Like, polarization is about the direction of the polarization

13:13 relative to the direction of motion of the light,

13:17 and so only these ones seem valid.

13:20 So all of the polarization we've talked

13:22 about so far is linearly polarized light,

13:24 which is always pointing in one direction.

13:27 But there's circularly polarized light,

13:28 which is which changes direction of polarisation as it travels forward.

13:33 So that is what this I represents and different

13:37 complex numbers would represent being more or less circularly polarised.

13:41 So yeah, for every qubit that you can write down,

13:44 when you translate that back into the physical system,

13:47 there is some way to get this object to behave

13:51 so that its state is actually equal to this.

13:54 And that is why it's so useful to abstract away and write

13:59 things like this, rather than always thinking about the actual physical system.

14:04 Because, no matter what architecture you use for quantum computing,

14:07 there you will always be able to write

14:10 states like this and manipulate them and guarantee

14:13 that there is some way that you can make this all happen in the real world.

14:19 And that is someone else's problem.