Building Blocks of Quantum Circuits — Programming on Quantum Computers Ep 4

– Now that you’ve seen that
this execution gives us the same result that we explained before, I want to show you two additional ways that people discuss the quantum state. (upbeat music) Hi, welcome back to “Coding with Qiskit.” This is Abe again. You might remember in our last video, we talked about how we
can build quantum circuits using Qiskit to manipulate qubits. Quantum circuits can be used to make qubits do all sorts
of interesting things. The key to understanding quantum circuits, whether complicated or simple, will be to understand
how quantum gates act on the individual qubits,
and how they work together. The point of this video
will be to enable you to learn how quantum
gates work on your own. Luckily, with the right tools, you can understand how all of
Qiskit’s quantum gates work. As open-source software,
Qiskit has a number of gates programmed into it already. So looking at my screen here,
I’m showing you an overview of the quantum gates available in Qiskit. The link to this page will be available in the description below. So you can see here this
list contains gates like the H gate, also called the Hadamard gate, CX, Identity, U3 gates, U2
gates, U1 gates, and so on. And in fact, in Qiskit you can even create your own custom gates. There are many different ways
to think about quantum gates. Some of them are more visual, and some of them are mathematical. So one of these gates, for
example, is the X gate. So let’s get right into it. Saying the X gate applied to state zero gives you the state one, and the X gate applied to state one gives you the state zero. So this is one representation
that you’ll see around. The other representation
is to take note of the fact that these gates are
actually unitary operations. And all that means is that
they can be represented as matrices, unitary
matrices in particular, and their operations can also be seen as operations on vectors,
which are the states. So by doing that, the X gate
is represented in this way. (laid-back music) The zero state is represented in this way. And if you write out the
matrix multiplication, you’ll find out that the
result is the following. So this is exactly equal
to that representation. The third and fourth ways
that you’ll see people talking about how a quantum circuit works, is in terms of both its
visual representation, and how the circuit works
as a result of measurements. To see these, we’ll go
straight into Qiskit. As always, I’ll start off by importing Qiskit into my workspace. So that can be done by typing out from Qiskit import everything. I will also be using the
visualization tools in Qiskit. So I can do that by writing out from, import plot, bloch, multivector. Okay, so now that we’ve imported Qiskit and the right tools that
we need to do our work, I’m going to do the following. So I’ll create a quantum circuit with one qubit, and one classical bit. So that quantum circuit will
have one operation on it. It’ll just be circuit.x
applied to that qubit. So what I’m going to do is take this quantum circuit and simulate it and see what the output of
that quantum circuit is. So the way to do that is to say the simulator is aer.get, backend. This time I’m going to use what’s called the state vector simulator. (laid-back music) If you’re curious, the
state vector is the vector that describes the quantum
state of our qubit. So, that’s where the name state
vector simulator comes from. And I’ll say, execute the circuit on the backend that I
choose to be the simulator, and I’ll say the results
from this execution will be stored in a
variable called result. And the state vector will be result.get state vector. And just so we can see
that this is working, I’ll say print state vector. All right, so as you can see, the state vector that’s
returned from the circuit, when it only has one X gate
on it, is the vector zero, with zero imaginary component, and one with zero imaginary component. This state vector is exactly what you saw in our representation before, where we wrote out X
applied on the state zero is the state one. So, let’s look at our quantum circuit. So what I’m going to
do is say circuit.draw, and I’ll be using the mat plot lib output, and I’ll be saying, just before that, mat plot lib inline. So what we’re saying is the
output of this quantum circuit is the state vector zero, one. Now that you’ve seen that this execution gives us the same result
that we explained before, I want to show you two additional ways that people discuss the quantum state. So the next way that I’m going to show you is a bit more visual, and tends to be the
favorite of a lot of people. What I can do is say
plot, bloch, multivector. And the thing I’m going
to describe with this is the state vector. (laid-back music) And this is what the
state vector looks like when it’s plotted on a bloch sphere. So, one thing you’ll notice is that people refer to
operations on quantum states as rotations on the sphere. And in fact, what that X gate has done is take us from the state
zero to the state one. And that’s why that
vector is pointing down in the direction of the state one. So it turns out that quantum
states for individual qubits can be represented on the sphere, as any point throughout
the surface of the sphere. And in particular, the one
that we’re working on here, the state one, is the very
bottom point of the sphere. And finally, remember that you can always run this quantum circuit,
execute measurements, and find out what the results
of those measurements are. So in Qiskit, this is done as follows. So what I’m going to do is add
a measurement to the circuit, and I’m going to measure qubit zero, and put that result in classical bit zero, and I’m going to say, backend is aer.get
backend, qasm simulator, and I’m going to say execute
this circuit on the backend, and I would like to do 1,024 shots. And the result of this
execution, I’d like to take out, and put in a variable called result. And I would like to take
the counts from this result, (laid-back music) and plot them out by using
the plot histogram tool, which is available from, import plot histogram. (laid-back music) And there you have it. The result is that for
100% of the results, you get the result one. And that’s because we have
the state one represented by the output of the circuit. And just real quick, I
also want to show you how to get the matrix
representation of a circuit. And the way to do this is very simple. What I’ll do is go back here, and instead of state vector simulator, what I’m going to do is say
I want the unitary simulator. So I’ll just copy over
this section of code, (laid-back music) and I’ll say unitary
simulator here, and I’ll say, the unitary is the output, and I’ll change get state
vector to get unitary. And I’ll print out the unitary that results from this operation. And as you see here, the
matrix that’s printed out is zero, one, one, zero. And this is exactly
the matrix that we used when we described the second method of talking about this quantum circuit, which is that the X gate is represented by the matrix zero, one, one, zero. And if you’re curious, the
meaning of these items, zero, J, is simply because each
of these matrix elements is actually a complex number. So it has a real part
and an imaginary part. And the number that you
see here, for example, zero plus zero J means
the real part is zero and the imaginary part is also zero. This particular number has a real part one and an imaginary part zero. So, that corresponds to the one that we’ve been seeing
in our matrices before. So we showed four different ways to talk about a quantum gate. One way is to describe its operation by writing down, as we saw before, for example, for the X gate, by saying, X applied on the state zero
gives you the state one. The second way that we used was to describe the linear algebra
behind how this gate works, by saying the matrix for the X gate, which is zero, one, one, zero, acting on the vector for the
zero state, which is one, zero, gives you the vector for the
one state, which is zero, one. The third way that we used to describe the operation of the X gate was by showing you something more visual, by plotting out the state vector that results from applying
that gate on a bloch vector. And the fourth way that we
used to describe its operation is by running a circuit
through a measurement, and seeing what the
measurement outcomes are. So hopefully with all these tools, you’re now equipped to answer the question of how a quantum gate works. And every time you encounter a new one, you can ask yourself the question of how it’s actually working
in a quantum circuit. So now that you know how to understand all of
Qiskit’s quantum gates, it’s important for you to know
that you’re ready to move on to building more complicated
quantum circuits. Next week we’ll be covering
your first quantum algorithm. And this will be quantum teleportation. Let us know which method of talking about quantum
gates made more sense to you. For me personally, I like
to see the quantum circuit represented as a matrix. So I like to write code in Qiskit, and find out what the matrix corresponding to a quantum gate looks like. Thanks again for watching, and we’ll see you in the next video. (upbeat music)

15 thoughts on “Building Blocks of Quantum Circuits — Programming on Quantum Computers Ep 4

  1. Which explanation made the most sense? Was there a 5th choice? 😉

    Pretty much none of this makes any real sense to me, but I'm still trying to follow the bouncing ball and appreciate the efforts in providing these videos. Thanks.

  2. nice video sir ,as the levels get high I also recommend you to give us some ,assignments or a problem to code .And you can discuss the solution in next class sir 🙂 cheers !

  3. I preferred the bra-ket representation, it was easy to follow along, but the matrix method feels a bit more detailed (helpful) and would like to understand it a bit better. I also liked the bloch sphere cos it reminds me of an electron having an up or down spin and is also a nice graphic.
    can't wait for the algos and quantum teleportation. Yet another banger by Abe and the team.

  4. IMHO I have preferences in following order: 1. Matrix method. 2. Bloch sphere 3. bra-ket representation 4. measurement. Also for newcomers in quantum computing, I do suggest to watch series which starts from this video . It is like quantum computing 101 but without oversimplification.

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