Qubits, Quantum Gates and Quantum Circuits – a Computer Science Perspective: Part 2 Quantum Gates and Quantum Circuits

Content of Part 2

1. Quantum gates and quantum circuits

- Exercises for students
- Definitions
- Assembling a quantum circuit
- Example: Flipping the state of a qubit using the Pauli-X gate
- Using Qiskit

2. Elementary quantum gates

- The identity gate
- The Hadamard gate

1. Quantum gates and quantum circuits

Exercises for students

Exercise 1: Define the classical logic gates “AND”, “OR”, “XOR” and “NOT”. List the corresponding truth tables. Research the results of “NOT (A AND B)” or “NOT (A OR B)”.

Exercise 2: Compare handling the logic gates “AND” and “OR” with the multiplication and addition of numbers.

(Classical logic gates are covered in the lesson Classical Computing – Introduction to Logic Gates)

Definitions

Just as classical gates are used in classical computers to perform elementary operations on classical bits, quantum gates are used in quantum computers to manipulate qubits.
To run a programme on a quantum computer, the appropriate elementary quantum gates are selected and applied to one or more qubits. This can be illustrated in a quantum circuit that shows how the various components of the programme – qubits and gates – are assembled.

Definition of a quantum gate

In quantum computers, operations are performed using quantum gates. Quantum gates act on one or more qubits and can be represented by a unitary matrix. When qubits are manipulated using quantum gates, this corresponds to physical changes in these qubits. Examples: The qubit flips from one basis state to another, or it is brought into a superposition of the basis states.

Definition of a quantum circuit

A quantum circuit is a visual representation of a quantum algorithm: it can be used to represent computing code for a quantum computer. A quantum circuit consists of qubits as well as of quantum gates that are applied to these qubits. In a quantum circuit, time flows from left to right. At the beginning – on the left – are the qubits. The quantum gates are then applied one after the other, from left to right, to the qubits. Measurements and their results are also represented in a quantum circuit. They are usually located on the far right, or sometimes in between.

Assembling a quantum circuit

Quantum circuits are very useful when it comes to programming a quantum computer. The steps are:

1. Prepare one or several qubit(s) in an initial state (e. g. ${\displaystyle |0⟩}$).
2. Apply one or several quantum gates on the qubit(s).
3. Perform a measurement and obtain a result.

One can assemble a quantum circuit visually using a graphical interface or by using a programming language, like for instance Qiskit.

A quantum circuit is usually run many times because quantum computers are susceptible to noise: even if the probability of measuring a certain result is 100% "0", the result is sometimes not "0" – due to noise. Therefore, quantum circuits are usually run a number of times called “shots” (for instance 1000 shots). The larger the number of shots, the clearer the "correct" result stands out – for example, that 1 + 1 indeed equals 2.

Example: Flipping the state of a qubit using the Pauli-X gate

The Pauli-X gate changes the state of a qubit from $|0⟩$ to $|1⟩$, or from $|1⟩$ to $|0⟩$: it flips the state of a qubit. Because of its similarity with the NOT operation on classical bits, it is also called a NOT (quantum) gate.

The Pauli-X gate can be represented by the matrix $X=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$:

${\displaystyle X\cdot <!--\; \cdot \; -->|0⟩<!--\; ⟩\; -->=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\cdot <!--\; \cdot \; -->|0⟩<!--\; ⟩\; -->=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\cdot <!--\; \cdot \; -->\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}0\cdot <!--\; \cdot \; -->1+1\cdot <!--\; \cdot \; -->0\\ 1\cdot <!--\; \cdot \; -->1+0\cdot <!--\; \cdot \; -->0\end{array}\right]=\left[\begin{array}{c}0\\ 1\end{array}\right]=|1⟩<!--\; ⟩\; -->}$ and

${\displaystyle X\cdot <!--\; \cdot \; -->|1⟩<!--\; ⟩\; -->=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\cdot <!--\; \cdot \; -->|1⟩<!--\; ⟩\; -->=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\cdot <!--\; \cdot \; -->\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}0\cdot <!--\; \cdot \; -->0+1\cdot <!--\; \cdot \; -->1\\ 1\cdot <!--\; \cdot \; -->0+0\cdot <!--\; \cdot \; -->1\end{array}\right]=\left[\begin{array}{c}1\\ 0\end{array}\right]=|0⟩<!--\; ⟩\; -->}$.

This is described in concrete terms and step by step below using the above example (a Pauli-X gate is applied to a qubit in the state $|0⟩$).

The quantum circuit of a Pauli-X gate applied to a qubit looks like this:

Fig. 1: In this quantum circuit, a Pauli-X gate is applied to a qubit and a measurement is performed.

In Figure 1, $q_0$ is the quantum register – just imagine it as saying, "a qubit is prepared in a specific initial state so that one or more quantum gates can be applied to it". The “q” in $q_0$ stands for “quantum”, and the “0” means that this is the first (and in this case only) qubit. A second qubit could be prepared in the register $q_1$, a third qubit in $q_2$, and so on. The blue square with the “X” symbolises the Pauli-X gate. The grey square (depicting a kind of "measuring device") indicates that a measurement is performed. The arrow indicates that a measurement is performed on the qubit $q_0$. The measurement yields a real number (either 0 or 1 in this case), so the register in which the result of the mesurement is stored is a classical register labelled $c_0$ (“c” stands for “classical”, and “0” means that it is the first and only register needed in this example).

Using Qiskit

Instead of assembling a programme visually using a quantum circuit, one can also write it using Qiskit, an open source framework developed by IBM for programming quantum computers. 

These lines of code only describe the actual quantum circuit. They must be inserted at the point marked in red in the following code:^[1]

The code above includes a so-called transpiler that optimises the quantum circuit. This is necessary because real quantum computers do not always offer all the desired gates. Some optimisations also reduce the quantum circuit’s noise and achieve better results.

The following is the result of the above programme run on a real quantum computer (the y-axis displays the number of shots):

Fig. 2: A Pauli-X quantum gate is applied to a qubit in the state | 0 ⟩ . The result shows that the qubit is now in the other basis state, the state | 1 ⟩ . The higher the number of shots (plotted on the y-axis), the more clearly the "correct" result stands out. The more the noise is suppressed, the smaller the left bar in Fig. 2.

2. Elementary quantum gates

The identity gate

Besides the Pauli-X gate, another elementary and very simple quantum gate is the identity gate. It does not change the state of the qubit. 

Its matrix is the identity matrix $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$:

$I\cdot |0⟩=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\cdot |0⟩=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\cdot \left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}1\cdot 1+0\cdot 0\\ 0\cdot 1+1\cdot 0\end{array}\right]=\left[\begin{array}{c}1\\ 0\end{array}\right]=|0⟩$ and

$I\cdot |1⟩=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\cdot |1⟩=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\cdot \left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}1\cdot 0+0\cdot 1\\ 0\cdot 0+1\cdot 1\end{array}\right]=\left[\begin{array}{c}0\\ 1\end{array}\right]=|1⟩$.

In combination with other gates, the identity gate – i.e. “do nothing” – can be useful in some cases.
The quantum circuit of an identity gate applied to a qubit looks like this:

Fig. 3: In this quantum circuit, an identity gate is applied to a qubit. Subsequently, a measurement is performed.

Fig. 4: Result of applying the identity gate to a qubit in the state ${\displaystyle |0⟩}$. It leaves the qubit unchanged.

The Hadamard gate

A special gate is the Hadamard gate: starting from the basis state $|0⟩$ (or the basis state $|1⟩$), it creates a superposition of $|0⟩$ und $|1⟩$, the two basis states of a qubit:

${\displaystyle H\cdot <!--\; \cdot \; -->|0⟩<!--\; ⟩\; -->=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->|0⟩<!--\; ⟩\; -->+\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->|1⟩<!--\; ⟩\; -->=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{c}1\\ 1\end{array}\right]}$.

The Hadamard gate is represented by the matrix ${\displaystyle H=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{cc}1& 1\\ 1& -<!--\; -\; -->1\end{array}\right]}$:

${\displaystyle H\cdot <!--\; \cdot \; -->|0⟩<!--\; ⟩\; -->=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{cc}1& 1\\ 1& -<!--\; -\; -->1\end{array}\right]\cdot <!--\; \cdot \; -->|0⟩<!--\; ⟩\; -->=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{cc}1& 1\\ 1& -<!--\; -\; -->1\end{array}\right]\cdot <!--\; \cdot \; -->\left[\begin{array}{c}1\\ 0\end{array}\right]=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{c}1\cdot <!--\; \cdot \; -->1+1\cdot <!--\; \cdot \; -->0\\ 1\cdot <!--\; \cdot \; -->1+(-<!--\; -\; -->1)\cdot <!--\; \cdot \; -->0\end{array}\right]}$

${\displaystyle =\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{c}1\\ 1\end{array}\right]=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{c}1\\ 0\end{array}\right]+\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{c}0\\ 1\end{array}\right]=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->|0⟩<!--\; ⟩\; -->+\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->|1⟩<!--\; ⟩\; -->}$

and

${\displaystyle H\cdot <!--\; \cdot \; -->|1⟩<!--\; ⟩\; -->=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{cc}1& 1\\ 1& -<!--\; -\; -->1\end{array}\right]\cdot <!--\; \cdot \; -->|1⟩<!--\; ⟩\; -->=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{cc}1& 1\\ 1& -<!--\; -\; -->1\end{array}\right]\cdot <!--\; \cdot \; -->\left[\begin{array}{c}0\\ 1\end{array}\right]=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{c}1\cdot <!--\; \cdot \; -->0+1\cdot <!--\; \cdot \; -->1\\ 1\cdot <!--\; \cdot \; -->0+(-<!--\; -\; -->1)\cdot <!--\; \cdot \; -->1\end{array}\right]}$

${\displaystyle =\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{c}1\\ -<!--\; -\; -->1\end{array}\right]=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{c}1\\ 0\end{array}\right]-<!--\; -\; -->\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->\left[\begin{array}{c}0\\ 1\end{array}\right]=\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->|0⟩<!--\; ⟩\; -->-<!--\; -\; -->\frac{1}{\sqrt{2}}\cdot <!--\; \cdot \; -->|1⟩<!--\; ⟩\; -->}$.

The two amplitudes (i.e. the two coefficients in front of the basis states) in the superposition state are equal, ${\displaystyle \frac{1}{\sqrt{2}}}$ in both cases. The squares of the amplitudes give the probabilities for finding the qubit in state ${\displaystyle |0⟩}$ resp. ${\displaystyle |1⟩}$(see Basics of Quantum Physics). This means that the probability for finding the qubit in state ${\displaystyle |0⟩}$ is ${\displaystyle {\left[\frac{1}{\sqrt{2}}\right]}^2=\frac{1}{2}=50\mathrm{\%<!--\; \%\; -->}}$, and the probability for finding the qubit in state ${\displaystyle |1⟩}$ is ${\displaystyle {\left[\frac{1}{\sqrt{2}}\right]}^2={\left[-<!--\; -\; -->\frac{1}{\sqrt{2}}\right]}^2=\frac{1}{2}=50\mathrm{\%<!--\; \%\; -->}}$ as well.

Fig. 5: In this quantum circuit, a Hadamard gate is applied to a qubit. A measurement is then performed.

Fig. 6: A Hadamard gate is applied to a qubit (in state ${\displaystyle |0⟩}$ or ${\displaystyle |1⟩}$, in both cases the same result is obtained). In approximately $50\%$ of cases, the qubit is measured in state ${\displaystyle |0⟩}$, and in $50\%$ of cases in state ${\displaystyle |1⟩}$.