Single-Qubit Gates

A single-qubit gate is a basic building block of quantum computers. It is a unitary transformation that can be applied to a single qubit, and it can be used to manipulate the state of the qubit in a variety of ways.

• It is a general one-qubit gate
• It has one input and one output
• It gives the states |0> and |1> of a qubit in coherent superposition

Coherent superposition is a property of quantum systems that allows them to exist in multiple states at the same time. In the case of a qubit, this means that the qubit can be in the |0⟩ state and the |1⟩ state at the same time.

General Single-Qubit Gate

A general single-qubit gate can be represented by a 2x2 unitary matrix. The matrix has the following form:

Where, θ and ϕ are real numbers. The values of θ and ϕ determine the effect of the gate on the state of the qubit.

The equations show how the state of a qubit is transformed by the Uθ,ϕ gate. The first equation shows that if the qubit is in the |0⟩ state before the gate is applied, it will be in a superposition of the |0⟩ and |1⟩ states after the gate is applied. The second equation shows that if the qubit is in the |1⟩ state before the gate is applied, it will be in a superposition of the |0⟩ and |1⟩ states after the gate is applied, but with a phase shift of eiϕ.

Hadamard Gate

This gate flips the state of the qubit with probability 1/2. If the qubit is in the |0⟩ state before the gate is applied, it will be in the |1⟩ state after the gate is applied. And vice versa.

If θ = 𝜋/2 and φ = 0, the single-qubit gate Uθ,ϕ becomes the Hadamard gate. The Hadamard gate is a special type of single-qubit gate that puts the qubit in a coherent superposition of the |0⟩ and |1⟩ states. This means that the qubit is equally likely to be found in the |0⟩ state or the |1⟩ state after the gate is applied.

The equations show how the state of a qubit is transformed by the Hadamard gate. The first equation shows that if the qubit is in the |0⟩ state before the gate is applied, it will be in a coherent superposition of the |0⟩ and |1⟩ states after the gate is applied. The second equation shows that if the qubit is in the |1⟩ state before the gate is applied, it will be in a coherent superposition of the |0⟩ and |1⟩ states after the gate is applied, but with a phase shift of -1.

Fig., Working of Hadamard gate when inputs are |0> and |1>

In matrix form, the Hadamard gate is given as:

The 1 and -1 in the matrix represent the binary values 0 and 1, respectively.

The 1 / √2 factor in front of the matrix is a normalization factor that ensures that the matrix has a determinant of 1. This is necessary because the Hadamard gate is a unitary gate, which means that it preserves the magnitude of the probability amplitudes.

Bit-Flip Gate (Pauli-X Gate)

The bit-flip gate, also known as the Pauli-X gate, is a single-qubit gate that flips the state of a qubit.

If θ = 𝜋 and φ = 0, the input state |0> of a qubit undergoes a transformation where it is converted to the state |1> as an output, while the state |1> is converted to the |0> state. This specific type of gate is commonly referred to as the bit-flip gate or the Pauli-X gate, and it is represented by the symbol X.

i.e.,

The gate is represented by the following matrix:

Phase-Flip Gate (Pauli-Z Gate)

When the values of θ and φ are set to 0 and 𝜋, respectively, in equations 1 and 2, the state |0> remains unchanged, while the state |1> is transformed into -|1>. This specific quantum gate is commonly referred to as the phase-flip gate or Pauli-Z gate, denoted by the symbol Z."

i.e.,

In matrix form, it is given as:

Other Single-Qubit Gates

In addition to the Hadamard gate, bit-flip gate, and phase-flip gate, there are many other single-qubit gates that can be used to manipulate the state of a qubit. Some of these gates include:

The T gate

It has a phase shift of π/4.

The phase gate

This gate has a phase shift of π/2. These gates do not change the probability of measuring the qubit as |0⟩ or |1⟩, but they do change the phase of the state.

In general, if the input state is |0⟩, the phase-shifter gate will not change the phase. However, if the input state is |1⟩, the phase will be multiplied by $e^{i\varphi }$, where $\varphi$ is the phase shift of the gate. The matrix representation of a phase-shifter gate can be written as follows:

Where different phase-shifter gates have different values for φ.

The controlled-NOT gate

This gate flips the state of a qubit if and only if another qubit is in a particular state.

Operations

These gates can be used to perform a variety of quantum operations, such as:

• Quantum logic operations, such as addition and multiplication.
• Quantum algorithms, such as Shor's algorithm for factoring large numbers.
• Quantum simulations, such as the simulation of molecules and materials.