Theory

1. What is Quantum Computing?

Quantum computing is a field of computing that uses principles of quantum mechanics to perform calculations. Unlike classical computers, which use bits as the smallest unit of information (0 or 1), quantum computers use qubits, which can exist in a combination (superposition) of both 0 and 1.

Quantum computing is powerful because:

• Qubits can exist in superpositions
• Multiple qubits can be entangled, sharing information instantly
• It enables new algorithms that are faster than classical ones for certain problems (e.g., Shor’s algorithm, Grover’s algorithm)

This experiment focuses on the basic concept of measuring a qubit and understanding what the outcome means through the expectation value.

2. What is a Qubit?

A qubit is the fundamental unit of quantum information. It is a two-level quantum system that can be in state |0⟩, |1⟩, or any linear combination of the two.

Mathematically, a qubit can be written as:

[ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle ]

Where:

• ( \alpha ) and ( \beta ) are complex numbers
• ( |\alpha|^2 + |\beta|^2 = 1 ), to satisfy normalization

In matrix form:

• ( |0\rangle = \begin{bmatrix}1 \ 0\end{bmatrix} )
• ( |1\rangle = \begin{bmatrix}0 \ 1\end{bmatrix} )

This superposition is a key difference from classical bits, which are either 0 or 1 — but not both.

3. Qubit on the Bloch Sphere

Every pure qubit state can be represented on the surface of a sphere called the Bloch Sphere.

The general form using spherical coordinates is:

[ |\psi\rangle = \cos\left(\frac{\theta}{2}\right)|0\rangle + e^{i\phi}\sin\left(\frac{\theta}{2}\right)|1\rangle ]

Where:

• ( \theta ) is the angle from the z-axis
• ( \phi ) is the angle in the x-y plane
• This form helps visualize the state in 3D

This is useful for understanding quantum operations as rotations on the sphere.

4. Measurement in Quantum Computing

Quantum measurement collapses the qubit to one of the basis states:

• Measuring in the standard basis means the outcomes will be either |0⟩ or |1⟩.
• The probability of getting |0⟩ is ( |\alpha|^2 ), and for |1⟩ is ( |\beta|^2 )

In practice, we often represent measurement using an operator or matrix, called an observable.

5. Observable Operators (Measurement Basis)

In quantum mechanics, observables are physical quantities we can measure, such as spin or energy. Mathematically, observables are represented using Hermitian matrices, which have real eigenvalues and orthogonal eigenstates.

What Are Eigenvalues and Eigenstates?

When an observable is applied to a quantum state, the only possible measurement outcomes are its eigenvalues. The system will collapse into the corresponding eigenstate after measurement.

Mathematically:

[ A|\psi\rangle = \lambda|\psi\rangle ]

• ( A ) is the observable (such as a Pauli matrix)
• ( |\psi\rangle ) is the eigenstate
• ( \lambda ) is the eigenvalue (the result you observe)

This means:

• If the system is already in an eigenstate of ( A ), measuring ( A ) will return ( \lambda ) with certainty.
• If the system is not in an eigenstate, the outcome will be probabilistic, and the state will collapse to one of the eigenstates.

Pauli-Z Operator (Z)

This is the standard measurement basis (computational basis). It measures whether the qubit is in the state |0⟩ or |1⟩.

[ Z = \begin{bmatrix}1 & 0 \ 0 & -1\end{bmatrix} ]

• If the qubit is in state |0⟩, measurement gives +1
• If in |1⟩, measurement gives -1
• Any superposition of |0⟩ and |1⟩ gives an average (expectation value) between -1 and +1

Z measures the spin along the Z-axis of the Bloch sphere.

Pauli-X Operator (X)

The Pauli-X matrix is often called the quantum NOT gate. It flips |0⟩ to |1⟩ and vice versa. When used as an observable, it measures in the X-basis (|+⟩ and |−⟩ states):

[ X = \begin{bmatrix}0 & 1 \ 1 & 0\end{bmatrix} ]

• ( |+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) ) → eigenvalue +1
• ( |-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle) ) → eigenvalue -1

So, measuring with X tells us whether the qubit is aligned along the X-axis.

Pauli-Y Operator (Y)

The Pauli-Y matrix introduces a phase difference and is related to rotation and spin in the Y direction:

[ Y = \begin{bmatrix}0 & -i \ i & 0\end{bmatrix} ]

Its eigenstates are complex combinations of |0⟩ and |1⟩:

• ( \frac{1}{\sqrt{2}}(|0\rangle + i|1\rangle) ) → eigenvalue +1
• ( \frac{1}{\sqrt{2}}(|0\rangle - i|1\rangle) ) → eigenvalue -1

Measuring with Y gives the projection along the Y-axis on the Bloch sphere.

Operator	Matrix	Measures Along
Pauli-Z	(\begin{bmatrix}1 & 0 \ 0 & -1\end{bmatrix})	Z-axis
Pauli-X	(\begin{bmatrix}0 & 1 \ 1 & 0\end{bmatrix})	X-axis
Pauli-Y	(\begin{bmatrix}0 & -i \ i & 0\end{bmatrix})	Y-axis

Each operator defines a different measurement basis and can give different expectation values depending on the state of the qubit.

6. Expectation Value

The expectation value of an observable A in a quantum state ( |\psi\rangle ) is the average value we expect after many measurements of the same system:

[ \langle A \rangle = \langle \psi | A | \psi \rangle ]

This gives a number between -1 and 1 when measuring a single qubit with Pauli matrices.

7. How to Calculate It

To calculate the expectation value:

1. Represent the state ( |\psi\rangle ) as a column vector.
2. Take its conjugate transpose to form ( \langle \psi | )
3. Multiply using the formula:
   ( \langle \psi | A | \psi \rangle )

Example:

If the qubit is in state:

[ |\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) = \begin{bmatrix} \frac{1}{\sqrt{2}} \ \frac{1}{\sqrt{2}} \end{bmatrix} ]

And the observable is Pauli-Z:

[ Z = \begin{bmatrix}1 & 0 \ 0 & -1\end{bmatrix} ]

Then the expectation value is:

[ \langle Z \rangle = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix}1 & 0 \ 0 & -1\end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}} \ \frac{1}{\sqrt{2}} \end{bmatrix} = 0 ]

So, even though each measurement gives +1 or -1, the average of many measurements is 0 — meaning equal probability of |0⟩ and |1⟩.

8. Interpretation of Expectation Value

The expectation value tells you about the balance between |0⟩ and |1⟩ in the qubit state.

• ( \langle Z \rangle = 1 ) → qubit is completely in |0⟩
• ( \langle Z \rangle = -1 ) → qubit is completely in |1⟩
• ( \langle Z \rangle = 0 ) → qubit is an equal mix of |0⟩ and |1⟩

This value is helpful to visualize the state on the Bloch Sphere (z-axis position).