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 (for example, Shor’s algorithm and 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 $|0\rangle$, $|1\rangle$, 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
• Normalization condition: $|\alpha|^2 + |\beta|^2 = 1$

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 never both at the same time.

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 representation using spherical coordinates is:

$$|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi} \sin(\theta/2)|1\rangle$$

Where:

• $\theta$ represents the angle from the Z-axis
• $\phi$ represents the angle in the X--Y plane

This representation helps visualize quantum states as points on a three-dimensional sphere.

4. Measurement in Quantum Computing

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

If we measure in the computational basis, the outcome will be either $|0\rangle$ or $|1\rangle$.

Probability of measuring the state $|0\rangle$:

$$P(0) = |\alpha|^2$$

Probability of measuring the state $|1\rangle$:

$$P(1) = |\beta|^2$$

In quantum mechanics, measurements are represented using observable operators, which are matrices acting on quantum states.

5. Observable Operators (Measurement Basis)

In quantum mechanics, every measurable physical quantity (like position, momentum, or spin) is represented by an observable—a mathematical operator that acts on quantum states.

Observables represent measurable quantities in a quantum system. They are represented mathematically using Hermitian matrices, which have the special property that:

• All eigenvalues are real numbers (required since measurement outcomes must be physically real)
• Eigenvectors form a complete orthonormal basis (allowing any quantum state to be expressed as a superposition of eigenstates)

Why Hermitian Matrices?

Observable operators must be Hermitian (self-adjoint) for two fundamental reasons:

1. Real Eigenvalues: Physical measurement outcomes are always real numbers. Hermitian matrices guarantee that their eigenvalues are real, ensuring that quantum measurements always produce real, physical results.

2. Orthogonal Eigenvectors: A Hermitian matrix has a complete set of orthogonal eigenvectors. This orthogonal basis is essential because it allows any quantum state to be decomposed into eigenstates, making the eigenvalue structure central to understanding measurement outcomes.

Eigenvalues and Eigenstates

When an observable acts on a quantum state, the measurement outcome corresponds to one of its eigenvalues:

$$A|\psi\rangle = \lambda|\psi\rangle$$

Where:

• $A$ is the observable operator (a Hermitian matrix)
• $|\psi\rangle$ is the eigenstate of $A$
• $\lambda$ is the eigenvalue (the measurement outcome)

Physical Interpretation:

If the system is in an eigenstate $|\psi\rangle$ of the observable, then measuring it will always yield the eigenvalue $\lambda$ with 100% certainty (deterministic result).

• If the system is in a superposition (not an eigenstate), the measurement outcome is probabilistic. The result will be one of the eigenvalues, with probability determined by the overlap of the state with each eigenstate.
• For a general state, the expected measurement outcome is the expectation value of the observable, which is a weighted average of all eigenvalues based on the state composition.

Pauli-Z Operator

The Pauli-Z operator measures the qubit along the Z-axis.

Matrix representation:

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

Measurement outcomes:

• $|0\rangle$ gives outcome $+1$
• $|1\rangle$ gives outcome $-1$

For a superposition state, the measurement produces an expectation value between −1 and +1.

Pauli-X Operator

The Pauli-X operator flips the state of a qubit and is often called the quantum NOT gate.

Matrix representation:

$$X = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}$$

Eigenstates:

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

Pauli-Y Operator

The Pauli-Y operator introduces a phase difference and measures along the Y-axis.

Matrix representation:

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

Eigenstates:

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

Operator	Matrix	Axis
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

6. Expectation Value

The expectation value of an observable expressed as $A$ for a quantum state $|\psi\rangle$ is the statistical average of measurement outcomes obtained from many repeated measurements on identically prepared quantum systems.

$$\langle A \rangle = \langle\psi|A|\psi\rangle$$

Connection to Eigenvalues: The expectation value can be understood as a weighted average of the observable's eigenvalues:

$$\langle A \rangle = \sum_i P_i \lambda_i$$

Where:

• $\lambda_i$ are the eigenvalues of observable $A$
• $P_i$ is the probability of measuring eigenvalue $\lambda_i$
• $P_i = |\langle \psi_i | \psi \rangle|^2$ where $|\psi_i\rangle$ is the eigenstate corresponding to $\lambda_i$
• $\sum_i P_i = 1$ (sum of all probabilities equals 1)

Key Points:

• For eigenstates: If $|\psi\rangle$ is an eigenstate of $A$ with eigenvalue $\lambda$ then $\langle A \rangle = \lambda$
• For superpositions: $\langle A \rangle$ lies between smallest and largest eigenvalues of $A$
• For Pauli operators: $-1 \leq \langle A \rangle \leq +1$

7. Example Calculation

Consider the qubit state:

$$|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$$

Vector representation:

$$|\psi\rangle = \begin{bmatrix} 1/\sqrt{2} \ 1/\sqrt{2} \end{bmatrix}$$

Observable:

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

Expectation value:

$$\langle Z \rangle = \begin{bmatrix} 1/\sqrt{2} & 1/\sqrt{2} \end{bmatrix} \begin{bmatrix}1&0\0&-1\end{bmatrix} \begin{bmatrix} 1/\sqrt{2} \ 1/\sqrt{2} \end{bmatrix}$$

Result:

$$\langle Z \rangle = 0$$

8. Interpretation of Expectation Value

The expectation value is fundamentally a measure of how much the quantum state aligns with the eigenstates of the observable being measured.

For the Pauli-Z operator: $\langle Z \rangle$ describes the balance between probabilities of measuring $(+1)$ and $(-1)$.

When the expectation value equals +1 ($\langle Z \rangle = +1$), the state is eigenstate $|0\rangle$ with 100% probability of outcome $+1$.

When the expectation value equals -1 ($\langle Z \rangle = -1$), the state is eigenstate $|1\rangle$ with 100% probability of outcome $-1$.

When the expectation value equals 0 ($\langle Z \rangle = 0$), it is an equal superposition (50% chance of $+1$, 50% chance of $-1$).

Geometric Interpretation (Bloch Sphere): On the Bloch sphere, the expectation value equals the z-component of the qubit state's position. This is because:

• The Bloch sphere's surface represents all possible quantum states
• Each measurement axis (X, Y, or Z) defines a direction in 3D space
• The expectation value of a Pauli operator equals the projection of the state onto that measurement axis
• This projection directly gives the weighted average of the eigenvalue outcomes

This geometric picture provides intuition:

• A state pointing "up" (toward $+Z$) has $\langle Z \rangle = +1$
• A state pointing "down" (toward $-Z$) has $\langle Z \rangle = -1$