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Module 4: Tensor Product and Multi-Qubit Gates in Quantum Computing

This lesson extends single-qubit reasoning into multi-qubit systems by introducing tensor products and the structure of multi-qubit gates.

Course Contents

Summary

Module 4 introduces the tensor product as the core operation for combining quantum systems and shows how multi-qubit gates act across larger state spaces. It frames the jump from isolated qubits to quantum circuits that can express richer behavior.

Learning Goals

Understand how composite quantum states are built, interpret multi-qubit state dimensions correctly, and reason about how gates such as controlled operations affect more than one qubit.

What Learners Obtain

Learners gain the ability to model multi-qubit systems, read more complex circuit structures, and prepare for later topics such as entanglement, measurement, and Bell states.

1. Conceptual Framework: Scaling Quantum Systems

When moving from a single isolated qubit to larger processors, we must combine separate quantum systems into one composite computational space.

  • The Scaling Law ($2^n$): Adding one qubit increases state-space size exponentially. An $n$-qubit system has a composite Hilbert space dimension of $2^n$.
  • State Coexistence: A two-qubit register spans four basis states simultaneously: $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$.

2. The Tensor Product ($\otimes$)

The tensor product combines individual qubit vectors (or gate matrices) into a single multi-qubit representation.

Constructing Multi-Qubit Basis Vectors

For two qubits both initialized in $|0\rangle$:

$$|q_0 q_1\rangle = |q_0\rangle \otimes |q_1\rangle = |0\rangle \otimes |0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \otimes \begin{bmatrix} 1 \\ 0 \end{bmatrix}$$

Distribute each scalar from the first vector across the second:

$$|0\rangle \otimes |0\rangle = \begin{bmatrix} 1 \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\ 0 \cdot \begin{bmatrix} 1 \\ 0 \end{bmatrix} \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} = |00\rangle$$

Applying the same method gives the full two-qubit basis set:

$$|01\rangle = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \quad |10\rangle = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \quad |11\rangle = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$$

3. Conditional Logic: Multi-Qubit Gates

The Controlled-NOT Gate (CNOT / CX)

CNOT uses a control qubit and a target qubit. It flips the target if and only if the control is in $|1\rangle$; otherwise it leaves the target unchanged.

Matrix Representation of CNOT

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

Matrix Multiplications in Action

  • Control is $|0\rangle$ (no flip):
    $$CX|01\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}\begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} = |01\rangle$$
  • Control is $|1\rangle$ (flip triggered):
    $$CX|10\rangle = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}\begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} = |11\rangle$$

4. Summary Matrix: CNOT Gate Transformations

Input Computational State Control State Target State Output Computational State Action Taken
$|00\rangle$ $|0\rangle$ $|0\rangle$ $|00\rangle$ No flip
$|01\rangle$ $|0\rangle$ $|1\rangle$ $|01\rangle$ No flip
$|10\rangle$ $|1\rangle$ $|0\rangle$ $|11\rangle$ Target flips
$|11\rangle$ $|1\rangle$ $|1\rangle$ $|10\rangle$ Target flips

Architectural note on bit order formatting: Always verify endianness conventions. Physics references often use big-endian notation, while software frameworks such as Qiskit frequently use little-endian index ordering.