Matrix Multiplication Essentials

1. Why Matrices? Gates as State Transformations

In your electronics studies, you've seen how components like amplifiers or filters transform an input signal into an output signal. Quantum gates do something similar: they transform an input quantum state (a vector) into an output quantum state (another vector). The mathematical tool for this transformation is a matrix.

Let's see this in action. The video below introduces the Hadamard gate, one of the most important single-qubit gates, and shows how it's applied to a qubit state.

As the video demonstrated, applying the Hadamard gate $H$ to the state $|0⟩$ is a matrix-vector multiplication:

To compute this, we multiply each row of the matrix by the column vector:

This resulting vector, $\left(\begin{array}{c}1/\sqrt{2}\\ 1/\sqrt{2}\end{array}\right)$, represents the |+⟩state, a perfect superposition of|0⟩and|1⟩`. This is the core mechanism: gates are matrices, states are vectors, and gate application is matrix-vector multiplication.

2. The Rules of Matrix Multiplication

Now, let's break down the mechanics of matrix multiplication. The process is systematic and follows a clear "row-by-column" rule.

Let's summarize the essential rules from the video:

1. Dimension Rule: To multiply matrix $A$ by matrix $B$ (i.e., to compute $AB$), the number of columns in $A$ must equal the number of rows in $B$. If $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, the product $AB$ will be an $m\times p$ matrix.

   m x n ⋅ n x p → m x p
   (The inner dimensions must match)

2. Calculation Rule: The element in row $i$ and column $j$ of the product matrix $AB$ is calculated by taking the dot product of row $i$ from matrix $A$ and column $j$ from matrix $B$.

3. Practice: Composing Single-Qubit Gates (2x2)

In quantum computing, we often apply a sequence of gates. This is equivalent to multiplying their matrices. Let's see what happens when we apply the Hadamard gate twice.

Problem: Calculate the matrix product $H\cdot H$.

Based on what we've learned, we need to compute:

First, we can multiply the scalar constants: $\frac{1}{\sqrt{2}}\times \frac{1}{\sqrt{2}}=\frac{1}{2}$. Now for the matrix part:

• Row 1, Col 1: $(1\cdot 1)+(1\cdot 1)=2$
• Row 1, Col 2: $(1\cdot 1)+(1\cdot -1)=0$
• Row 2, Col 1: $(1\cdot 1)+(-1\cdot 1)=0$
• Row 2, Col 2: $(1\cdot 1)+(-1\cdot -1)=2$

Putting it all together with the scalar:

The result is the identity matrix, $I$! This means applying the Hadamard gate twice is equivalent to doing nothing. This is an essential property and a demonstration of the reversibility of quantum gates.

4. Scaling Up: Multi-Qubit Gates (4x4)

The same rules apply to larger matrices, which represent gates acting on multiple qubits. A common two-qubit gate is the Controlled-NOT (CNOT) gate.

The CNOT gate is represented by the following 4x4 matrix:

Let's try the same exercise as before: what is the effect of applying a CNOT gate twice? We need to calculate $\text{CNOT}\cdot \text{CNOT}$.

The calculation is larger, but the process is identical. Let's compute one element as an example: the element in row 3, column 4 of the product.

• Take row 3 of the first CNOT matrix: [0, 0, 0, 1]
• Take column 4 of the second CNOT matrix: [0, 0, 1, 0]
• Calculate the dot product:
  $(0\cdot 0)+(0\cdot 0)+(0\cdot 1)+(1\cdot 0)=0$

If you continue this for all 16 elements, you will find that $\text{CNOT}\cdot \text{CNOT}=I$, the 4x4 identity matrix. Just like the Hadamard gate, the CNOT gate is its own inverse.

For larger multiplications like this, it can be helpful to use a calculator to check your work once you're comfortable with the process.

Conclusion

Today we've demystified how quantum operations are performed mathematically. This skill is the engine that drives the evolution of quantum states in a circuit.

Key Takeaways:

• Quantum gates are represented by matrices, and applying a gate to a state is performed via matrix-vector multiplication.
• A sequence of gates applied to a state is equivalent to a single gate represented by the matrix-matrix product of the individual gate matrices.
• The rule for matrix multiplication is that the element at (row i, col j) of the product is the dot product of row i from the first matrix and col j from the second.
• The dimensions must align: for a product $AB$, the number of columns in $A$ must equal the number of rows in $B$. For our quantum gates, this will always be the case (e.g., 2x2 times 2x2, or 4x4 times 4x4).

Preview of the Next Lesson:
We've now seen how to represent single-qubit states (2x1 vectors) and two-qubit gates (4x4 matrices). But how do we represent the state of a two-qubit system itself? It's not as simple as stacking two vectors. In the next lesson, we will learn about the tensor product, the mathematical operation used to combine single-qubit state vectors into a multi-qubit state vector.