Three Important Matrices To Know About When Quantum Computing

A lesson on Hermitian, Orthogonal and Unitary Matrices and how they are used in Quantum Computing.

Portrait of Paul Dirac, who formulated the Dirac equation, describing relativistic quantum mechanics and predicting the existence of antimatter. His work laid foundational principles for quantum mechanics, including bra-ket notation, which became essential in quantum computation and information theory. (Source)

Matrices and operations involving them are crucial to learn when quantum computing.

Here are three essential matrices you will encounter multiple times if you’re deep-diving into this field.

1. Hermitian Matrix

A Hermitian matrix is a square matrix that is equal to its conjugate transpose.

Consider a matrix A as follows:

Its transpose is calculated as:

Next, its complex conjugate completes the Conjugate transpose operation, leading to the following:

This is the same as A and hence A is a Hermitian Matrix.

Hermitian matrices represent observables (physical quantities that can be measured, such as spin, position, momentum, energy and more) in quantum mechanics.

This is because the eigenvalues of Hermitian matrices are all real numbers.

We will discuss this further in the upcoming lesson on Eigenvectors and Eigenvalues.

2. Unitary Matrix

A Unitary matrix is a square matrix whose conjugate transpose is its inverse.

It satisfies the following equation (we previously saw this when discussing the Identity matrix):

where:

• U is a square matrix
• U† is its inverse
• I is the identity matrix

Consider the following matrix U:

Its conjugate transform is calculated as:

To check if the matrix U is Unitary, we do the following operation:

We arrive at the following conclusion:

Hence, the matrix U is a Unitary matrix.

In Quantum computing, Quantum gates are represented by Unitary matrices.

Two important quantum gates are:

1. Hadamard Gate