Linear Algebra in Quantum Computing

Kinga

2020/12/30 08:26

Introduction

Linear Algebra is a language of Quantum computing. It is widely used to describe qubit states, quantum operations and to predict the response of quantum computers to a sequence of instructions. Linear algebra is also like a storage space. Since you primarily need to store your data somewhere else to compute, it stores your data in vectors, matrices and tensors and it operates according to the instructions given. In linear algebra you need to understand the four major Data structures or Mathematical objects and their properties and they are scalar, vector, matrices and tensor.

Scalar

It is a single number that can be either Real or Natural number. It is generally represented by an italic small case letter.

s ∈ R {where R is set of Real numbers}
s ∈ N {where N is set of Natural numbers}

Vector

It is a list of numbers, arranged in order and we can identify each element by its index. Vectors can be thought of as directions from the origin, or a point in space. The individual elements of the vector are scalars.

for example:

x=[x1, x2, x3]

Matrix

2-Dimensional(2-D) array of numbers, where elements are identified by two indices instead of one.

Tensor

Tensor is an array of numbers in more than two axes, in other words it is a 3-D array of numbers.

The figure below shows all the four data structures or mathematical objects of linear algebra.

Representation of 1 qubit and 2 qubit

The vector representation of a single qubit is:

The vectors v0 and v1 determine the probability of measuring a 0 or a 1, when measuring the state of the qubit.

The value zero is represented by

The value one is represented by

The combined state of two qubits is the tensor product of the two qubits. The tensor product is denoted by the symbol .

The vector representation of two qubits is:

The action of the gate on a specific quantum state is found by multiplying the vector which represents the state, by the matrix representing the gate. The result is a new quantum state .

.