Mathematical Toolkit 1: Matrices and Vector
We need to acquire some mathematical tools in order to proceed with constructing qubits and quantum circuits. We will need to learn basic linear algebra and complex numbers in order to build the framework for quantum computing.
These might seem like difficult topics, but linear algebra is just a fancy expression for condensing multiple equations into a more compact form. Complex numbers behave almost identically to ordinary numbers, with a few small differences.
We will keep adding these mathematical tools as and when we need them throughout the next few modules.
We can start by building a matrix out of things we have seen before. If you are familar with matrix algebra, please feel free to skip this section. If this is new to you, that is completely okay! Let's take two standard binomials:
2x + 5y = 3
3x +4 y = 8
These two equations are actually a matrix in disguise! There are three parts of the equations: coefficients, variables, and the answer. The matrix is essentially the coefficients all groups together that we will multiply by the variables. Let's separate it out and look at each part:
The coefficients are all in the matrix, in the same order that they appeared in the equations. We will put the variables (x and y) together in a column, which we call a vector. We will talk about vectors shortly.
More generally, a matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. The dimension of a matrix is given by number of rows x number of columns. Some examples of matrices are:
A random 2 × 2 matrix looks like this
A matrix 3 × 3 with only diagonal elements non-zero, looks like this
A random 2 × 3 matrix looks like this
A square matrix is the one with an equal number of rows and columns, as in example 1. If a square matrix has all diagonal elements as 1s and rest of the elements as 0, the matrix is called an identity matrix, as shown in example 2.
Like wise you have to learn about:
Transpose
Transpose of a matrix is essentially a reflection of a matrix involving an exchange of row elements and column elements.
Addition
Matrix addition can be performed between two matrices of same dimensions. Each number from the first matrix is added o the number in the same position in the second matrix to generate the resulting matrix.
Multiplication
If we want to multiply a single number and a matrix, we simply have to multiply that number with each individual element of the matrix.
Linear Algebra
Linear algebra is at the heart of quantum mechanics and quantum computing. Important concepts include:
Vectors: Represent quantum states (e.g., ∣0⟩|0\rangle and ∣1⟩|1\rangle).
Matrices: Represent quantum gates and operations (e.g., the Pauli matrices, Hadamard gate).
Matrix Multiplication: Describes the evolution of quantum states.
Eigenvalues and Eigenvectors: Related to measurement outcomes.
Inner Products: Measure overlaps between quantum states (e.g., ⟨ψ∣ϕ⟩\langle \psi | \phi \rangle).
Outer Products: Used for projection operators.
Hermitian and Unitary Matrices:
Hermitian matrices represent observable properties.
Unitary matrices preserve the norm of quantum states.
2. Complex Numbers
Quantum states and amplitudes often involve complex numbers.
Real and Imaginary Parts: z=a+biz = a + bi, where i=−1i = \sqrt{-1}.
Modulus (Magnitude): ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}.
Conjugates: z∗=a−biz^* = a - bi.
Euler’s Formula: eiθ=cosθ+isinθe^{i\theta} = \cos\theta + i\sin\theta, which is central in quantum phase manipulation.
3. Probability Theory
Quantum mechanics uses probabilities to predict measurement outcomes.
Probability Distributions: Probabilities of different states (e.g., ∣α∣2|\alpha|^2 and ∣β∣2|\beta|^2 for ∣ψ⟩=α∣0⟩+β∣1⟩|\psi\rangle = \alpha|0\rangle + \beta|1\rangle).
Normalization: The sum of probabilities must equal 1 (∣α∣2+∣β∣2=1|\alpha|^2 + |\beta|^2 = 1).
Expectation Value: The average value of repeated measurements.
4. Basic Set Theory and Functions
Understanding functions and sets helps grasp state spaces and mappings.
Set Operations: Union, intersection, and complements.
Functions: Linear mappings and transformations between vector spaces.
5. Basic Group Theory
Group theory underpins symmetry operations in quantum mechanics.
Groups and Subgroups: Used in analyzing quantum gates.
Commutativity: Non-commuting operators are key in quantum mechanics.
6. Calculus (Basics)
While not always critical for introductory quantum computing, calculus is useful for understanding the foundations of quantum mechanics.
Derivatives: Rates of change, useful in dynamic systems.
Integrals: Area under a curve, related to probabilities.
7. Dirac Notation (Bra-Ket Notation)
Unique to quantum mechanics, this is essential for quantum computing.
Ket: ∣ψ⟩|\psi\rangle represents a column vector (state).
Bra: ⟨ψ∣\langle\psi| represents the conjugate transpose (row vector).
Inner Product: ⟨ϕ∣ψ⟩\langle\phi|\psi\rangle.
Outer Product: ∣ψ⟩⟨ϕ∣|\psi\rangle\langle\phi|.
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