Quantum Computation

Mathematical Foundations

Quantum States

Kets & Bras

A state of a quantum system can be represented as a vector $\psi$ in a Hilbert space $\H$,

the dual state of a quantum state $|\psi\rangle$ is denoted as $\langle\psi|$, $$ \langle\psi| = |\psi\rangle^\dagger $$

The inner product between a ket and a bra is defined as:

$$ \begin{aligned} \langle\psi|\phi\rangle &= \underbrace{\sum \alpha_i^* \beta_i}_{{\color{blue}\text{discrete case}}} \quad \text{or} \quad \underbrace{\int \psi^*(x) \phi(x) dx}_{{\color{forestgreen}\text{continuous case}}} \end{aligned} $$ for any quantum state $|\psi\rangle$ $$\|\psi\|^2=\langle\psi|\psi\rangle=1$$
  1. In a single qubit system, the state is a vector $|\psi\rangle\in \C^2$, $$ |\psi\rangle = \frac{i}{\sqrt{3}}|0\rangle - \frac{2}{\sqrt{3}}|1\rangle = \sum_{i=0}^1 = \begin{pmatrix} \frac{i}{\sqrt{3}} \\ -\frac{2}{\sqrt{3}} \end{pmatrix} $$
  2. In a $n$ qubit system, the state is a vector $|\psi\rangle\in \C^{2^n}$, $$ |\psi\rangle = \sum_{i=1}^{2^n} \alpha_i|i\rangle= \begin{pmatrix} \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_{2^n} \end{pmatrix} $$

Measure & Collapse

$$ \psi = \begin{pmatrix} \frac{i}{\sqrt{3}} \\ -\frac{2}{\sqrt{3}} \end{pmatrix},\qquad A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = |0\rangle\langle0| $$ $$A|\psi\rangle = |0\rangle\langle0|\psi\rangle = \frac{i}{\sqrt{3}}|0\rangle$$ $$ \langle\psi|A|\psi\rangle =\langle\psi |0\rangle\langle0|\psi\rangle = (\frac{i}{\sqrt{3}})^2 = \frac{1}{3} $$ $$ \langle0|A|0\rangle = \langle0|0\rangle\langle0|0\rangle = 1 $$
Hermitian Operators

Hermitian operators $H = H^{†}$ represent observable quantities. They generate unitary evolutions via:

$$ U = e^{-iHt} $$

An observable A can be decomposed as:

$$ H = \sum_k \lambda_k |\psi_k\rangle\langle\psi_k| $$

where λₖ are eigenvalues and |ψₖ⟩ are the corresponding eigenstates.

Unitary Operators

A linear operator U is unitary if:

$$ U^\dagger U = I $$

where U† is its conjugate transpose and I is the identity operator.

Quantum Gates

Quantum gates are linear operators on quantum states. $$ U|\psi(t)\rangle=\int U\psi(x, t)|x\rangle d x $$

Wave Functions

A wave function $\psi(x,t)$ is a complex-valued function that describes the quantum state of an isolated physical system. $$\psi(x,t): \H \times \R \to \C$$ The wave function has the following properties:
  1. It is continuous and single-valued everywhere
  2. It is normalized such that: $$\int_{-\infty}^{\infty} |\psi(x,t)|^2 dx = 1$$
  3. The quantity $|\psi(x,t)|^2$ represents the probability density of the system at time $t$.

Quantum states at time $t$ can be written as

$$ |\psi(t)\rangle=\int \psi(x, t)|x\rangle d x $$

The wave function $\psi(x,t)$ satisfies the Schrödinger equation:

$$ i\hbar \frac{\partial}{\partial t}\psi(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi(x,t) + V(x)\psi(x,t) $$

For a free particle $(V(x)=0)$, the solution takes the form:

$$ \psi(x,t) = Ae^{i(kx-\omega t)} $$

Linear Operators and Matrix Representations

Particles and Spin

In 3-d space, a particle can be classified into two categories:

In 2-d space, particles can exhibit a continuous range of statistical behavior, these are called anyons.

Quantum Gates and Operations

Quantum Gates

Common single-qubit gates include:

Hadamard gate:

$$ H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} $$

Phase gate:

$$ S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} $$

NOT gate (Pauli-X):

$$ X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$

Multi-Qubit Gates

The CNOT gate:

$$ CNOT = |0\rangle\langle0| \otimes I + |1\rangle\langle1| \otimes X $$

4. Quantum Entanglement

Bell States

The maximally entangled Bell state:

$$ |\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) $$

Schmidt Decomposition

Any bipartite pure state can be written as:

$$ |\psi\rangle = \sum_i \sqrt{\lambda_i} |u_i\rangle \otimes |v_i\rangle $$

Entanglement Entropy

Quantified as:

$$ S = -\sum_i \lambda_i^2 \log(\lambda_i^2) $$

5. Quantum Algorithms

Quantum Fourier Transform

The QFT acts on basis states as:

$$ QFT|j\rangle = \frac{1}{\sqrt{N}} \sum_k e^{2\pi ijk/N} |k\rangle $$

Grover's Algorithm

The Grover operator is:

$$ \mathcal{Q} = (2|s\rangle\langle s| - I) \Omega $$

where |s⟩ is the marked state and Ω is the oracle.

Phase Estimation

Used in Shor's algorithm and other quantum algorithms to extract eigenvalues of unitary operators.