Qubit

The qubit is the fundamental unit of quantum information, represented by a normalized two-dimensional complex vector \(|\psi\rangle \in \mathbb{C}^2, \| |\psi\rangle \|_2 = 1\).

Mathematically, we can define two orthonormal basis vectors called computational states:
\(|0\rangle = \begin{bmatrix}1\\0\end{bmatrix}, \quad |1\rangle = \begin{bmatrix}0\\1\end{bmatrix}\),
and express a qubit as a linear combination of the basis vectors:
\(|\psi\rangle = \alpha |0\rangle + \beta |1\rangle, \quad \alpha, \beta \in \mathbb{C}, \quad |\alpha|^2 + |\beta|^2 = 1\).

In column vector form, this corresponds to:
\(|\psi\rangle = \begin{bmatrix}\alpha\\\beta\end{bmatrix} = \begin{bmatrix}a+ib\\c+id\end{bmatrix}, \quad a,b,c,d \in \mathbb{R}, \quad a^2+b^2+c^2+d^2=1\).

Qubit Measurement

When a qubit \(|\psi\rangle = \alpha |0\rangle + \beta |1\rangle \in \mathbb{C}^2\) is measured in the computational basis \(\{|0\rangle, |1\rangle\}\), the state collapses to one of the basis vectors with probabilities determined by the amplitudes:

\(|0\rangle \text{ with probability } |\alpha|^2 = |\langle 0|\psi\rangle|^2,\)
\(|1\rangle \text{ with probability } |\beta|^2 = |\langle 1|\psi\rangle|^2\).

In other words, a qubit exists in a superposition of classical states before measurement. Upon measuring, it collapses probabilistically: \(|\psi\rangle = \alpha |0\rangle + \beta |1\rangle \to |0\rangle \text{ or } |1\rangle\) with probabilities \(|\alpha|^2\) and \(|\beta|^2\), respectively. This process is also called the collapse of the wave function.

Multi-Qubit Systems

Multi-qubit gates are unitary operators acting on \(\mathbb{C}^{2^n}\) and can generate entanglement by coupling qubits. Measurement generalizes directly: the probability of observing a specific computational basis state is given by the squared magnitude of its corresponding coefficient, regardless of whether the state is separable or entangled.

Multi-Qubit Systems, Entanglement

When multiple qubits \(|\psi_1\rangle, \dots, |\psi_n\rangle\) are considered jointly, their combined state is represented by the tensor (Kronecker) product of the individual states:
\(|\psi\rangle = |\psi_1\rangle \otimes |\psi_2\rangle \otimes \cdots \otimes |\psi_n\rangle \in \mathbb{C}^{2^n}\).
Such a collection of qubits forms a quantum register. A convenient shorthand notation is \(|\psi_1\psi_2\cdots\psi_n\rangle\).

Two-Qubit Example

For two qubits \( |\psi_1\rangle = \alpha|0\rangle + \beta|1\rangle \) and \( |\psi_2\rangle = \gamma|0\rangle + \delta|1\rangle \), the joint state is
\(|\psi\rangle = |\psi_1\rangle \otimes |\psi_2\rangle = \begin{bmatrix} \alpha\gamma \\ \alpha\delta \\ \beta\gamma \\ \beta\delta \end{bmatrix}\).
Product states of this form are called separable; they correspond to rank-1 tensors and can be decomposed into individual qubit states.

Entanglement

Multi-qubit systems are not restricted to separable states. A state that cannot be written as a tensor product of single-qubit states is called entangled. A canonical example is the two-qubit Bell (EPR) state:
\( |\psi\rangle = \frac{1}{\sqrt{2}} (|01\rangle + |10\rangle) = \frac{1}{\sqrt{2}} \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}\).
No coefficients \(\alpha,\beta,\gamma,\delta\) exist such that this state can be factorized into \( |\psi_1\rangle \otimes |\psi_2\rangle \); hence it is genuinely entangled.

Quantum State Evolution

The state of an \(n\)-qubit quantum system can be actively manipulated over time using controlled external interactions. Let \(|\psi(0)\rangle\) denote the initial state of the system. Its time evolution is governed by the system Hamiltonian \(H(t) \in \mathbb{C}^{2^n \times 2^n}\), which encodes the energies and couplings induced by the experimental setup.

Schrödinger Equation

The time evolution of the quantum state \(|\psi(t)\rangle\) is described by the Schrödinger equation:

\(i\hbar \frac{d}{dt} |\psi(t)\rangle = H(t)\,|\psi(t)\rangle\).

Here, \(i\) denotes the imaginary unit, \(\hbar\) is the reduced Planck constant, and the Hamiltonian \(H(t)\) is a Hermitian operator acting on the \(2^n\)-dimensional Hilbert space.

Hamiltonian

Put simply, and in analogy to classical computing, a Hamiltonian can be viewed as an energy function—a mathematical expression describing how energy is distributed across a quantum system. A time-dependent Hamiltonian defines an evolving energy landscape that governs the system’s dynamics through the Schrödinger equation.
The specific structure of the Hamiltonian, together with how this evolution is realized or approximated in time, fundamentally determines the quantum computing paradigm. In gate-based quantum computing, evolution is discretized into sequences of unitary operations (quantum gates), whereas in adiabatic quantum computing (AQC) the system is steered continuously by slowly varying the Hamiltonian toward the ground state encoding the solution.

Gate-Based Quantum Computing (GQC)

Following the Schrödinger evolution of a quantum system, one of the primary ways to manipulate qubits is via quantum gates. These gates are unitary operators applied to one or multiple qubits, defining the discrete-time evolution of the system. Quantum circuits are sequences of such gates that perform computations analogous to classical logical circuits, but in a reversible and linear-algebraic manner.

Adiabatic Quantum Computing (AQC) & Quantum Annealing

Unlike gate-based QC, AQC evolves a quantum system continuously under a time-dependent Hamiltonian \(H(t)\). The system is prepared in the ground state of an initial Hamiltonian \(H_I\) and evolves slowly to a problem Hamiltonian \(H_P\), encoding the solution to an optimization problem.