Wavefunction

The wavefunction is the central mathematical object in quantum mechanics, encoding everything that can be known about a quantum system. For a system of $n$ qubits, the wavefunction is a vector in a $2^n$-dimensional complex Hilbert space, and its components determine the probability of every possible measurement outcome via the Born rule.

Mathematical form

For a single qubit, the wavefunction is a vector in $\mathbb{C}^2$:

$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$

where $\alpha, \beta \in \mathbb{C}$ and $|\alpha|^2 + |\beta|^2 = 1$ (normalization).

For $n$ qubits, the wavefunction is a vector in $\mathbb{C}^{2^n}$:

$|\psi\rangle = \sum_{x \in \{0,1\}^n} c_x |x\rangle$

with $\sum_x |c_x|^2 = 1$. Each coefficient $c_x$ is a complex number called an amplitude. The probability of measuring the bitstring $x$ is $|c_x|^2$.

The wavefunction is written in Dirac notation as a ket $|\psi\rangle$. Its conjugate transpose is the bra $\langle\psi|$. The inner product $\langle\phi|\psi\rangle$ gives the overlap between two states.

Evolution

The wavefunction evolves in two distinct ways:

Unitary evolution: Between measurements, the wavefunction evolves according to the Schrodinger equation. In the circuit model, this is a sequence of unitary operators (quantum gates):

$|\psi(t)\rangle = U|\psi(0)\rangle$

Unitary evolution is deterministic, linear, and reversible.

Measurement: When a measurement is performed, the wavefunction collapses to an eigenstate of the measured observable. For a computational basis measurement, the state $|\psi\rangle = \sum_x c_x |x\rangle$ collapses to $|x\rangle$ with probability $|c_x|^2$. This process is probabilistic and irreversible.

Wavefunction vs. density matrix

The wavefunction describes a pure state: a system about which we have maximal knowledge. When a system is in a statistical mixture of pure states (due to decoherence or entanglement with an environment), the wavefunction description is insufficient and the density matrix formalism is needed.

For a pure state, the density matrix is simply $\rho = |\psi\rangle\langle\psi|$, a rank-1 projector. The density matrix formalism is strictly more general: it handles both pure and mixed states.

Simulation cost

Storing the wavefunction of $n$ qubits requires $2^n$ complex numbers. At 16 bytes per complex number (double precision), 30 qubits require about 16 GB, 40 qubits require about 16 TB, and 50 qubits require about 16 PB. This exponential scaling is why classical simulation of quantum circuits becomes intractable beyond roughly 40 to 50 qubits, depending on available memory and circuit structure.

Specialized simulation techniques (tensor networks, stabilizer simulation for Clifford circuits, matrix product states for low-entanglement states) can handle certain circuits with many more qubits by exploiting structure, but general wavefunction simulation hits this exponential wall.

Why it matters for learners

The wavefunction is the foundation of all quantum computation. Every quantum gate transforms the wavefunction, and every measurement extracts information from it. Understanding the wavefunction, its exponential size, its complex-valued nature, and the role of normalization and phase, is the prerequisite for understanding quantum algorithms, error correction, and the fundamental limits of quantum simulation.

See also

• Hilbert Space
• Born Rule
• Superposition
• Density Matrix
• Bra-Ket Notation