Quantum State Preparation

A Qiskit implementation of arbitrary quantum state preparation: given a normalized complex vector $\psi \in \mathbb{C}^{2^n}$, build a unitary $U$ with

$$U|0\rangle^{\otimes n} = \sum_{x=0}^{2^n - 1} \psi_x,|x\rangle.$$

The circuit uses only single-qubit gates and multi-controlled $R_Z$ rotations, with no measurements. The output is a pure unitary, usable as a subroutine that can be composed, controlled, or inverted.

Approach

The construction loops over a binary tree, with data loaded as follows:

• Magnitudes are loaded by conditional-probability $R_Y$ rotations: at each node the rotation splits the remaining probability mass between its two subtrees.
• Phases are loaded by an argument-averaging $R_Z$ tree: each node applies the phase difference between its children and carries their average upward. The leftover root average is the overall global phase; it is unobservable on its own, but is recorded on the circuit so that a controlled version remains correct.

The function state_preparation_data returns the tree as a plain list of operation dictionaries together with that global phase, and build_state_prep_circuit translates them into gates.

Two implementation notes:

• Multi-controlled $R_Y$ is realized as a multi-controlled $R_Z$ conjugated by single-qubit gates ($S H R_Z H S^\dagger = R_Y$).
• Controls on an arbitrary bit-pattern are implemented by applying $X$ gates to set all qubits equal to $1$, and reversing them afterwards.

Amplitudes are indexed in Qiskit's standard ordering (qubit 0 least significant), so the statevector index equals the integer $x$.

See walkthrough.ipynb for the full derivation and a worked example.

Repository

• state_prep.py — the implementation (angle helper, tree builder, circuit builder).
• walkthrough.ipynb — a derivation of the method, with an $n=3$ demonstration.
• testing_suite.ipynb — the test suite (correctness, error isolation, path coverage, contract).

Usage

uv sync
uv run jupyter lab

Then open walkthrough.ipynb. To prepare a state directly:

import numpy as np
from state_prep import state_preparation_data, build_state_prep_circuit

psi = np.array([...], dtype=complex)        # length 2^n, will be normalized
n = len(psi).bit_length() - 1
ops, global_phase = state_preparation_data(psi)
qc = build_state_prep_circuit(n, ops, global_phase)

License

MIT.