A Rigorous and Self--Contained Proof of the Grover--Rudolph State Preparation Algorithm

TL;DR

The paper provides a rigorous, self-contained proof of the Grover–Rudolph state preparation algorithm for amplitude-encoded distributions. It formalizes a dyadic probability tree, derives a precise angle map via conditional masses, and proves by induction that a circuit with $N-1$ stages prepares the target state with amplitudes $\sqrt{p_k}$, i.e., $U\ket{0}^{\otimes n}=\sum_k\sqrt{p_k}\,|b_n(k)\rangle$. Beyond correctness, it makes the connection to uniformly controlled rotations explicit and shows how each stage can be transpiled ancilla-free into a gate dictionary $\{\mathrm{R}_y(\cdot),X,\mathrm{CNOT}\}$ via a Gray-code ladder and a Walsh–Hadamard transform, facilitating practical implementations. The work also provides a numerical example illustrating the construction and discusses accuracy and hardware considerations, thereby bridging rigorous theory and circuit-level realizations. Overall, the results offer a complete, verifiable pathway from a classical density to an exact, efficiently synthesizable quantum state preparation circuit with explicit gate-level decompositions.

Abstract

Preparing quantum states whose amplitudes encode classical probability distributions is a fundamental primitive in quantum algorithms based on amplitude encoding and amplitude estimation. Given a probability distribution $\{p_k\}_{k=0}^{2^n-1}$, the Grover--Rudolph procedure constructs an $n$-qubit state $\ketψ=\sum_{k=0}^{2^n-1}\sqrt{p_k}\ket{k}$ by recursively applying families of controlled one-qubit rotations determined by a dyadic refinement of the target distribution. Despite its widespread use, the algorithm is often presented with informal correctness arguments and implicit conventions on the underlying dyadic tree. In this work we give a rigorous and self-contained analysis of the Grover--Rudolph construction: we formalize the dyadic probability tree, define the associated angle map via conditional masses, derive the resulting trigonometric factorizations, and prove by induction that the circuit prepares exactly the desired measurement law in the computational basis. As a complementary circuit-theoretic contribution, we show that each Grover--Rudolph stage is a uniformly controlled $\RY$ rotation on an active register and provide an explicit ancilla-free transpilation into the gate dictionary $\{\RY(\cdot),X,\CNOT(\cdot\to\cdot)\}$ using Gray-code ladders and a Walsh--Hadamard angle transform.

Figures (7)

• Figure 1: The probability density function $\varrho(x)$.
• Figure 2: The integral $\int_{0}^{1} \varrho(x)\,dx$ (blue) and $\int_{0}^{1/2} \varrho(x)\,dx$ (dark blue).
• Figure 3: The integrals $\int_{0}^{1/2}\varrho(x)\,dx$ (blue) and $\int_{1/2}^{1}\varrho(x)\,dx$ (green), and their dyadic refinement.
• Figure 4: Integral regions used to compute $\theta_{00}$, $\theta_{01}$, $\theta_{10}$, and $\theta_{11}$.
• Figure 5: Quantum circuit $\mathsf{U}$ implementing Theorem \ref{['theorem:Grover-Rudolph']} for $n=3$.

Theorems & Definitions (15)

• Lemma 2.1: Binary shift identities
• proof
• Corollary 2.2: Dyadic refinement
• Lemma 2.3: Suffix extraction via scaling
• proof
• Theorem 2.4: Grover--Rudolph state preparation
• Proposition 3.1: Trigonometric factorization
• proof
• Lemma 3.2: Inductive amplitude form
• proof