PyEncode: An Open-Source Library for Structured Quantum State Preparation

2026-03-30

AI Summary

This work addresses the inefficiency of generic quantum state preparation methods in encoding classically structured vectors by proposing a unified framework that exploits intrinsic vector structures—such as sparsity, step functions, Fourier, geometric, and Walsh patterns—to enable exact amplitude encoding via closed-form quantum circuits, thereby avoiding explicit instantiation or approximation. For the first time, we open-source a toolkit supporting closed-form preparation for multiple classes of structured vectors and integrate the Linear Combination of Unitaries (LCU) protocol to exactly synthesize piecewise-structured states. Built on Qiskit, our approach fuses quantum Fourier transforms, constant adders, and LCU techniques to drastically reduce circuit complexity: sparse, step, and Walsh vectors require only O(m) gates; geometric vectors need no two-qubit gates; and interval and Fourier vectors scale as O(m²), substantially outperforming the O(2^m) cost of generic methods.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Quantum algorithms require encoding classical vectors as quantum states, a step known as amplitude encoding. General-purpose state preparation routines accept any input vector of length $N = 2^m$ and produce circuits with $\bigO{2^m}$ gates. However, vectors arising in scientific and engineering applications often exhibit mathematical structure that admits far more efficient encoding. Recent theoretical work has established closed-form circuits for several structured vector classes, but without open-source implementations. We present PyEncode, an open-source Python library that implements this body of theory in a unified, immediately deployable framework. The library covers sparse, step, square (general interval), Walsh, geometric, and Fourier patterns, and supports weighted superpositions of pattern states via the linear combination of unitaries (LCU) protocol, enabling exact preparation of piecewise-structured vectors such as multi-interval Hamiltonians. PyEncode exposes a single function encode(VectorObj, N) that maps a typed parameter declaration directly to a verified Qiskit circuit, with no vector materialization and no approximation. Sparse, step, and Walsh vectors require only $\bigO{m}$ gates; geometric (exponential-decay) vectors require $\bigO{m}$ gates with zero two-qubit gates; square (general interval) vectors require $\bigO{m^2}$ gates via a QFT-based constant adder, with $\bigO{m}$ special cases; Fourier (sinusoidal) vectors require $\bigO{m^2}$ gates via the inverse Quantum Fourier Transform -- all exponentially fewer than the $\bigO{2^m}$ cost of general state preparation. LCU combines $r$ component circuits whose total gate cost is the sum of individual component costs, with success probability $p \in (0,1]$ determined analytically. The library is available at https://github.com/UW-ERSL/PyEncode.

Problem

• quantum state preparation
• amplitude encoding
• structured vectors
• efficient encoding
• quantum algorithms

Innovation

• structured quantum state preparation
• amplitude encoding
• linear combination of unitaries (LCU)
• quantum circuit optimization
• PyEncode