Nonlinear quantum computation by amplified encodings

TL;DR

This paper introduces a new high-dimensional nonlinear quantum computing framework using amplified tensor product encodings, enabling efficient polynomial evaluation and nonlinear equation solving with potential quantum advantage.

Contribution

It proposes a novel quantum encoding framework for nonlinear computation that preserves quantum advantage and compares fixed-point and Newton's methods for practical implementation.

Findings

Quantum advantage with logarithmic complexity scaling.

Newton's method achieves near-optimal theoretical complexity.

Fixed-point iteration is more suitable for noisy hardware, supported by numerical experiments.

Abstract

This paper presents a novel framework for high-dimensional nonlinear quantum computation that exploits tensor products of amplified vector and matrix encodings to efficiently evaluate multivariate polynomials. The approach enables the solution of nonlinear equations by quantum implementations of the fixed-point iteration and Newton's method, with quantitative runtime bounds derived in terms of the error tolerance. These results show that a quantum advantage, characterized by a logarithmic scaling of complexity with the dimension of the problem, is preserved. While Newton's method attains near-optimal theoretical complexity, the fixed-point iteration may be better suited to near-term noisy hardware, as supported by our numerical experiments.