On the Schrödingerization method for linear non-unitary dynamics with optimal dependence on matrix queries

TL;DR

This work advances Schrödingerization as a universal method for simulating linear non-unitary dynamics by embedding the problem into a higher-dimensional unitary Schrödinger-type system via a warped phase transformation. It identifies non-smooth initial auxiliary data as the source of suboptimal scaling in matrix queries and introduces three criteria plus four smooth initializations to restore optimal or near-optimal precision scaling, notably an erf-based construction that achieves optimal log(1/ε) queries. The authors provide a rigorous complexity framework, detailed error analyses for truncation and spectral discretization, and concrete constructions (cut-off, high-order interpolation, Fourier-kernel) to realize near-optimal performance. By bridging smooth-approximation theory with quantum Hamiltonian simulation, the paper demonstrates that optimal or near-optimal matrix-query scaling is attainable for a broad class of non-unitary linear dynamics, with practical implications for quantum PDE solvers. The results also clarify the limitations of mollifier-based smoothings and connect to recent generalized LCHS techniques, offering a path toward matching quantum lower bounds on query complexity.

Abstract

The Schrödingerization method converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schrödinger-type equations with unitary evolution. It does so via the so-called warped phase transformation that maps the original equation into a Schrödinger-type equation in one higher dimension \cite{Schrshort,JLY22SchrLong}. The original proposal used a particular initial function in the auxiliary space that did not achieve optimal scaling in precision. Here we show that, by choosing smoother initial functions in auxiliary space, Schrödingerization \textit{can} in fact achieve near optimal and even optimal scaling in matrix queries. We construct three necessary criteria that the initial auxiliary state must satisfy to achieve optimality. This paper presents detailed implementation of four smooth initializations for the Schrödingerization method: (a) the error function and related functions, (b) the cut-off function, (c) the higher-order polynomial interpolation, and (d) Fourier transform methods. Method (a) achieves optimality and methods (b), (c) and (d) can achieve near-optimality. A detailed analysis of key parameters affecting time complexity is conducted.

Figures (7)

• Figure 1: Quantum circuit for Schrödingerization of \ref{['generalSchr']}, where $\bm{\psi}_h = \sum_{k\in [N_p]} \psi(p_k)| k \rangle$.
• Figure 2: A smooth extension of $\mathrm{e}^{-p}$
• Figure 3: A snapshot of the domain of the cut-off function in $\mathbb{R}^2$
• Figure 4: The cut-off function and the resulting smooth extension
• Figure 5: The smooth initial data of $\psi(p)$ by using high-order interpolation.

Theorems & Definitions (24)

• Theorem 2.1
• Theorem 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2: cut-off function
• Theorem 3.1
• proof
• Theorem 4.1
• proof