Transmutation based Quantum Simulation for Non-unitary Dynamics

TL;DR

The paper introduces a Kannai-transform–driven quantum algorithm to simulate non-unitary dissipative diffusion generated by $A=L^\dagger L$ by recasting diffusion as a Gaussian-weighted superposition of unitary wave propagators. It combines a first-order unitary embedding with high-order Gaussian quadrature and LCU to achieve a query complexity of $\tilde{O}(\sqrt{\|A\|T\log(1/\varepsilon)})$ and polylogarithmic dependence on the precision, improving over generic Hamiltonian-simulation methods. The authors instantiate the framework for the heat equation, biharmonic diffusion, and viscous Hamilton–Jacobi surrogates, and also develop a long-time regime linear solver with complexity $\tilde{O}(\kappa^{3/2}\log^2(1/\varepsilon))$, illustrating practical quantum advantages for parabolic PDEs. The approach extends to non-normal generators and provides a suite of transmutation-based reductions, including spherical means and transport–heat averaging, broadening the toolbox for non-unitary quantum simulations. Overall, the work offers a versatile, efficient pathway to non-unitary diffusion on quantum hardware with favorable scaling in time, norm, and accuracy.

Abstract

We present a quantum algorithm for simulating dissipative diffusion dynamics generated by positive semidefinite operators of the form $A=L^\dagger L$, a structure that arises naturally in standard discretizations of elliptic operators. Our main tool is the Kannai transform, which represents the diffusion semigroup $e^{-TA}$ as a Gaussian-weighted superposition of unitary wave propagators. This representation leads to a linear-combination-of-unitaries implementation with a Gaussian tail and yields query complexity $\tilde{\mathcal{O}}(\sqrt{\|A\| T \log(1/\varepsilon)})$, up to standard dependence on state-preparation and output norms, improving the scaling in $\|A\|, T$ and $\varepsilon$ compared with generic Hamiltonian-simulation-based methods. We instantiate the method for the heat equation and biharmonic diffusion under non-periodic physical boundary conditions, and we further use it as a subroutine for constant-coefficient linear parabolic surrogates arising in entropy-penalization schemes for viscous Hamilton--Jacobi equations. In the long-time regime, the same framework yields a structured quantum linear solver for $A\mathbf{x}=\mathbf{b}$ with $A=L^\dagger L$, achieving $\tilde{\mathcal{O}}(κ^{3/2}\log^2(1/\varepsilon))$ queries and improving the condition-number dependence over standard quantum linear-system algorithms in this factorized setting.

Figures (6)

• Figure 1: Quantum circuit for the homogeneous term.
• Figure 2: Kernel truncation behaviour for several unitary-superposition representations. Left: truncation error $\varepsilon(R):=2\int_{R}^{\infty}\!\lvert K_i(s)\rvert\,\mathrm{d}s$ as a function of the truncation length $R$ (logarithmic scale in $\varepsilon$). Right: kernel magnitudes $\lvert K_i(s)\rvert$ on $s\in[-10,10]$ for a fixed target precision $\varepsilon=10^{-6}$ and $T=1$ .
• Figure 3: Circuit for the normal case \ref{['eq:final_commuting_formula']}.
• Figure : (a) Case 1: Dirichlet $u(t,0)=u(t,1)=1$.
• Figure : (a) Case 1: Dirichlet $u(t,0)=u(t,1)=1$.

Theorems & Definitions (22)

• Lemma 2.1
• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Corollary 4.1
• Corollary 4.2
• Corollary 5.1
• proof
• Corollary 5.2