The Koopman-von Neumann framework is applied to Bayesian inverse problems with Gaussian priors and observation noise, reducing posterior-state preparation to the preparation of a structured Hamiltonian's ground state and introducing a qubit encoding that supports efficient block encodings of the truncated operators.
Abstract
The Koopman-von Neumann framework has been proposed to design quantum algorithms for non-linear dynamics. It maps a non-linear ordinary differential equation to a linear partial differential equation (PDE) governing a probability amplitude. Previous works represents this amplitude in the Hermite-function basis, equivalently as a bosonic state, and truncates the total Hermite degree to obtain a representation over $\Theta(m\log N)$ qubits, where $N$ is the number of variables and $m$ the truncation order. We extend this approach to a broader class of linear PDEs whose differential operators have a structured polynomial form. We prove convergence of the truncation for both time-dependent dynamics and gapped ground-state problems under explicit regularity and stability assumptions. We then introduce a qubit encoding that supports efficient block encodings of the truncated operators. Finally, we apply the framework to Bayesian inverse problems with Gaussian priors and observation noise, reducing posterior-state preparation to the preparation of a structured Hamiltonian's ground state.
We connect different degrees of smoothness of real-valued periodic functions to the number of qubits required for their high-precision Fourier-basis amplitude encodings as quantum states. Our resulting central observation is that the Gevrey hierarchy, which stratifies the space between smooth and analytic functions, provides a natural class for high-precision quantum algorithms. We then specialize to solving general linear partial differential equations (PDEs) with periodic boundary conditions, showing how our Fourier methods do so efficiently at varying target precisions on a quantum computer. This also demonstrates how our framework enables passage from query-complexity results to explicit elementary gate counts. As an application, we introduce a hierarchy of many-body quantum simulation pipelines that harness these high-precision algorithms to probe the linear response of atomistic systems in first quantization. Each level of the hierarchy unlocks a further polynomial-degree quantum speedup, yielding a gradual improvement in simulation efficiency as quantum computers scale.
Pooya Ronagh, Mariia Sobchuk, Xiaoran Li et al.· 0 citations
The biharmonic equation is a prototypical fourth-order partial differential equation whose high-dimensional discretization suffers from rapidly growing degrees of freedom and severe ill-conditioning. We develop QSVT--VTAA quantum linear-system algorithms by constructing explicit block-encodings tailored to periodic, simply supported, and Dirichlet--Neumann boundary conditions. For periodic and simply supported problems, Fourier and sine-transform diagonalizations yield augmented Poisson systems with the condition-number scaling of a second-order operator. For Dirichlet--Neumann problems, we introduce a second-order boundary-corrected finite-difference discretization, establish mesh-independent stability, and construct an explicit block-encoding of the resulting nonsymmetric matrix. We also formulate a coupled-Laplace system with additional boundary unknowns and characterize its complexity in terms of the condition number of the complete augmented matrix. The analysis covers discretization error, block-encoding normalization, gate complexity, and solution extraction under an amplitude-input and quantum-state-output model. Numerical experiments validate the proposed discretizations and the corresponding linear solves.
Can a quantum PDE algorithm avoid the polynomial cost of resolving a fine spatial mesh? For standard fixed-order discretizations, direct classical methods require work polynomial in $h^{-1}$, or equivalently in the number of spatial degrees of freedom $N_h=\Theta(h^{-d})$. Direct quantum implementations of a parabolic semigroup still have coherent complexity $\widetilde{\mathcal O}(\sqrt{T}/h)$, and gradient-dependent observables such as heat flux and dissipation introduce additional mesh dependence. Decay of the solution norm will further suppress the postselection probability for preparing a normalized final state. We develop a multilevel quantum algorithm that estimates linear and quadratic observables $directly$ and places the fine--coarse cancellation inside the circuit before measurement. A contour-based LCU reconstructs each target-time correction from a coherent family of shifted resolvent differences. Rather than block encoding the fine and coarse inverses separately, we encode their difference through a shifted Ritz--Schur factorization, exposing its $\mathcal O(h_\ell^2)$ two-grid normalization. For Fourier hierarchies, the corresponding SELECT oracle consists of a quantum Fourier or sine transform, a spectral-band selector, and reversible diagonal arithmetic. We also give a non-Fourier realization based on energy-orthogonal dyadic midpoint details in one dimension, together with structured tensor-product extensions under fixed-rank coefficient and access assumptions. For readouts with derivative order $0\le\chi\le2$, optimized amplitude estimation removes $all$ polynomial dependence on the finest mesh size. Under the stated access assumptions, both linear and quadratic observables can be estimated with complexity $\widetilde{\mathcal O}(1+(T\epsilon)^{-1})$, with only polylogarithmic dependence on $h^{-1}$.
We present a variational quantum linear solver (VQLS) for the Poisson equation built on an exact block encoding of the discrete Laplacian, and demonstrate its performance on physically motivated benchmarks. Unlike LCU-based VQLS where the number of distinct circuits required per cost-function evaluation is $\mathcal{O}(L^2)$, where $L$ is the number of terms in the LCU decomposition of the discrete Laplacian operator, this approach requires only a single circuit for cost evaluation. We further empirically demonstrate that the choice of classical optimizer materially affects where the variational optimization ceases to make progress. The solver is benchmarked on three problems: a Poisson equation with sinusoidal forcing and a steady-state heat conduction problem with a localized Gaussian source, both with Dirichlet boundaries, and the pressure-Poisson equation of a two-dimensional lid-driven cavity flow, in which the solver is invoked once per time step under Neumann boundary conditions.
Viraj Dsouza, A. Singhal, Alex Khan et al.· 0 citations
We develop a quantum algorithm for solving the dynamics of the nonlinear viscous Burgers equation. We apply the Carleman linearization procedure on the spatially discretized equation, followed by a padding scheme that allows implementation on qubit registers. Existing Carleman-based quantum algorithms commonly formulate the lifted linear differential equation in an oracle model. Here we decompose the padded generator into diagonal masks and reversible arithmetic permutations using the Permutation Matrix Representation (PMR), which we show to be naturally compatible with the Linear Combination of Hamiltonian Simulations (LCHS) algorithm. Under the assumptions required by LCHS - most importantly positive semidefiniteness of the Hermitian part of the linear generator, possibly after a stabilizing shift - the algorithm prepares a normalized quantum state proportional to the solution of the truncated lifted system; the stabilizing shift introduces an exponential postselection overhead, which we quantify and mitigate through a rescaling scheme. We show that our algorithm scales with the off-diagonal norm of the Carleman generator instead of the matrix norm, which can be advantageous for other generators that are diagonally dominant. We also extend the PMR scheme to general fluid equations that may contain higher-order derivatives or nonlinear terms, or may involve multiple fluid variables or spatial dimensions. The construction illustrates how PMR can serve as a convenient Hamiltonian-simulation primitive for a broader class of LCU-based algorithms.
Hriday Sabharwal, A. Kalev, Itay Hen· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
