Matthias Deiml

Matthias Deiml, Oliver Hüttenhofer, Ram Mosco et al.

We introduce Unitaria, a Python library that brings the simplicity of classical linear algebra toolkits such as NumPy and SciPy to the implementation of quantum algorithms based on block encodings, a general-purpose abstraction in which a matrix is embedded as a sub-block of a larger unitary operator. Their implementation has so far required deep knowledge of low-level circuit construction, which Unitaria aims to eliminate. The library provides a composable, array-like interface through which users can define block encodings of matrices and vectors, combine them through standard operations such as addition, multiplication, tensor products, and the Quantum Singular Value Transformation, and extract the resulting quantum circuits automatically. A key feature is a matrix-arithmetic evaluation path in which every operation can be computed directly on encoded vectors and matrices without dependence on ancilla qubits or circuit simulation. This enables correctness verification and classical simulation that scale well beyond what state vector simulation permits and also allows resource estimation, including gate counts, qubit counts, and normalization constants, without executing any circuit. Together, these capabilities allow researchers to develop, verify, and analyze quantum linear algebra algorithms today, ahead of the availability of error-corrected hardware. Unitaria is open source and available at https://github.com/tequilahub/unitaria.

Loïc Balazi, Matthias Deiml, Daniel Peterseim

We propose a numerical homogenization method for scalar linear partial differential equations with rough coefficients, that integrates classical coarse-scale solvers with quantum subroutines for fine-scale corrections. Inspired by the Localized Orthogonal Decomposition, we employ quantum local problem solvers to capture fine-scale features efficiently. Crucially, the approach does not rely on the periodicity of the problem, and the integration of the quantum computation within a coarse model requires only selected measurements of the quantum representative volume elements, overcoming the information bottleneck of quantum interfaces that could eliminate the speed-up. We demonstrate that the local quantum solver can achieve solutions with sufficient accuracy, with a number of operations that scales only logarithmically with the fine-scale resolution, determined by the smallest length scale encoded in the diffusion coefficient. The potential of the approach is illustrated through two-dimensional test cases, using a classical simulation of the local quantum solver.

Matthias Deiml, Daniel Peterseim

Quantum linear system solvers typically realize the inverse map as a polynomial transformation of the spectrum, so their practical cost hinges on implementing this transformation at a low polynomial degree. We introduce constrained optimal polynomials as a framework for this task, drawing on classical Krylov subspace theory. Within this framework, we develop three classes of polynomial solvers. Baseline quantum Chebyshev-type iterations provide general-purpose polynomials based on spectral bounds. Constrained Uniform Polynomial (CUP) solvers optimize the tradeoff between approximation accuracy and block encoding normalization under a uniform spectral model consistent with the available bounds. Constrained Adaptive Polynomial (CAP) solvers retain this structure but replace the uniform model with a probability measure reconstructed from spectral moments via a maximum entropy ansatz, where the moments are extracted from QSVT measurements. Numerical experiments under hardware and stochastic noise show that these methods achieve lower error than standard QSVT-based inversion at a comparable polynomial degree, up to an order of magnitude in noise-limited regimes. CUP offers robust performance under generic spectra, while CAP provides further improvement when the spectral structure can be exploited.