We present a framework for linear combination of unitaries (LCU) style decomposition tailored to structured sparse matrices, which frequently arise in the numerical solution of partial differential equations (PDEs). While LCU is a foundational primitive in both variational and fault-tolerant quantum algorithms, conventional approaches based on the Pauli basis can require a number of terms that scale quadratically with matrix size. We introduce the Sigma basis, a compact set of simple, nonunitary operators that can better capture sparsity and structure, enabling decompositions with only polylogarithmic scaling in the number of terms. We develop both numerical and semianalytical methods for computing Sigma-basis decompositions of arbitrary matrices. Given this new basis is comprised of nonunitary operators, we leverage the concept of unitary completion to design efficient quantum circuits for evaluating observables in variational quantum algorithms and for constructing block encodings in fault-tolerant quantum algorithms. We compare our framework with related techniques such as unitary dilation, and other sparsity and structure exploiting block-encoding methods, and demonstrate its efficacy and generality. We illustrate our framework on several PDE examples and show an exponential improvement in decomposition size while retaining circuit efficiency. We also prescribe how our framework can be readily generalized by incorporating other unitary and nonunitary operators or “things” to expand or modify the Sigma basis and tailor the decomposition for specific problems to gain maximum efficiency.

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• Quantum algorithms
• Quantum algorithms & computation