Sparse Block-Encodings for Linear Combinations of Ladder Operators

Block-encodings provide access to information about non-unitary operators within quantum algorithms. Two frameworks, Linear Combination of Unitaries (LCU) and sparse oracle block-encodings, have resulted in compiled block-encoding circuits for various operators. In this work, we detail a framework that we refer to as LOBE (Ladder Operator Block-Encoding) which gives compiled block-encodings of linear combinations of products of ladder operators that act on (anti)fermionic and bosonic momentum modes. This framework builds upon similar sparse oracle block-encodings of ladder operators acting only on fermionic modes. We establish a clear connection between these sparse oracle block-encodings and LCU. This connection allows for the use of techniques arising in both frameworks to give efficient compilations with reduced rescaling factors. Notably, LOBE is a clear extension of LCU as the terms in this linear combination are not restricted to unitaries and we believe it can be extended to include linear combinations of other non-unitary operators. Block-encodings of operators acting on (anti)fermions and bosons pave the way for quantum simulation of systems that are not purely fermionic such as those that arise in models of high-energy physics and quantum field theories.