Linear Combination of Non-unitaries (LCNU)

Abstract

This tutorial provides a comprehensive exposure to the linear combination of non-unitaries (LCNU), a new framework for more efficient linear combination of unitaries (LCU) decomposition for quantum linear algebra applications. Naive LCU strategies, such as Pauli decomposition, often fail to exploit the symmetry and structure of the underlying matrix and produce a large number of decomposition terms that may nullify any quantum advantage. The LCNU framework addresses this challenge by using an alternative set of simple non-unitary operators that better exploit such underlying matrix properties to achieve a scaling that is polylogarithmic with matrix size. Given that LCU is a fundamental component of many variational and fully fault-tolerant quantum algorithms, replacing LCU with LCNU has been shown to benefit many quantum linear algebra applications, including numerical methods for solving ordinary/partial differential equations. Furthermore, LCNU's flexibility — allowing incorporation of custom operators to tailor decompositions for specific problems — positions it as a versatile "linear combination of things" with broad applicability in quantum computing. The tutorial gives an accessible exposure to the LCNU framework, including the latest theoretical and computational developments, end-to-end case studies, and hands-on coding exercises. After the tutorial, attendees are expected to obtain a rigorous understanding of the LCNU framework, its advantages over alternative methods, and a defined scope of where it can be beneficially applied.

Agenda

References

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