On generating direct powers of dynamical Lie algebras

TL;DR

This paper explores how to generate direct powers of dynamical Lie algebras efficiently, enabling the construction of larger algebraic structures with minimal additional resources, which is useful for quantum algorithm design.

Contribution

It introduces methods to produce direct sums of DLAs using logarithmic additional qubits and minimal generator increases, focusing on cyclic DLAs like Pauli and QAOA-MaxCut.

Findings

Efficient construction of direct powers of DLAs with logarithmic qubits

Methods applicable to cyclic DLAs including Pauli and QAOA-MaxCut

Minimal increase in generators for direct sum formation

Abstract

The expressibility and trainability of parameterized quantum circuits has been shown to be intimately related to their associated dynamical Lie algebras (DLAs). From a quantum algorithm design perspective, given a set $A$ of DLA generators, two natural questions arise: (i) what is the DLA $\mathfrak{g}_{A}$ generated by ${A}$; and (ii) how does modifying the generator set lead to changes in the resulting DLA. While the first question has been the subject of significant attention, much less has been done regarding the second. In this work we focus on the second question, and show how modifying ${A}$ can result in a generator set ${A}'$ such that $\mathfrak{g}_{{A}'}\cong \bigoplus_{j=1}^{K}\mathfrak{g}_{A}$, for some $K \ge 1$. In other words, one generates the direct sum of $K$ copies of the original DLA. In particular, we give qubit- and parameter-efficient ways of achieving this, using only $\log K$ additional qubits, and only a constant factor increase in the number of DLA generators. For cyclic DLAs, which include Pauli DLAs and QAOA-MaxCut DLAs as special cases, this can be done with $\log K $ additional qubits and the same number of DLA generators as ${A}$.