A Unified Framework for Optimizing Uniformly Controlled Structures in Quantum Circuits

TL;DR

This paper introduces a unified algebraic framework for optimizing uniformly controlled structures in quantum circuits, significantly reducing their complexity and depth, and unifying various circuit types under one model.

Contribution

The paper develops a general algebraic model called rUCGs, providing systematic methods to reduce circuit size and depth for uniformly controlled gates, unifying diverse circuit structures.

Findings

Reduces gate complexity from O(n2^n) to O(2^n)

Reduces circuit depth from O(2^n log n) to O(2^n log n / n)

Empirical results confirm significant reductions in depth and size for QAOA circuits.

Abstract

Quantum unitaries of the form $\Sigma_{c}\ket{c}\bra{c}\otimes U_{c}$ are ubiquitous in quantum algorithms. This class encompasses not only standard uniformly controlled gates (UCGs) but also a wide range of circuits with uniformly controlled structures. However, their circuit-depth and gate-count complexities have not been systematically analyzed within a unified framework. In this work, we study the general decomposition problem for UCG and UCG-like structure. We then introduce the restricted Uniformly Controlled Gates (rUCGs) as a unified algebraic model, defined by a 2-divisible Abelian group that models the controlled gate set. This model captures uniformly controlled rotations, multi-qubit uniformly controlled gates, and diagonal unitaries. Furthermore, this model also naturally incorporates k-sparse version (k-rUCGs), where only a subset of control qubits participate in each multi-qubit gate. Building on this algebraic model, we develop a general framework. For an n-control rUCG, the framework reduce the gate complexity from ${O(n2^n)}$ to ${O(2^n})$ and the circuit depth from ${O(2^n\log n)}$ to ${O(2^n\log n/n)}$. The framework further provides systematic size and depth bounds for k-rUCGs by exploiting sparsity in the control space, with same optimization coefficient as rUCG, respectively. Empirical evaluations on representative QAOA circuits confirm reductions in depth and size, which highlight that the rUCG model and its associated decomposition framework unify circuits previously considered structurally distinct under a single, asymptotically optimal synthesis paradigm.