A Unified Framework for Optimizing Uniformly Controlled Structures in Quantum Circuits

TL;DR

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.

Abstract

Quantum unitaries of the form $Σ_{c}\ket{c}\bra{c}\otimes U_{c}$ 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.

Figures (25)

• Figure 1: Overview of the proposed rUCG model that unifies traditional UCGs and UCG-like operator families within a single model, together with the goal of unified decomposition framework for size and depth.
• Figure 2: Uniformly controlled gate (UCG) $L_n[\mathbf{U}]$ implemented as a sequence of $N=2^n$ conditioned unitaries $C_n^i[U_i]$. Each $U_i$ acts on the $m$-qubit target register $|t\rangle$ when the control register is in control state $|i\rangle$.
• Figure 3: Restricted uniformly controlled gate(rUCG) $\mathrm{LU}(\bm{\chi})$, where $\bm{\chi}$ is called a target vector determining the target operators. For brevity, $\chi_i$ denotes $U(\chi_i)$ in a gate.
• Figure 6: Decomposition of rUCG with optimal number of gates for $n=3$, within GP$^*(3)$.
• Figure 7: A circuit framework to implement a n-qubit unitary diagonal matrix, $\Lambda_{n}$, where $r_t = \lfloor n/2 \rfloor$, $r_c = n - r_t = \lceil n/2 \rceil$, and $\ell_{k_t} \leq {2\binom{r_t}{k_t}}/{r_t}+2$. The depth of the operator $\mathcal{G}_{k_t}$ is $O(\frac{1}{n}\binom n{k_t})$ for each ${k_t} \in [k]$, and the depth of the operator $R$ is $O(r_t / \log r_t)$.

Theorems & Definitions (46)

• Definition 1
• Definition 2: Restricted UCG, rUCG
• Definition 3
• Corollary 1
• proof
• Definition 4
• Definition 5: Standard k-rUCG
• Definition 6: General k-rUCG
• Lemma 1
• proof