Quantum Circuit Optimization by Graph Coloring

TL;DR

This work addresses depth optimization for quantum circuits composed of pairwise commuting gates by proving an exact equivalence to the graph coloring problem, with the minimum depth equal to the chromatic number $\chi(G)$ of a gate-graph. It provides a constructive reduction via two transformations (circuit to graph and colored graph to circuit) and demonstrates that any graph-coloring solver can yield low-depth circuits, effectively mapping quantum optimization to classical graph algorithms. The authors apply the method to finite field multiplication and QFT-based addition, achieving substantial depth reductions (e.g., Toffoli depth in $\mathbb{F}_{2^n}$ designs from $4n-4$ to $2n-1$) and illustrating depth-qubit tradeoffs controlled by colorings. Overall, the paper delivers a practical optimization strategy for commuting quantum circuits and highlights a bridge between quantum circuit design and classical graph-coloring techniques, with potential extensions to broader circuit classes.

Abstract

Depth optimization of a quantum circuit consisting of commuting operations is shown to be reducible to the vertex coloring problem in graph theory. The reduction immediately leads to an algorithm for circuit optimization of commuting gates utilizing any coloring solver. To examine its applicability, known quantum circuits from the literature are optimized.

Figures (6)

• Figure 1: Depth optimization of multiplication in $\mathbb{F}_{2^n}$ by solving graph coloring. Red dots are obtained by using $\mathsf{DSatur}$ algorithm for graph coloring, black dashed line is the lower bound, and blue solid line is Maslov et al's original design.
• Figure 2: Illustration of the multiplication. The terms in each triangle are to be summed where the summing order does not affect the result.
• Figure 3: Depth-qubit tradeoffs for QFT-based addition circuits (a) $\lvert a\rangle\lvert b\rangle \mapsto \lvert a\rangle\lvert b+a\rangle$ and (b) $\lvert a\rangle\lvert b\rangle\lvert c\rangle\mapsto \lvert a\rangle\lvert b\rangle\lvert c+ab\rangle$. The circled and squared points are the most efficient circuit designs with respect to the cost metrics in Eq. (\ref{['eq:metric']}), respectively (see, Table \ref{['tab:cost-comparison']}).
• Figure 4: QFT-based addition $\lvert a\rangle\lvert b\rangle \mapsto \lvert a\rangle\lvert b+a\rangle$ is shown in (a) and its (trivially) optimized circuit is given in (b). Reducing the depth by extra work qubit is illustrated in (c). A circuit for $\lvert a\rangle\lvert b\rangle\lvert c\rangle\mapsto \lvert a\rangle\lvert b\rangle\lvert c+ab\rangle$ is given in (d).
• Figure 5: An example of the parallelization by extra qubits. Introducing one extra qubit gives rise to altering the graph such that it becomes sparser (in edge density).

Theorems & Definitions (7)

• definition thmcounterdefinition: Commuting operations
• definition thmcounterdefinition: Depth of a circuit
• definition thmcounterdefinition: Gate ordering
• definition thmcounterdefinition: Proper coloring
• definition thmcounterdefinition: Vertex coloring problem
• proof
• proof