On the uniqueness of compiling graphs under the parity transformation

TL;DR

The paper formalizes the parity transformation as a map between equivalence classes of hypergraphs, enabling parity-encoded compilation of optimization problems for quantum devices. It introduces loop labelings and the induced graphs $G_{H,\\ell}$ to characterize the preimage under the parity map, yielding injectivity criteria on graph domains. While the transformation is shown to be non-unique in general, the authors identify a practically relevant class—rectangular plaquette layouts—where the preimage is uniquely determined and provide a polynomial-time algorithm to decide applicability and generate the corresponding layout. The results bridge graph-theoretic concepts with quantum hardware constraints, offering concrete methods to efficiently map problems onto hardware with local four-body interactions. This contributes to reliable, scalable parity-based compilation strategies for quantum optimization on common grid-like architectures.

Abstract

In this article, we establish a mathematical framework that utilizes concepts from graph theory to formalize the parity transformation, an encoding strategy for compiling optimization problems on quantum devices. We introduce the transformation as a mapping that encompasses all possible compiled hypergraphs and investigate its uniqueness properties in more detail. Specifically, by introducing so-called loop labelings, we derive an alternative expression of the preimage of any set of compiled hypergraphs under this encoding procedure when all equivalence classes of graphs are being considered. We then deduce equivalent conditions for the injectivity of the parity transformation on any subset of all equivalences classes of graphs. Through concrete examples, we demonstrate that the parity transformation is not an injective mapping, and also introduce an important class of physical layouts and their corresponding set of constraints whose preimage is uniquely determined. In addition, we provide an algorithm which is based on classical algorithms from theoretical computer science and computes a compiled physical layout in this class in polynomial time.

Figures (17)

• Figure 1: Left: Visualization of the original problem as a graph. Right: Illustration of the constraints of the parity-encoded problem which are represented as three- and four-body plaquettes.
• Figure 2: Cycles in the loop space of $H$ in Example \ref{['examp:cycles']}.
• Figure 3: Left: Constraint $C_1$ in Example \ref{['examp:constraints']}. Right: Illustration of $C_2\in\mathcal{P}(E)$ in Example \ref{['examp:constraints']} which is not a constraint.
• Figure 4: Illustration of the three compiled hypergraphs in Example \ref{['examp:comphypergraphs']}. Left: $\mathcal{P}_{B_1}$. Center: $\mathcal{P}_{B_2}$. Right: $\mathcal{P}_{B_3}$.
• Figure 5: Illustration of the induced graphs of the two labelings in Example \ref{['examp:looplabelings']}.

Theorems & Definitions (55)

• Definition 2.1: Hypergraph
• Definition 2.2: Isomorphic hypergraphs
• Remark 2.3
• Definition 2.4: Walk, path and cycles in graphs
• Definition 2.5
• Definition 2.6
• Example 2.7
• Remark 2.8
• Theorem 2.9
• Definition 2.10: Constraint space of hypergraphs