Multi-controlled Phase Gate Synthesis with ZX-calculus applied to Neutral Atom Hardware

TL;DR

This work introduces a ZX-calculus–based method for synthesizing quantum circuits toward single-qubit gates and arbitrary-sized multi-controlled phase gates $C_nP(\varphi)$ to leverage neutral-atom hardware capabilities. By encoding circuits as graph-like ZX-diagrams and applying phase-gadget decompositions, the authors derive an exact extraction procedure that identifies and implements MCP gates even when not explicitly present in the original circuit, preserving gflow throughout. The approach is specialized to neutral-atom platforms, where native $C_nP$ gates and global single-qubit operations can dramatically reduce execution time, as demonstrated by benchmarks against Qiskit. The results show notable reductions in total execution time for several circuit families, while also highlighting cases where the ZX strategy may be less beneficial and where gadget insertions can incur higher worst-case costs. Overall, the paper offers a principled, exact pathway to hardware-aware synthesis with MCP gates and motivates further exploration of heuristics and alternative representations for MCP optimization across quantum architectures.

Abstract

Quantum circuit synthesis describes the process of converting arbitrary unitary operations into a gate sequence of a fixed universal gate set, usually defined by the operations native to a given hardware platform. Most current synthesis algorithms are designed to synthesize towards a set of single qubit rotations and an additional entangling two qubit gate, such as CX, CZ, or the Molmer Sorensen gate. However, with the emergence of neutral atom based hardware and their native support for gates with more than two qubits, synthesis approaches tailored to these new gate sets become necessary. In this work, we present an approach to synthesize multi controlled phase gates using ZX calculus. By representing quantum circuits as graph like ZX diagrams, one can utilize the distinct graph structure of diagonal gates to identify multi controlled phase gates inherently present in some quantum circuits even if none were explicitly defined in the original circuit. We evaluate the approach on a wide range of benchmark circuits and compare them to the standard Qiskit synthesis regarding its circuit execution time for neutral atom based hardware with native support of multi controlled gates. Our results show possible advantages for current state of the art hardware and represent the first exact synthesis algorithm supporting arbitrary sized multi controlled phase gates.

Figures (3)

• Figure 1: Translation of a two-qubit Grover search into a graph like ZX-diagram. Gates are replaced by their ZX-calculus counterpart and the diagram is made graph-like by repeated application of ($f$) and ($h$).
• Figure 2: Illustration of the neutral atom gate capabilities and the process of synthesizing and scheduling quantum circuits to the hardware. (a) Native (multi-) controlled phase gates ($\mathrm{C}_{n} \mathrm{P}(\varphi)$), here shown for three qubits. (b) Global single-qubit rotations in the XY-plane. (c) Individually addressable Z-rotations ($\mathrm{R}_{z}(\gamma)$). (d) Synthesis to alternating single- and multi-qubit layers: First, the synthesis of multi-qubit gates to $\mathrm{C}_{n} \mathrm{P}$ gates. Second, the synthesis and scheduling of single-qubit gates into global XY-rotations and individually addressable Z-rotations according to the transversal decomposition of nottinghamDecomposingRoutingQuantum2023.
• Figure 3: Illustration of the possible combinations and their contribution in \ref{['eq:lemma-n-ueq-l']} for $n=3$. For each possible value of $l=0,\, \dots \, , 3$ the combinations for the possible $k \leq n$ and $j \leq l$ are illustrated as three-circle circles, and their contribution to the sum is computed. The final row shows that the sums of the contributions fulfill the condition of \ref{['lemma:multi-controlled-phase-gate']}.

Theorems & Definitions (8)

• Theorem 1: Multi-controlled phase gates
• proof
• Lemma 1: Insertion of $YZ$ measurements on outputs
• proof
• Corollary 1
• proof
• Lemma 2: Multi-controlled phase gate
• proof