Promise of Graph Sparsification and Decomposition for Noise Reduction in QAOA: Analysis for Trapped-Ion Compilations

TL;DR

This work provides the first provable guarantees using sparsification and decomposition to improve quantum noise resilience and reduce quantum circuit complexity for edge-by-edge QAOA compilations.

Abstract

We develop new approximate compilation schemes that significantly reduce the expense of compiling the Quantum Approximate Optimization Algorithm (QAOA) for solving the Max-Cut problem. Our main focus is on compilation with trapped-ion simulators using Pauli-$X$ operations and all-to-all Ising Hamiltonian $H_\text{Ising}$ evolution generated by Molmer-Sorensen or optical dipole force interactions, though some of our results also apply to standard gate-based compilations. Our results are based on principles of graph sparsification and decomposition; the former reduces the number of edges in a graph while maintaining its cut structure, while the latter breaks a weighted graph into a small number of unweighted graphs. Though these techniques have been used as heuristics in various hybrid quantum algorithms, there have been no guarantees on their performance, to the best of our knowledge. This work provides the first provable guarantees using sparsification and decomposition to improve quantum noise resilience and reduce quantum circuit complexity. For quantum hardware that uses edge-by-edge QAOA compilations, sparsification leads to a direct reduction in circuit complexity. For trapped-ion quantum simulators implementing all-to-all $H_\text{Ising}$ pulses, we show that for a $(1-ε)$ factor loss in the Max-Cut approximation ($ε>0)$, our compilations improve the (worst-case) number of $H_\text{Ising}$ pulses from $O(n^2)$ to $O(n\log(n/ε))$ and the (worst-case) number of Pauli-$X$ bit flips from $O(n^2)$ to $O\left(\frac{n\log(n/ε)}{ε^2}\right)$ for $n$-node graphs. We demonstrate significant reductions in noise are obtained in our new compilation approaches using theory and numerical calculations for trapped-ion hardware. We anticipate these approximate compilation techniques will be useful tools in a variety of future quantum computing experiments.

Figures (9)

• Figure 1: Left to right: graph sparsification, with the original unweighted graph $G$ (left, 397 edges) and two weighted graph sparsifiers $H_1$ (middle, 136 edges) and $H_2$ (right, 48 edges) of $G$ (all graphs have the same number of vertices). The Max-Cut in $H_1$ gives a cut in $G$ with cut value that is $90\%$ of the Max-Cut in $G$, i.e., is a $0.9$-approximation to Max-Cut in $G$. The Max-Cut in $H_2$ is a $0.82$-approximation to Max-Cut in $G$.
• Figure 2: The graph decomposition process: The weighted graph $G$ on the left can be written as the weighted sum $G_1 + 2 G_2$ of two unweighted graphs $G_1, G_2$ on the right.
• Figure 3: Reduction in $N(H_{\textsc{Ising}})$ (total number of $H_{\mathrm{Ising}}$ pulses) vs Max-Cut approximation for various runs of our experiment on weighted graphs in MQLib, colored by parameter $q$. Each data point is a single run. Points on the lower right have high reductions and high Max-Cut approximation.
• Figure 4: Reduction in $N(\textsc{Total}_{\textsc{Ops}})$ (total number of $H_{\mathrm{Ising}}$ pulses and bit flip gates) vs Max-Cut approximation for various runs of our experiment on weighted graphs in MQLib. Each data point is a single run. Points on the lower right have high reductions and high Max-Cut approximation.
• Figure 5: Reduction in the total length or time of pulses vs Max-Cut approximation for various runs of our experiment on weighted graphs in MQLib. Each data point is a single run.

Theorems & Definitions (10)

• Lemma 1
• Definition 1: Graph compilation
• Lemma 2: rajakumar_generating_2022, Lemma 1
• Theorem 1: rajakumar_generating_2022 Theorems 5, 6
• Lemma 3
• Theorem 2: batson_twice-ramanujan_2014, Theorem 1
• Theorem 3
• Theorem 4
• proof : Proof of Theorem \ref{['thm: faster-union-of-stars-weighted-graphs']}.
• proof : Proof of Theorem \ref{['thm: second-graph-coupling-number']}