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\title{Supplementary Information: Nonlinear Quantum Optimization Algorithms via Parallel and Dense Ising Models}

\author{Taylor L. Patti}
\email[]{taylorpatti@g.harvard.edu}
\affiliation{Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA}
\affiliation{NVIDIA, Santa Clara, California 95051, USA}

\author{Jean Kossaifi}
\affiliation{NVIDIA, Santa Clara, California 95051, USA}

\author{Anima Anandkumar}
\affiliation{Department of Computing + Mathematical Sciences (CMS), California Institute of Technology (Caltech), Pasadena, CA 91125 USA}
\affiliation{NVIDIA, Santa Clara, California 95051, USA}

\author{Susanne F. Yelin}
\affiliation{Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA}

\maketitle

As an initialization procedure for both the NP-VQA and ND-VQA methods, we restrict the amount of entanglement between groups of qubits and the untrained weights. We heuristically note that this leads to mildly improved performance and hypothesize that it may be due to the deleterious effects of random entanglement on barren plateaus and problem convexity, as explained in the main text. In particular, for the $n=100$, $512$ ($n=8$, $20$) cases, we pre-train up to five blocks of two qubits each by minimizing their entanglement with the remainder of the circuit until it is less than $15\%$ ($0.05\%$) of its maximal value, which can be done experimentally through state tomography or balanced single-qubit measurements, or until some maximal pre-train step limit is exceeded. Judiciously setting $\hat{\theta}$ is an alternative method that also satisfies our constraint. Despite this initial entanglement minimization, the qubits become rapidly entangled during the learning process, as illustrated in Fig. 3a.

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