Complex vector gain-based annealer for minimizing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>X</mml:mi> <mml:mi>Y</mml:mi> </mml:math> Hamiltonians

. We present the complex vector gain-based annealer (CVGA), an analog computing platform designed to overcome energy barriers in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>X</a:mi> <a:mi>Y</a:mi> </a:math> Hamiltonians through a higher-dimensional representation. Traditional gain-based solvers using optical or photonic hardware typically represent each <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>X</c:mi> <c:mi>Y</c:mi> </c:math> spin with a single complex field. These solvers often struggle with large energy barriers in complex landscapes, leading to relaxation into excited states. CVGA addresses these limitations by using two complex fields to represent each <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>X</e:mi> <e:mi>Y</e:mi> </e:math> spin and dynamically evolving the energy landscape through time-dependent annealing. Operating in a higher-dimensional space, CVGA bridges energy barriers in this expanded space during the continuous phase evolution, thus avoiding entrapment in local minima. We introduce several graph structures that pose challenges for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>X</g:mi> <g:mi>Y</g:mi> </g:math> minimization and use them to benchmark CVGA against single-dimension <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>X</i:mi> <i:mi>Y</i:mi> </i:math> solvers, highlighting the benefits of higher-dimensional operation.