Warm-Started QAOA with Custom Mixers Provably Converges and Computationally Beats Goemans-Williamson&amp;apos;s Max-Cut at Low Circuit Depths

We generalize the Quantum Approximate Optimization Algorithm (QAOA) of Farhi et al. (2014) to allow for arbitrary separable initial states with corresponding mixers such that the starting state is the most excited state of the mixing Hamiltonian. We demonstrate this version of QAOA, which we call <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi><mml:mi>A</mml:mi><mml:mi>O</mml:mi><mml:mi>A</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mi>w</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>m</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi></mml:math>, by simulating Max-Cut on weighted graphs. We initialize the starting state as a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>w</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>m</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>t</mml:mi></mml:math> using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn></mml:math>-dimensional approximations obtained using randomized projections of solutions to Max-Cut's semi-definite program, and define a warm-start dependent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>c</mml:mi><mml:mi>u</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>m</mml:mi><mml:mi>m</mml:mi><mml:mi>i</mml:mi><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi></mml:math>. We show that these warm-starts initialize the QAOA circuit with constant-factor approximations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0.658</mml:mn></mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math>-dimensional and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0.585</mml:mn></mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn></mml:math>-dimensional warm-starts for graphs with non-negative edge weights, improving upon previously known trivial (i.e., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0.5</mml:mn></mml:math> for standard initialization) worst-case bounds at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>. These factors in fact lower bound the approximation achieved for Max-Cut at higher circuit depths, since we also show that QAOA-warmest with any separable initial state converges to Max-Cut under the adiabatic limit as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo stretchy="false">&amp;#x2192;</mml:mo><mml:mi mathvariant="normal">&amp;#x221E;</mml:mi></mml:math>. However, the choice of warm-starts significantly impacts the rate of convergence to Max-Cut, and we show empirically that our warm-starts achieve a faster convergence compared to existing approaches. Additionally, our numerical simulations show higher quality cuts compared to standard QAOA, the classical Goemans-Williamson algorithm, and a warm-started QAOA without custom mixers for an instance library of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1148</mml:mn></mml:math> graphs (upto <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>11</mml:mn></mml:math> nodes) and depth <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mn>8</mml:mn></mml:math>. We further show that QAOA-warmest outperforms the standard QAOA of Farhi et al. in experiments on current IBM-Q and Quantinuum hardware.