Quantum algorithms for escaping from saddle points

We initiate the study of quantum algorithms for escaping from saddle points with provable guarantee. Given a function<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow></mml:math>, our quantum algorithm outputs an<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϵ</mml:mi></mml:math>-approximate second-order stationary point using<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>O</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>log</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>ϵ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1.75</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>queries to the quantum evaluation oracle (i.e., the zeroth-order oracle). Compared to the classical state-of-the-art algorithm by Jin et al. with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi>O</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>log</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>6</mml:mn></mml:mrow></mml:msup><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>ϵ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1.75</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>queries to the gradient oracle (i.e., the first-order oracle), our quantum algorithm is polynomially better in terms of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>n</mml:mi></mml:math>and matches its complexity in terms of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>ϵ</mml:mi></mml:math>. Technically, our main contribution is the idea of replacing the classical perturbations in gradient descent methods by simulating quantum wave equations, which constitutes the improvement in the quantum query complexity with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>n</mml:mi></mml:math>factors for escaping from saddle points. We also show how to use a quantum gradient computation algorithm due to Jordan to replace the classical gradient queries by quantum evaluation queries with the same complexity. Finally, we also perform numerical experiments that support our theoretical findings.