On the energy landscape of symmetric quantum signal processing
Jiasu Wang, Yulong Dong, Lin Lin
Abstract
Symmetric quantum signal processing provides a parameterized representation of a real polynomial, which can be translated into an efficient quantum circuit for performing a wide range of computational tasks on quantum computers. For a given polynomial <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math>, the parameters (called phase factors) can be obtained by solving an optimization problem. However, the cost function is non-convex, and has a very complex energy landscape with numerous global and local minima. It is therefore surprising that the solution can be robustly obtained in practice, starting from a fixed initial guess <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi mathvariant="normal">&#x03A6;</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math> that contains no information of the input polynomial. To investigate this phenomenon, we first explicitly characterize all the global minima of the cost function. We then prove that one particular global minimum (called the maximal solution) belongs to a neighborhood of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi mathvariant="normal">&#x03A6;</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math>, on which the cost function is strongly convex under the condition <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow><mml:mo symmetric="true">&#x2016;</mml:mo><mml:mi>f</mml:mi><mml:mo symmetric="true">&#x2016;</mml:mo></mml:mrow></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">&#x221E;</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>d</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="normal">e</mml:mi><mml:mi mathvariant="normal">g</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. Our result provides a partial explanation of the aforementioned success of optimization algorithms.