Quantum algorithms and approximating polynomials for composed functions with shared inputs

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math>be an<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>-bit Boolean function and consider an<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-bit function<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math>obtained by applying<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math>to conjunctions of possibly overlapping subsets of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>variables. If<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math>has quantum query complexity<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, we give an algorithm for evaluating<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math>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:msqrt><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>⋅</mml:mo><mml:mi>n</mml:mi></mml:msqrt><mml:mo stretchy="false">)</mml:mo></mml:math>quantum queries. This improves on the bound of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>Q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>⋅</mml:mo><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt><mml:mo stretchy="false">)</mml:mo></mml:math>that follows by treating each conjunction independently, and our bound is tight for worst-case choices of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math>. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math>.By recursively applying our composition theorems, we obtain a nearly optimal<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>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>upper bound on the quantum query complexity and approximate degree of linear-size depth-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>AC<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi /><mml:mn>0</mml:mn></mml:msup></mml:math>circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi /><mml:mn>0</mml:mn></mml:msup></mml:math>circuits.As an additional consequence, we show that AC<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi /><mml:mn>0</mml:mn></mml:msup><mml:mo>∘</mml:mo><mml:mo>⊕</mml:mo></mml:math>circuits of depth<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>require size<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>≥</mml:mo><mml:mi>ω</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>to compute the Inner Product function even on average. The previous best size lower bound was<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mn>4</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>and only held in the worst case (Cheraghchi et al., JCSS 2018).