Quantum algorithms and approximating polynomials for composed functions with shared inputs

<jats:p>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).</jats:p>

[1]  Ronald de Wolf,et al.  Bounds for small-error and zero-error quantum algorithms , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[2]  Rocco A. Servedio,et al.  Learning DNF in time 2Õ(n1/3) , 2004, J. Comput. Syst. Sci..

[3]  Rahul Santhanam,et al.  On the Limits of Sparsification , 2012, ICALP.

[4]  Alexander A. Razborov,et al.  The Sign-Rank of AC0 , 2010, SIAM J. Comput..

[5]  Peter L. Bartlett,et al.  On efficient agnostic learning of linear combinations of basis functions , 1995, COLT '95.

[6]  Andrew M. Childs,et al.  Discrete-Query Quantum Algorithm for NAND Trees , 2009, Theory Comput..

[7]  Ben Reichardt,et al.  Reflections for quantum query algorithms , 2010, SODA '11.

[8]  Andris Ambainis,et al.  Any AND-OR Formula of Size N Can Be Evaluated in Time N1/2+o(1) on a Quantum Computer , 2010, SIAM J. Comput..

[9]  Linda Sellie,et al.  Toward efficient agnostic learning , 1992, Annual Conference Computational Learning Theory.

[10]  Alexander A. Sherstov.  Algorithmic polynomials , 2018, Electron. Colloquium Comput. Complex..

[11]  Gil Cohen,et al.  The Complexity of DNF of Parities , 2016, Electron. Colloquium Comput. Complex..

[12]  Adam Bouland,et al.  On the Power of Statistical Zero Knowledge , 2016, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[13]  Ronald de Wolf,et al.  Robust Polynomials and Quantum Algorithms , 2003, Theory of Computing Systems.

[14]  Alexander A. Sherstov.  Making polynomials robust to noise , 2012, Symposium on the Theory of Computing.

[15]  Justin Thaler,et al.  The polynomial method strikes back: tight quantum query bounds via dual polynomials , 2017, Electron. Colloquium Comput. Complex..

[16]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[17]  Avi Wigderson,et al.  Linear-Size Constant-Depth Polylog-Treshold Circuits , 1991, Inf. Process. Lett..

[18]  Rocco A. Servedio,et al.  Agnostically learning halfspaces , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[19]  Alexander A. Sherstov.  The Intersection of Two Halfspaces Has High Threshold Degree , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[20]  Andrew M. Childs,et al.  The Quantum Query Complexity of Read-Many Formulas , 2012, ESA.

[21]  Ronald de Wolf,et al.  Quantum lower bounds by polynomials , 2001, JACM.

[22]  Mark Bun,et al.  Guest Column , 2020, SIGACT News.

[23]  Alexander A. Sherstov.  Strong direct product theorems for quantum communication and query complexity , 2010, STOC '11.

[24]  Justin Thaler,et al.  Quantum algorithms and approximating polynomials for composed functions with shared inputs , 2018, Electron. Colloquium Comput. Complex..

[25]  Edward Farhi,et al.  A Quantum Algorithm for the Hamiltonian NAND Tree , 2008, Theory Comput..

[26]  Troy Lee,et al.  Lower Bounds in Communication Complexity , 2009, Found. Trends Theor. Comput. Sci..

[27]  Russell Impagliazzo,et al.  The Complexity of Satisfiability of Small Depth Circuits , 2009, IWPEC.

[28]  Alexander A. Sherstov.  The Pattern Matrix Method , 2009, SIAM J. Comput..

[29]  Leslie G. Valiant,et al.  A theory of the learnable , 1984, CACM.

[30]  Jehoshua Bruck,et al.  Polynomial threshold functions, AC functions and spectrum norms , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[31]  Alon Rosen,et al.  Candidate weak pseudorandom functions in AC0 ○ MOD2 , 2014, ITCS.

[32]  Avishay Tal,et al.  Formula lower bounds via the quantum method , 2017, STOC.

[33]  Michal Koucký,et al.  Circuit Complexity of Regular Languages , 2007, Theory of Computing Systems.

[34]  Troy Lee,et al.  Negative weights make adversaries stronger , 2007, STOC '07.

[35]  Jaikumar Radhakrishnan,et al.  Deterministic restrictions in circuit complexity , 1996, STOC '96.

[36]  Michael Ben-Or,et al.  A theorem on probabilistic constant depth Computations , 1984, STOC '84.