Quantum lower bounds for approximate counting via laurent polynomials

We study quantum algorithms that are given access to trusted and untrusted quantum witnesses. We establish strong limitations of such algorithms, via new techniques based on Laurent polynomials (i.e., polynomials with positive and negative integer exponents). Specifically, we resolve the complexity of approximate counting, the problem of multiplicatively estimating the size of a nonempty set S ⊆ [N], in two natural generalizations of quantum query complexity. Our first result holds in the standard Quantum Merlin-Arthur (QMA) setting, in which a quantum algorithm receives an untrusted quantum witness. We show that, if the algorithm makes T quantum queries to S, and also receives an (untrusted) m-qubit quantum witness, then either [MATH HERE]. This is optimal, matching the straightforward protocols where the witness is either empty, or specifies all the elements of S. As a corollary, this resolves the open problem of giving an oracle separation between SBP, the complexity class that captures approximate counting, and QMA. In our second result, we ask what if, in addition to a membership oracle for S, a quantum algorithm is also given "QSamples"- i.e., copies of the state [MATH HERE] - or even access to a unitary transformation that enables QSampling? We show that, even then, the algorithm needs either [MATH HERE] queries or else [MATH HERE] QSamples or accesses to the unitary. Our lower bounds in both settings make essential use of Laurent polynomials, but in different ways.

[1]  Adam D. Bookatz.  QMA-complete problems , 2012, Quantum Inf. Comput..

[2]  Lov K. Grover,et al.  Creating superpositions that correspond to efficiently integrable probability distributions , 2002, quant-ph/0208112.

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

[4]  Silvio Micali,et al.  The Knowledge Complexity of Interactive Proof Systems , 1989, SIAM J. Comput..

[5]  Gilles Brassard,et al.  Quantum Cryptanalysis of Hash and Claw-Free Functions , 1998, LATIN.

[6]  Justin Thaler,et al.  Hardness Amplification and the Approximate Degree of Constant-Depth Circuits , 2013, ICALP.

[7]  D. Newman.  Rational approximation to | x , 1964 .

[8]  Larry J. Stockmeyer,et al.  On Approximation Algorithms for #P , 1985, SIAM J. Comput..

[9]  Ryan O'Donnell,et al.  Analysis of Boolean Functions , 2014, ArXiv.

[10]  Chris Marriott,et al.  Quantum Arthur–Merlin games , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[11]  Rocco A. Servedio,et al.  Attribute-Efficient Learning and Weight-Degree Tradeoffs for Polynomial Threshold Functions , 2012, COLT.

[12]  H. Buhrman,et al.  Complexity measures and decision tree complexity: a survey , 2002, Theor. Comput. Sci..

[13]  Daniele Micciancio,et al.  Statistical Zero-Knowledge Proofs with Efficient Provers: Lattice Problems and More , 2003, CRYPTO.

[14]  Elliott Ward Cheney,et al.  A Comparison of Uniform Approximations on an Interval and a Finite Subset Thereof , 1966 .

[15]  T. Sanders,et al.  Analysis of Boolean Functions , 2012, ArXiv.

[16]  Scott Aaronson,et al.  Impossibility of succinct quantum proofs for collision-freeness , 2011, Quantum Inf. Comput..

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

[18]  Gilles Brassard,et al.  Strengths and Weaknesses of Quantum Computing , 1997, SIAM J. Comput..

[19]  N. K. Vereschchagin.  On the power of PP , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[20]  Yongmei Shi.  Approximating linear restrictions of Boolean functions , 2002 .

[21]  Justin Thaler,et al.  Dual lower bounds for approximate degree and Markov-Bernstein inequalities , 2013, Inf. Comput..

[22]  Hartmut Klauck,et al.  Quantum and classical strong direct product theorems and optimal time-space tradeoffs , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[23]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, WG.

[24]  Christian Glaßer,et al.  Error-Bounded Probabilistic Computations between MA and AM , 2003, MFCS.

[25]  Scott Aaronson,et al.  Quantum lower bound for the collision problem , 2001, STOC '02.

[26]  Scott Aaronson,et al.  Quantum lower bounds for the collision and the element distinctness problems , 2004, JACM.

[27]  Alexander A. Sherstov,et al.  Vanishing-Error Approximate Degree and QMA Complexity , 2019, Electron. Colloquium Comput. Complex..

[28]  Scott Aaronson,et al.  Quantum computing, postselection, and probabilistic polynomial-time , 2004, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[29]  Amnon Ta-Shma,et al.  Adiabatic quantum state generation and statistical zero knowledge , 2003, STOC '03.

[30]  Karl Zeller,et al.  Schwankung von Polynomen zwischen Gitterpunkten , 1964 .

[31]  Nikolai K. Vereshchagin.  On The Power of PP , 1992, Computational Complexity Conference.

[32]  T. J. Rivlin,et al.  The growth of polynomials bounded at equally spaced points , 1992 .

[33]  Greg Kuperberg,et al.  How Hard Is It to Approximate the Jones Polynomial? , 2009, Theory Comput..

[34]  Felix Wu,et al.  The quantum query complexity of approximating the median and related statistics , 1998, STOC '99.

[35]  Vinod Vaikuntanathan,et al.  Noninteractive Statistical Zero-Knowledge Proofs for Lattice Problems , 2008, CRYPTO.

[36]  Martin E. Dyer,et al.  A random polynomial-time algorithm for approximating the volume of convex bodies , 1991, JACM.

[37]  Gilles Brassard,et al.  Quantum Counting , 1998, ICALP.

[38]  Ramamohan Paturi,et al.  On the degree of polynomials that approximate symmetric Boolean functions (preliminary version) , 1992, STOC '92.

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

[40]  Ronald de Wolf,et al.  Rational approximations and quantum algorithms with postselection , 2015, Quantum Inf. Comput..

[41]  Mark Zhandry,et al.  How to Construct Quantum Random Functions , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[42]  Eric Vigoda,et al.  A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries , 2001, STOC '01.

[43]  Oded Goldreich,et al.  On the Limits of Nonapproximability of Lattice Problems , 2000, J. Comput. Syst. Sci..

[44]  Ansis Rosmanis,et al.  Tight Quantum Lower Bound for Approximate Counting with Quantum States , 2020, 2002.06879.

[45]  Scott Aaronson,et al.  Limitations of quantum advice and one-way communication , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[46]  William Kretschmer.  Lower Bounding the AND-OR Tree via Symmetrization , 2019, ArXiv.

[47]  Peter Bro Miltersen,et al.  Derandomizing Arthur–Merlin Games using Hitting Sets , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[48]  S. Lloyd,et al.  Quantum principal component analysis , 2013, Nature Physics.

[49]  Eyal Kushilevitz,et al.  A perfect zero-knowledge proof system for a problem equivalent to the discrete logarithm , 1993, Journal of Cryptology.

[50]  Robert Spalek,et al.  A Dual Polynomial for OR , 2008, ArXiv.

[51]  Scott Aaronson,et al.  Quantum Approximate Counting, Simplified , 2019, SOSA.

[52]  Shachar Lovett,et al.  Rectangles Are Nonnegative Juntas , 2015, SIAM J. Comput..

[53]  Justin Thaler,et al.  Lower Bounds for the Approximate Degree of Block-Composed Functions , 2014, Electron. Colloquium Comput. Complex..

[54]  Ran Raz,et al.  On the power of quantum proofs , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[55]  Maris Ozols,et al.  Hamiltonian simulation with optimal sample complexity , 2016, npj Quantum Information.

[56]  Alexander A. Sherstov.  Optimal bounds for sign-representing the intersection of two halfspaces by polynomials , 2009, STOC '10.

[57]  G. Brassard,et al.  Quantum Amplitude Amplification and Estimation , 2000, quant-ph/0005055.

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

[59]  Mark Zhandry,et al.  Quantum Lightning Never Strikes the Same State Twice. Or: Quantum Money from Cryptographic Assumptions , 2017, Journal of Cryptology.

[60]  Eric Vigoda,et al.  A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries , 2004, JACM.