 Award ID(s):
 1845125
 NSFPAR ID:
 10326343
 Date Published:
 Journal Name:
 Quantum
 Volume:
 5
 ISSN:
 2521327X
 Page Range / eLocation ID:
 543
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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We prove two new results about the inability of lowdegree polynomials to uniformly approximate constantdepth circuits, even to slightlybetterthantrivial error. First, we prove a tight Omega~(n^{1/2}) lower bound on the threshold degree of the SURJECTIVITY function on n variables. This matches the best known threshold degree bound for any AC^0 function, previously exhibited by a much more complicated circuit of larger depth (Sherstov, FOCS 2015). Our result also extends to a 2^{Omega~(n^{1/2})} lower bound on the signrank of an AC^0 function, improving on the previous best bound of 2^{Omega(n^{2/5})} (Bun and Thaler, ICALP 2016). Second, for any delta>0, we exhibit a function f : {1,1}^n > {1,1} that is computed by a circuit of depth O(1/delta) and is hard to approximate by polynomials in the following sense: f cannot be uniformly approximated to error epsilon=12^{Omega(n^{1delta})}, even by polynomials of degree n^{1delta}. Our recent prior work (Bun and Thaler, FOCS 2017) proved a similar lower bound, but which held only for error epsilon=1/3. Our result implies 2^{Omega(n^{1delta})} lower bounds on the complexity of AC^0 under a variety of basic measures such as discrepancy, margin complexity, and threshold weight. This nearly matches the trivial upper bound of 2^{O(n)} that holds for every function. The previous best lower bound on AC^0 for these measures was 2^{Omega(n^{1/2})} (Sherstov, FOCS 2015). Additional applications in learning theory, communication complexity, and cryptography are described.more » « less

Abstract We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in
and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathsf {Quasi}\text {}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$ $\mathrm{Quasi}\mathrm{NP}=\mathrm{NTIME}\left[{n}^{{\left(\mathrm{log}n\right)}^{O\left(1\right)}}\right]$ , by showing that$\mathcal { C}$ $C$ admits nontrivial satisfiability and/or$\mathcal { C}$ $C$# SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a nontrivial# SAT algorithm for a circuit class . Say that a symmetric Boolean function${\mathcal C}$ $C$f (x _{1},…,x _{n}) issparse if it outputs 1 onO (1) values of . We show that for every sparse${\sum }_{i} x_{i}$ ${\sum}_{i}{x}_{i}$f , and for all “typical” , faster$\mathcal { C}$ $C$# SAT algorithms for circuits imply lower bounds against the circuit class$\mathcal { C}$ $C$ , which may be$f \circ \mathcal { C}$ $f\circ C$stronger than itself. In particular:$\mathcal { C}$ $C$# SAT algorithms forn ^{k}size circuits running in 2^{n}/$\mathcal { C}$ $C$n ^{k}time (for allk ) implyN E X P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$# SAT algorithms for size$2^{n^{{\varepsilon }}}$ ${2}^{{n}^{\epsilon}}$ circuits running in$\mathcal { C}$ $C$ time (for some$2^{nn^{{\varepsilon }}}$ ${2}^{n{n}^{\epsilon}}$ε > 0) implyQ u a s i N P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$Applying
# SAT algorithms from the literature, one immediate corollary of our results is thatQ u a s i N P does not haveE M A J ∘A C C ^{0}∘T H R circuits of polynomial size, whereE M A J is the “exact majority” function, improving previous lower bounds againstA C C ^{0}[Williams JACM’14] andA C C ^{0}∘T H R [Williams STOC’14], [MurrayWilliams STOC’18]. This is the first nontrivial lower bound against such a circuit class. 
Abstract We prove that
depth local random quantum circuits with two qudit nearestneighbor gates on a$${{\,\textrm{poly}\,}}(t) \cdot n^{1/D}$$ $\phantom{\rule{0ex}{0ex}}\text{poly}\phantom{\rule{0ex}{0ex}}\left(t\right)\xb7{n}^{1/D}$D dimensional lattice withn qudits are approximatet designs in various measures. These include the “monomial” measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was due to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for$${{\,\textrm{poly}\,}}(t)\cdot n$$ $\phantom{\rule{0ex}{0ex}}\text{poly}\phantom{\rule{0ex}{0ex}}\left(t\right)\xb7n$ . We also improve the “scrambling” and “decoupling” bounds for spatially local random circuits due to Brown and Fawzi (Scrambling speed of random quantum circuits, 2012). One consequence of our result is that assuming the polynomial hierarchy ($$D=1$$ $D=1$ ) is infinite and that certain counting problems are$${{\,\mathrm{\textsf{PH}}\,}}$$ $\phantom{\rule{0ex}{0ex}}\mathrm{PH}\phantom{\rule{0ex}{0ex}}$ hard “on average”, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constantdepth quantum circuits was known to be hard for classical computers under these assumptions. However the standard strategy for extending this hardness result to approximate sampling requires the quantum circuits to have a property called “anticoncentration”, meaning roughly that the output has nearmaximal entropy. Unitary 2designs have the desired anticoncentration property. Our result improves the required depth for this level of anticoncentration from linear depth to a sublinear value, depending on the geometry of the interactions. This is relevant to a recent experiment by the Google Quantum AI group to perform such a sampling task with 53 qubits on a twodimensional lattice (Arute in Nature 574(7779):505–510, 2019; Boixo et al. in Nate Phys 14(6):595–600, 2018) (and related experiments by USTC), and confirms their conjecture that$$\#{\textsf{P}}$$ $\#P$ depth suffices for anticoncentration. The proof is based on a previous construction of$$O(\sqrt{n})$$ $O\left(\sqrt{n}\right)$t designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasiorthogonality of permutation operators developed by Brandão et al. (2016). Different versions of the approximate design condition correspond to different norms, and part of our contribution is to introduce the norm corresponding to anticoncentration and to establish equivalence between these various norms for lowdepth circuits. For random circuits with longrange gates, we use different methods to show that anticoncentration happens at circuit size corresponding to depth$$O(n\ln ^2 n)$$ $O\left(n{ln}^{2}n\right)$ . We also show a lower bound of$$O(\ln ^3 n)$$ $O\left({ln}^{3}n\right)$ for the size of such circuit in this case. We also prove that anticoncentration is possible in depth$$\Omega (n \ln n)$$ $\Omega (nlnn)$ (size$$O(\ln n \ln \ln n)$$ $O(lnnlnlnn)$ ) using a different model.$$O(n \ln n \ln \ln n)$$ $O(nlnnlnlnn)$ 
null (Ed.)The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. The approximate degree of f is known to be a lower bound on the quantum query complexity of f (Beals et al., FOCS 1998 and J. ACM 2001). We find tight or nearly tight bounds on the approximate degree and quantum query complexities of several basic functions. Specifically, we show the following. kDistinctness: For any constant k, the approximate degree and quantum query complexity of the kdistinctness function is Ω(n3/4−1/(2k)). This is nearly tight for large k, as Belovs (FOCS 2012) has shown that for any constant k, the approximate degree and quantum query complexity of kdistinctness is O(n3/4−1/(2k+2−4)). Image size testing: The approximate degree and quantum query complexity of testing the size of the image of a function [n]→[n] is Ω~(n1/2). This proves a conjecture of Ambainis et al. (SODA 2016), and it implies tight lower bounds on the approximate degree and quantum query complexity of the following natural problems. kJunta testing: A tight Ω~(k1/2) lower bound for kjunta testing, answering the main open question of Ambainis et al. (SODA 2016). Statistical distance from uniform: A tight Ω~(n1/2) lower bound for approximating the statistical distance of a distribution from uniform, answering the main question left open by Bravyi et al. (STACS 2010 and IEEE Trans. Inf. Theory 2011). Shannon entropy: A tight Ω~(n1/2) lower bound for approximating Shannon entropy up to a certain additive constant, answering a question of Li and Wu (2017). Surjectivity: The approximate degree of the surjectivity function is Ω~(n3/4). The best prior lower bound was Ω(n2/3). Our result matches an upper bound of O~(n3/4) due to Sherstov (STOC 2018), which we reprove using different techniques. The quantum query complexity of this function is known to be Θ(n) (Beame and Machmouchi, Quantum Inf. Comput. 2012 and Sherstov, FOCS 2015). Our upper bound for surjectivity introduces new techniques for approximating Boolean functions by lowdegree polynomials. Our lower bounds are proved by significantly refining techniques recently introduced by Bun and Thaler (FOCS 2017).more » « less

Recently, Bravyi, Gosset, and Konig (Science, 2018) exhibited a search problem called the 2D Hidden Linear Function (2D HLF) problem that can be solved exactly by a constantdepth quantum circuit using bounded fanin gates (or QNC^0 circuits), but cannot be solved by any constantdepth classical circuit using bounded fanin AND, OR, and NOT gates (or NC^0 circuits). In other words, they exhibited a search problem in QNC^0 that is not in NC^0. We strengthen their result by proving that the 2D HLF problem is not contained in AC^0, the class of classical, polynomialsize, constantdepth circuits over the gate set of unbounded fanin AND and OR gates, and NOT gates. We also supplement this worstcase lower bound with an averagecase result: There exists a simple distribution under which any AC^0 circuit (even of nearly exponential size) has exponentially small correlation with the 2D HLF problem. Our results are shown by constructing a new problem in QNC^0, which we call the Parity Halving Problem, which is easier to work with. We prove our AC^0 lower bounds for this problem, and then show that it reduces to the 2D HLF problem.more » « less