We present a simple method based on a variant of Hölder’s inequality to lower-bound the trace norm of Boolean matrices. As the main result, we obtain an exponential separation between the randomized decision tree depth and the spectral norm (i.e. the Fourier $$L_1$$ -norm) of a Boolean function. This answers an open question of Cheung, Hatami, Hosseini, and Shirley (CCC 2023). As immediate consequences, we obtain the following results.
We present a simple method based on a variant of Hölder’s inequality to lower-bound the trace norm of Boolean matrices. As the main result, we obtain an exponential separation between the randomized decision tree depth and the spectral norm (i.e. the Fourier L1 -norm) of a Boolean function. This answers an open question of Cheung, Hatami, Hosseini, and Shirley (CCC 2023). As immediate consequences, we obtain the following results.
We present a new method for deriving lower bounds to the expected number of queries made by noisy decision trees computing Boolean functions. The new method has the feature that expectations are taken with respect to a uniformly distributed random input, as well as with respect to the random noise, thus yielding stronger lower bounds. It also applies to many more functions than do previous results. The method yields a simple proof of the result (previously established by Reischuck and Schmeltz) that almost all Boolean functions of n arguments require Ω(n log n) queries and strengthens this bound from the worst-case over inputs to the average over inputs. The method also yields bounds for specific Boolean functions in terms of their spectra (their Fourier transforms). The simplest instance of this spectral bound yields the result (previously established by Feige, Peleg, Raghavan and Upfal) that the parity function of n arguments requires Ω(n log n) queries, and again strengthens this bound from the worst-case over inputs to the average over inputs. In its full generality, the spectral bound applies to the "highly resilient" functions introduced by Chor, Friedman, Goldreigh, Hastad, Rudich and Smolensky, and it yields non-linear lower bounds whenever the resiliency is asymptotic to the number of arguments.
We present a new method for deriving lower bounds to the expected number of queries made by noisy decision trees computing Boolean functions. The new method has the feature that expectations are taken with respect to a uniformly distributed random input, as well as with respect to the random noise, thus yielding stronger lower bounds. It also applies to many more functions than do previous results. The method yields a simple proof of the result (previously established by Reischuk and Schmeltz) that almost all Boolean functions of n arguments require $\Me(n \log n)$ queries, and strengthens this bound from the worst-case over inputs to the average over inputs. The method also yields bounds for specific Boolean functions in terms of their spectra (their Fourier transforms). The simplest instance of this spectral bound yields the result (previously established by Feige, Peleg, Raghavan, and Upfal) that the parity function of n arguments requires $\Me(n \log n)$ queries and again strengthens this bound from the worst-case over inputs to the average over inputs. In its full generality, the spectral bound applies to the "highly resilient" functions introduced by Chor, Friedman, Goldreich, Hastad, Rudich, and Smolensky, and it yields nonlinear lower bounds whenever the resiliency is asymptotic to the number of arguments.
We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric d×d matrices A1,…,An each with ∥Ai∥op≤1 and rank at most n/log3n, one can efficiently find ±1 signs x1,…,xn such that their signed sum has spectral norm ∥∑i=1nxiAi∥op=O(n). This result also implies a logn−Ω(loglogn) qubit lower bound for quantum random access codes encoding n classical bits with advantage ≫1/n. Our proof uses the recent refinement of the non-commutative Khintchine inequality in [Bandeira, Boedihardjo, van Handel, 2022] for random matrices with correlated Gaussian entries.
We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order ℓ≥1 sum to at most cℓ(ℓd)(1+logn)ℓ−1, where n is the number of variables, d is the tree depth, and c>0 is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal (arxiv 2019; FOCS 2020). The bounds prior to our work degraded rapidly with ℓ, becoming trivial already at ℓ=d. As an application, we obtain, for every integer k≥1, a partial Boolean function on n bits that has bounded-error quantum query complexity at most k and randomized query complexity Ω~(n1−2k1). This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis (STOC 2015) and Bravyi, Gosset, Grier, and Schaeffer (2021). Prior to our work, the best known separation was polynomially weaker: O(1) versus Ω(n2/3−ε) for any ε>0 (Tal, FOCS 2020). As another application, we obtain an essentially optimal separation of O(logn) versus Ω(n1−ε) for bounded-error quantum versus randomized communication complexity, for any ε>0. The best previous separation was polynomially weaker: O(logn) versus Ω(n2/3−ε) (implicit in Tal, FOCS 2020).
In a breakthrough result, Razborov (2003) gave optimal lower bounds on the communication complexity of every function f of the form f(x,y)=D(|x AND y|) for some D:{0,1,...,n}->{0,1}, in the bounded-error quantum model with and without prior entanglement. This was proved by the multidimensional discrepancy method. We give an entirely different proof of Razborov's result, using the original, one-dimensional discrepancy method. This refutes the commonly held intuition (Razborov 2003) that the original discrepancy method fails for functions such as DISJOINTNESS. More importantly, our communication lower bounds hold for a much broader class of functions for which no methods were available. Namely, fix an arbitrary function f:{0,1}n/4->{0,1} and let A be the Boolean matrix whose columns are each an application of f to some subset of the variables x1,x2,...,xn. We prove that the communication complexity of A in the bounded-error quantum model with and without prior entanglement is Omega(d), where d is the approximate degree of f. From this result, Razborov's lower bounds follow easily. Our result also establishes a large new class of total Boolean functions whose quantum communication complexity (regardless of prior entanglement) is at best polynomially smaller than their classical complexity. Our proof method is a novel combination of two ingredients. The first is a certain equivalence of approximation and orthogonality in Euclidean n-space, which follows by linear-programming duality. The second is a new construction of suitably structured matrices with low spectral norm, the pattern matrices, which we realize using matrix analysis and the Fourier transform over (Z2)n. The method of this paper has recently inspired important progress in multiparty communication complexity.