Quantum Discrepancy: A Non-Commutative Version of Combinatorial Discrepancy

Combinatorial Discrepancy: A Tutorial

Notes taken by Karel Král, Martin Balko and Vojta Tůma at REU 2013. Discrepancy theory grew out of the study of irregularities of distributions and number sequences, or, in layman’s terms, trying to be an egalitarian economist. From its number theoretic origins in early nineteenth century, discrepancy theory has developed into a rich and beautiful field over the last three decades. Besides bein...

On the discrepancy of combinatorial rectangles

Let B n denote the family which consists of all subsets S1 × · · · × Sd, where Si ⊆ [n], and Si 6= ∅, for i = 1, . . . , d. We compute the L2–discrepancy of B n and give estimates for the Lp– discrepancy of B n for 1 ≤ p ≤ ∞.

Non-commutative combinatorial algebra

A Discrepancy-Based Proof of Razborov’s Quantum Lower Bounds

In a breakthrough result, Razborov (2003) gave optimal lower bounds on the quantum communication complexity Q1/3( f ) of every function f (x, y) = D(|x ∧ y|), where D : {0, 1, . . . , n} → {0, 1}. Namely, he showed that Q1/3( f ) = Ω ( `1(D) + √ n `0(D) ) , where `0(D), `1(D) ∈ {0, 1, . . . , dn/2e} are the smallest integers such that D is constant in the range [`0(D), n − `1(D)]. This was prov...

Combinatorial Discrepancy for Boxes via the γ2 Norm

The γ2 norm of a realm×n matrix A is the minimum number t such that the column vectors of A are contained in a 0-centered ellipsoid E ⊆ R that in turn is contained in the hypercube [−t, t]. This classical quantity is polynomial-time computable and was proved by the second author and Talwar to approximate the hereditary discrepancy herdiscA as follows: γ2(A)/O(logm) ≤ herdiscA ≤ γ2(A) · O( √ log...