اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدLower Bounding the AND-OR Tree via Symmetrization
67
0
0.0
(
0
)
تأليف
William Kretschmer
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove a simple, nearly tight lower bound on the approximate degree of the two-level $mathsf{AND}$-$mathsf{OR}$ tree using symmetrization arguments. Specifically, we show that $widetilde{mathrm{deg}}(mathsf{AND}_m circ mathsf{OR}_n) = widetilde{Omega}(sqrt{mn})$. We prove this lower bound via reduction to the $mathsf{OR}$ function through a series of symmetrization steps, in contrast to most other proofs that involve formulating approximate degree as a linear program [BT13, She13, BDBGK18]. Our proof also demonstrates the power of a symmetrization technique involving Laurent polynomials (polynomials with negative exponents) that was previously introduced by Aaronson, Kothari, Kretschmer, and Thaler [AKKT19].
قيم البحث
اقرأ أيضاً
We prove a query complexity lower bound for $mathsf{QMA}$ protocols that solve approximate counting: estimating the size of a set given a membership oracle. This gives rise to an oracle $A$ such that $mathsf{SBP}^A otsubset mathsf{QMA}^A$, resolving
an open problem of Aaronson [2]. Our proof uses the polynomial method to derive a lower bound for the $mathsf{SBQP}$ query complexity of the $mathsf{AND}$ of two approximate counting instances. We use Laurent polynomials as a tool in our proof, showing that the Laurent polynomial method can be useful even for problems involving ordinary polynomials.
We show a nearly quadratic separation between deterministic communication complexity and the logarithm of the partition number, which is essentially optimal. This improves upon a recent power 1.5 separation of Goos, Pitassi, and Watson (FOCS 2015). I
n query complexity, we establish a nearly quadratic separation between deterministic (and even randomized) query complexity and subcube partition complexity, which is also essentially optimal. We also establish a nearly power 1.5 separation between quantum query complexity and subcube partition complexity, the first superlinear separation between the two measures. Lastly, we show a quadratic separation between quantum query complexity and one-sided subcube partition complexity. Our query complexity separations use the recent cheat sheet framework of Aaronson, Ben-David, and the author. Our query functions are built up in stages by alternating function composition with the cheat sheet construction. The communication complexity separation follows from lifting the query separation to communication complexity.
We show that any quantum circuit of treewidth $t$, built from $r$-qubit gates, requires at least $Omega(frac{n^{2}}{2^{O(rcdot t)}cdot log^4 n})$ gates to compute the element distinctness function. Our result generalizes a near-quadratic lower bound
for quantum formula size obtained by Roychowdhury and Vatan [SIAM J. on Computing, 2001]. The proof of our lower bound follows by an extension of Nev{c}iporuks method to the context of quantum circuits of constant treewidth. This extension is made via a combination of techniques from structural graph theory, tensor-network theory, and the connected-component counting method, which is a classic tool in algebraic geometry.
The problem of distinguishing between a random function and a random permutation on a domain of size $N$ is important in theoretical cryptography, where the security of many primitives depend on the problems hardness. We study the quantum query compl
exity of this problem, and show that any quantum algorithm that solves this problem with bounded error must make $Omega(N^{1/5}/log N)$ queries to the input function. Our lower bound proof uses a combination of the Collision Problem lower bound and Ambainiss adversary theorem.
We show that most arithmetic circuit lower bounds and relations between lower bounds naturally fit into the representation-theoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique (Nisan-Wigders
on), the results of Razborov and Smolensky on $AC^0[p]$, multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, Baur-Strassen), the connected components technique (Ben-Or), depth 3 arithmetic circuit lower bounds over finite fields (Grigoriev-Karpinski), lower bounds on permanent versus determinant (Mignon-Ressayre, Landsberg-Manivel-Ressayre), lower bounds on matrix multiplication (B{u}rgisser-Ikenmeyer) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta-Kayal-Kamath-Saptharishi; Agrawal-Vinay; Koiran), matrix rigidity (Valiant) and others. That is, the original proofs, with what is often just a little extra work, already provide representation-theoretic obstructions in the sense of GCT for their respective lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a natural unification and broad generalization of known results. It also shows that the framework of GCT is at least as powerful as known methods, and gives many new proofs-of-concept that GCT can indeed provide significant asymptotic lower bounds. This new viewpoint also opens up the possibility of fruitful two-way interactions between previous results and the new methods of GCT; we provide several concrete suggestions of such interactions. For example, the representation-theoretic viewpoint of GCT naturally provides new properties to consider in the search for new lower bounds.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
