scholarly journals Impossibility of succinct quantum proofs for collision-freeness

2012 ◽  
Vol 12 (1&2) ◽  
pp. 21-28 ◽  
Author(s):  
Scott Aaronson
Keyword(s):  
Lower Bound ◽  
Quantum Algorithm ◽  
Simple Extension ◽  
Class A ◽  
Collision Problem ◽  
Open Question

We show that any quantum algorithm to decide whether a function f[n] \rightarrow [n] is a permutation or far from a permutation\ must make \Omega( n^{1/3}/w) queries to f, even if the algorithm is given a w-qubit quantum witness in support of f being a permutation. This implies that there exists an oracle A such that {SZK}^{A}\not \subset {QMA}^{A}, answering an eight-year-old open question of the author. \ Indeed, we show that relative to some oracle, {SZK} is not in the counting class {A}_0{PP} defined by Vyalyi. The proof is a fairly simple extension of the quantum lower bound for the collision problem.

scholarly journals A quantum lower bound for distinguishing random functions from random permutations

2014 ◽  
Vol 14 (13&14) ◽  
pp. 1089-1097
Author(s):  
Henry Yuen
Keyword(s):  
Lower Bound ◽  
Quantum Algorithm ◽  
Input Function ◽  
Random Permutation ◽  
Query Complexity ◽  
Random Permutations ◽  
Random Functions ◽  
Collision Problem ◽  
Bounded Error ◽  
Quantum Query

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 problem's hardness. We study the quantum query complexity of this problem, and show that any quantum algorithm that solves this problem with bounded error must make $\Omega(N^{1/5}/\polylog N)$ queries to the input function. Our lower bound proof uses a combination of the Collision Problem lower bound and Ambainis's adversary theorem.

Quantum lower bound for recursive Fourier sampling

2003 ◽  
Vol 3 (2) ◽  
pp. 165-174
Author(s):  
S. Aaronson
Keyword(s):  
Lower Bound ◽  
Quantum Algorithm ◽  
Independent Interest ◽  
Classical Algorithm ◽  
Fourier Sampling ◽  
Adversary Method

We revisit the oft-neglected `recursive Fourier sampling' (RFS) problem, introduced by Bernstein and Vazirani to prove an oracle separation between BPP and BQP. We show that the known quantum algorithm for RFS is essentially optimal, despite its seemingly wasteful need to uncompute information. This implies that, to place \mathsf{BQP} outside of PH[\log] relative to an oracle, one would need to go outside the RFS framework. Our proof argues that, given any variant of RFS, either the adversary method of Ambainis yields a good quantum lower bound, or else there is an efficient classical algorithm. This technique may be of independent interest.

Lower Bounds on OBDD Proofs with Several Orders

2021 ◽  
Vol 22 (4) ◽  
pp. 1-30
Author(s):  
Sam Buss ◽  
Dmitry Itsykson ◽  
Alexander Knop ◽  
Artur Riazanov ◽  
Dmitry Sokolov
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Multiparty Communication ◽  
Constant Degree ◽  
Polynomial Size ◽  
Explicit Constant ◽  
Open Question ◽  
System 1 ◽  
Exponential Size

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε > 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε > 0 is a constant. The lower bound is based on multiparty communication complexity.

scholarly journals Class A PBPs have a distinct and unique role in the construction of the pneumococcal cell wall

2020 ◽  
Vol 117 (11) ◽  
pp. 6129-6138 ◽  
Author(s):  
Daniel Straume ◽  
Katarzyna Wiaroslawa Piechowiak ◽  
Silje Olsen ◽  
Gro Anita Stamsås ◽  
Kari Helene Berg ◽  
...  
Keyword(s):  
Cell Wall ◽  
Peptidoglycan Hydrolase ◽  
Unique Role ◽  
Penicillin Binding Proteins ◽  
The Core ◽  
Class A ◽  
Peptidoglycan Synthesis ◽  
Resistant Form ◽  
Class B ◽  
Open Question

In oval-shapedStreptococcus pneumoniae, septal and longitudinal peptidoglycan syntheses are performed by independent functional complexes: the divisome and the elongasome. Penicillin-binding proteins (PBPs) were long considered the key peptidoglycan-synthesizing enzymes in these complexes. Among these were the bifunctional class A PBPs, which are both glycosyltransferases and transpeptidases, and monofunctional class B PBPs with only transpeptidase activity. Recently, however, it was established that the monofunctional class B PBPs work together with transmembrane glycosyltransferases (FtsW and RodA) from the shape, elongation, division, and sporulation (SEDS) family to make up the core peptidoglycan-synthesizing machineries within the pneumococcal divisome (FtsW/PBP2x) and elongasome (RodA/PBP2b). The function of class A PBPs is therefore now an open question. Here we utilize the peptidoglycan hydrolase CbpD that targets the septum ofS. pneumoniaecells to show that class A PBPs have an autonomous role during pneumococcal cell wall synthesis. Using assays to specifically inhibit the function of PBP2x and FtsW, we demonstrate that CbpD attacks nascent peptidoglycan synthesized by the divisome. Notably, class A PBPs could process this nascent peptidoglycan from a CbpD-sensitive to a CbpD-resistant form. The class A PBP-mediated processing was independent of divisome and elongasome activities. Class A PBPs thus constitute an autonomous functional entity which processes recently formed peptidoglycan synthesized by FtsW/PBP2×. Our results support a model in which mature pneumococcal peptidoglycan is synthesized by three functional entities, the divisome, the elongasome, and bifunctional PBPs. The latter modify existing peptidoglycan but are probably not involved in primary peptidoglycan synthesis.

scholarly journals The quantum query complexity of AC0

2012 ◽  
Vol 12 (7&8) ◽  
pp. 670-676
Author(s):  
Paul Beame ◽  
Widad Machmouchi
Keyword(s):  
Lower Bound ◽  
Quantum Algorithm ◽  
Input Function ◽  
Query Complexity ◽  
Quantum Query Complexity ◽  
Quantum Query

We show that any quantum algorithm deciding whether an input function $f$ from $[n]$ to $[n]$ is 2-to-1 or almost 2-to-1 requires $\Theta(n)$ queries to $f$. The same lower bound holds for determining whether or not a function $f$ from $[2n-2]$ to $[n]$ is surjective. These results yield a nearly linear $\Omega(n/\log n)$ lower bound on the quantum query complexity of $\cl{AC}^0$. The best previous lower bound known for any $\cl{AC^0}$ function was the $\Omega ((n/\log n)^{2/3})$ bound given by Aaronson and Shi's $\Omega(n^{2/3})$ lower bound for the element distinctness problem.

scholarly journals Quantum algorithms for search with wildcards and combinatorial group testing

2014 ◽  
Vol 14 (5&6) ◽  
pp. 439-453
Author(s):  
Andris Ambainis ◽  
Ashley Montanaro
Keyword(s):  
Lower Bound ◽  
Quantum Algorithms ◽  
Group Testing ◽  
Quantum Algorithm ◽  
Quantum Walk ◽  
Combinatorial Problems ◽  
Special Items ◽  
State Discrimination ◽  
Combinatorial Group Testing ◽  
Combinatorial Group

We consider two combinatorial problems. The first we call ``search with wildcards'': given an unknown $n$-bit string $x$, and the ability to check whether any subset of the bits of $x$ is equal to a provided query string, the goal is to output $x$. We give a nearly optimal $O(\sqrt{n} \log n)$ quantum query algorithm for search with wildcards, beating the classical lower bound of $\Omega(n)$ queries. Rather than using amplitude amplification or a quantum walk, our algorithm is ultimately based on the solution to a state discrimination problem. The second problem we consider is combinatorial group testing, which is the task of identifying a subset of at most $k$ special items out of a set of $n$ items, given the ability to make queries of the form ``does the set $S$ contain any special items?''\ for any subset $S$ of the $n$ items. We give a simple quantum algorithm which uses $O(k)$ queries to solve this problem, as compared with the classical lower bound of $\Omega(k \log(n/k))$ queries.

scholarly journals Quantum Algorithm for the Collision Problem

2016 ◽  
pp. 1662-1664 ◽  
Author(s):  
Gilles Brassard ◽  
Peter Høyer ◽  
Alain Tapp
Keyword(s):  
Quantum Algorithm ◽  
Collision Problem

scholarly journals A tight lower bound for semi-synchronous collaborative grid exploration

2020 ◽  
Vol 33 (6) ◽  
pp. 471-484
Author(s):  
Sebastian Brandt ◽  
Jara Uitto ◽  
Roger Wattenhofer
Keyword(s):  
Lower Bound ◽  
Finite Automaton ◽  
Finite Automata ◽  
Grid Cell ◽  
Grid Cells ◽  
Initial State ◽  
Selected Subset ◽  
Open Question ◽  
Agent Protocol

Abstract Recently, there has been a growing interest in grid exploration by agents with limited capabilities. We show that the grid cannot be explored by three semi-synchronous finite automata, answering an open question by Emek et al. (Theor Comput Sci 608:255–267, 2015) in the negative. In the setting we consider, time is divided into discrete steps, where in each step, an adversarially selected subset of the agents executes one look–compute–move cycle. The agents operate according to a shared finite automaton, where every agent is allowed to have a distinct initial state. The only means of communication is to sense the states of the agents sharing the same grid cell. The agents are equipped with a global compass and whenever an agent moves, the destination cell of the movement is chosen by the agent’s automaton from the set of neighboring grid cells. In contrast to the four agent protocol by Emek et al., we show that three agents do not suffice for grid exploration.

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