arXiv subjects

Gatis Midrijanis

Publications and source records attributed to Gatis Midrijanis.

7 recordsLinked to original sources

arXiv2009

On parallel composition of zero-knowledge proofs with black-box quantum simulators

Let L be a language decided by a constant-round quantum Arthur-Merlin (QAM) protocol with negligible soundness error and all but possibly the last message being classical. We prove that if this protocol is zero knowledge with a black-box, quantum simulator S, then L in BQP. Our result also applies to any language having a three-round quantum interactive proof (QIP), with all but possibly the last message being classical, with negligible soundness error and a black-box quantum simulator. These results in particular make it unlikely that certain protocols can be composed in parallel in order to reduce soundness error, while maintaining zero knowledge with a black-box quantum simulator. They generalize analogous classical results of Goldreich and Krawczyk (1990). Our proof goes via a reduction to quantum black-box search. We show that the existence of a black-box quantum simulator for such protocols when L notin BQP would imply an impossibly-good quantum search algorithm.

quant-ph↗

arXiv2005

On Randomized and Quantum Query Complexities

We study randomized and quantum query (a.k.a. decision tree) complexity for all total Boolean functions, with emphasis to derandomization and dequantization (removing quantumness from algorithms). Firstly, we show that $D(f) = O(Q_1(f)^3)$ for any total function $f$, where $D(f)$ is the minimal number of queries made by a deterministic query algorithm and $Q_1(f)$ is the number of queries made by any quantum query algorithm (decision tree analog in quantum case) with one-sided constant error; both algorithms compute function $f$. Secondly, we show that for all total Boolean functions $f$ holds $R_0(f)=O(R_2(f)^2 \log N)$, where $R_0(f)$ and $R_2(f)$ are randomized zero-sided (a.k.a Las Vegas) and two-sided (a.k.a. Monte Carlo) error query complexities.

quant-ph↗

arXiv2005

Three lines proof of the lower bound for the matrix rigidity

The rigidity of a matrix describes the minimal number of entries one has to change to reduce matrix's rank to r. We give very simple combinatorial proof of the lower bound for the rigidity of Sylvester (special case of Hadamard) matrix that matches the best known result by de Wolf(2005) for Hadamard matrices proved by quantum information theoretical arguments.

cs.CC↗

arXiv2004

Quantum lower bounds for the set equality problems

The set equality problem is to decide whether two sets $A$ and $B$ are equal or disjoint, under the promise that one of these is the case. Some other problems, like the Graph Isomorphism problem, is solvable by reduction to the set quality problem. It was an open problem to find any $w(1)$ query lower bound when sets $A$ and $B$ are given by quantum oracles with functions $a$ and $b$. We will prove $Ω(\frac{n^{1/3}}{\log^{1/3} n})$ lower bound for the set equality problem when the set of the preimages are very small for every element in $A$ and $B$.

quant-ph↗

arXiv2004

A polynomial quantum query lower bound for the set equality problem

The set equality problem is to tell whether two sets $A$ and $B$ are equal or disjoint under the promise that one of these is the case. This problem is related to the Graph Isomorphism problem. It was an open problem to find any $ω(1)$ query lower bound when sets $A$ and $B$ are given by quantum oracles. We will show that any error-bounded quantum query algorithm that solves the set equality problem must evaluate oracles $Ω(\sqrt[5]{\frac{n}{\ln n}})$ times, where $n=|A|=|B|$.

quant-ph↗

arXiv2004

Exact quantum query complexity for total Boolean functions

We will show that if there exists a quantum query algorithm that exactly computes some total Boolean function f by making T queries, then there is a classical deterministic algorithm A that exactly computes f making O(T^3) queries. The best know bound previously was O(T^4) due to Beals et al.

quant-ph↗

arXiv2003

The Complexity of Probabilistic versus Quantum Finite Automata

We present a language $L_n$ which is recognizable by a probabilistic finite automaton (PFA) with probability $1 - ε$ for all $ε> 0$ with $O(log^2n)$ states, with a deterministic finite automaton (DFA) with O(n) states, but a quantum finite automaton (QFA) needs at least $2^{Ω(n/ \log n)}$ states.

quant-ph↗