On Computational Power of Quantum Read-Once Branching Programs

In this paper we review our current results concerning the computational power of quantum read-once branching programs. First of all, based on the circuit presentation of quantum branching programs and our variant of quantum fingerprinting technique, we show that any Boolean function with linear polynomial presentation can be computed by a quantum read-once branching program using a relatively small (usually logarithmic in the size of input) number of qubits. Then we show that the described class of Boolean functions is closed under the polynomial projections.Comment: In Proceedings HPC 2010, arXiv:1103.226

Quantum vs. Classical Read-once Branching Programs

The paper presents the first nontrivial upper and lower bounds for (non-oblivious) quantum read-once branching programs. It is shown that the computational power of quantum and classical read-once branching programs is incomparable in the following sense: (i) A simple, explicit boolean function on 2n input bits is presented that is computable by error-free quantum read-once branching programs of size O(n^3), while each classical randomized read-once branching program and each quantum OBDD for this function with bounded two-sided error requires size 2^{\Omega(n)}. (ii) Quantum branching programs reading each input variable exactly once are shown to require size 2^{\Omega(n)} for computing the set-disjointness function DISJ_n from communication complexity theory with two-sided error bounded by a constant smaller than 1/2-2\sqrt{3}/7. This function is trivially computable even by deterministic OBDDs of linear size. The technically most involved part is the proof of the lower bound in (ii). For this, a new model of quantum multi-partition communication protocols is introduced and a suitable extension of the information cost technique of Jain, Radhakrishnan, and Sen (2003) to this model is presented.Comment: 35 pages. Lower bound for disjointness: Error in application of info theory corrected and regularity of quantum read-once BPs (each variable at least once) added as additional assumption of the theorem. Some more informal explanations adde

On computational power of classical and quantum Branching programs

We present a classical stochastic simulation technique of quantum Branching programs. This technique allows to prove the following relations among complexity classes: PrQP-BP ⊆ PP-BP and BQP-BP ⊆ PP-BP. Here BPP-BP and PP-BP stands for the classes of functions computable with bounded error and unbounded error respectively by stochastic branching program of polynomial size. BQP-BP and PrQP-BP stands the classes of functions computable with bounded error and unbounded error respectively by quantum branching program of polynomial size. Second. We present two different types. of complexity lower bounds for quantum nonuniform automata (OBDDs). We call them "metric" and "entropic" lower bounds in according to proof technique used. We present explicit Boolean functions that show that these lower bounds are tight enough. We show that when considering "almost all Boolean functions" on n variables our entropic lower bounds gives exponential (2c(δ)(n-logn)) lower bound for the width of quantum OBDDs depending on the error δ allowed

Classical and quantum computations with restricted memory

© 2018, Springer Nature Switzerland AG. Automata and branching programs are known models of computation with restricted memory. These models of computation were in focus of a large number of researchers during the last decades. Streaming algorithms are a modern model of computation with restricted memory. In this paper, we present recent results on the comparative computational power of quantum and classical models of branching programs and streaming algorithms. In addition to comparative complexity results, we present a quantum branching program for computing a practically important quantum function (quantum hash function) and prove optimality of this algorithm

Comparative power of quantum and classical computation models

In the talk we present results on comparitve power of classical and quantum computational models. We focus on two well known in Computer Science models: finite automata which is known as uniform computational model and branching programs which is known as nonuniform computational model

Algorithms for Quantum Branching Programs Based on Fingerprinting

In the paper we develop a method for constructing quantum algorithms for computing Boolean functions by quantum ordered read-once branching programs (quantum OBDDs). Our method is based on fingerprinting technique and representation of Boolean functions by their characteristic polynomials. We use circuit notation for branching programs for desired algorithms presentation. For several known functions our approach provides optimal QOBDDs. Namely we consider such functions as Equality, Palindrome, and Permutation Matrix Test. We also propose a generalization of our method and apply it to the Boolean variant of the Hidden Subgroup Problem