scholarly journals Quantum and Randomised Algorithms for Non-linearity Estimation

2021 ◽  
Vol 2 (2) ◽  
pp. 1-27
Author(s):  
Debajyoti Bera ◽  
Sapv Tharrmashastha
Keyword(s):  
Boolean Function ◽  
Linear Function ◽  
Lower Bounds ◽  
Quantum Algorithm ◽  
Walsh Spectrum ◽  
Absolute Value ◽  
Additive Error ◽  
Amplitude Estimation ◽  
Walsh Coefficient ◽  
Amplitude Amplification

Non-linearity of a Boolean function indicates how far it is from any linear function. Despite there being several strong results about identifying a linear function and distinguishing one from a sufficiently non-linear function, we found a surprising lack of work on computing the non-linearity of a function. The non-linearity is related to the Walsh coefficient with the largest absolute value; however, the naive attempt of picking the maximum after constructing a Walsh spectrum requires Θ (2 n ) queries to an n -bit function. We improve the scenario by designing highly efficient quantum and randomised algorithms to approximate the non-linearity allowing additive error, denoted λ, with query complexities that depend polynomially on λ. We prove lower bounds to show that these are not very far from the optimal ones. The number of queries made by our randomised algorithm is linear in n , already an exponential improvement, and the number of queries made by our quantum algorithm is surprisingly independent of n . Our randomised algorithm uses a Goldreich-Levin style of navigating all Walsh coefficients and our quantum algorithm uses a clever combination of Deutsch-Jozsa, amplitude amplification and amplitude estimation to improve upon the existing quantum versions of the Goldreich-Levin technique.

scholarly journals Testing Boolean Functions Properties

2021 ◽  
Vol 182 (4) ◽  
pp. 321-344
Author(s):  
Xie Zhengwei ◽  
Qiu Daowen ◽  
Cai Guangya ◽  
Jozef Gruska ◽  
Paulo Mateus
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Randomized Algorithm ◽  
Quantum Algorithm ◽  
Black Box ◽  
Query Complexity ◽  
Testing Problem ◽  
Amplification Technique ◽  
Query Algorithm ◽  
Amplitude Amplification

The goal in the area of functions property testing is to determine whether a given black-box Boolean function has a particular given property or is ɛ-far from having that property. We investigate here several types of properties testing for Boolean functions (identity, correlations and balancedness) using the Deutsch-Jozsa algorithm (for the Deutsch-Jozsa (D-J) problem) and also the amplitude amplification technique. At first, we study here a particular testing problem: namely whether a given Boolean function f, of n variables, is identical with a given function g or is ɛ-far from g, where ɛ is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity O(1ε). Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is Θ(1ε). Secondly, we consider the D-J problem from the perspective of functional correlations and let C(f, g) denote the correlation of f and g. We propose an exact quantum algorithm for making distinction between |C(f, g)| = ɛ and |C(f, g)| = 1 using six queries, while the classical deterministic query complexity for this problem is Θ(2n) queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus ɛ-far balanced using O(1ε) queries. We also prove here that our quantum algorithm for balancedness testing is optimal. At the same time, for this balancedness testing problem we present a classical randomized algorithm with query complexity of O(1/ɛ2). Also this randomized algorithm is optimal. Besides, we link the problems considered here together and generalize them to the general case.

MÖBIUS TRANSFORMATIONS IN QUANTUM AMPLITUDE AMPLIFICATION WITH GENERALIZED PHASES

2010 ◽  
Vol 08 (06) ◽  
pp. 923-935 ◽  
Author(s):  
CÉSAR BAUTISTA-RAMOS ◽  
NORA CASTILLO-TÉPOX
Keyword(s):  
Error Probability ◽  
Quantum Algorithm ◽  
Linear Transformations ◽  
Möbius Transformations ◽  
Good State ◽  
Quantum Amplitude ◽  
Initial States ◽  
Phase Angles ◽  
Amplitude Amplification ◽  
Mobius Transformations

The iteration of the operators employed in quantum amplitude amplification with generalized phases is analyzed by using elementary properties (geometric and algebraic) of the Möbius transformations (fractional linear transformations). It is shown that, for a given quantum algorithm without measurement, which produces a good state with probability a of success, if the phase angles φ and ϕ which mark the good and initial states respectively satisfy φ = ϕ with a small enough, then, for a number n of iterations with [Formula: see text] we get an error probability that is at most O(aϕ2).

scholarly journals Enhancing the Quantum Linear Systems Algorithm Using Richardson Extrapolation

2022 ◽  
Vol 3 (1) ◽  
pp. 1-37
Author(s):  
Almudena Carrera Vazquez ◽  
Ralf Hiptmair ◽  
Stefan Woerner
Keyword(s):  
Necessary Conditions ◽  
Linear Equations ◽  
Circuit Complexity ◽  
Quantum Algorithm ◽  
Richardson Extrapolation ◽  
Classical Simulation ◽  
Hamiltonian Simulation ◽  
Systems Of Linear Equations ◽  
Amplitude Amplification ◽  
Theoretical Results

We present a quantum algorithm to solve systems of linear equations of the form Ax = b , where A is a tridiagonal Toeplitz matrix and b results from discretizing an analytic function, with a circuit complexity of O (1/√ε, poly (log κ, log N )), where N denotes the number of equations, ε is the accuracy, and κ the condition number. The repeat-until-success algorithm has to be run O (κ/(1-ε)) times to succeed, leveraging amplitude amplification, and needs to be sampled O (1/ε 2 ) times. Thus, the algorithm achieves an exponential improvement with respect to N over classical methods. In particular, we present efficient oracles for state preparation, Hamiltonian simulation, and a set of observables together with the corresponding error and complexity analyses. As the main result of this work, we show how to use Richardson extrapolation to enhance Hamiltonian simulation, resulting in an implementation of Quantum Phase Estimation (QPE) within the algorithm with 1/√ε circuits that can be run in parallel each with circuit complexity 1/√ ε instead of 1/ε. Furthermore, we analyze necessary conditions for the overall algorithm to achieve an exponential speedup compared to classical methods. Our approach is not limited to the considered setting and can be applied to more general problems where Hamiltonian simulation is approximated via product formulae, although our theoretical results would need to be extended accordingly. All the procedures presented are implemented with Qiskit and tested for small systems using classical simulation as well as using real quantum devices available through the IBM Quantum Experience.

scholarly journals Lower Bounds for the Average Genus of a CF-graph

10.37236/422 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Yichao Chen
Keyword(s):  
Lower Bound ◽  
Linear Function ◽  
Lower Bounds ◽  
Betti Number ◽  
Maximum Genus ◽  
Simple Graphs ◽  
Structure Theorems ◽  
Average Genus

CF-graphs form a class of multigraphs that contains all simple graphs. We prove a lower bound for the average genus of a CF-graph which is a linear function of its Betti number. A lower bound for average genus in terms of the maximum genus and some structure theorems for graphs with a given average genus are also provided.

scholarly journals Exact quantum algorithms have advantage for almost all Boolean functions

2015 ◽  
pp. 435-452
Author(s):  
Andris Ambainis ◽  
Jozef Gruska ◽  
Shenggen Zheng
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Almost All ◽  
Quantum Query

It has been proved that almost all n-bit Boolean functions have exact classical query complexity n. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all n-bit Boolean functions can be computed by an exact quantum algorithm with less than n queries. More exactly, we prove that ANDn is the only n-bit Boolean function, up to isomorphism, that requires n queries.

scholarly journals EXPONENTIAL IMPROVEMENT IN PRECISION FOR SIMULATING SPARSE HAMILTONIANS

2017 ◽  
Vol 5 ◽  
Author(s):  
DOMINIC W. BERRY ◽  
ANDREW M. CHILDS ◽  
RICHARD CLEVE ◽  
ROBIN KOTHARI ◽  
ROLANDO D. SOMMA
Keyword(s):  
Discrete Model ◽  
Quantum Algorithm ◽  
Small Error ◽  
Query Complexity ◽  
Input State ◽  
Time Varying ◽  
Correction Procedure ◽  
Complex Fault ◽  
New Form ◽  
Amplitude Amplification

We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a $d$-sparse Hamiltonian $H$ acting on $n$ qubits can be simulated for time $t$ with precision $\unicode[STIX]{x1D716}$ using $O(\unicode[STIX]{x1D70F}(\log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716})/\log \log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716})))$ queries and $O(\unicode[STIX]{x1D70F}(\log ^{2}(\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716})/\log \log (\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D716}))n)$ additional 2-qubit gates, where $\unicode[STIX]{x1D70F}=d^{2}\Vert H\Vert _{\max }t$. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuous- and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also simplify the analysis of this conversion, avoiding the need for a complex fault-correction procedure. Our simplification relies on a new form of ‘oblivious amplitude amplification’ that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.

scholarly journals Following Forrelation – quantum algorithms in exploring Boolean functions' spectra

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Suman Dutta ◽  
Subhamoy Maitra ◽  
Chandra Sekhar Mukherjee
Keyword(s):  
Boolean Functions ◽  
Cross Correlation ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Sampling Technique ◽  
Linear Functions ◽  
Query Complexity ◽  
Walsh Spectrum ◽  
Correlation Spectrum ◽  
The Cross

<p style='text-indent:20px;'>Here we revisit the quantum algorithms for obtaining Forrelation [Aaronson et al., 2015] values to evaluate some of the well-known cryptographically significant spectra of Boolean functions, namely the Walsh spectrum, the cross-correlation spectrum, and the autocorrelation spectrum. We introduce the existing 2-fold Forrelation formulation with bent duality-based promise problems as desirable instantiations. Next, we concentrate on the 3-fold version through two approaches. First, we judiciously set up some of the functions in 3-fold Forrelation so that given oracle access, one can sample from the Walsh Spectrum of <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula>. Using this, we obtain improved results than what one can achieve by exploiting the Deutsch-Jozsa algorithm. In turn, it has implications in resiliency checking. Furthermore, we use a similar idea to obtain a technique in estimating the cross-correlation (and thus autocorrelation) value at any point, improving upon the existing algorithms. Finally, we tweak the quantum algorithm with the superposition of linear functions to obtain a cross-correlation sampling technique. This is the first cross-correlation sampling algorithm with constant query complexity to the best of our knowledge. This also provides a strategy to check if two functions are uncorrelated of degree <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula>. We further modify this using Dicke states so that the time complexity reduces, particularly for constant values of <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>.</p>

scholarly journals The extremal pentagon-chain polymers with respect to permanental sum

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Tingzeng Wu ◽  
Hongge Wang ◽  
Shanjun Zhang ◽  
Kai Deng
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Organic Polymers ◽  
Absolute Value ◽  
Important Type ◽  
Computing Complexity ◽  
Permanental Polynomial ◽  
Stability Of Structure

Abstract The permanental sum of a graph G can be defined as the sum of absolute value of coefficients of permanental polynomial of G. It is closely related to stability of structure of a graph, and its computing complexity is #P-complete. Pentagon-chain polymers is an important type of organic polymers. In this paper, we determine the upper and lower bounds of permanental sum of pentagon-chain polymers, and the corresponding pentagon-chain polymers are also determined.

Algorithms Based on Amplitude Amplification

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Search Algorithm ◽  
Quantum Algorithm ◽  
Mathematical Structure ◽  
Large Integer ◽  
Problem Instance ◽  
Wide Range ◽  
Naive Algorithm ◽  
Long Time ◽  
Speed Up ◽  
Amplitude Amplification

In this section, we discuss a broadly applicable quantum algorithm that provides a polynomial speed-up over the best-known classical algorithms for a wide class of important problems. The quantum search algorithm performs a generic search for a solution to a very wide range of problems. Consider any problem where one can efficiently recognize a good solution and wishes to search through a list of potential solutions in order to find a good one. For example, given a large integer N, one can efficiently recognize whether an integer p is a non-trivial factor of N, and thus one naive strategy for finding non-trivial factors of N is to simply search through the set {2, 3, 4, . . . , ⌊√N⌋} until a factor is found. The factoring algorithm we described in Chapter 7 is not such a naive algorithm, as it makes profound use of the structure of the problem. However, for many interesting problems, there are no known techniques that make much use of the structure of the problem, and the best-known algorithm for solving these problems is to naively search through the potential solutions until one is found. Typically the number of potential solutions is exponential in the size of the problem instance, and so the naive algorithm is not efficient. Often the best-known classical search makes some very limited use of the structure of the problem, perhaps to rule out some obviously impossible candidates, or to prioritize some more likely candidates, but the overall complexity of the search is still exponential. Quantum searching is a tool for speeding up these sorts of generic searches through a space of potential solutions. It is worth noting that having a means of recognizing a solution to a problem, and knowing the set of possible solutions, means that in some sense one ‘knows’ the solution. However, one cannot necessarily efficiently produce the solution. For example, it is easy to recognize the factors of a number, but finding those factors can take a long time. We give this problem a more general mathematical structure as follows.

Sign in / Sign up

Export Citation Format

Share Document