Algorithms With Superpolynomial Speed-Up

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Quantum Algorithms ◽  
Discrete Logarithm ◽  
Quantum Algorithm ◽  
Estimation Algorithm ◽  
Discrete Logarithm Problem ◽  
Phase Estimation ◽  
Post Process ◽  
Classical Algorithm ◽  
Speed Up ◽  
Hadamard Gate

In this chapter we examine one of two main classes of algorithms: quantum algorithms that solve problems with a complexity that is superpolynomially less than the complexity of the best-known classical algorithm for the same problem. That is, the complexity of the best-known classical algorithm cannot be bounded above by any polynomial in the complexity of the quantum algorithm. The algorithms we will detail all make use of the quantum Fourier transform (QFT). We start off the chapter by studying the problem of quantum phase estimation, which leads us naturally to the QFT. Section 7.1 also looks at using the QFT to find the period of periodic states, and introduces some elementary number theory that is needed in order to post-process the quantum algorithm. In Section 7.2, we apply phase estimation in order to estimate eigenvalues of unitary operators. Then in Section 7.3, we apply the eigenvalue estimation algorithm in order to derive the quantum factoring algorithm, and in Section 7.4 to solve the discrete logarithm problem. In Section 7.5, we introduce the hidden subgroup problem which encompasses both the order finding and discrete logarithm problem as well as many others. This chapter by no means exhaustively covers the quantum algorithms that are superpolynomially faster than any known classical algorithm, but it does cover the most well-known such algorithms. In Section 7.6, we briefly discuss other quantum algorithms that appear to provide a superpolynomial advantage. To introduce the idea of phase estimation, we begin by noting that the final Hadamard gate in the Deutsch algorithm, and the Deutsch–Jozsa algorithm, was used to get at information encoded in the relative phases of a state. The Hadamard gate is self-inverse and thus does the opposite as well, namely it can be used to encode information into the phases. To make this concrete, first consider H acting on the basis state |x⟩ (where x ∊ {0, 1}). It is easy to see that You can think about the Hadamard gate as having encoded information about the value of x into the relative phases between the basis states |0⟩ and |1⟩.

scholarly journals Research on Palmprint Identification Method Based on Quantum Algorithms

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Hui Li ◽  
Zhanzhan Zhang
Keyword(s):  
Fourier Transform ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Palmprint Recognition ◽  
Quantum Image ◽  
Grover Algorithm ◽  
Filtering Algorithm ◽  
Classical Algorithm ◽  
Set Operations ◽  
Speed Up

Quantum image recognition is a technology by using quantum algorithm to process the image information. It can obtain better effect than classical algorithm. In this paper, four different quantum algorithms are used in the three stages of palmprint recognition. First, quantum adaptive median filtering algorithm is presented in palmprint filtering processing. Quantum filtering algorithm can get a better filtering result than classical algorithm through the comparison. Next, quantum Fourier transform (QFT) is used to extract pattern features by only one operation due to quantum parallelism. The proposed algorithm exhibits an exponential speed-up compared with discrete Fourier transform in the feature extraction. Finally, quantum set operations and Grover algorithm are used in palmprint matching. According to the experimental results, quantum algorithm only needs to apply square ofNoperations to find out the target palmprint, but the traditional method needsNtimes of calculation. At the same time, the matching accuracy of quantum algorithm is almost 100%.

scholarly journals A Phase Estimation Algorithm for Quantum Speed-Up Multi-Party Computing

2021 ◽  
Vol 67 (1) ◽  
pp. 241-252
Author(s):  
Wenbin Yu ◽  
Hao Feng ◽  
Yinsong Xu ◽  
Na Yin ◽  
Yadang Chen ◽  
...  
Keyword(s):  
Estimation Algorithm ◽  
Phase Estimation ◽  
Speed Up ◽  
Phase Estimation Algorithm ◽  
Quantum Speed Up

scholarly journals Phase estimation using an approximate eigenstate

2016 ◽  
Vol 16 (9&10) ◽  
pp. 803-812
Author(s):  
Avatar Tulsi
Keyword(s):  
Building Block ◽  
Unitary Operator ◽  
Quantum Algorithms ◽  
Estimation Algorithm ◽  
Single Copy ◽  
Phase Estimation ◽  
Basic Building Block ◽  
Multiple Copies ◽  
Phase Estimation Algorithm ◽  
Spatial Resources

A basic building block of many quantum algorithms is the Phase Estimation algorithm (PEA). It finds an eigenphase φ of a unitary operator using a copy of the corresponding eigenstate |φi. Suppose, in place of |φi, we have a copy of an approximate eigenstate |ψi whose component in |φi is at least p 2/3. Then the PEA fails with a constant probability. Using multiple copies of |ψi, this probability can be made to decrease exponentially with the number of copies. Here we show that a single copy is sufficient to find φ if we can selectively invert the |ψi state. As an application, we consider the eigenpath traversal problem (ETP) where the goal is to travel a path of non-degenerate eigenstates of n different operators. The fastest algorithm for ETP is due to Boixo, Knill and Somma (BKS) which needs Θ(ln n) copies of the eigenstates. Using our method, the BKS algorithm can work with just a single copy but its running time Q increases to O(Q ln2 Q). This tradeoff is beneficial if the spatial resources are more constrained than the temporal resources.

scholarly journals A panoply of quantum algorithms

2008 ◽  
Vol 8 (8&9) ◽  
pp. 834-859
Author(s):  
B. Furrow
Keyword(s):  
Quantum Physics ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Breadth First Search ◽  
Basic Tool ◽  
Grover’S Algorithm ◽  
Classical Algorithm ◽  
Grover's Algorithm ◽  
Programming Techniques ◽  
New Algorithms

This paper's aim is to explore improvements to, and applications of, a fundamental quantum algorithm invented by Grover\cite{grover}. Grover's algorithm is a basic tool that can be applied to a large number of problems in computer science, creating quantum algorithms that are polynomially faster than fastest known and fastest possible classical algorithms that solve the same problems. Our goal in this paper is to make these techniques readily accessible to those without a strong background in quantum physics: we achieve this by providing a set of tools, each of which makes use of Grover's algorithm or similar techniques, which can be used as subroutines in many quantum algorithms.}{The tools we provide are carefully constructed: they are easy to use, and in many cases they are asymptotically faster than the best tools previously available. The tools we build on include algorithms by Boyer, Brassard, Hoyer and Tapp, Buhrman, Cleve, de Witt and Zalka and Durr and Hoyer.}{After creating our tools, we create several new quantum algorithms, each of which is faster than the fastest known deterministic classical algorithm that accomplishes the same aim, and some of which are faster than the fastest possible deterministic classical algorithm. These algorithms solve problems from the fields of graph theory and computational geometry, and some employ dynamic programming techniques. We discuss a breadth-first search that is faster than $\Theta(\text{edges})$ (the classical limit) in a dense graph, maximum-points-on-a-line in $O(N^{3/2}\lg N)$ (faster than the fastest classical algorithm known), as well as several other algorithms that are similarly illustrative of solutions in some class of problem. Through these new algorithms we illustrate the use of our tools, working to encourage their use and the study of quantum algorithms in general.

Hidden symmetry detection on a quantum computer

2007 ◽  
Vol 7 (1&2) ◽  
pp. 83-92
Author(s):  
R. Schutzhold ◽  
W.G. Unruh
Keyword(s):  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Computers ◽  
Self Similarity ◽  
Hidden Subgroup Problem ◽  
Hidden Symmetry ◽  
Speed Up ◽  
Subgroup Problem ◽  
Discrete Scale

The fastest quantum algorithms (for the solution of classical computational tasks) known so far are basically variations of the hidden subgroup problem with {$f(U[x])=f(x)$}. Following a discussion regarding which tasks might be solved efficiently by quantum computers, it will be demonstrated by means of a simple example, that the detection of more general hidden (two-point) symmetries {$V\{f(x),f(U[x])\}=0$} by a quantum algorithm can also admit an exponential speed-up. E.g., one member of this class of symmetries {$V\{f(x),f(U[x])\}=0$} is discrete self-similarity (or discrete scale invariance).

Classical and Quantum Algorithms for Assembling a Text from a Dictionary

2021 ◽  
Vol 24 (3) ◽  
pp. 207-221
Author(s):  
Kamil Khadiev ◽  
Vladislav Remidovskii
Keyword(s):  
Deoxyribonucleic Acid ◽  
Quantum Algorithms ◽  
Randomized Algorithm ◽  
Quantum Algorithm ◽  
Deterministic Algorithm ◽  
Constant Length ◽  
Running Time ◽  
Assembly Method ◽  
Speed Up ◽  
Quantum Speed Up

We study algorithms for solving the problem of assembling a text (long string) from a dictionary (a sequence of small strings). The problem has an application in bioinformatics and has a connection with the sequence assembly method for reconstructing a long deoxyribonucleic-acid (DNA) sequence from small fragments. The problem is assembling a string t of length n from strings s1,...,sm. Firstly, we provide a classical (randomized) algorithm with running time Õ(nL0.5 + L) where L is the sum of lengths of s1,...,sm. Secondly, we provide a quantum algorithm with running time Õ(nL0.25 + √mL). Thirdly, we show the lower bound for a classical (randomized or deterministic) algorithm that is Ω(n+L). So, we obtain the quadratic quantum speed-up with respect to the parameter L; and our quantum algorithm have smaller running time comparing to any classical (randomized or deterministic) algorithm in the case of non-constant length of strings in the dictionary.

scholarly journals Quantum algorithm for multivariate polynomial interpolation

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170480 ◽  
Author(s):  
Jianxin Chen ◽  
Andrew M. Childs ◽  
Shih-Han Hung
Keyword(s):  
Open Problem ◽  
Polynomial Interpolation ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Large Field ◽  
Multivariate Polynomial ◽  
Multivariate Polynomial Interpolation ◽  
Special Cases ◽  
Speed Up

How many quantum queries are required to determine the coefficients of a degree- d polynomial in n variables? We present and analyse quantum algorithms for this multivariate polynomial interpolation problem over the fields F q , R and C . We show that k C and 2 k C queries suffice to achieve probability 1 for C and R , respectively, where k C = ⌈ ( 1 / ( n + 1 ) ) ( n + d d ) ⌉ except for d =2 and four other special cases. For F q , we show that ⌈( d /( n + d ))( n + d d ) ⌉ queries suffice to achieve probability approaching 1 for large field order q . The classical query complexity of this problem is ( n + d d ) , so our result provides a speed-up by a factor of n +1, ( n +1)/2 and ( n + d )/ d for C , R and F q , respectively. Thus, we find a much larger gap between classical and quantum algorithms than the univariate case, where the speedup is by a factor of 2. For the case of F q , we conjecture that 2 k C queries also suffice to achieve probability approaching 1 for large field order q , although we leave this as an open problem.

scholarly journals Quantum Phase Estimation Algorithm for Finding Polynomial Roots

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 839-849
Author(s):  
Theerapat Tansuwannont ◽  
Surachate Limkumnerd ◽  
Sujin Suwanna ◽  
Pruet Kalasuwan
Keyword(s):  
Quantum Algorithm ◽  
Estimation Algorithm ◽  
Quantum Circuit ◽  
Quantum Systems ◽  
Phase Estimation ◽  
Companion Matrix ◽  
Polynomial Roots ◽  
Mathematical Problems ◽  
Phase Estimation Algorithm ◽  
Computational Resources

AbstractQuantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum algorithm for finding the roots of nth degree polynomials where n is any positive integer. In classical algorithm, the resources required for solving this problem increase drastically when n increases and it would be impossible to practically solve the problem when n is large. It was found that any polynomial can be rearranged into a corresponding companion matrix, whose eigenvalues are roots of the polynomial. This leads to a possibility to perform a quantum algorithm where the number of computational resources increase as a polynomial of n. In this study, we construct a quantum circuit representing the companion matrix and use eigenvalue estimation technique to find roots of polynomial.

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