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 Quantifying computational advantage of Grover’s algorithm with the trace speed

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Valentin Gebhart ◽  
Luca Pezzè ◽  
Augusto Smerzi
Keyword(s):  
Quantum Coherence ◽  
Search Algorithm ◽  
Quantum Algorithms ◽  
Physical Origin ◽  
Computational Advantage ◽  
Speed Up ◽  
Quantum Speed Up ◽  
Quantum Statistical ◽  
Grover’S Search Algorithm ◽  
General Physical

AbstractDespite intensive research, the physical origin of the speed-up offered by quantum algorithms remains mysterious. No general physical quantity, like, for instance, entanglement, can be singled out as the essential useful resource. Here we report a close connection between the trace speed and the quantum speed-up in Grover’s search algorithm implemented with pure and pseudo-pure states. For a noiseless algorithm, we find a one-to-one correspondence between the quantum speed-up and the polarization of the pseudo-pure state, which can be connected to a wide class of quantum statistical speeds. For time-dependent partial depolarization and for interrupted Grover searches, the speed-up is specifically bounded by the maximal trace speed that occurs during the algorithm operations. Our results quantify the quantum speed-up with a physical resource that is experimentally measurable and related to multipartite entanglement and quantum coherence.

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⟩.

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).

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 advantage with shallow circuits

Science ◽  
2018 ◽  
Vol 362 (6412) ◽  
pp. 308-311 ◽  
Author(s):  
Sergey Bravyi ◽  
David Gosset ◽  
Robert König
Keyword(s):  
Quadratic Forms ◽  
Nearest Neighbor ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Nonlocality ◽  
Binary Quadratic Forms ◽  
Time Period ◽  
Speed Up ◽  
Active Debate ◽  
Near Term

Quantum effects can enhance information-processing capabilities and speed up the solution of certain computational problems. Whether a quantum advantage can be rigorously proven in some setting or demonstrated experimentally using near-term devices is the subject of active debate. We show that parallel quantum algorithms running in a constant time period are strictly more powerful than their classical counterparts; they are provably better at solving certain linear algebra problems associated with binary quadratic forms. Our work gives an unconditional proof of a computational quantum advantage and simultaneously pinpoints its origin: It is a consequence of quantum nonlocality. The proposed quantum algorithm is a suitable candidate for near-future experimental realizations, as it requires only constant-depth quantum circuits with nearest-neighbor gates on a two-dimensional grid of qubits (quantum bits).

scholarly journals Quantum algorithms and lower bounds for convex optimization

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 221 ◽  
Author(s):  
Shouvanik Chakrabarti ◽  
Andrew M. Childs ◽  
Tongyang Li ◽  
Xiaodi Wu
Keyword(s):  
Convex Body ◽  
Convex Optimization ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Computers ◽  
Semidefinite Programs ◽  
Membership Queries ◽  
Speed Up ◽  
General Convex ◽  
Quantum Complexity

While recent work suggests that quantum computers can speed up the solution of semidefinite programs, little is known about the quantum complexity of more general convex optimization. We present a quantum algorithm that can optimize a convex function over an n-dimensional convex body using O~(n) queries to oracles that evaluate the objective function and determine membership in the convex body. This represents a quadratic improvement over the best-known classical algorithm. We also study limitations on the power of quantum computers for general convex optimization, showing that it requires Ω~(n) evaluation queries and Ω(n) membership queries.

scholarly journals Solving Burgers’ equation with quantum computing

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Furkan Oz ◽  
Rohit K. S. S. Vuppala ◽  
Kursat Kara ◽  
Frank Gaitan
Keyword(s):  
Numerical Simulation ◽  
Direct Numerical Simulation ◽  
Burgers Equation ◽  
Exact Analytical Solution ◽  
Quantum Algorithms ◽  
Turbulence Models ◽  
Quantum Algorithm ◽  
Spatial Dimensions ◽  
Speed Up ◽  
Computational Fluid Dynamics Cfd

AbstractComputational fluid dynamics (CFD) simulations are a vital part of the design process in the aerospace industry. Although reliable CFD results can be obtained with turbulence models, direct numerical simulation of complex bodies in three spatial dimensions (3D) is impracticable due to the massive amount of computational elements. For instance, a 3D direct numerical simulation of a turbulent boundary-layer over the wing of a commercial jetliner that resolves all relevant length scales using a serial CFD solver on a modern digital computer would take approximately 750 million years or roughly 20% of the earth’s age. Over the past 25 years, quantum computers have become the object of great interest worldwide as powerful quantum algorithms have been constructed for several important, computationally challenging problems that provide enormous speed-up over the best-known classical algorithms. In this paper, we adapt a recently introduced quantum algorithm for partial differential equations to Burgers’ equation and develop a quantum CFD solver that determines its solutions. We used our quantum CFD solver to verify the quantum Burgers’ equation algorithm to find the flow solution when a shockwave is and is not present. The quantum simulation results were compared to: (i) an exact analytical solution for a flow without a shockwave; and (ii) the results of a classical CFD solver for flows with and without a shockwave. Excellent agreement was found in both cases, and the error of the quantum CFD solver was comparable to that of the classical CFD solver.

scholarly journals LIMITATIONS OF SOME SIMPLE ADIABATIC QUANTUM ALGORITHMS

2008 ◽  
Vol 06 (03) ◽  
pp. 419-426 ◽  
Author(s):  
LAWRENCE M. IOANNOU ◽  
MICHELE MOSCA
Keyword(s):  
Search Algorithm ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Running Time ◽  
Computational Basis

Let H(t) = (1 - t/T)H0 + (t/T)H1, t ∈ [0,T], be the Hamiltonian governing an adiabatic quantum algorithm, where H0 is diagonal in the Hadamard basis and H1 is diagonal in the computational basis. We prove that H0 and H1 must each have at least two large mutually-orthogonal eigenspaces if the algorithm's running time is to be subexponential in the number of qubits. We also reproduce the optimality proof of Farhi and Gutmann's search algorithm in the context of this adiabatic scheme; because we only consider initial Hamiltonians that are diagonal in the Hadamard basis, our result is slightly stronger than the original.

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%.

MAPPING, PROGRAMMABILITY AND SCALABILITY OF PROBLEMS FOR QUANTUM SPEED-UP

2003 ◽  
Vol 14 (05) ◽  
pp. 587-600
Author(s):  
E. V. KRISHNAMURTHY
Keyword(s):  
Optimization Problems ◽  
Quantum Algorithm ◽  
Curse Of Dimensionality ◽  
Quantum Entropy ◽  
Quantum Computers ◽  
Computational Algorithms ◽  
Stringent Condition ◽  
Practical Utility ◽  
Speed Up ◽  
Quantum Speed Up

This paper explores the reasons as to why the quantum paradigm is not so easy to extend to all of the classical computational algorithms. We also explain the failure of programmability and scalability in quantum speed-up. Due to the presence of quantum entropy, quantum algorithm cannot obviate the curse of dimensionality encountered in solving many complex numerical and optimization problems. Finally, the stringent condition that quantum computers have to be interaction-free, leave them with little versatility and practical utility.

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