scholarly journals On the Amplitude Amplification of Quantum States Corresponding to the Solutions of the Partition Problem

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2027
Author(s):  
Mauro Mezzini ◽  
Jose J. Paulet ◽  
Fernando Cuartero ◽  
Hernan I. Cruz ◽  
Fernando L. Pelayo
Keyword(s):  
Quantum Algorithm ◽  
Quantum States ◽  
Partition Problem ◽  
Amplitude Amplification

In this paper we investigate the effects of a quantum algorithm which increases the amplitude of the states corresponding to the solutions of the partition problem by a factor of almost two. The study is limited to one iteration.

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 Quantum Generative Adversarial Networks for learning and loading random distributions

2019 ◽  
Vol 5 (1) ◽  
Author(s):  
Christa Zoufal ◽  
Aurélien Lucchi ◽  
Stefan Woerner
Keyword(s):  
Quantum State ◽  
Quantum Channel ◽  
Quantum Algorithms ◽  
Probability Distributions ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Simulation ◽  
Quantum States ◽  
Generative Adversarial Networks ◽  
Adversarial Networks

AbstractQuantum algorithms have the potential to outperform their classical counterparts in a variety of tasks. The realization of the advantage often requires the ability to load classical data efficiently into quantum states. However, the best known methods require $${\mathcal{O}}\left({2}^{n}\right)$$O2n gates to load an exact representation of a generic data structure into an $$n$$n-qubit state. This scaling can easily predominate the complexity of a quantum algorithm and, thereby, impair potential quantum advantage. Our work presents a hybrid quantum-classical algorithm for efficient, approximate quantum state loading. More precisely, we use quantum Generative Adversarial Networks (qGANs) to facilitate efficient learning and loading of generic probability distributions - implicitly given by data samples - into quantum states. Through the interplay of a quantum channel, such as a variational quantum circuit, and a classical neural network, the qGAN can learn a representation of the probability distribution underlying the data samples and load it into a quantum state. The loading requires $${\mathcal{O}}\left(poly\left(n\right)\right)$$Opolyn gates and can thus enable the use of potentially advantageous quantum algorithms, such as Quantum Amplitude Estimation. We implement the qGAN distribution learning and loading method with Qiskit and test it using a quantum simulation as well as actual quantum processors provided by the IBM Q Experience. Furthermore, we employ quantum simulation to demonstrate the use of the trained quantum channel in a quantum finance application.

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 Enhanced quantum signature scheme using quantum amplitude amplification operators

PLoS ONE ◽  
2021 ◽  
Vol 16 (10) ◽  
pp. e0258091
Author(s):  
Basma Elias ◽  
Ahmed Younes
Keyword(s):  
Quantum Computing ◽  
Quantum Signature ◽  
Quantum States ◽  
Bell States ◽  
Signature Scheme ◽  
Signature Schemes ◽  
Diffusion Operator ◽  
Quantum Amplitude ◽  
Preparation Phase ◽  
Amplitude Amplification

Quantum signature is the use of the principles of quantum computing to establish a trusted communication between two parties. In this paper, a quantum signature scheme using amplitude amplification techniques will be proposed. To secure the signature, the proposed scheme uses a partial diffusion operator and a diffusion operator to hide/unhide certain quantum states during communication. The proposed scheme consists of three phases, preparation phase, signature phase and verification phase. To confuse the eavesdropper, the quantum states representing the signature might be hidden, not hidden or encoded in Bell states. It will be shown that the proposed scheme is more secure against eavesdropping when compared with relevant quantum signature schemes.

scholarly journals Quantum algorithm for estimating α -Renyi entropies of quantum states

2021 ◽  
Vol 104 (2) ◽  
Author(s):  
Sathyawageeswar Subramanian ◽  
Min-Hsiu Hsieh
Keyword(s):  
Quantum Algorithm ◽  
Quantum States ◽  
Rényi Entropies ◽  
Renyi Entropies

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.

scholarly journals Operating Quantum States in Single Magnetic Molecules: Implementation of Grover’s Quantum Algorithm

2017 ◽  
Vol 119 (18) ◽  
Author(s):  
C. Godfrin ◽  
A. Ferhat ◽  
R. Ballou ◽  
S. Klyatskaya ◽  
M. Ruben ◽  
...  
Keyword(s):  
Quantum Algorithm ◽  
Quantum States ◽  
Magnetic Molecules

scholarly journals EXACT QUANTUM FOURIER TRANSFORMS AND DISCRETE LOGARITHM ALGORITHMS

2004 ◽  
Vol 02 (01) ◽  
pp. 91-100 ◽  
Author(s):  
MICHELE MOSCA ◽  
CHRISTOF ZALKA
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Quantum Algorithms ◽  
Discrete Logarithm ◽  
Quantum Algorithm ◽  
Quantum Fourier Transform ◽  
The Family ◽  
The Fourier Transform ◽  
Quantum Fourier Transforms ◽  
Amplitude Amplification

We show how the Quantum Fast Fourier Transform (QFFT) can be made exact for arbitrary orders (first showing it for large primes). Most quantum algorithms only need a good approximation of the quantum Fourier transform of order 2n to succeed with high probability, and this QFFT can in fact be done exactly. Kitaev1 showed how to approximate the Fourier transform for any order. Here we show how his construction can be made exact by using the technique known as "amplitude amplification". Although unlikely to be of any practical use, this construction allows one to make Shor's discrete logarithm quantum algorithm exact. Thus we have the first example of an exact non black box fast quantum algorithm, thereby giving more evidence that "quantum" need not be probabilistic. We also show that in a certain sense the family of circuits for the exact QFFT is uniform. Namely, the parameters of the gates can be approximated efficiently.

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.

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