amplitude amplification
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2022 ◽  
Vol 3 (1) ◽  
pp. 1-37
Author(s):  
Almudena Carrera Vazquez ◽  
Ralf Hiptmair ◽  
Stefan Woerner

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.


2021 ◽  
Vol 104 (6) ◽  
Author(s):  
Hyeokjea Kwon ◽  
Joonwoo Bae

2021 ◽  
Vol 182 (4) ◽  
pp. 321-344
Author(s):  
Xie Zhengwei ◽  
Qiu Daowen ◽  
Cai Guangya ◽  
Jozef Gruska ◽  
Paulo Mateus

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.


PLoS ONE ◽  
2021 ◽  
Vol 16 (10) ◽  
pp. e0258091
Author(s):  
Basma Elias ◽  
Ahmed Younes

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.


Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2027
Author(s):  
Mauro Mezzini ◽  
Jose J. Paulet ◽  
Fernando Cuartero ◽  
Hernan I. Cruz ◽  
Fernando L. Pelayo

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.


2021 ◽  
Vol 3 (2) ◽  
Author(s):  
A. Hamann ◽  
V. Dunjko ◽  
S. Wölk

AbstractIn recent years, quantum-enhanced machine learning has emerged as a particularly fruitful application of quantum algorithms, covering aspects of supervised, unsupervised and reinforcement learning. Reinforcement learning offers numerous options of how quantum theory can be applied, and is arguably the least explored, from a quantum perspective. Here, an agent explores an environment and tries to find a behavior optimizing some figure of merit. Some of the first approaches investigated settings where this exploration can be sped-up, by considering quantum analogs of classical environments, which can then be queried in superposition. If the environments have a strict periodic structure in time (i.e. are strictly episodic), such environments can be effectively converted to conventional oracles encountered in quantum information. However, in general environments, we obtain scenarios that generalize standard oracle tasks. In this work, we consider one such generalization, where the environment is not strictly episodic, which is mapped to an oracle identification setting with a changing oracle. We analyze this case and show that standard amplitude-amplification techniques can, with minor modifications, still be applied to achieve quadratic speed-ups. In addition, we prove that an algorithm based on Grover iterations is optimal for oracle identification even if the oracle changes over time in a way that the “rewarded space” is monotonically increasing. This result constitutes one of the first generalizations of quantum-accessible reinforcement learning.


Cybersecurity ◽  
2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Hui Liu ◽  
Li Yang

AbstractThe quantum security of lightweight block ciphers is receiving more and more attention. However, the existing quantum attacks on lightweight block ciphers only focused on the quantum exhaustive search, while the quantum attacks combined with classical cryptanalysis methods haven’t been well studied. In this paper, we study quantum key recovery attack on SIMON32/64 using Quantum Amplitude Amplification algorithm in Q1 model. At first, we reanalyze the quantum circuit complexity of quantum exhaustive search on SIMON32/64. We estimate the Clifford gates count more accurately and reduce the T gate count. Also, the T-depth and full depth is reduced due to our minor modifications. Then, using four differentials given by Biryukov in FSE 2014 as our distinguisher, we give our quantum key recovery attack on 19-round SIMON32/64. We treat the two phases of key recovery attack as two QAA instances separately, and the first QAA instance consists of four sub-QAA instances. Then, we design the quantum circuit of these two QAA instances and estimate their corresponding quantum circuit complexity. We conclude that the quantum circuit of our quantum key recovery attack is lower than quantum exhaustive search. Our work firstly studies the quantum dedicated attack on SIMON32/64. And this is the first work to study the complexity of quantum dedicated attacks from the perspective of quantum circuit complexity, which is a more fine-grained analysis of quantum dedicated attacks’ complexity.


2021 ◽  
Vol 2 (2) ◽  
pp. 1-27
Author(s):  
Debajyoti Bera ◽  
Sapv Tharrmashastha

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.


Algorithms ◽  
2020 ◽  
Vol 13 (11) ◽  
pp. 305
Author(s):  
Giuseppe Di Molfetta ◽  
Basile Herzog

We provide numerical evidence that the nonlinear searching algorithm introduced by Wong and Meyer, rephrased in terms of quantum walks with effective nonlinear phase, can be extended to the finite 2-dimensional grid, keeping the same computational advantage with respect to the classical algorithms. For this purpose, we have considered the free lattice Hamiltonian, with linear dispersion relation introduced by Childs and Ge The numerical simulations showed that the walker finds the marked vertex in O(N1/4log3/4N) steps, with probability O(1/logN), for an overall complexity of O(N1/4log5/4N), using amplitude amplification. We also proved that there exists an optimal choice of the walker parameters to avoid the time measurement precision affecting the complexity searching time of the algorithm.


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