scholarly journals Quantum and classical algorithms for approximate submodular function minimization

2019 ◽  
Vol 19 (15&16) ◽  
pp. 1325-1349
Author(s):  
Yassine Hamoudi ◽  
Patrick Rebentrost ◽  
Ansis Rosmanis ◽  
Miklos Santha
Keyword(s):  
Quantum Algorithms ◽  
Optimization Theory ◽  
Quantum Algorithm ◽  
Strongly Polynomial Algorithm ◽  
Discrete Probability ◽  
Submodular Optimization ◽  
Support Size ◽  
Important Field ◽  
The Cost ◽  
Submodular Function Minimization

Submodular functions are set functions mapping every subset of some ground set of size n into the real numbers and satisfying the diminishing returns property. Submodular minimization is an important field in discrete optimization theory due to its relevance for various branches of mathematics, computer science and economics. The currently fastest strongly polynomial algorithm for exact minimization~\cite{LSW15} runs in time \so{n^3 \cdot \eo + n^4} where \eo denotes the cost to evaluate the function on any set. For functions with range [-1,1], the best \eps-additive approximation algorithm~\cite{CLSW17} runs in time \so{n^{5/3}/\eps^{2} \cdot \eo}. In this paper we present a classical and a quantum algorithm for approximate submodular minimization. Our classical result improves on the algorithm of \cite{CLSW17} and runs in time \so{n^{3/2}/\eps^2 \cdot \eo}. Our quantum algorithm is, up to our knowledge, the first attempt to use quantum computing for submodular optimization. The algorithm runs in time \so{n^{5/4}/\eps^{5/2} \cdot \log(1/\eps) \cdot \eo}. The main ingredient of the quantum result is a new method for sampling with high probability T independent elements from any discrete probability distribution of support size n in time \bo{\sqrt{Tn}}. Previous quantum algorithms for this problem were of complexity \bo{T\sqrt{n}}.

scholarly journals Smooth input preparation for quantum and quantum-inspired machine learning

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Zhikuan Zhao ◽  
Jack K. Fitzsimons ◽  
Patrick Rebentrost ◽  
Vedran Dunjko ◽  
Joseph F. Fitzsimons
Keyword(s):  
Machine Learning ◽  
Quantum Algorithms ◽  
Machine Learning Algorithms ◽  
Low Rank ◽  
Smoothed Analysis ◽  
Machine Learning Applications ◽  
Data Points ◽  
Input Model ◽  
The Cost ◽  
Input Perturbation

AbstractMachine learning has recently emerged as a fruitful area for finding potential quantum computational advantage. Many of the quantum-enhanced machine learning algorithms critically hinge upon the ability to efficiently produce states proportional to high-dimensional data points stored in a quantum accessible memory. Even given query access to exponentially many entries stored in a database, the construction of which is considered a one-off overhead, it has been argued that the cost of preparing such amplitude-encoded states may offset any exponential quantum advantage. Here we prove using smoothed analysis that if the data analysis algorithm is robust against small entry-wise input perturbation, state preparation can always be achieved with constant queries. This criterion is typically satisfied in realistic machine learning applications, where input data is subjective to moderate noise. Our results are equally applicable to the recent seminal progress in quantum-inspired algorithms, where specially constructed databases suffice for polylogarithmic classical algorithm in low-rank cases. The consequence of our finding is that for the purpose of practical machine learning, polylogarithmic processing time is possible under a general and flexible input model with quantum algorithms or quantum-inspired classical algorithms in the low-rank cases.

scholarly journals Practical Quantum Computing

2021 ◽  
Vol 2 (1) ◽  
pp. 1-35
Author(s):  
Adrien Suau ◽  
Gabriel Staffelbach ◽  
Henri Calandra
Keyword(s):  
Wave Equation ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Equation Solving ◽  
Small Quantum ◽  
Quantum Wave ◽  
Variational Algorithms ◽  
The One

In the last few years, several quantum algorithms that try to address the problem of partial differential equation solving have been devised: on the one hand, “direct” quantum algorithms that aim at encoding the solution of the PDE by executing one large quantum circuit; on the other hand, variational algorithms that approximate the solution of the PDE by executing several small quantum circuits and making profit of classical optimisers. In this work, we propose an experimental study of the costs (in terms of gate number and execution time on a idealised hardware created from realistic gate data) associated with one of the “direct” quantum algorithm: the wave equation solver devised in [32]. We show that our implementation of the quantum wave equation solver agrees with the theoretical big-O complexity of the algorithm. We also explain in great detail the implementation steps and discuss some possibilities of improvements. Finally, our implementation proves experimentally that some PDE can be solved on a quantum computer, even if the direct quantum algorithm chosen will require error-corrected quantum chips, which are not believed to be available in the short-term.

scholarly journals Deep neural networks for quantum circuit mapping

2021 ◽  
Author(s):  
Giovanni Acampora ◽  
Roberto Schiattarella
Keyword(s):  
Neural Networks ◽  
Deep Neural Networks ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Machine Learning Techniques ◽  
Quantum Computers ◽  
Physical Constraints ◽  
Proper Mapping ◽  
Correct Execution

AbstractQuantum computers have become reality thanks to the effort of some majors in developing innovative technologies that enable the usage of quantum effects in computation, so as to pave the way towards the design of efficient quantum algorithms to use in different applications domains, from finance and chemistry to artificial and computational intelligence. However, there are still some technological limitations that do not allow a correct design of quantum algorithms, compromising the achievement of the so-called quantum advantage. Specifically, a major limitation in the design of a quantum algorithm is related to its proper mapping to a specific quantum processor so that the underlying physical constraints are satisfied. This hard problem, known as circuit mapping, is a critical task to face in quantum world, and it needs to be efficiently addressed to allow quantum computers to work correctly and productively. In order to bridge above gap, this paper introduces a very first circuit mapping approach based on deep neural networks, which opens a completely new scenario in which the correct execution of quantum algorithms is supported by classical machine learning techniques. As shown in experimental section, the proposed approach speeds up current state-of-the-art mapping algorithms when used on 5-qubits IBM Q processors, maintaining suitable mapping accuracy.

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 A Short Path Quantum Algorithm for Exact Optimization

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 78 ◽  
Author(s):  
M. B. Hastings
Keyword(s):  
Hybrid Algorithm ◽  
Optimal Solution ◽  
Quantum Algorithm ◽  
Low Energy ◽  
Weighted Sum ◽  
Natural Hybrid ◽  
Sum Of Products ◽  
Time Required ◽  
The Cost ◽  
Exact Optimization

We give a quantum algorithm to exactly solve certain problems in combinatorial optimization, including weighted MAX-2-SAT as well as problems where the objective function is a weighted sum of products of Ising variables, all terms of the same degree D; this problem is called weighted MAX-ED-LIN2. We require that the optimal solution be unique for odd D and doubly degenerate for even D; however, we expect that the algorithm still works without this condition and we show how to reduce to the case without this assumption at the cost of an additional overhead. While the time required is still exponential, the algorithm provably outperforms Grover's algorithm assuming a mild condition on the number of low energy states of the target Hamiltonian. The detailed analysis of the runtime dependence on a tradeoff between the number of such states and algorithm speed: fewer such states allows a greater speedup. This leads to a natural hybrid algorithm that finds either an exact or approximate solution.

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 A quantum algorithm for uniform sampling of models of propositional logic based on quantum probability

2016 ◽  
Vol 16 (1) ◽  
pp. 57-65
Author(s):  
Radhakrishnan Balu ◽  
Dale Shires ◽  
Raju Namburu
Keyword(s):  
Fixed Points ◽  
Propositional Logic ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Probability ◽  
Stochastic Flows ◽  
Von Neumann ◽  
Uniform Sampling ◽  
Successful Technique ◽  
D Wave

We describe a class of quantum algorithms to generate models of propositional logic with equal probability. We consider quantum stochastic flows that are the quantum analogues of classical Markov chains and establish a relation between fixed points on the two flows. We construct chains inspired by von Neumann algorithms using uniform measures as fixed points to construct the corresponding irreversible quantum stochastic flows. We formulate sampling models of propositions in the framework of adiabatic quantum computing and solve the underlying satisfiability instances. Satisfiability formulation is an important and successful technique in modeling the decision theoretic problems in a classical context. We discuss some features of the proposed algorithms tested on an existing quantum annealer D-Wave II extending the simulation of decision theoretic problems to a quantum context.

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 Quantum algorithms for Second-Order Cone Programming and Support Vector Machines

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 427
Author(s):  
Iordanis Kerenidis ◽  
Anupam Prakash ◽  
Dániel Szilágyi
Keyword(s):  
Quantum Algorithms ◽  
Matrix Multiplication ◽  
Optimal Solution ◽  
Quantum Algorithm ◽  
Second Order ◽  
Support Vector ◽  
Scaling Exponent ◽  
Cone Programming ◽  
Second Order Cone Programming ◽  
Second Order Cone

We present a quantum interior-point method (IPM) for second-order cone programming (SOCP) that runs in time O~(nrζκδ2log⁡(1/ϵ)) where r is the rank and n the dimension of the SOCP, δ bounds the distance of intermediate solutions from the cone boundary, ζ is a parameter upper bounded by n, and κ is an upper bound on the condition number of matrices arising in the classical IPM for SOCP. The algorithm takes as its input a suitable quantum description of an arbitrary SOCP and outputs a classical description of a δ-approximate ϵ-optimal solution of the given problem.Furthermore, we perform numerical simulations to determine the values of the aforementioned parameters when solving the SOCP up to a fixed precision ϵ. We present experimental evidence that in this case our quantum algorithm exhibits a polynomial speedup over the best classical algorithms for solving general SOCPs that run in time O(nω+0.5) (here, ω is the matrix multiplication exponent, with a value of roughly 2.37 in theory, and up to 3 in practice). For the case of random SVM (support vector machine) instances of size O(n), the quantum algorithm scales as O(nk), where the exponent k is estimated to be 2.59 using a least-squares power law. On the same family random instances, the estimated scaling exponent for an external SOCP solver is 3.31 while that for a state-of-the-art SVM solver is 3.11.

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.

Sign in / Sign up

Export Citation Format

Share Document