Variational Quantum Optimization with Multi-Basis Encodings

Abstract Despite extensive research efforts, few quantum algorithms for classical optimization demonstrate realizable advantage. The utility of many quantum algorithms is limited by high requisite circuit depth and nonconvex optimization landscapes. We tackle these challenges to quantum advantage with two new variational quantum algorithms, which utilize multi-basis graph encodings and nonlinear activation functions to outperform existing methods with remarkably shallow quantum circuits. Both algorithms provide a polynomial reduction in measurement complexity and either a factor of two speedup a factor of two reduction in quantum resources. Typically, the classical simulation of such algorithms with many qubits is impossible due to the exponential scaling of traditional quantum formalism and the limitations of tensor networks. Nonetheless, the shallow circuits and moderate entanglement of our algorithms, combined with efficient tensor method-based simulation, enable us to successfully optimize the MaxCut of high-connectivity global graphs with up to 512 nodes (qubits) on a single GPU.

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Applying Quantum Optimization Algorithms for Linear Programming

10.20944/preprints201703.0238.v1 ◽

2017 ◽

Author(s):

Mert Side ◽

Volkan Erol

Keyword(s):

Resource Allocation ◽

Linear Programming ◽

Production Scheduling ◽

Optimization Problem ◽

Optimization Problems ◽

Quantum Algorithms ◽

Quantum Algorithm ◽

Quantum Computers ◽

Quantum Optimization ◽

Special Case

Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we looked at the duality of resource allocation problems. First, we chose a real world optimization problem and looked at its solution with linear programming. Then, we restudied this problem with a quantum algorithm in order to understand whether if there is a speedup of the solution. The improvement in computation is analysed and some interesting results are reported.

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Warm-starting quantum optimization

Quantum ◽

10.22331/q-2021-06-17-479 ◽

2021 ◽

Vol 5 ◽

pp. 479

Author(s):

Daniel J. Egger ◽

Jakub Mareček ◽

Stefan Woerner

Keyword(s):

Combinatorial Optimization ◽

Optimization Problems ◽

Quantum Algorithms ◽

Quantum Algorithm ◽

Randomized Rounding ◽

Initial State ◽

Warm Start ◽

Performance Guarantees ◽

Unique Games ◽

Quantum Optimization

There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.

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Applying Quantum Optimization Algorithms for Linear Programming

10.20944/preprints201703.0238.v3 ◽

2017 ◽

Author(s):

Mert Side ◽

Volkan Erol

Keyword(s):

Resource Allocation ◽

Linear Programming ◽

Production Scheduling ◽

Optimization Problem ◽

Optimization Problems ◽

Quantum Algorithms ◽

Quantum Algorithm ◽

Quantum Computers ◽

Quantum Optimization ◽

Special Case

Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we looked at the duality of resource allocation problems. First, we chose a real world optimization problem and looked at its solution with linear programming. Then, we restudied this problem with a quantum algorithm in order to understand whether if there is a speedup of the solution. The improvement in computation is analysed and some interesting results are reported.

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AN OBSERVER-BASED DE-QUANTISATION OF DEUTSCH'S ALGORITHM

International Journal of Foundations of Computer Science ◽

10.1142/s0129054111007940 ◽

2011 ◽

Vol 22 (01) ◽

pp. 191-201

Author(s):

CRISTIAN S. CALUDE ◽

MATTEO CAVALIERE ◽

RADU MARDARE

Keyword(s):

Quantum Computing ◽

Quantum Measurement ◽

Quantum Algorithms ◽

Quantum Algorithm ◽

Classical Solutions ◽

Finite State Automata ◽

Computational Problem ◽

Classical Simulation ◽

Finite State ◽

Classical Computing

Deutsch's problem is the simplest and most frequently examined example of computational problem used to demonstrate the superiority of quantum computing over classical computing. Deutsch's quantum algorithm has been claimed to be faster than any classical ones solving the same problem, only to be discovered later that this was not the case. Various de-quantised solutions for Deutsch's quantum algorithm—classical solutions which are as efficient as the quantum one—have been proposed in the literature. These solutions are based on the possibility of classically simulating "superpositions", a key ingredient of quantum algorithms, in particular, Deutsch's algorithm. The de-quantisation proposed in this note is based on a classical simulation of the quantum measurement achieved with a model of observed system. As in some previous de-quantisations of Deutsch's quantum algorithm, the resulting de-quantised algorithm is deterministic. Finally, we classify observers (as finite state automata) that can solve the problem in terms of their "observational power".

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Applying Quantum Optimization Algorithms for Linear Programming

10.20944/preprints201703.0238.v2 ◽

2017 ◽

Author(s):

Mert Side ◽

Volkan Erol

Keyword(s):

Resource Allocation ◽

Linear Programming ◽

Production Scheduling ◽

Optimization Problem ◽

Optimization Problems ◽

Quantum Algorithms ◽

Quantum Algorithm ◽

Quantum Computers ◽

Quantum Optimization ◽

Special Case

Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we looked at the duality of resource allocation problems. First, we chose a real world optimization problem and looked at its solution with linear programming. Then, we restudied this problem with a quantum algorithm in order to understand whether if there is a speedup of the solution. The improvement in computation is analysed and some interesting results are reported.

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Simulating quantum computers with probabilistic methods

Quantum Information and Computation ◽

10.26421/qic11.9-10-5 ◽

2011 ◽

Vol 11 (9&10) ◽

pp. 784-812 ◽

Cited By ~ 1

Author(s):

Maarten Van den Nest

Keyword(s):

Fourier Spectrum ◽

Quantum Algorithms ◽

Quantum Algorithm ◽

Probabilistic Methods ◽

Quantum Circuits ◽

Quantum Computers ◽

Partition Functions ◽

Classical Simulation ◽

Last Stage ◽

Standard Techniques

We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate new classes of classically simulatable quantum circuits where standard techniques relying on the exact computation of measurement probabilities fail to provide efficient simulations. For example, we show how various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform and other circuits are classically simulatable. We also prove that sparse quantum circuits as well as circuits composed of CNOT and $\exp[{i\theta X}]$ gates can be simulated classically. In a second part, we apply our results to the simulation of quantum algorithms. It is shown that a recent quantum algorithm, concerned with the estimation of Potts model partition functions, can be simulated efficiently classically. Finally, we show that the exponential speed-ups of Simon's and Shor's algorithms crucially depend on the very last stage in these algorithms, dealing with the classical postprocessing of the measurement outcomes. Specifically, we prove that both algorithms would be classically simulatable if the function classically computed in this step had a sufficiently peaked Fourier spectrum.

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Bang-bang control as a design principle for classical and quantum optimization algorithms

Quantum Information and Computation ◽

10.26421/qic19.5-6-4 ◽

2019 ◽

Vol 19 (5&6) ◽

pp. 424-446

Author(s):

Aniruddha Bapat ◽

Stephen Jordan

Keyword(s):

Simulated Annealing ◽

Optimization Algorithm ◽

Control Strategy ◽

Design Principle ◽

Quantum Algorithms ◽

Optimization Algorithms ◽

Successful Control ◽

Problem Instances ◽

Quantum Optimization ◽

Bang Bang Control

Physically motivated classical heuristic optimization algorithms such as simulated annealing (SA) treat the objective function as an energy landscape, and allow walkers to escape local minima. It has been argued that quantum properties such as tunneling may give quantum algorithms advantage in finding ground states of vast, rugged cost landscapes. Indeed, the Quantum Adiabatic Algorithm (QAO) and the recent Quantum Approximate Optimization Algorithm (QAOA) have shown promising results on various problem instances that are considered classically hard. Here, building on previous observations from \cite{mcclean2016, Yang2017}, we argue that the type of control strategy used by the optimization algorithm may be crucial to its success. Along with SA, QAO, and QAOA, we define a new, bang-bang version of simulated annealing, BBSA, and study the performance of these algorithms on two well-studied problem instances from the literature. Both classically and quantumly, the successful control strategy is found to be bang-bang, exponentially outperforming the quasistatic analogues on the same instances. Lastly, we construct O(1)-depth QAOA protocols for a class of symmetric cost functions, and provide an accompanying physical picture.

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Practical Quantum Computing

ACM Transactions on Quantum Computing ◽

10.1145/3430030 ◽

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.

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Deep neural networks for quantum circuit mapping

Neural Computing and Applications ◽

10.1007/s00521-021-06009-3 ◽

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.

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