Quantum Integer Programming: an Annealing approach to the Job Shop Scheduling problem

openIn this thesis i developed a complete workflow to get optimal solutions of the JSS problem. During this work i deeply analized many of the steps required in the usage of quantum annealers. In detail i tested 2 different approaches to the minor-embedding procedure and compared them with the current default euristic implemented by Dwave. I also tested an hybrid algorithm wich extract the elements of the Graver basis of the problem and then augment an initial feasible solution to obtain an optimal one. The obtained results shows that the quantum annealers can get to optimal solutions in a competitive time but, due to the limited numer of working qubits and the sparse connectivity among them, only small instances can be efficiently solved with the current hardware.In this thesis i developed a complete workflow to get optimal solutions of the JSS problem. During this work i deeply analized many of the steps required in the usage of quantum annealers. In detail i tested 2 different approaches to the minor-embedding procedure and compared them with the current default euristic implemented by Dwave. I also tested an hybrid algorithm wich extract the elements of the Graver basis of the problem and then augment an initial feasible solution to obtain an optimal one. The obtained results shows that the quantum annealers can get to optimal solutions in a competitive time but, due to the limited numer of working qubits and the sparse connectivity among them, only small instances can be efficiently solved with the current hardware

Benchmarking Advantage and D-Wave 2000Q quantum annealers with exact cover problems

We benchmark the quantum processing units of the largest quantum annealers to date, the 5000+ qubit quantum annealer Advantage and its 2000+ qubit predecessor D-Wave 2000Q, using tail assignment and exact cover problems from aircraft scheduling scenarios. The benchmark set contains small, intermediate, and large problems with both sparsely connected and almost fully connected instances. We find that Advantage outperforms D-Wave 2000Q for almost all problems, with a notable increase in success rate and problem size. In particular, Advantage is also able to solve the largest problems with 120 logical qubits that D-Wave 2000Q cannot solve anymore. Furthermore, problems that can still be solved by D-Wave 2000Q are solved faster by Advantage. We find, however, that D-Wave 2000Q can achieve better success rates for sparsely connected problems that do not require the many new couplers present on Advantage, so improving the connectivity of a quantum annealer does not per se improve its performance.Comment: new experiments to test the conjecture about unused couplers (appendix B

Benchmarking quantum annealing with maximum cardinality matching problems

We benchmark Quantum Annealing (QA) vs. Simulated Annealing (SA) with a focus on the impact of the embedding of problems onto the different topologies of the D-Wave quantum annealers. The series of problems we study are especially designed instances of the maximum cardinality matching problem that are easy to solve classically but difficult for SA and, as found experimentally, not easy for QA either. In addition to using several D-Wave processors, we simulate the QA process by numerically solving the time-dependent Schrödinger equation. We find that the embedded problems can be significantly more difficult than the unembedded problems, and some parameters, such as the chain strength, can be very impactful for finding the optimal solution. Thus, finding a good embedding and optimal parameter values can improve the results considerably. Interestingly, we find that although SA succeeds for the unembedded problems, the SA results obtained for the embedded version scale quite poorly in comparison with what we can achieve on the D-Wave quantum annealers

Embedding of Complete Graphs in Broken Chimera Graphs

In order to solve real world combinatorial optimization problems with a D-Wave quantum annealer it is necessary to embed the problem at hand into the D-Wave hardware graph, namely Chimera or Pegasus. Most hard real world problems exhibit a strong connectivity. For the worst case scenario of a complete graph, there exists an efficient solution for the embedding into the ideal Chimera graph. However, since real machines almost always have broken qubits it is necessary to find an embedding into the broken hardware graph. We present a new approach to the problem of embedding complete graphs into broken Chimera graphs. This problem can be formulated as an optimization problem, more precisely as a matching problem with additional linear constraints. Although being NP-hard in general it is fixed parameter tractable in the number of inaccessible vertices in the Chimera graph. We tested our exact approach on various instances of broken hardware graphs, both related to real hardware as well as randomly generated. For fixed runtime, we were able to embed larger complete graphs compared to previous, heuristic approaches. As an extension, we developed a fast heuristic algorithm which enables us to solve even larger instances. We compared the performance of our heuristic and exact approaches.Comment: 26 pages, 9 figures, 2 table

Quantum Radio Astronomy: Quantum Linear Solvers for Redundant Baseline Calibration

The computational requirements of future large scale radio telescopes are expected to scale well beyond the capabilities of conventional digital resources. Current and planned telescopes are generally limited in their scientific potential by their ability to efficiently process the vast volumes of generated data. To mitigate this problem, we investigate the viability of emerging quantum computers for radio astronomy applications. In this a paper we demonstrate the potential use of variational quantum linear solvers in Noisy Intermediate Scale Quantum (NISQ) computers and combinatorial solvers in quantum annealers for a radio astronomy calibration pipeline. While we demonstrate that these approaches can lead to satisfying results when integrated in calibration pipelines, we show that current restrictions of quantum hardware limit their applicability and performance

Advanced unembedding techniques for quantum annealers

The D-Wave quantum annealers make it possible to obtain high quality solutions of NP-hard problems by mapping a problem in a QUBO (quadratic unconstrained binary optimization) or Ising form to the physical qubit connectivity structure on the D-Wave chip. However, the latter is restricted in that only a fraction of all pairwise couplers between physical qubits exists. Modeling the connectivity structure of a given problem instance thus necessitates the computation of a minor embedding of the variables in the problem specification onto the logical qubits, which consist of several physical qubits "chained" together to act as a logical one. After annealing, it is however not guaranteed that all chained qubits get the same value (-1 or +1 for an Ising model, and 0 or 1 for a QUBO), and several approaches exist to assign a final value to each logical qubit (a process called "unembedding"). In this work, we present tailored unembedding techniques for four important NP-hard problems: the Maximum Clique, Maximum Cut, Minimum Vertex Cover, and Graph Partitioning problems. Our techniques are simple and yet make use of structural properties of the problem being solved. Using Erd\H{o}s-R\'enyi random graphs as inputs, we compare our unembedding techniques to three popular ones (majority vote, random weighting, and minimize energy). We demonstrate that our proposed algorithms outperform the currently available ones in that they yield solutions of better quality, while being computationally equally efficient

Skipper: Improving the Reach and Fidelity of Quantum Annealers by Skipping Long Chains

Quantum Annealers (QAs) operate as single-instruction machines, lacking a SWAP operation to overcome limited qubit connectivity. Consequently, multiple physical qubits are chained to form a program qubit with higher connectivity, resulting in a drastically diminished effective QA capacity by up to 33x. We observe that in QAs: (a) chain lengths exhibit a power-law distribution, a few dominant chains holding substantially more qubits than others; and (b) about 25% of physical qubits remain unused, getting isolated between these chains. We propose Skipper, a software technique that enhances the capacity and fidelity of QAs by skipping dominant chains and substituting their program qubit with two readout results. Using a 5761-qubit QA, we demonstrate that Skipper can tackle up to 59% (Avg. 28%) larger problems when eleven chains are skipped. Additionally, Skipper can improve QA fidelity by up to 44% (Avg. 33%) when cutting five chains (32 runs). Users can specify up to eleven chain cuts in Skipper, necessitating about 2,000 distinct quantum executable runs. To mitigate this, we introduce Skipper-G, a greedy scheme that skips sub-problems less likely to hold the global optimum, executing a maximum of 23 quantum executables with eleven chain trims. Skipper-G can boost QA fidelity by up to 41% (Avg. 29%) when cutting five chains (11 runs)