Experimental Analysis of Quantum Annealers and Hybrid Solvers Using Benchmark Optimization Problems
Abstract
:1. Introduction
1.1. Related Work
1.2. Organization
2. The Standard QUBO Formulation
- asserts that every vertex must appear in exactly one position of the tour;
- states that each position of the tour is occupied by precisely one vertex;
- requires that the tour is comprised of edges that “really” exist. If the tour, mistakenly, contains a “phantom” edge, i.e., an edge belonging to , then this will incur an energy penalty;
- computes the cost or weight of the tour. A tour minimizing this Hamiltonian is an optimal tour.
- The Hamiltonians and , as given in (13) and (14), require a constant term, namely, n, linear constraints, and quadratic constraints, which brings the total number of constraints to . The linear constraints are the same in both and , but the quadratic constraints are different;
- If the graph is complete, then does not add any new constraints. To be precise, involves quadratic constraints, which are also present in the Hamiltonian. Hence, only enhances the relative weight of existing constraints;
- If the graph is not complete, then creates, for each missing edge that is not a self-loop, i.e., , n additional constraints. If k such edges are missing, there will be additional quadratic constraints in total. This fact is experimentally validated from the results presented in Section 5.3;
- In a complete graph with n nodes, introduces quadratic constraints, where stands for the number of edges in the graph. This formula can be generalized further to the case where the graph is not complete and has m edges. In such a case, will add quadratic constraints.
- In a complete graph, requires binary variables and constraints in total (recall that the linear constraints are the same). If the graph is not complete, it requires more constraints: specifically, , where k is the number of missing edges;
- For a complete graph, requires binary variables, quadratic constraints, and constraints in total. If the graph is not complete, it requires , where m is the number of existing edges and k is the number of missing edges;
- In the typical and practically important case where we have a simple graph, that is, a graph with no self-loops and no parallel edges, then . The previous formulas now give quadratic constraints and total constraints. In this case, the number of constraints required for the Hamiltonian is equal to the number of constraints required when the graph is complete.
3. The Matrix QUBO Formulation
4. Normalizing Graph Weights
Algorithm 1: Min-max normalization. |
|
5. Experimenting on D-Wave’s QPU
5.1. HCP Using QPU
5.2. TSP Using QPU
5.3. The Effect of Graph Connectivity on Qubit Usage
6. Experimenting on D-Wave’s Leap’s Hybrid Solvers
6.1. HCP Using LeapHybridSampler
6.2. TSP Using LeapHybridSampler
7. Discussion and Conclusions
- D-Wave’s Advantage_system4.1 is more efficient than the Advantage_system1.1 in its use of qubits for solving a problem and provides more consistently correct solutions;
- It is possible to run the Burma14 instance of the TSPLIB library on a quantum annealer using the aforementioned methods, although it cannot provide a “correct” solution because of the annealer’s limitations;
- The more connected a graph is, the fewer qubits are needed for it to be solved by the quantum annealers, as fewer constraints need to be imposed;
- Hybrid solvers always provide a correct solution to a problem and never break the constraints of a QUBO model, even for arbitrarily big problems.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Number of Binary Variables: | |||
---|---|---|---|
Number of Constraints | |||
Hamiltonian | Linear | Quadratic | Total |
0 | 0 | 0 | |
0 | |||
Number of Binary Variables: | |||
---|---|---|---|
Number of Constraints | |||
Hamiltonian | Linear | Quadratic | Total |
0 | |||
0 | |||
Number of Constraints when | |||
---|---|---|---|
Hamiltonian | Linear | Quadratic | Total |
Nodes | Solver | Runs | QAT (μs) | Qubits | Success Rate (%) |
---|---|---|---|---|---|
6 | Advantage_system1.1 | 20 | 126,643.89 | 126.83 | 100.0 |
6 | Advantage_system4.1 | 20 | 140,461.42 | 120.27 | 100.0 |
8 | Advantage_system1.1 | 20 | 147,772.21 | 388.25 | 15.0 |
8 | Advantage_system4.1 | 20 | 160,148.34 | 372.60 | 35.0 |
10 | Advantage_system1.1 | 20 | 168,190.20 | 1010.20 | 0.0 |
10 | Advantage_system4.1 | 20 | 209,123.62 | 910.15 | 0.0 |
12 | Advantage_system1.1 | 20 | 170,374.85 | 2091.70 | 0.0 |
12 | Advantage_system4.1 | 20 | 243,015.98 | 1924.95 | 0.0 |
14 | Advantage_system1.1 | 20 | 188,910.52 | 3913.45 | 0.0 |
14 | Advantage_system4.1 | 20 | 272,583.67 | 3413.00 | 0.0 |
Nodes | Solver | Runs | QAT (μs) | Qubits | Cost | Success Rate (%) |
---|---|---|---|---|---|---|
14 | Advantage_system4.1 | 20 | 282,016.86 | 4145.25 | 4737.70 | 0.0 |
Nodes | Solver | Runs | QAT (μs) | Run Time (μs) | Success Rate (%) |
---|---|---|---|---|---|
6 | hybrid_binary_quadratic_model_version2 | 20 | 66,339.20 | 2,994,650.00 | 100.0 |
8 | hybrid_binary_quadratic_model_version2 | 20 | 64,866.25 | 2,992,684.90 | 100.0 |
10 | hybrid_binary_quadratic_model_version2 | 20 | 63,423.32 | 2,994,797.68 | 100.0 |
12 | hybrid_binary_quadratic_model_version2 | 20 | 49,859.00 | 3,003,757.63 | 100.0 |
14 | hybrid_binary_quadratic_model_version2 | 20 | 35,612.10 | 3,008,348.10 | 100.0 |
Nodes | Solver | Runs | QAT (μs) | Run Time (μs) | Cost | Success Rate (%) |
---|---|---|---|---|---|---|
17 | hybrid_binary_quadratic_model_version2 | 20 | 14,217.95 | 2,995,196.1 | 2417.15 | 100.0 |
21 | hybrid_binary_quadratic_model_version2 | 20 | 7181.00 | 3,003,955.9 | 3775.75 | 100.0 |
120 | hybrid_binary_quadratic_model_version2 | 5 | 107,764.40 | 75,028,024.6 | 25,871.20 | 100.0 |
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Stogiannos, E.; Papalitsas, C.; Andronikos, T. Experimental Analysis of Quantum Annealers and Hybrid Solvers Using Benchmark Optimization Problems. Mathematics 2022, 10, 1294. https://doi.org/10.3390/math10081294
Stogiannos E, Papalitsas C, Andronikos T. Experimental Analysis of Quantum Annealers and Hybrid Solvers Using Benchmark Optimization Problems. Mathematics. 2022; 10(8):1294. https://doi.org/10.3390/math10081294
Chicago/Turabian StyleStogiannos, Evangelos, Christos Papalitsas, and Theodore Andronikos. 2022. "Experimental Analysis of Quantum Annealers and Hybrid Solvers Using Benchmark Optimization Problems" Mathematics 10, no. 8: 1294. https://doi.org/10.3390/math10081294
APA StyleStogiannos, E., Papalitsas, C., & Andronikos, T. (2022). Experimental Analysis of Quantum Annealers and Hybrid Solvers Using Benchmark Optimization Problems. Mathematics, 10(8), 1294. https://doi.org/10.3390/math10081294