# Quantum Annealing in the NISQ Era: Railway Conflict Management

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## Abstract

**:**

## 1. Introduction

## 2. Railway Dispatching Problem on Single-Track Lines

#### 2.1. Problem Description

#### 2.2. Existing Algorithms

- Order and precedence variables prescribe the order in which a machine processes jobs, i.e., the order of trains passing a given block section in the railway dispatching problem on single-track lines.
- Discrete time units, in which the decision variables belong to discretized time instants; the binary variables describe whether the event happens at a given time.

#### 2.3. Quantum Annealing and Related Methods

#### 2.3.1. Ising-Based Solvers

#### 2.3.2. Quantum Annealing

#### 2.3.3. Classical Algorithms for Solving Ising Problems

## 3. Our Model

#### 3.1. Integer Formulation of the Constaints

**The minimum passing time condition**ensures that no block sections are passed by any train faster than allowed:

**The single-block occupation**ensures that at most one train can be present in a block section at a time.

**The deadlock condition**ensures that no pairs of trains heading in the opposite direction will be waiting for each other to pass the same blocks:

**The rolling stock circulation condition**ensures the minimal technological time $R(j,{j}^{\prime})$ for a given train set arriving as train j at its terminating station ${s}_{j,\mathrm{end}}$ before operating again as train ${j}^{\prime}$:

#### 3.2. 0-1 Formulation

**The minimum passing time condition**defined in Equation (9) becomes

**The single-block occupation condition**from Equation (10) follows that

**The deadlock condition**is to be addressed for two trains heading in the opposite direction; from Equation (11), it follows that

**The rolling stock circulation condition**is defined in Equation (12) and can be rewritten as

#### 3.3. QUBO Formulation: Penalties

## 4. Results

- Railway line No. 216 (Nidzica–Olsztynek section);
- Railway line No. 191 (Goleszów–Wisła Uzdrowisko section).

#### 4.1. The Studied Network Segment

- 1.
- A moderate delay of the Inter-City train setting off from station block 1; see Figure S1a of the Supplemental Materials.
- 2.
- A moderate delay of all trains setting off from station block 1; see Figure S1b.
- 3.
- A significant delay of some trains setting off from station block 1; see Figure S1c.
- 4.
- A large delay of the Inter-City train setting off from station block 1; see Figure S1d.

#### 4.2. Simple Heuristics

#### 4.3. Quantum and Calculated QUBO Solutions

#### 4.3.1. Exact Calculation of the Low-Energy Spectrum

#### 4.3.2. Classical Algorithms for the Linear (Integer Programming) IP Model and QUBO

#### 4.3.3. Quantum Annealing on the D-Wave Machine

#### 4.4. Initial Studies on the D-Wave Advantage Machine

## 5. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Raymer, M.G.; Monroe, C. The US National Quantum Initiative. Quantum Sci. Technol.
**2019**, 4, 020504. [Google Scholar] [CrossRef] - Riedel, M.; Kovacs, M.; Zoller, P.; Mlynek, J.; Calarco, T. Europe’s Quantum Flagship initiative. Quantum Sci. Technol.
**2019**, 4, 020501. [Google Scholar] [CrossRef] - Yamamoto, Y.; Sasaki, M.; Takesue, H. Quantum information science and technology in Japan. Quantum Sci. Technol.
**2019**, 4, 020502. [Google Scholar] [CrossRef] - Sussman, B.; Corkum, P.; Blais, A.; Cory, D.; Damascelli, A. Quantum Canada. Quantum Sci. Technol.
**2019**, 4, 020503. [Google Scholar] [CrossRef] - Roberson, T.M.; White, A.G. Charting the Australian quantum landscape. Quantum Sci. Technol.
**2019**, 4, 020505. [Google Scholar] [CrossRef] - Sanders, B.C. How to Build a Quantum Computer; IOP Publishing: Bristol, UK, 2017; pp. 2399–2891. [Google Scholar] [CrossRef]
- Aiello, C.D.; Awschalom, D.D.; Bernien, H.; Brower, T.; Brown, K.R.; Brun, T.A.; Caram, J.R.; Chitambar, E.; Felice, R.D.; Edmonds, K.M.; et al. Achieving a quantum smart workforce. Quantum Sci. Technol.
**2021**, 6, 030501. [Google Scholar] [CrossRef] - Roberson, T.; Leach, J.; Raman, S. Talking about public good for the second quantum revolution: Analysing quantum technology narratives in the context of national strategies. Quantum Sci. Technol.
**2021**, 6, 025001. [Google Scholar] [CrossRef] - Arute, F.; Arya, K.; Babbush, R.; Bacon, D.; Bardin, J.C.; Barends, R.; Biswas, R.; Boixo, S.; Brandao, F.G.S.L.; Buell, D.A.; et al. Quantum supremacy using a programmable superconducting processor. Nature
**2019**, 574, 505–510. [Google Scholar] [CrossRef] [Green Version] - Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum
**2018**, 2, 79. [Google Scholar] [CrossRef] - Dattani, N.; Szalay, S.; Chancellor, N. Pegasus: The second connectivity graph for large-scale quantum annealing hardware. arXiv
**2019**, arXiv:1901.07636. [Google Scholar] - Więckowski, A.; Deffner, S.; Gardas, B. Disorder-assisted graph coloring on quantum annealers. Phys. Rev. A
**2019**, 100, 062304. [Google Scholar] [CrossRef] [Green Version] - Sax, I.; Feld, S.; Zielinski, S.; Gabor, T.; Linnhoff-Popien, C.; Mauerer, W. Approximate approximation on a quantum annealer. In Proceedings of the 17th ACM International Conference on Computing Frontiers, Catania, Italy, 11–13 May 2020; pp. 108–117. [Google Scholar]
- Stollenwerk, T.; O’Gorman, B.; Venturelli, D.; Mandrà, S.; Rodionova, O.; Ng, H.; Sridhar, B.; Rieffel, E.G.; Biswas, R. Quantum Annealing Applied to De-Conflicting Optimal Trajectories for Air Traffic Management. IEEE Trans. Intell. Transp. Syst.
**2020**, 21, 285–297. [Google Scholar] [CrossRef] [Green Version] - Domino, K.; Koniorczyk, M.; Krawiec, K.; Jałowiecki, K.; Gardas, B. Quantum computing approach to railway dispatching and conflict management optimization on single-track railway lines. arXiv
**2020**, arXiv:2010.08227. [Google Scholar] - Grozea, C.; Hans, R.; Koch, M.; Riehn, C.; Wolf, A. Optimising Rolling Stock Planning including Maintenance with Constraint Programming and Quantum Annealing. arXiv
**2021**, arXiv:2109.07212. [Google Scholar] - Yarkoni, S.; Huck, A.; Schülldorf, H.; Speitkamp, B.; Tabrizi, M.S.; Leib, M.; Bäck, T.; Neukart, F. Solving the Shipment Rerouting Problem with Quantum Optimization Techniques. In Lecture Notes in Computer Science; Springer International Publishing: Cham, Switzerland, 2021; pp. 502–517. [Google Scholar] [CrossRef]
- Cacchiani, V.; Huisman, D.; Kidd, M.; Kroon, L.; Toth, P.; Veelenturf, L.; Wagenaar, J. An overview of recovery models and algorithms for real-time railway rescheduling. Transp. Res. Part B Methodol.
**2014**, 63, 15–37. [Google Scholar] [CrossRef] [Green Version] - Törnquist, J.; Persson, J.A. N-tracked railway traffic re-scheduling during disturbances. Transp. Res. Part B Methodol.
**2007**, 41, 342–362. [Google Scholar] [CrossRef] - Lamorgese, L.; Mannino, C.; Pacciarelli, D.; Krasemann, J.T. Handbook of Optimization in the Railway Industry. In Train Dispatching; Springer International Publishing: Cham, Switzerland, 2018; pp. 265–283. [Google Scholar] [CrossRef]
- Jensen, J.; Nielsen, O.; Prato, C. Passenger Perspectives in Railway Timetabling: A Literature Review. Transp. Rev.
**2016**, 36, 500–526. [Google Scholar] [CrossRef] - Wen, C.; Huang, P.; Li, Z.; Lessan, J.; Fu, L.; Jiang, C.; Xu, X. Train Dispatching Management With Data- Driven Approaches: A Comprehensive Review and Appraisal. IEEE Access
**2019**, 7, 114547–114571. [Google Scholar] [CrossRef] - Van Leeuwen, J. Handbook of Theoretical Computer Science (vol. A) Algorithms and Complexity; MIT Press: Cambridge, MA, USA, 1991. [Google Scholar]
- Cai, X.; Goh, C.J. A fast heuristic for the train scheduling problem. Comput. Oper. Res.
**1994**, 21, 499–510. [Google Scholar] [CrossRef] - Szpigel, B. Optimal train scheduling on a single line railway. Oper. Res.
**1973**, 72, 343–352. [Google Scholar] - Pinedo, M.L. Scheduling: Theory, Algorithms, and Systems, 3rd ed.; Springer Publishing Company, Incorporated: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Cordeau, J.F.; Toth, P.; Vigo, D. A Survey of Optimization Models for Train Routing and Scheduling. Transp. Sci.
**1998**, 32, 380–404. [Google Scholar] [CrossRef] - Törnquist, J. Computer-based decision support for railway traffic scheduling and dispatching: A review of models and algorithms. In Proceedings of the 5th Workshop on Algorithmic Methods and Models for Optimization of Railways (ATMOS’05), Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Palma de Mallorca, Spain, 14 September 2005. [Google Scholar]
- Dollevoet, T.; Huisman, D.; Schmidt, M.; Schöbel, A. Delay Propagation and Delay Management in Transportation Networks. In Handbook of Optimization in the Railway Industry; Springer International Publishing: Cham, Switzerland, 2018; pp. 285–317. [Google Scholar] [CrossRef]
- Corman, F.; Meng, L. A Review of Online Dynamic Models and Algorithms for Railway Traffic Management. IEEE Trans. Intell. Transp. Syst.
**2015**, 16, 1274–1284. [Google Scholar] [CrossRef] - Cacchiani, V.; Toth, P. Nominal and robust train timetabling problems. Eur. J. Oper. Res.
**2012**, 219, 727–737. [Google Scholar] [CrossRef] - Hansen, I. State-of-the-art of railway operations research. In Timetable Planning and Information Quality; WIT Press: Wessex, UK, 2010; pp. 35–47. [Google Scholar]
- Lange, J.; Werner, F. Approaches to modeling train scheduling problems as job-shop problems with blocking constraints. J. Sched.
**2018**, 21, 191–207. [Google Scholar] [CrossRef] - Mascis, A.; Pacciarelli, D. Job-shop scheduling with blocking and no-wait constraints. Eur. J. Oper. Res.
**2002**, 143, 498–517. [Google Scholar] [CrossRef] - D’Ariano, A.; Pacciarelli, D.; Pranzo, M. A branch and bound algorithm for scheduling trains in a railway network. Eur. J. Oper. Res.
**2007**, 183, 643–657. [Google Scholar] [CrossRef] - Venturelli, D.; Marchand, D.J.J.; Rojo, G. Quantum Annealing Implementation of Job-Shop Scheduling. arXiv
**2015**, arXiv:1506.08479. [Google Scholar] - Zhou, X.; Zhong, M. Single-track train timetabling with guaranteed optimality: Branch-and-bound algorithms with enhanced lower bounds. Transp. Res. Part B Methodol.
**2007**, 41, 320–341. [Google Scholar] [CrossRef] - Harrod, S. Modeling Network Transition Constraints with Hypergraphs. Transp. Sci.
**2011**, 45, 81–97. [Google Scholar] [CrossRef] - Meng, L.; Zhou, X. Simultaneous train rerouting and rescheduling on an N-track network: A model reformulation with network-based cumulative flow variables. Transp. Res. Part B Methodol.
**2014**, 67, 208–234. [Google Scholar] [CrossRef] - Kadowaki, T.; Nishimori, H. Quantum annealing in the transverse Ising model. Phys. Rev. E
**1998**, 58, 5355–5363. [Google Scholar] [CrossRef] - Aharonov, D.; van Dam, W.; Kempe, J.; Landau, Z.; Lloyd, S.; Regev, O. Adiabatic quantum computation is equivalent to standard quantum computation. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, Rome, Italy, 17–19 October 2004; pp. 42–51. [Google Scholar] [CrossRef] [Green Version]
- Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information: 10th Anniversary Edition; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar] [CrossRef] [Green Version]
- Biamonte, J.D.; Love, P.J. Realizable Hamiltonians for universal adiabatic quantum computers. Phys. Rev. A
**2008**, 78, 012352. [Google Scholar] [CrossRef] [Green Version] - Lucas, A. Ising formulations of many NP problems. Front. Phys.
**2014**, 2, 5. [Google Scholar] [CrossRef] [Green Version] - Albash, T.; Lidar, D.A. Adiabatic quantum computation. Rev. Mod. Phys.
**2018**, 90, 015002. [Google Scholar] [CrossRef] [Green Version] - Lanting, T.; Przybysz, A.J.; Smirnov, A.Y.; Spedalieri, F.M.; Amin, M.H.; Berkley, A.J.; Harris, R.; Altomare, F.; Boixo, S.; Bunyk, P.; et al. Entanglement in a quantum annealing processor. Phys. Rev. X
**2014**, 4, 021041. [Google Scholar] [CrossRef] [Green Version] - Wu, F.Y. The Potts model. Rev. Mod. Phys.
**1982**, 54, 235–268. [Google Scholar] [CrossRef] - Castellani, T.; Cavagna, A. Spin-glass theory for pedestrians. J. Stat. Mech.
**2005**, 2005, P05012. [Google Scholar] [CrossRef] [Green Version] - Glover, F.; Kochenberger, G.; Du, Y. Quantum Bridge Analytics I: A tutorial on formulating and using QUBO models. Ann. J. Oper. Res.
**2019**, 17, 335–371. [Google Scholar] [CrossRef] - Aramon, M.; Rosenberg, G.; Valiante, E.; Miyazawa, T.; Tamura, H.; Katzgraber, H.G. Physics-Inspired Optimization for Quadratic Unconstrained Problems Using a Digital Annealer. Front. Phys.
**2019**, 7, 48. [Google Scholar] [CrossRef] - Pierangeli, D.; Rafayelyan, M.; Conti, C.; Gigan, S. Scalable spin-glass optical simulator. arXiv
**2020**, arXiv:2006.00828. [Google Scholar] [CrossRef] - Yamamoto, Y.; Aihara, K.; Leleu, T.; Kawarabayashi, K.; Kako, S.; Fejer, M.; Inoue, K.; Takesue, H. Coherent Ising machines—Optical neural networks operating at the quantum limit. Npj Quantum Inf.
**2017**, 3, 49. [Google Scholar] [CrossRef] [Green Version] - Fukushima-Kimura, B.H.; Handa, S.; Kamakura, K.; Kamijima, Y.; Sakai, A. Mixing time and simulated annealing for the stochastic cellular automata. arXiv
**2020**, arXiv:2007.11287. [Google Scholar] - Cai, F.; Kumar, S.; Van Vaerenbergh, T.; Sheng, X.; Liu, R.; Li, C.; Liu, Z.; Foltin, M.; Yu, S.; Xia, Q.; et al. Power-efficient combinatorial optimization using intrinsic noise in memristor Hopfield neural networks. Nat. Electron.
**2020**, 3, 409–418. [Google Scholar] [CrossRef] - Avron, J.E.; Elgart, A. Adiabatic Theorem without a Gap Condition. Commun. Math. Phys.
**1999**, 203, 445–463. [Google Scholar] [CrossRef] [Green Version] - Ozfidan, I.; Deng, C.; Smirnov, A.Y.; Lanting, T.; Harris, R.; Swenson, L.; Whittaker, J.; Altomare, F.; Babcock, M.; Baron, C.; et al. Demonstration of nonstoquastic Hamiltonian in coupled superconducting flux qubits. arXiv
**2019**, arXiv:1903.06139. [Google Scholar] [CrossRef] [Green Version] - Choi, V. Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Process.
**2008**, 7, 193–209. [Google Scholar] [CrossRef] [Green Version] - Hamerly, R.; Inagaki, T.; McMahon, P.L.; Venturelli, D.; Marandi, A.; Onodera, T.; Ng, E.; Langrock, C.; Inaba, K.; Honjo, T.; et al. Experimental investigation of performance differences between coherent Ising machines and a quantum annealer. Sci. Adv.
**2019**, 5, aau0823. [Google Scholar] [CrossRef] [Green Version] - King, A.D.; Bernoudy, W.; King, J.; Berkley, A.J.; Lanting, T. Emulating the coherent Ising machine with a mean-field algorithm. arXiv
**2018**, arXiv:1806.08422v1. [Google Scholar] - Onodera, T.; Ng, E.; McMahon, P.L. A quantum annealer with fully programmable all-to-all coupling via Floquet engineering. arXiv
**2019**, arXiv:math-ph/0409035. [Google Scholar] [CrossRef] - Childs, A.M.; Farhi, E.; Preskill, J. Robustness of adiabatic quantum computation. Phys. Rev. A
**2001**, 65, 012322. [Google Scholar] [CrossRef] [Green Version] - Katzgraber, H.G.; Hamze, F.; Zhu, Z.; Ochoa, A.J.; Munoz-Bauza, H. Seeking Quantum Speedup Through Spin Glasses: The Good, the Bad, and the Ugly. Phys. Rev. X
**2015**, 5, 031026. [Google Scholar] [CrossRef] - Sachdev, S. Quantum Phase Transitions; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Dziarmaga, J. Dynamics of a quantum phase transition: Exact solution of the quantum Ising model. Phys. Rev. Lett.
**2005**, 95, 245701. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dziarmaga, J. Dynamics of a quantum phase transition and relaxation to a steady state. Adv. Phys.
**2010**, 59, 1063–1189. [Google Scholar] [CrossRef] [Green Version] - Kibble, T.W.B. Topology of cosmic domains and strings. J. Phys. A Math. Gen.
**1976**, 9, 1387. [Google Scholar] [CrossRef] - Kibble, T.W.B. Some implications of a cosmological phase transition. Phys. Rep.
**1980**, 67, 183–199. [Google Scholar] [CrossRef] - Zurek, W.H. Cosmological experiments in superfluid helium? Nature
**1985**, 317, 505. [Google Scholar] [CrossRef] [Green Version] - Francuz, A.; Dziarmaga, J.; Gardas, B.; Zurek, W.H. Space and time renormalization in phase transition dynamics. Phys. Rev. B
**2016**, 93, 075134. [Google Scholar] [CrossRef] [Green Version] - Deffner, S. Kibble-Zurek scaling of the irreversible entropy production. Phys. Rev. E
**2017**, 96, 052125. [Google Scholar] [CrossRef] [Green Version] - Gardas, B.; Dziarmaga, J.; Zurek, W.H. Dynamics of the quantum phase transition in the one-dimensional Bose-Hubbard model: Excitations and correlations induced by a quench. Phys. Rev. B
**2017**, 95, 104306. [Google Scholar] [CrossRef] [Green Version] - Gardas, B.; Dziarmaga, J.; Zurek, W.H.; Zwolak, M. Defects in Quantum Computers. Sci. Rep.
**2018**, 8, 4539. [Google Scholar] [CrossRef] [Green Version] - Venuti, L.C.; Albash, T.; Lidar, D.A.; Zanardi, P. Adiabaticity in open quantum systems. Phys. Rev. A
**2016**, 93, 032118. [Google Scholar] [CrossRef] - Schollwöck, U. The density-matrix renormalization group. Rev. Mod. Phys.
**2005**, 77, 259–315. [Google Scholar] [CrossRef] [Green Version] - Verstraete, F.; Cirac, J.I. Matrix product states represent ground states faithfully. Phys. Rev. B
**2006**, 73, 094423. [Google Scholar] [CrossRef] [Green Version] - Rams, M.M.; Mohseni, M.; Eppens, D.; Jałowiecki, K.; Gardas, B. Approximate optimization, sampling, and spin-glass droplet discovery with tensor networks. Phys. Rev. E
**2021**, 104, 025308. [Google Scholar] [CrossRef] [PubMed] - Czartowski, J.; Szymański, K.; Gardas, B.; Fyodorov, Y.V.; Życzkowski, K. Separability gap and large-deviation entanglement criterion. Phys. Rev. A
**2019**, 100, 042326. [Google Scholar] [CrossRef] [Green Version] - Jałowiecki, K.; Więckowski, A.; Gawron, P.; Gardas, B. Parallel in time dynamics with quantum annealers. Sci. Rep.
**2020**, 10, 13534. [Google Scholar] [CrossRef] - Gardas, B.; Rams, M.M.; Dziarmaga, J. Quantum neural networks to simulate many-body quantum systems. Phys. Rev. B
**2018**, 98, 184304. [Google Scholar] [CrossRef] [Green Version] - Jałowiecki, K.; Rams, M.M.; Gardas, B. Brute-forcing spin-glass problems with CUDA. Comput. Phys. Commun.
**2021**, 260. [Google Scholar] [CrossRef] - Luenberger, D.; Ye, Y. Linear and Nonlinear Programming; International Series in Operations Research & Management Science; Springer International Publishing: Berlin/Heidelberg, Germany, 2015. [Google Scholar]
- Hen, I.; Spedalieri, F.M. Quantum Annealing for Constrained Optimization. Phys. Rev. Appl.
**2016**, 5, 034007. [Google Scholar] [CrossRef] [Green Version] - DWave Ocean Software Documentation. Available online: https://docs.ocean.dwavesys.com/en/stable (accessed on 29 June 2020).
- Gusmeroli, N.; Wiegele, A. EXPEDIS: An exact penalty method over discrete sets. Discret. Optim.
**2022**, 44, 100622. [Google Scholar] [CrossRef] - PKP Polskie Linie Kolejowe, S.A. Public Procurement Website. Available online: https://zamowienia.plk-sa.pl/ (accessed on 3 February 2020).
- CPLEX Optimizer. Available online: https://www.ibm.com/analytics/cplex-optimizer (accessed on 29 June 2020).
- Optimization with PuLP. Available online: https://coin-or.github.io/pulp (accessed on 15 February 2021).
- Zbinden, S.; Bärtschi, A.; Djidjev, H.; Eidenbenz, S. Embedding Algorithms for Quantum Annealers with Chimera and Pegasus Connection Topologies. In Proceedings of the International Conference on High Performance Computing, Frankfurt, Germany, 22–25 June 2020; pp. 187–206. [Google Scholar]
- Pelofske, E.; Hahn, G.; Djidjev, H. Decomposition Algorithms for Solving NP-hard Problems on a Quantum Annealer. J. Signal Process. Syst.
**2021**, 93, 405–420. [Google Scholar] [CrossRef] - Endo, S.; Cai, Z.; Benjamin, S.C.; Yuan, X. Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation. J. Phys. Soc. Jpn.
**2021**, 90, 032001. [Google Scholar] [CrossRef] - Ding, Y.; Chen, X.; Lamata, L.; Solano, E.; Sanz, M. Implementation of a Hybrid Classical-Quantum Annealing Algorithm for Logistic Network Design. SN Comput. Sci.
**2021**, 2, 68. [Google Scholar] [CrossRef]

**Figure 1.**

**D-Wave processor specification**. Left: An example of the Chimera topology, here composed of $4\times 4$ (${C}_{4}$) grid consisting of clusters (units cells) of 8 qubits each. The total number of variables (vertices) for this graph is $N=4\xb74\xb78=128$. A graph’s edges indicate possible interactions between qubits. The maximum number of qubits is ${N}_{\mathrm{max}}=2048$ for the Chimera ${C}_{16}$ topology, whereas the total number of connections between them is limited to $6000\ll {N}_{\mathrm{max}}^{2}$. Right: A typical annealing schedule controlling the evolution of a quantum processor, where T denotes the time to complete one annealing cycle (the annealing time). It ranges from microseconds (∼2 μs) to milliseconds (∼2000 μs). The parameters g and Δ are used in Equation (6).

**Figure 2.**The railway line segments and their initial (undisturbed) timetables addressed in our calculations. The train diagrams in subfigures (

**c**,

**d**) represent train paths by connecting characteristic points of the location of trains at certain times by straight lines. Subfigures (

**a**,

**b**) represent the network topologies. The lines are the railway tracks. Their numbers represent blocks (as used, e.g., in the vertical axes of the train diagrams) and their upper indices in parentheses refer to the sidings (i.e., parallel tracks of stations). The rectangles represent the passenger platforms, circles represent the block boundaries (white: between a station and a line block, blue filled: between two line blocks). (

**a**) Nidzica–Olsztynek section of railway line No. 216. (

**b**) Goleszów–Wisła Uzdrowisko section of railway line No. 191. (

**c**) Train diagram for the timetable of the line in (

**a**). (

**d**) Train diagram for the timetable of the line in (

**b**).

**Figure 3.**A possible solution of the conflict on line No. 216. (

**a**) The conflicted diagram. All the three trains would meet in block 4 as it can be seen from the intersecting train paths. (

**b**) The solution; FCFS, FLFS, and AMCC give the same outcome with a maximum seconday delay of 4 min.

**Figure 4.**Spectra of the low-energy solutions for two penalty strategies of the brute-force (exact) solution. The black line separates the phase in which only feasible solutions appear. Observe the mixing phase, in which both feasible and unfeasible solutions occur. Here, ${p}_{\mathrm{pair}}$ and ${p}_{\mathrm{sum}}$ are penalties of the unconstrained problem expressed in the “logical” variables. The term ${p}_{\mathrm{sum}}={\left(\right)}^{{\sum}_{i\in {\mathcal{V}}_{\mathrm{s}}}}2$, cf. Equation (24), ensures that each train leaves a station only once, whereas ${p}_{\mathrm{pair}}={\sum}_{(i,j)\in {\mathcal{V}}_{\mathrm{p}}}({x}_{i}{x}_{j}+{x}_{j}{x}_{i})$, cf. Equation (25), imposes the following: minimal passing time constrain, single block occupation constrain, deadlock constrain, and rolling stock circulation constrain. (

**a**) ${p}_{\mathrm{pair}}=2.7,{p}_{\mathrm{sum}}=2.2$. (

**b**) ${p}_{\mathrm{pair}}={p}_{\mathrm{sum}}=1.75$.

**Figure 5.**Distribution of the energies corresponding to the states (solutions), which are sampled by the D-Wave 2000Q quantum annealer of 48 logical quantum bits instance of line No. 216. In particular, 1000 samples were taken for each annealing time, and the strength of embedding was set to $css=2.0$. This device is still very noisy and prone to errors, so the sample contains excited states. (

**a**) QUBO param.: ${p}_{\mathrm{pair}}=2.7,{p}_{\mathrm{sum}}=2.2$. (

**b**) ${p}_{\mathrm{pair}}={p}_{\mathrm{sum}}=1.75$.

**Figure 6.**Embedding of a simple, six-qubit problem. (

**Left**) graph of the original problem. (

**Right**) problem embedded into a unit cell of Chimera. Here, different colors correspond to different logical variables. Apparently, the original problem does not map directly onto Chimera as it contains cycles of length 3. Therefore, two chains have to be introduced. Couplings corresponding to inner-chain penalties are marked with the same color as the variable to which they correspond.

**Figure 7.**Train diagrams of the best D-Wave solutions, the lowest energies of the quantum annealing on the D-Wave machine (green: feasible, red: not feasible), and the optimal tensor network solution. The raw computational time on the D-Wave (n.o. runs × annealing time) was in the range $5\times {10}^{-3}$–2 s. (

**a**) The optimal solution from (

**c**). (

**b**) The optimal solution from (

**d**). (

**c**) ${p}_{\mathrm{pair}}=2.7,{p}_{\mathrm{sum}}=2.2$. (

**d**) ${p}_{\mathrm{pair}}={p}_{\mathrm{sum}}=1.75$.

**Figure 8.**Line No. 216, with the minimal energy from the D-Wave quantum annealer, using 1000 runs. Green dots indicates the feasible solutions, while the red dots denote the unfeasible ones. In general, the energy rises as the css strength rises. We do not observe that the different settings of ${p}_{\mathrm{pair}}$ and ${p}_{\mathrm{sum}}$ improve the feasibility; see (

**a**) The minimal energies vs. css for ${p}_{\mathrm{pair}}=2.2,{p}_{\mathrm{sum}}=2.7$. (

**b**) The minimal energies vs. css for ${p}_{\mathrm{pair}}=1.75,{p}_{\mathrm{sum}}=1.75$.

**Figure 9.**Line No. 191, with the minimal energy from the D-Wave annealer at 250 k runs, css = $2.0$, and ${p}_{\mathrm{pair}}=1.75,{p}_{\mathrm{sum}}=1.75$. The output does not dependent on the annealing time (in the investigated range) and is still far from the ground state. The raw computational time on the D-Wave (n.o. runs × annealing time) was in the range 200–350 s. (

**a**) Best D-Wave solutions (these are the lowest excited states we have recorded). Red dots indicate that the solutions are not feasible. (

**b**) Comparison of the objective and hard penalty for the D-Wave outcome and the optimal solution calculated with CPLEX.

**Figure 10.**The best solutions obtained from the D-Wave quantum annealer for line No. 191. For case 1 (

**a**), the annealing time is $t=1400$. The solution is unfeasible since the stay of Ks3 at station 7 is below 1 min. If the solution is corrected (i.e., the stay is introduced), it loses its optimailty and reflects a dispatching situation different from those obtained from FCFS, FLFS, AMCC, CPLEX, or the tensor network. For case 2 (

**b**), $t=1200$ is used. The solution is unfeasible as Ks3 does not stop at station 7; hence, Ks3 and IC2 are supposed to meet and pass between stations 7 and 10. It can, however, be amended to an optimal solution: shortening the stay of Ks3 at station 3 and shortening the stay of IC2 at station 7 (and 3 if necessary) result in a meet-and-pass situation at station 10, and this is optimal. (

**a**) Case 1. (

**b**) Case 2.

**Figure 11.**The CPLEX QUBO solution, coinciding with the linear model’s solution of the 18-train problem.

**Table 1.**The maximum secondary delays, in minutes, resulting from simple heuristics. Observe that for each case, there are solutions far below ${d}_{\mathrm{max}}=10$.

Heuristics | Case 1 | Case 2 | Case 3 | Case 4 |
---|---|---|---|---|

FLFS | 6 | 13 | 4 | 2 |

FCFS | 5 | 5 | 5 | 2 |

AMCC | 5 | 5 | 4 | 2 |

**Table 2.**The values of the objective function $f\left(\mathbf{x}\right)$ for the solutions obtained by the classical calculation of the QUBO, linear integer programming approach, and all the heuristics. The blue color denotes equivalence from the dispatching point of view with the ground state of the QUBO or the output of the linear integer programming. The equivalence concerns the same order of trains at each station.

Method | Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|---|

QUBO approach | CPLEX | 0.54 | 1.40 | 0.73 | 0.20 |

tensor network | 0.54 | 1.40 | 1.65 | 0.29 | |

linear integer programming | 0.54 | 1.40 | 0.73 | 0.20 | |

Simple heuristics | AMCC | 0.77 | 1.30 | 0.73 | 0.20 |

FLFS | 0.54 | 1.71 | 0.73 | 0.20 | |

FCFS | 0.77 | 1.30 | 0.95 | 0.20 |

**Table 3.**Line No. 216, with the objective functions and penalties for violating the hard constraints: see Equation (S39) of Supplemental Materials. Output from the D-Wave quantum annealer for the annealing time of 2000 μs. If ${f}^{\prime}\left(\mathbf{x}\right)>0$, the solution is not feasible. The ${p}_{\mathrm{sum}}={p}_{\mathrm{pair}}=1.75$ policy gives lower objectives.

css | ${\mathit{p}}_{\mathbf{sum}},{\mathit{p}}_{\mathbf{pair}}$ | Hard Constraints’ Penalty ${\mathit{f}}^{\prime}\left(\mathit{x}\right)$ | $\mathit{f}\left(\mathit{x}\right)$ |
---|---|---|---|

2 | $1.75,1.75$ | $0.0$ | $1.36$ |

2 | $2.2,2.7$ | $0.0$ | $1.57$ |

4 | $1.75,1.75$ | $0.0$ | $1.93$ |

4 | $2.2,2.7$ | $2.2$ | $2.07$ |

6 | $1.75,1.75$ | $5.25$ | $0.43$ |

6 | $2.2,2.7$ | $6.6$ | $0.86$ |

**Table 4.**Graph densities for various problems. Since case 3 is supposed to be the most complicated one of cases 1–4, it has the largest graph density, #—number of.

Features | Line 216 | Line 191 | ||||
---|---|---|---|---|---|---|

Case 1 | Case 2 | Case 3 | Case 4 | Enlarged | ||

problem size (# logical bits) | 48 | 198 | 198 | 198 | 198 | 594 |

# edges | 395 | 1851 | 2038 | 2180 | 1831 | 5552 |

density (vs. full graph) | $0.35$ | $0.095$ | $0.104$ | $0.111$ | $0.094$ | $0.032$ |

embedding into | Chimera | Chimera | Chimera | Ideal Chimera | Chimera | Pegasus |

approximate # physical bits | 373 | <2048 | <2048 | ≈ 2048 | <2048 | <5760 |

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**MDPI and ACS Style**

Domino, K.; Koniorczyk, M.; Krawiec, K.; Jałowiecki, K.; Deffner, S.; Gardas, B.
Quantum Annealing in the NISQ Era: Railway Conflict Management. *Entropy* **2023**, *25*, 191.
https://doi.org/10.3390/e25020191

**AMA Style**

Domino K, Koniorczyk M, Krawiec K, Jałowiecki K, Deffner S, Gardas B.
Quantum Annealing in the NISQ Era: Railway Conflict Management. *Entropy*. 2023; 25(2):191.
https://doi.org/10.3390/e25020191

**Chicago/Turabian Style**

Domino, Krzysztof, Mátyás Koniorczyk, Krzysztof Krawiec, Konrad Jałowiecki, Sebastian Deffner, and Bartłomiej Gardas.
2023. "Quantum Annealing in the NISQ Era: Railway Conflict Management" *Entropy* 25, no. 2: 191.
https://doi.org/10.3390/e25020191