Variational Amplitude Amplification for Solving QUBO Problems
Abstract
:1. Introduction
Layout
2. QUBO Definitions
Linear QUBO
3. Amplitude Amplification
Algorithm 1 Amplitude Amplification Algorithm 

3.1. Solution Space Distribution
3.2. Cost Oracle ${U}_{\mathrm{c}}$
3.3. Scaling Parameter ${p}_{\mathrm{s}}$
4. Gaussian Amplitude Amplification
4.1. Achievable Probabilities
4.2. Solution Space Skewness
4.3. Sampling for ${p}_{\mathrm{s}}$
5. Variational Amplitude Amplification
5.1. Boosting NearOptimal Solutions
5.2. Constant Iterations
5.3. Information through Measurements
5.4. Quantum Verification
6. Hybrid Solving
Supporting Greedy Algorithms
7. More Oracle Problems
7.1. Weighted and Unweighted MaxCut
7.2. Graph Coloring
7.3. Subset Sum
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. QUBO Data
$\phantom{\rule{9.95863pt}{0ex}}\mathit{N}\phantom{\rule{9.95863pt}{0ex}}$  # of QUBOs Studied 

17  5000 
18  3000 
19  2000 
20  1500 
21  1200 
22  1000 
23  1000 
24  600 
25  500 
26  400 
27  100 
Appendix B. Linear Regression
Appendix C. MaxCut Circuit
References
 Grover, L.K. A fast quantum mechanical algorithm for database search. arXiv 1996, arXiv:9605043. [Google Scholar]
 Boyer, M.; Brassard, G.; Hoyer, P.; Tapp, A. Tight bounds on quantum searching. Fortschritte Phys. 1998, 46, 493–506. [Google Scholar] [CrossRef]
 Bennett, C.H.; Bernstein, E.; Brassard, G.; Vazirani, U. Strengths and weaknesses of quantum computing. SIAM J. Comput. 1997, 26, 1510–1523. [Google Scholar] [CrossRef]
 Farhi, E.; Gutmann, S. Analog analogue of a digital quantum computation. Phys. Rev. A 1998, 57, 2403. [Google Scholar] [CrossRef]
 Brassard, G.; Hoyer, P.; Tapp, A. Quantum counting. In Proceedings of the 25th International Colloquium on Automata, Languages and Programming (ICALP), Aalborg, Denmark, 13–17 July 1998; Volume 1443, pp. 820–831. [Google Scholar]
 Brassard, G.; Hoyer, P.; Mosca, M.; Tapp, A. Quantum amplitude amplification and estimation. Quantum Comput. Quantum Inf. AMS Contemp. Math. 2002, 305, 53–74. [Google Scholar]
 Childs, A.M.; Goldstone, J. Spatial search by quantum walk. Phys. Rev. A 2004, 70, 022314. [Google Scholar] [CrossRef]
 Ambainis, A. Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. arXiv 2010, arXiv:1010.4458. [Google Scholar]
 Singleton, R.L., Jr.; Rogers, M.L.; Ostby, D.L. Grover’s algorithm with diffusion and amplitude steering. arXiv 2021, arXiv:2110.11163. [Google Scholar]
 Farhi, E.; Goldstone, J.; Gutmann, S. A quantum approximate optimization algorithm. arXiv 2014, arXiv:1411.4028. [Google Scholar]
 Hadfield, S.; Wang, Z.; O’Gorman, B.; Rieffel, E.G.; Venturelli, D.; Biswas, R. From the quantum approximate optimization algorithm to a quantum alternating operator ansatz. Algorithms 2019, 12, 34. [Google Scholar] [CrossRef]
 Peruzzo, A.; McClean, J.; Shadbolt, P.; Yung, M.H.; Zhou, X.Q.; Love, P.J.; AspuruGuzik, A.; O’Brien, J.L. A variational eigenvalue solver on a quantum processor. Nat. Commun. 2014, 5, 4213. [Google Scholar] [CrossRef]
 Date, P.; Patton, R.; Schuman, C.; Potok, T. Efficiently embedding QUBO problems on adiabatic quantum computers. Quantum Inf. Process. 2019, 18, 117. [Google Scholar] [CrossRef]
 UshijimaMwesigwa, H.; Negre, C.F.A.; Mniszewski, S.M. Graph partitioning using quantum annealing on the DWave system. arXiv 2017, arXiv:1705.03082. [Google Scholar]
 Pastorello, D.; Blanzieri, E. Quantum annealing learning search for solving QUBO problems. Quantum Inf. Process. 2019, 18, 10. [Google Scholar] [CrossRef]
 CruzSantos, W.; VenegasAndraca, S.E.; Lanzagorta, M. A QUBO formulation of minimum multicut problem instances in trees for DWave quantum annealers. Sci. Rep. 2019, 9, 17216. [Google Scholar] [CrossRef]
 Gilliam, A.; Woerner, S.; Gonciulea, C. Grover adaptive search for constrained polynomial binary optimization. Quantum 2021, 5, 428. [Google Scholar] [CrossRef]
 Seidel, R.; Becker, C.K.U.; Bock, S.; Tcholtchev, N.; GheorgePop, I.D.; Hauswirth, M. Automatic generation of grover quantum oracles for arbitrary data structures. arXiv 2021, arXiv:2110.07545. [Google Scholar] [CrossRef]
 Koch, D.; Cutugno, M.; Karlson, S.; Patel, S.; Wessing, L.; Alsing, P.M. Gaussian amplitude amplification for quantum pathfinding. Entropy 2022, 24, 963. [Google Scholar] [CrossRef] [PubMed]
 Lloyd, S. Quantum search without entanglement. Phys. Rev. A 1999, 61, 010301. [Google Scholar] [CrossRef]
 Viamontes, G.F.; Markov, I.L.; Hayes, J.P. Is quantum search practical? arXiv 2004, arXiv:0405001. [Google Scholar] [CrossRef]
 Regev, O.; Schiff, L. Impossibility of a quantum speedup with a faulty oracle. arXiv 2012, arXiv:1202.1027. [Google Scholar]
 Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2000; p. 249. [Google Scholar]
 Bang, J.; Yoo, S.; Lim, J.; Ryu, J.; Lee, C.; Lee, J. Quantum heuristic algorithm for traveling salesman problem. J. Korean Phys. Soc. 2012, 61, 1944. [Google Scholar] [CrossRef]
 Satoh, T.; Ohkura, Y.; Meter, R.V. Subdivided phase oracle for NISQ search algorithms. IEEE Trans. Quantum Eng. 2020, 1, 3100815. [Google Scholar] [CrossRef]
 Benchasattabuse, N.; Satoh, T.; Hajdušek, M.; Meter, R.V. Amplitude amplification for optimization via subdivided phase oracle. arXiv 2022, arXiv:2205.00602. [Google Scholar]
 Shyamsundar, P. Nonboolean quantum amplitude amplification and quantum mean estimation. arXiv 2021, arXiv:2102.04975. [Google Scholar]
 Long, G.L.; Zhang, W.L.; Li, Y.S.; Niu, L. Arbitrary phase rotation of the marked state cannot be used for Grover’s quantum search algorithm. Commun. Theor. Phys. 1999, 32, 335. [Google Scholar]
 Long, G.L.; Li, Y.S.; Zhang, W.L.; Niu, L. Phase matching in quantum searching. Phys. Lett. A 1999, 262, 27–34. [Google Scholar] [CrossRef]
 Hoyer, P. Arbitrary phases in quantum amplitude amplification. Phys. Rev. A 2000, 62, 052304. [Google Scholar] [CrossRef]
 Younes, A. Towards more reliable fixed phase quantum search algorithm. Appl. Math. Inf. Sci. 2013, 1, 10. [Google Scholar] [CrossRef]
 Li, T.; Bao, W.; Lin, W.Q.; Zhang, H.; Fu, X.Q. Quantum search algorithm based on multiphase. Chin. Phys. Lett. 2014, 31, 050301. [Google Scholar] [CrossRef]
 Guo, Y.; Shi, W.; Wang, Y.; Hu, J. Qlearningbased adjustable fixedphase quantum Grover search algorithm. J. Phys. Soc. Jpn. 2017, 86, 024006. [Google Scholar] [CrossRef]
 Song, P.H.; Kim, I. Computational leakage: Grover’s algorithm with imperfections. Eur. Phys. J. D 2000, 23, 299–303. [Google Scholar] [CrossRef]
 Pomeransky, A.A.; Zhirov, O.V.; Shepelyansky, D.L. Phase diagram for the Grover algorithm with static imperfections. Eur. Phys. J. D 2004, 31, 131–135. [Google Scholar] [CrossRef]
 Janmark, J.; Meyer, D.A.; Wong, T.G. Global symmetry is unnecessary for fast quantum search. Phys. Rev. Lett. 2014, 112, 210502. [Google Scholar] [CrossRef]
 Jong, K.D. Learning with genetic algorithms: An overview. Mach. Lang. 1988, 3, 121–139. [Google Scholar] [CrossRef]
 Forrest, S. Genetic algorithms: Principles of natural selection applied to computation. Science 1993, 261, 5123. [Google Scholar] [CrossRef]
 Kochenberger, G.; Hao, J.K.; Glover, F.; Lewis, M.; Lu, Z.; Wang, H.; Wang, Y. The unconstrained binary quadratic programming problem: A survey. J. Comb. Optim. 2014, 28, 58–81. [Google Scholar] [CrossRef]
 Lucas, A. Ising formulations of many NP problems. Front. Phys. 2014, 12, 2. [Google Scholar] [CrossRef]
 Glover, F.; Kochenberger, G.; Du, Y. A tutorial on formulating and using QUBO models. arXiv 2018, arXiv:1811.11538. [Google Scholar]
 Date, P.; Arthur, D.; PuseyNazzaro, L. QUBO formulations for training machine learning models. Sci. Rep. 2021, 11, 10029. [Google Scholar] [CrossRef] [PubMed]
 Herman, D.; Googin, C.; Liu, X.; Galda, A.; Safro, I.; Sun, Y.; Pistoia, M.; Alexeev, Y. A survey of quantum computing for finance. arXiv 2022, arXiv:2201.02773. [Google Scholar]
 Guerreschi, G.G.; Matsuura, A.Y. QAOA for maxcut requires hundreds of qubits for quantum speedup. Sci. Rep. 2019, 9, 6903. [Google Scholar] [CrossRef]
 Guerreschi, G.G. Solving quadratic unconstrained binary optimization with divideandconquer and quantum algorithms. arXiv 2021, arXiv:2101.07813. [Google Scholar]
 Streif, M.; Leib, M. Comparison of QAOA with quantum and simulated annealing. arXiv 2019, arXiv:1901.01903. [Google Scholar]
 Gabor, T.; Rosenfeld, M.L.; Feld, S.; LinnhoffPopien, C. How to approximate any objective function via quadratic unconstrained binary optimization. arXiv 2022, arXiv:2204.11035. [Google Scholar]
 Pelofske, E.; Bartschi, A.; Eidenbenz, S. Quantum annealing vs. QAOA: 127 qubit higherorder ising problems on NISQ computers. arXiv 2023, arXiv:2301.00520. [Google Scholar]
 Bernoulli, J. Ars Conjectandi; Thurnisiorum: Basileae, Switzerland, 1713. [Google Scholar]
 Laplace, P.S. Mémoire sur les approximations des formules qui sont fonctions de très grands nombres et sur leur application aux probabilités. Mém. Acad. R. Sci. Paris 1810, 10, 353–415. [Google Scholar]
 Gauss, C.F. Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium; Friedrich Perthes and I.H. Besser: Hamburg, Germany, 1809. [Google Scholar]
 Srinivas, M.; Patnaik, L.M. Genetic algorithms: A survey. IEEE Comput. 1994, 27, 17–26. [Google Scholar] [CrossRef]
 Parsons, R.J.; Forrest, S.; Burks, C. Genetic algorithms, operators, and DNA fragment assembly. Mach. Learn. 1995, 21, 11–33. [Google Scholar] [CrossRef]
 Finnila, A.B.; Gomez, M.A.; Sebenik, C.; Stenson, C.; Doll, J.D. Quantum annealing: A new method for minimizing multidimensional functions. Chem. Phys. Lett. 1994, 219, 343–348. [Google Scholar] [CrossRef]
 Koshka, Y.; Novotny, M.A. Comparison of DWave quantum annealing and classical simulated annealing for local minima determination. IEEE J. Sel. Areas Inf. Theory 2020, 1, 2. [Google Scholar] [CrossRef]
 Wierichs, D.; Gogolin, C.; Kastoryano, M. Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer. Phys. Rev. Res. 2020, 2, 043246. [Google Scholar] [CrossRef]
 RiveraDean, J.; Huembeli, P.; Acin, A.; Bowles, J. Avoiding local minima in variational quantum algorithms with neural networks. arXiv 2021, arXiv:2104.02955. [Google Scholar]
 Sack, S.H.; Serbyn, M. Quantum annealing initialization of the quantum approximate optimization algorithm. Quantum 2021, 5, 491. [Google Scholar] [CrossRef]
 Eisert, J.; Hangleiter, D.; Walk, N.; Roth, I.; Markham, D.; Parekh, R.; Chabaud, U.; Kashefi, E. Quantum certification and benchmarking. Nat. Rev. Phys. 2020, 2, 382–390. [Google Scholar] [CrossRef]
 Willsch, D.; Willsch, M.; Calaza, C.D.G.; Jin, F.; Raedt, H.D.; Svensson, M.; Michielsen, K. Benchmarking advantage and DWave 2000Q quantum annealers with exact cover problems. Quantum Inf. Process. 2022, 21, 141. [Google Scholar] [CrossRef]
 Noiri, A.; Takeda, K.; Nakajima, T.; Kobayashi, T.; Sammak, A.; Scappucci, G.; Tarucha, S. Fast universal quantum gate above the faulttolerance threshold in silicon. Nature 2022, 601, 338–342. [Google Scholar] [CrossRef]
 Zhang, Z.; Schwartz, S.; Wagner, L.; Miller, W. A greedy algorithm for aligning DNA sequences. J. Comp. Biol. 2004, 7, 203–214. [Google Scholar] [CrossRef]
 Lin, L.; Cao, L.; Wang, J.; Zhang, C. The applications of genetic algorithms in stock market data mining optimisation. WIT Trans. Inf. Commun. Technol. 2004, 33. [Google Scholar]
 Korte, B.; Lovasz, L. Mathematical structures underlying greedy algorithms. In Fundamentals of Computation Theory; Springer: Berlin/Heidelberg, Germany, 1981. [Google Scholar]
 BangJensen, J.; Gutin, G.; Yeo, A. When the greedy algorithm fails. Discret. Optim. 2004, 1, 121–127. [Google Scholar] [CrossRef]
 Glover, F.; Gutin, G.; Yeo, A.; Zverovich, A. Construction heuristics for the asymmetric TSP. Eur. J. Oper. Res. 2001, 129, 3. [Google Scholar] [CrossRef]
 Festa, P.; Pardalos, P.M.; Resende, M.G.C.; Ribeiro, C.C. Randomized heuristics for the MaxCut problem. Optim. Methods Softw. 2002, 17, 6. [Google Scholar] [CrossRef]
 Karp, R. Reducibility among combinatorial problems. In Proceedings of the Symposium on the Complexity of Computer Computations, Yorktown Heights, NY, USA, 20–22 March 1972. [Google Scholar]
 Garey, M.R.; Johnson, D.S.; Stockmeyer, L. Some simplified NPcomplete graph problems. Theor. Comput. Sci. 1976, 1, 237–267. [Google Scholar] [CrossRef]
 Wang, Y.; Hu, Z.; Sanders, B.C.; Kais, S. Qudits and highdimensional quantum computing. Front. Phys. 2020, 10, 8. [Google Scholar] [CrossRef]
 Lanyon, B.P.; Barbieri, M.; Almeida, M.P.; Jennewein, T.; Ralph, T.C.; Resch, K.J.; Pryde, G.J.; O’Brien, J.L.; Gilchrist, A.; White, A.G. Quantum computing using shortcuts through higher dimensions. Nat. Phys. 2009, 5, 134. [Google Scholar] [CrossRef]
 Luo, M.X.; Wang, X.J. Universal quantum computation with qudits. Sci. China Phys. Mech. Astron. 2014, 57, 1712–1717. [Google Scholar] [CrossRef]
 Niu, M.Y.; Chuang, I.L.; Shapiro, J.H. QuditBasis Universal Quantum Computation Using χ^{2} Interactions. Phys. Rev. Lett. 2018, 120, 160502. [Google Scholar] [CrossRef]
M  100  500  1000  2000 

Average ${\tilde{p}}_{\mathrm{s}}$ Error  7.28%  6.37%  6.31%  6.29% 
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Koch, D.; Cutugno, M.; Patel, S.; Wessing, L.; Alsing, P.M. Variational Amplitude Amplification for Solving QUBO Problems. Quantum Rep. 2023, 5, 625658. https://doi.org/10.3390/quantum5040041
Koch D, Cutugno M, Patel S, Wessing L, Alsing PM. Variational Amplitude Amplification for Solving QUBO Problems. Quantum Reports. 2023; 5(4):625658. https://doi.org/10.3390/quantum5040041
Chicago/Turabian StyleKoch, Daniel, Massimiliano Cutugno, Saahil Patel, Laura Wessing, and Paul M. Alsing. 2023. "Variational Amplitude Amplification for Solving QUBO Problems" Quantum Reports 5, no. 4: 625658. https://doi.org/10.3390/quantum5040041