Calculating the expected value function of a two-stage stochastic optimization program with a quantum algorithm (2402.15029v2)
Abstract: Two-stage stochastic programming is a problem formulation for decision-making under uncertainty. In the first stage, the actor makes a best "here and now" decision in the presence of uncertain quantities that will be resolved in the future, represented in the objective function as the expected value function. This function is a multi-dimensional integral of the second stage optimization problem, which must be solved over all possible future scenarios. This work uses a quantum algorithm to estimate the expected value function with a polynomial speedup. Our algorithm gains its advantage through the two following observations. First, by encoding the probability distribution as a quantum wavefunction in an auxilliary register, and using this register as control logic for a phase-separation unitary, Digitized Quantum Annealing (DQA) can converge to the minimium of each scenario in the random variable in parallel. Second, Quantum Amplitude Estimation (QAE) on DQA can calculate the expected value of this per-scenario optimized wavefunction, producing an estimate for the expected value function. Quantum optimization is theorized to have a polynomial speedup for combinatorial optimization problems, and estimation error from QAE is known to converge inverse-linear in the number of samples (as opposed to the best case inverse of a square root in classical Monte Carlo). Therefore, assuming the probability distribution wavefunction can be prepared efficiently, we conclude our method has a polynomial speedup (of varying degree, depending on the optimization problem) over classical methods for estimating the expected value function. We conclude by demonstrating this algorithm on a stochastic programming problem inspired by operating the power grid under weather uncertainty.
- Willem Klein Haneveld and Maarten Vlerk “Optimizing electricity distribution using two-stage integer recourse models” In University of Groningen, Research Institute SOM (Systems, Organisations and Management), Research Report 54, 2000 DOI: 10.1007/978-1-4757-6594-6˙7
- GILBERT LAPORTE, FRANÇOIS LOUVEAUX and HÉLÈNE MERCURE “The Vehicle Routing Problem with Stochastic Travel Times” In Transportation Science 26.3 INFORMS, 1992, pp. 161–170 URL: http://www.jstor.org/stable/25768536
- “Analytical Evaluation of Hierarchical Planning Systems” In Operations Research 29.4, 1981, pp. 707–716
- Alexander Shapiro, Darinka Dentcheva and Andrzej Ruszczynski “Lectures on stochastic programming: modeling and theory” SIAM, 2021
- Paul Glasserman, Philip Heidelberger and Perwez Shahabuddin “Efficient Monte Carlo Methods for Value-at-Risk” In Master. Risk 2, 2000
- “A tutorial on stochastic programming”, 2007
- “Computational complexity of stochastic programming problems” In Mathematical Programming 106.3, 2006, pp. 423–432 DOI: 10.1007/s10107-005-0597-0
- Can Li and Ignacio E. Grossmann “A Review of Stochastic Programming Methods for Optimization of Process Systems Under Uncertainty” In Frontiers in Chemical Engineering 2, 2021 DOI: 10.3389/fceng.2020.622241
- Alexander Shapiro “Monte Carlo Sampling Methods” In Stochastic Programming 10, Handbooks in Operations Research and Management Science Elsevier, 2003, pp. 353–425 DOI: https://doi.org/10.1016/S0927-0507(03)10006-0
- “Neur2SP: Neural Two-Stage Stochastic Programming”, 2022 arXiv:2205.12006 [math.OC]
- “Learning Fast Optimizers for Contextual Stochastic Integer Programs” In Conference on Uncertainty in Artificial Intelligence, 2018 URL: https://api.semanticscholar.org/CorpusID:54002167
- “A Learning-Based Algorithm to Quickly Compute Good Primal Solutions for Stochastic Integer Programs” In Integration of Constraint Programming, Artificial Intelligence, and Operations Research Cham: Springer International Publishing, 2020, pp. 99–111
- Edward Farhi, Jeffrey Goldstone and Sam Gutmann “A Quantum Approximate Optimization Algorithm”, 2014 arXiv:1411.4028 [quant-ph]
- “From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz” In Algorithms 12.2 MDPI AG, 2019, pp. 34 DOI: 10.3390/a12020034
- Jonathan Wurtz and Peter J. Love “Counterdiabaticity and the quantum approximate optimization algorithm” In Quantum 6 Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften, 2022, pp. 635 DOI: 10.22331/q-2022-01-27-635
- “Quantum amplitude amplification and estimation” In Quantum Computation and Information American Mathematical Society, 2002, pp. 53–74 DOI: 10.1090/conm/305/05215
- “Residual Energies after Slow Quantum Annealing” In Journal of the Physical Society of Japan 74.6 Physical Society of Japan, 2005, pp. 1649–1652 DOI: 10.1143/jpsj.74.1649
- Rolando D. Somma, Daniel Nagaj and Mária Kieferová “Quantum Speedup by Quantum Annealing” In Physical Review Letters 109.5 American Physical Society (APS), 2012 DOI: 10.1103/physrevlett.109.050501
- “Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices” In Phys. Rev. X 10 American Physical Society, 2020, pp. 021067 DOI: 10.1103/PhysRevX.10.021067
- “QAOA for Max-Cut requires hundreds of qubits for quantum speed-up” In Scientific Reports 9.1, 2019, pp. 6903 DOI: 10.1038/s41598-019-43176-9
- Ashley Montanaro “Quantum speedup of Monte Carlo methods” In Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471.2181 The Royal Society, 2015, pp. 20150301 DOI: 10.1098/rspa.2015.0301
- “Low depth algorithms for quantum amplitude estimation” In Quantum 6 Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften, 2022, pp. 745 DOI: 10.22331/q-2022-06-27-745
- “Creating superpositions that correspond to efficiently integrable probability distributions”, 2002 arXiv:quant-ph/0208112 [quant-ph]
- “Automated detection of symmetry-protected subspaces in quantum simulations” In Phys. Rev. Res. 5 American Physical Society, 2023, pp. 033082 DOI: 10.1103/PhysRevResearch.5.033082
- “Customized Quantum Annealing Schedules” In Phys. Rev. Appl. 17 American Physical Society, 2022, pp. 044005 DOI: 10.1103/PhysRevApplied.17.044005
- “Quantum annealing in the transverse Ising model” In Phys. Rev. E 58 American Physical Society, 1998, pp. 5355–5363 DOI: 10.1103/PhysRevE.58.5355
- “Quantum annealing for industry applications: introduction and review” In Reports on Progress in Physics 85.10 IOP Publishing, 2022, pp. 104001 DOI: 10.1088/1361-6633/ac8c54
- “An Analog Analogue of a Digital Quantum Computation”, 1996 arXiv:quant-ph/9612026 [quant-ph]
- “Quantum Computation by Adiabatic Evolution”, 2000 arXiv:quant-ph/0001106 [quant-ph]
- Jérémie Roland and Nicolas J. Cerf “Adiabatic quantum search algorithm for structured problems” In Phys. Rev. A 68 American Physical Society, 2003, pp. 062312 DOI: 10.1103/PhysRevA.68.062312
- Tameem Albash and Daniel A. Lidar “Adiabatic quantum computation” In Reviews of Modern Physics 90.1 American Physical Society (APS), 2018 DOI: 10.1103/revmodphys.90.015002
- “Alignment between initial state and mixer improves QAOA performance for constrained optimization” In npj Quantum Information 9.1 Springer ScienceBusiness Media LLC, 2023 DOI: 10.1038/s41534-023-00787-5
- “Constraint Preserving Mixers for the Quantum Approximate Optimization Algorithm” In Algorithms 15.6 MDPI AG, 2022, pp. 202 DOI: 10.3390/a15060202
- “Constrained optimization via quantum Zeno dynamics” In Communications Physics 6.1, 2023, pp. 219 DOI: 10.1038/s42005-023-01331-9
- “Subspace Correction for Constraints”, 2023 arXiv:2310.20191 [quant-ph]
- “Quantum advantage in learning from experiments” In Science 376.6598 American Association for the Advancement of Science (AAAS), 2022, pp. 1182–1186 DOI: 10.1126/science.abn7293
- “Loading Probability Distributions in a Quantum circuit”, 2022 arXiv:2208.13372 [quant-ph]
- “Quantum-state preparation with universal gate decompositions” In Physical Review A 83.3 American Physical Society (APS), 2011 DOI: 10.1103/physreva.83.032302
- Christa Zoufal, Aurélien Lucchi and Stefan Woerner “Quantum Generative Adversarial Networks for learning and loading random distributions” In npj Quantum Information 5.1 Springer ScienceBusiness Media LLC, 2019 DOI: 10.1038/s41534-019-0223-2
- “On the approximability of random-hypergraph MAX-3-XORSAT problems with quantum algorithms”, 2024 arXiv:2312.06104 [quant-ph]
- I. Hen “Period finding with adiabatic quantum computation” In EPL (Europhysics Letters) 105.5 IOP Publishing, 2014, pp. 50005 DOI: 10.1209/0295-5075/105/50005
- “Amplitude estimation without phase estimation” In Quantum Information Processing 19.2 Springer ScienceBusiness Media LLC, 2020 DOI: 10.1007/s11128-019-2565-2
- Almudena Carrera Vazquez and Stefan Woerner “Efficient State Preparation for Quantum Amplitude Estimation” In Phys. Rev. Appl. 15 American Physical Society, 2021, pp. 034027 DOI: 10.1103/PhysRevApplied.15.034027
- A.Yu. Kitaev “Quantum measurements and the Abelian Stabilizer Problem”, 1995 arXiv:quant-ph/9511026 [quant-ph]
- “Option Pricing using Quantum Computers” In Quantum 4 Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften, 2020, pp. 291 DOI: 10.22331/q-2020-07-06-291
- Stefan Woerner and Daniel J. Egger “Quantum risk analysis” In npj Quantum Information 5.1, 2019, pp. 15 DOI: 10.1038/s41534-019-0130-6
- “Quantum Algorithms and Circuits for Scientific Computing”, 2015 arXiv:1511.08253 [quant-ph]
- Vlatko Vedral, Adriano Barenco and Artur Ekert “Quantum networks for elementary arithmetic operations” In Phys. Rev. A 54 American Physical Society, 1996, pp. 147–153 DOI: 10.1103/PhysRevA.54.147
- “A Threshold for Quantum Advantage in Derivative Pricing” In Quantum 5 Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften, 2021, pp. 463 DOI: 10.22331/q-2021-06-01-463
- D. Coppersmith “An approximate Fourier transform useful in quantum factoring”, 2002 arXiv:quant-ph/0201067 [quant-ph]
- Mark R. Jerrum, Leslie G. Valiant and Vijay V. Vazirani “Random generation of combinatorial structures from a uniform distribution” In Theoretical Computer Science 43, 1986, pp. 169–188 DOI: https://doi.org/10.1016/0304-3975(86)90174-X
- T. Popoviciu “Sur les équations algébriques ayant toutes leurs racines réelles” In Mathematica (Cluj) 9, 1935
- Martin Håberg “Fundamentals and recent developments in stochastic unit commitment” In International Journal of Electrical Power & Energy Systems 109, 2019, pp. 38–48 DOI: https://doi.org/10.1016/j.ijepes.2019.01.037
- “Scenario creation and power-conditioning strategies for operating power grids with two-stage stochastic economic dispatch” In 2020 IEEE Power & Energy Society General Meeting (PESGM), 2020, pp. 1–5 DOI: 10.1109/PESGM41954.2020.9281609
- “Deterministic Preparation of Dicke States” In Lecture Notes in Computer Science Springer International Publishing, 2019, pp. 126–139 DOI: 10.1007/978-3-030-25027-0˙9
- Qiskit contributors “Qiskit: An Open-source Framework for Quantum Computing”, 2023 DOI: 10.5281/zenodo.2573505
- “Author Correction: Analog errors in quantum annealing: doom and hope” In npj Quantum Information 6.1 Springer ScienceBusiness Media LLC, 2020 DOI: 10.1038/s41534-020-00297-8