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Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods



Benjamin Zanger1, Christian B. Mendl1,2, Martin Schulz1,3, and Martin Schreiber1

1Technical University of Munich, Department of Informatics, Boltzmannstraße 3, 85748 Garching, Germany
2TUM Institute for Advanced Study, Lichtenbergstraße 2a, 85748 Garching, Germany
3Leibniz Supercomputing Centre, Boltzmannstraße 1, 85748 Garching, Germany

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Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6$^{mathrm{th}}$ order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an “oracle” within quantum search algorithms for inverse problems.

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► References

[1] Héctor Abraham, AduOffei, Rochisha Agarwal, Ismail Yunus Akhalwaya, Gadi Aleksandrowicz, et al. Qiskit: An open-source framework for quantum computing. 2019. 10.5281/​zenodo.2562110.

[2] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, et al. Quantum supremacy using a programmable superconducting processor. Nature, 574: 505–510, 2019. 10.1038/​s41586-019-1666-5.

[3] Samuel L. Braunstein and Peter van Loock. Quantum information with continuous variables. Rev. Mod. Phys., 77: 513–577, 2005. 10.1103/​RevModPhys.77.513.

[4] John C Butcher. Coefficients for the study of Runge-Kutta integration processes. J. Austral. Math. Soc., 3 (2): 185–201, 1963. 10.1017/​S1446788700027932.

[5] Jun Cai, William G. Macready, and Aidan Roy. A practical heuristic for finding graph minors. arXiv:1406.2741, 2014. URL https:/​/​​abs/​1406.2741.

[6] Chia Cheng Chang, Arjun Gambhir, Travis S. Humble, and Shigetoshi Sota. Quantum annealing for systems of polynomial equations. Scientific Reports, 9 (1): 10258, Jul 2019. ISSN 2045-2322. 10.1038/​s41598-019-46729-0.

[7] P. C. S. Costa, S. Jordan, and A. Ostrander. Quantum algorithm for simulating the wave equation. Phys. Rev. A, 99: 012323, 2019. 10.1103/​PhysRevA.99.012323.

[8] Thomas G Draper. Addition on a quantum computer. arXiv:quant-ph/​0008033, 2000. URL https:/​/​​abs/​quant-ph/​0008033.

[9] Dale R. Durran. Numerical Methods for Fluid Dynamics, volume 32. Springer New York, 2010. ISBN 978-1-4419-6411-3. 10.1007/​978-1-4419-6412-0.

[10] Alok Dutt, Leslie Greengard, and Vladimir Rokhlin. Spectral deferred correction methods for ordinary differential equations. BIT Numerical Mathematics, 40: 241–266, 1998. 10.1023/​A:1022338906936.

[11] Matthew Emmett and Michael Minion. Toward an efficient parallel in time method for partial differential equations. Comm. App. Math. Comp. Sci., 7: 105–132, 2012. 10.2140/​camcos.2012.7.105.

[12] Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. Quantum computation by adiabatic evolution. arXiv e-prints, art. quant-ph/​0001106, January 2000. URL https:/​/​​abs/​quant-ph/​0001106.

[13] Martin J. Gander. 50 years of time parallel time integration. In Multiple Shooting and Time Domain Decomposition. Springer, 2015. 10.1007/​978-3-319-23321-5_3.

[14] Fred Glover, Gary Kochenberger, and Yu Du. Quantum bridge analytics i: a tutorial on formulating and using QUBO models. 4OR, 17 (4): 335–371, nov 2019. 10.1007/​s10288-019-00424-y.

[15] Lov K. Grover. Quantum computers can search arbitrarily large databases by a single query. Phys. Rev. Lett., 79: 4709–4712, 1997. 10.1103/​PhysRevLett.79.4709.

[16] Ernst Hairer and Gerhard Wanner. Solving ordinary differential equations II: Stiff and differential-algebraic problems, volume 14. Springer, Berlin, Heidelberg, 1996. 10.1007/​978-3-642-05221-7.

[17] Ernst Hairer, Syvert P. Nørsett, and Gerhard Wanner. Solving ordinary differential equations I: Nonstiff problems. Springer, Berlin, Heidelberg, 1993. 10.1007/​978-3-540-78862-1.

[18] A. W. Harrow, A. Hassidim, and S. Lloyd. Quantum algorithm for linear systems of equations. Phys. Rev. Lett., 103: 150502, 2009. 10.1103/​PhysRevLett.103.150502.

[19] Ilon Joseph. Koopman–von neumann approach to quantum simulation of nonlinear classical dynamics. Phys. Rev. Research, 2: 043102, Oct 2020. 10.1103/​PhysRevResearch.2.043102.

[20] Tadashi Kadowaki and Hidetoshi Nishimori. Quantum annealing in the transverse Ising model. Phys. Rev. E, 58 (5): 5355, 1998. 10.1103/​PhysRevE.58.5355.

[21] Wolfgang Lechner, Philipp Hauke, and Peter Zoller. A quantum annealing architecture with all-to-all connectivity from local interactions. Sci. Adv., 1: e1500838, 2015. 10.1126/​sciadv.1500838.

[22] Jin-Peng Liu, Herman Øie Kolden, Hari K. Krovi, Nuno F. Loureiro, Konstantina Trivisa, and Andrew M. Childs. Efficient quantum algorithm for dissipative nonlinear differential equations. arXiv e-prints, art. arXiv:2011.03185, Nov 2020. URL https:/​/​​abs/​2011.03185.

[23] Seth Lloyd and Samuel L Braunstein. Quantum computation over continuous variables. In Quantum Information with Continuous Variables, pages 9–17. Springer, 1999a. 10.1007/​978-94-015-1258-9_2.

[24] Seth Lloyd and Samuel L. Braunstein. Quantum computation over continuous variables. Phys. Rev. Lett., 82: 1784–1787, 1999b. 10.1103/​PhysRevLett.82.1784.

[25] Seth Lloyd, Giacomo De Palma, Can Gokler, Bobak Kiani, Zi-Wen Liu, Milad Marvian, Felix Tennie, and Tim Palmer. Quantum algorithm for nonlinear differential equations. arXiv e-prints, art. arXiv:2011.06571, Nov 2020. URL https:/​/​​abs/​2011.06571.

[26] Michael Lubasch, Jaewoo Joo, Pierre Moinier, Martin Kiffner, and Dieter Jaksch. Variational quantum algorithms for nonlinear problems. Phys. Rev. A, 101: 010301, Jan 2020. 10.1103/​PhysRevA.101.010301.

[27] Jarrod R. McClean, Jonathan Romero, Ryan Babbush, and Alan Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New J. Phys., 18: 023023, 2016. 10.1088/​1367-2630/​18/​2/​023023.

[28] Michael L. Minion. A hybrid parareal spectral deferred corrections method. Comm. App. Math. Comp. Sci., 5: 265–301, 2010. 10.2140/​camcos.2010.5.265.

[29] D. O’Malley and V. V. Vesselinov. Toq.jl: A high-level programming language for d-wave machines based on julia. In 2016 IEEE High Performance Extreme Computing Conference (HPEC), pages 1–7, 2016. 10.1109/​HPEC.2016.7761616.

[30] Ivo G Rosenberg. Reduction of bivalent maximization to the quadratic case. Cahiers du Centre d’études de Recherche Operationnelle, 1975.

[31] Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput., 26: 1484–1509, 1997. 10.1137/​S0097539795293172.

[32] Chi-Wang Shu and Stanley Osher. Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys., 77: 439–471, 1988. 10.1016/​0021-9991(88)90177-5.

[33] Daniel R. Simon. On the power of quantum computation. SIAM J. Comput., 26 (5): 1474–1483, 1997. 10.1137/​S0097539796298637.

[34] Andrew Staniforth and John Thuburn. Horizontal grids for global weather and climate prediction models: A review. Q. J. R. Meteorol. Soc., 138 (662): 1–26, 2012. 10.1002/​qj.958.

[35] Kotaro Tanahashi, Shinichi Takayanagi, Tomomitsu Motohashi, and Shu Tanaka. Application of ising machines and a software development for ising machines. Journal of the Physical Society of Japan, 88 (6): 061010, 2019. 10.7566/​JPSJ.88.061010.

[36] Vlatko Vedral, Adriano Barenco, and Artur Ekert. Quantum networks for elementary arithmetic operations. Phys. Rev. A, 54: 147–153, 1996. 10.1103/​PhysRevA.54.147.

[37] Guillaume Verdon, Jason Pye, and Michael Broughton. A universal training algorithm for quantum deep learning. arXiv:1806.09729, 2018. URL https:/​/​​abs/​1806.09729.

[38] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd. Gaussian quantum information. Rev. Mod. Phys., 84: 621–669, 2012. 10.1103/​RevModPhys.84.621.

[39] Tao Xin, Shijie Wei, Jianlian Cui, Junxiang Xiao, Iñigo Arrazola, Lucas Lamata, Xiangyu Kong, Dawei Lu, Enrique Solano, and Guilu Long. Quantum algorithm for solving linear differential equations: Theory and experiment. Phys. Rev. A, 101: 032307, 2020. 10.1103/​PhysRevA.101.032307.

Cited by

[1] Martin Knudsen and Christian B. Mendl, “Solving Differential Equations via Continuous-Variable Quantum Computers”, arXiv:2012.12220.

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Quantum walk-based portfolio optimisation



N. Slate, E. Matwiejew, S. Marsh, and J. B. Wang

Department of Physics, The University of Western Australia, Perth WA 6009, Australia

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This paper proposes a highly efficient quantum algorithm for portfolio optimisation targeted at near-term noisy intermediate-scale quantum computers. Recent work by Hodson et al. (2019) explored potential application of hybrid quantum-classical algorithms to the problem of financial portfolio rebalancing. In particular, they deal with the portfolio optimisation problem using the Quantum Approximate Optimisation Algorithm and the Quantum Alternating Operator Ansatz. In this paper, we demonstrate substantially better performance using a newly developed Quantum Walk Optimisation Algorithm in finding high-quality solutions to the portfolio optimisation problem.

► BibTeX data

► References

[1] P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Sci. Comput., 26 (5): 1484–1509, 1997. 10.1137/​s0097539795293172.

[2] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, 2010. 10.1017/​CBO9780511976667.

[3] J. Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2: 79, 2018. 10.22331/​q-2018-08-06-79.

[4] P. Rebentrost and S. Lloyd. Quantum computational finance: quantum algorithm for portfolio optimization, 2018. URL https:/​/​​abs/​1811.03975.

[5] P. Rebentrost, B. Gupt, and T. R. Bromley. Quantum computational finance: Monte Carlo pricing of financial derivatives. Phys. Rev. A, 98: 022321, 2018. 10.1103/​PhysRevA.98.022321.

[6] R. Orús, S. Mugel, and E. Lizaso. Quantum computing for finance: Overview and prospects. Rev. Phys., 4: 100028, 2019. 10.1016/​j.revip.2019.100028.

[7] S. Woerner and D. J. Egger. Quantum risk analysis. npj Quantum Inf., 5 (1), 2019. 10.1038/​s41534-019-0130-6.

[8] M. Hodson, B. Ruck, H. Ong, D. Garvin, and S. Dulman. Portfolio rebalancing experiments using the Quantum Alternating Operator Ansatz, 2019. URL https:/​/​​abs/​1911.05296.

[9] H. Markowitz. Portfolio selection. J. Finance, 7 (1): 77–91, 1952. 10.1111/​j.1540-6261.1952.tb01525.x.

[10] A. Palczewski. LP algorithms for portfolio optimization: The PortfolioOptim package. R J., 10: 308–327, 2018. 10.32614/​RJ-2018-028.

[11] R. Mansini and M. G. Speranza. Heuristic algorithms for the portfolio selection problem with minimum transaction lots. Eur. J. Oper. Res., 114 (2): 219–233, 1999. 10.1016/​S0377-2217(98)00252-5.

[12] T. F. Coleman, Y. Li, and J. Henniger. Minimizing tracking error while restricting the number of assets. J. Risk, 8: 33, 2006. 10.21314/​JOR.2006.134.

[13] J. Cook, S. Eidenbenz, and A. Bärtschi. The Quantum Alternating Operator Ansatz on max-k vertex cover, 2019. URL https:/​/​​abs/​1910.13483. 10.1109/​QCE49297.2020.00021.

[14] E. Farhi and A. W. Harrow. Quantum supremacy through the Quantum Approximate Optimization Algorithm, 2016. URL https:/​/​​abs/​1602.07674.

[15] S. Marsh and J. B. Wang. Combinatorial optimization via highly efficient quantum walks. Phys. Rev. Research, 2: 023302, 2020. 10.1103/​PhysRevResearch.2.023302.

[16] M. C. Steinbach. Markowitz revisited: Mean-variance models in financial portfolio analysis. SIAM Rev., 43 (1): 31–85, 2001. 10.1137/​S0036144500376650.

[17] K. P. Anagnostopoulos and G. Mamanis. The mean–variance cardinality constrained portfolio optimization problem: An experimental evaluation of five multiobjective evolutionary algorithms. Expert Syst. Appl., 38 (11): 14208–14217, 2011. 10.1016/​j.eswa.2011.04.233.

[18] M. Bióna. Handbook of enumerative combinatorics. CRC Press, Boca Raton, 2015. 10.1201/​b18255.

[19] S. Hadfield, Z. Wang, B. O’Gorman, E. Rieffel, D. Venturelli, and R. Biswas. From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz. Algorithms, 12 (2): 34, 2019. 10.3390/​a12020034.

[20] E. Farhi and S. Gutmann. Quantum computation and decision trees. Phys. Rev. A, 58: 915–928, 1998. 10.1103/​PhysRevA.58.915.

[21] K. Manouchehri and J. B. Wang. Physical implementation of quantum walks. Springer, Heidelberg, 2014. 10.1007/​978-3-642-36014-5.

[22] X. Qiang, T. Loke, A. Montanaro, K. Aungskunsiri, X. Zhou, J. L. O’Brien, J. B. Wang, and J. C. F. Matthews. Efficient quantum walk on a quantum processor. Nat. Commun., 7 (1): 11511, 2016. 10.1038/​ncomms11511.

[23] A. Mahasinghe and J. B. Wang. Efficient quantum circuits for Toeplitz and Hankel matrices. J. Phys. A, 49 (27): 275301, 2016. 10.1088/​1751-8113/​49/​27/​275301.

[24] S. S. Zhou, T. Loke, J. A. Izaac, and J. B. Wang. Quantum Fourier transform in computational basis. Quantum Inf. Process., 16 (3): 82, 2017. 10.1007/​s11128-017-1515-0.

[25] S. S. Zhou and J. B. Wang. Efficient quantum circuits for dense circulant and circulant-like operators. R. Soc. Open Sci., 4 (5, May): 160906, 12, 2017. 10.1098/​rsos.160906.

[26] T. Loke and J. B. Wang. Efficient quantum circuits for Szegedy quantum walks. Ann. Phys., 382: 64–84, 2017a. 10.1016/​j.aop.2017.04.006.

[27] T. Loke and J. B. Wang. Efficient quantum circuits for continuous-time quantum walks on composite graphs. J. Phys. A, 50 (5): 055303, 2017b. 10.1088/​1751-8121/​aa53a9.

[28] X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews. Large-scale silicon quantum photonics implementing arbitrary two-qubit processing. Nat. Photonics, 12 (9): 534–539, 2018. 10.1038/​s41566-018-0236-y.

[29] C.-H. Yu, F. Gao, C. Liu, D. Huynh, M. Reynolds, and J. Wang. Quantum algorithm for visual tracking. Phys. Rev. A, 99: 022301, 2019. 10.1103/​PhysRevA.99.022301.

[30] R. Cleve and J. Watrous. Fast parallel circuits for the quantum Fourier transform. In Proceedings of SFCS 41, pages 526–536, 2000. 10.1109/​SFCS.2000.892140.

[31] G. R. Ahokas. Improved algorithms for approximate quantum Fourier transforms and sparse Hamiltonian simulations. Master’s thesis, University of Calgary, 2004. URL https:/​/​​10.11575/​PRISM/​22839.

[32] E. Matwiejew. QuOp_MPI: Parallel distributed memory simulation of Quantum Approximate Optimization Algorithms, 2020. URL https:/​/​​10.5281/​zenodo.3681801.

[33] E. Matwiejew and J. B. Wang. QSW_MPI: A framework for parallel simulation of quantum stochastic walks. Comput. Phys. Commun., 260: 107724, 2021. 10.1016/​j.cpc.2020.107724.

[34] W. McKinney. Data Structures for Statistical Computing in Python. In Proceedings of the 9th Python in Science Conference, pages 56–61, 2010. 10.25080/​Majora-92bf1922-00a.

[35] L. Han and M. Neumann. Effect of dimensionality on the Nelder–Mead simplex method. Optim. Methods Softw., 21 (1): 1–16, 2006. 10.1080/​10556780512331318290.

[36] C. Zalka. Grover’s quantum searching algorithm is optimal. Phys. Rev. A, 60: 2746–2751, 1999. 10.1103/​PhysRevA.60.2746.

[37] A. M. Childs and J. Goldstone. Spatial search by quantum walk. Phys. Rev. A, 70: 022314, 2004. 10.1103/​PhysRevA.70.022314.

[38] J. Roland and N. J. Cerf. Quantum-circuit model of Hamiltonian search algorithms. Phys. Rev. A, 68: 062311, 2003. 10.1103/​PhysRevA.68.062311.

[39] D. L. Applegate, R. E. Bixby, V. Chvatal, and W. J. Cook. The Traveling Salesman Problem: A Computational Study. Princeton University Press, USA, 2007. 10.1515/​9781400841103.

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An efficient quantum algorithm for the time evolution of parameterized circuits



Stefano Barison, Filippo Vicentini, and Giuseppe Carleo

Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

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We introduce a novel hybrid algorithm to simulate the real-time evolution of quantum systems using parameterized quantum circuits. The method, named “projected – Variational Quantum Dynamics” (p-VQD) realizes an iterative, global projection of the exact time evolution onto the parameterized manifold. In the small time-step limit, this is equivalent to the McLachlan’s variational principle. Our approach is efficient in the sense that it exhibits an optimal linear scaling with the total number of variational parameters. Furthermore, it is global in the sense that it uses the variational principle to optimize all parameters at once. The global nature of our approach then significantly extends the scope of existing efficient variational methods, that instead typically rely on the iterative optimization of a restricted subset of variational parameters. Through numerical experiments, we also show that our approach is particularly advantageous over existing global optimization algorithms based on the time-dependent variational principle that, due to a demanding quadratic scaling with parameter numbers, are unsuitable for large parameterized quantum circuits.

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► References

[1] Frank Arute “Quantum supremacy using a programmable superconducting processor” Nature 574, 505-510 (2019).

[2] LeeAnn M. Sager, Scott E. Smart, and David A. Mazziotti, “Preparation of an exciton condensate of photons on a 53-qubit quantum computer” Physical Review Research 2 (2020).

[3] P.W. Shor “Algorithms for quantum computation: discrete logarithms and factoring” Proceedings 35th Annual Symposium on Foundations of Computer Science (1994).

[4] D. Coppersmith “An approximate Fourier transform useful in quantum factoring” (1994).

[5] Giuseppe E Santoroand Erio Tosatti “Optimization using quantum mechanics: quantum annealing through adiabatic evolution” Journal of Physics A: Mathematical and General 39, R393–R431 (2006).

[6] Ivan Kassal, Stephen P. Jordan, Peter J. Love, Masoud Mohseni, and Alán Aspuru-Guzik, “Polynomial-time quantum algorithm for the simulation of chemical dynamics” Proceedings of the National Academy of Sciences 105, 18681–18686 (2008).

[7] I. M. Georgescu, S. Ashhab, and Franco Nori, “Quantum simulation” Rev. Mod. Phys. 86, 153–185 (2014).

[8] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O’Brien, “A variational eigenvalue solver on a photonic quantum processor” Nature Communications 5 (2014).

[9] Ying Liand Simon C. Benjamin “Efficient Variational Quantum Simulator Incorporating Active Error Minimization” Phys. Rev. X 7, 021050 (2017).

[10] Pauline J. Ollitrault, Abhinav Kandala, Chun-Fu Chen, Panagiotis Kl. Barkoutsos, Antonio Mezzacapo, Marco Pistoia, Sarah Sheldon, Stefan Woerner, Jay M. Gambetta, and Ivano Tavernelli, “Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor” Physical Review Research 2 (2020).

[11] Mario Motta, Chong Sun, Adrian T. K. Tan, Matthew J. O’Rourke, Erika Ye, Austin J. Minnich, Fernando G. S. L. Brandão, and Garnet Kin-Lic Chan, “Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution” Nature Physics 16, 205–210 (2019).

[12] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, “Variational Quantum Algorithms” (2020).

[13] Jacob L. Beckey, M. Cerezo, Akira Sone, and Patrick J. Coles, “Variational Quantum Algorithm for Estimating the Quantum Fisher Information” (2020).

[14] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd, “Quantum machine learning” Nature 549, 195–202 (2017).

[15] Sima E. Borujeni, Saideep Nannapaneni, Nam H. Nguyen, Elizabeth C. Behrman, and James E. Steck, “Quantum circuit representation of Bayesian networks” (2020).

[16] Jonathan Romero, Jonathan P Olson, and Alan Aspuru-Guzik, “Quantum autoencoders for efficient compression of quantum data” Quantum Science and Technology 2, 045001 (2017).

[17] Iordanis Kerenidis, Jonas Landman, Alessandro Luongo, and Anupam Prakash, “q-means: A quantum algorithm for unsupervised machine learning” Advances in Neural Information Processing Systems 32, 4134–4144 (2019).

[18] Maria Schuldand Nathan Killoran “Quantum Machine Learning in Feature Hilbert Spaces” Phys. Rev. Lett. 122, 040504 (2019).

[19] Maria Schuld, Alex Bocharov, Krysta M. Svore, and Nathan Wiebe, “Circuit-centric quantum classifiers” Physical Review A 101 (2020).

[20] Vojtěch Havlíček, Antonio D. Córcoles, Kristan Temme, Aram W. Harrow, Abhinav Kandala, Jerry M. Chow, and Jay M. Gambetta, “Supervised learning with quantum-enhanced feature spaces” Nature 567, 209–212 (2019).

[21] Mohammad H. Amin, Evgeny Andriyash, Jason Rolfe, Bohdan Kulchytskyy, and Roger Melko, “Quantum Boltzmann Machine” Phys. Rev. X 8, 021050 (2018).

[22] Iris Cong, Soonwon Choi, and Mikhail D. Lukin, “Quantum convolutional neural networks” Nature Physics 15, 1273–1278 (2019).

[23] P. J. J. O’Malley “Scalable Quantum Simulation of Molecular Energies” Phys. Rev. X 6, 031007 (2016).

[24] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta, “Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets” Nature 549, 242–246 (2017).

[25] Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin-Lic Chan, “Quantum Algorithms for Quantum Chemistry and Quantum Materials Science” Chemical Reviews 120, 12685–12717 (2020).

[26] H. F. Trotter “On the product of semi-groups of operators” Proc. Amer. Math. Soc. 10, 545–551 (1959).

[27] Masuo Suzuki “General theory of fractal path integrals with applications to many‐body theories and statistical physics” Journal of Mathematical Physics 32, 400–407 (1991).

[28] Daniel S. Abramsand Seth Lloyd “Simulation of Many-Body Fermi Systems on a Universal Quantum Computer” Phys. Rev. Lett. 79, 2586–2589 (1997).

[29] G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, “Quantum algorithms for fermionic simulations” Phys. Rev. A 64, 022319 (2001).

[30] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C. Benjamin, “Theory of variational quantum simulation” Quantum 3, 191 (2019).

[31] Cristina Cîrstoiu, Zoë Holmes, Joseph Iosue, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, “Variational fast forwarding for quantum simulation beyond the coherence time” npj Quantum Information 6 (2020).

[32] Benjamin Commeau, M. Cerezo, Zoë Holmes, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, “Variational Hamiltonian Diagonalization for Dynamical Quantum Simulation” (2020).

[33] Kishor Bhartiand Tobias Haug “Quantum Assisted Simulator” (2020).

[34] P. A. M. Dirac “Note on Exchange Phenomena in the Thomas Atom” Mathematical Proceedings of the Cambridge Philosophical Society 26, 376–385 (1930).

[35] Jacov Frenkel “Wave Mechanics: Advanced General Theory” Oxford University Press (1934).

[36] A.D. McLachlan “A variational solution of the time-dependent Schrodinger equation” Molecular Physics 8, 39–44 (1964).

[37] Jutho Haegeman, J. Ignacio Cirac, Tobias J. Osborne, Iztok Pižorn, Henri Verschelde, and Frank Verstraete, “Time-Dependent Variational Principle for Quantum Lattices” Phys. Rev. Lett. 107, 070601 (2011).

[38] Jutho Haegeman, Christian Lubich, Ivan Oseledets, Bart Vandereycken, and Frank Verstraete, “Unifying time evolution and optimization with matrix product states” Phys. Rev. B 94, 165116 (2016).

[39] Giuseppe Carleo, Federico Becca, Marco Schiro, and Michele Fabrizio, “Localization and Glassy Dynamics Of Many-Body Quantum Systems” Scientific Reports 2, 243 (2012).

[40] Giuseppe Carleo, Federico Becca, Laurent Sanchez-Palencia, Sandro Sorella, and Michele Fabrizio, “Light-cone effect and supersonic correlations in one- and two-dimensional bosonic superfluids” Phys. Rev. A 89, 031602 (2014).

[41] Michael Kolodrubetz, Dries Sels, Pankaj Mehta, and Anatoli Polkovnikov, “Geometry and non-adiabatic response in quantum and classical systems” Physics Reports 697, 1–87 (2017).

[42] Marin Bukov, Dries Sels, and Anatoli Polkovnikov, “Geometric Speed Limit of Accessible Many-Body State Preparation” Phys. Rev. X 9, 011034 (2019).

[43] Marcello Benedetti, Mattia Fiorentini, and Michael Lubasch, “Hardware-efficient variational quantum algorithms for time evolution” (2020).

[44] Lucas Slattery, Benjamin Villalonga, and Bryan K. Clark, “Unitary Block Optimization for Variational Quantum Algorithms” (2021).

[45] F. Barratt, James Dborin, Matthias Bal, Vid Stojevic, Frank Pollmann, and A. G. Green, “Parallel quantum simulation of large systems on small NISQ computers” npj Quantum Information 7 (2021).

[46] Sheng-Hsuan Lin, Rohit Dilip, Andrew G. Green, Adam Smith, and Frank Pollmann, “Real- and Imaginary-Time Evolution with Compressed Quantum Circuits” PRX Quantum 2 (2021).

[47] Matthew Otten, Cristian L. Cortes, and Stephen K. Gray, “Noise-Resilient Quantum Dynamics Using Symmetry-Preserving Ansatzes” (2019).

[48] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo, “Quantum Natural Gradient” Quantum 4, 269 (2020).

[49] Daniel Gottesmanand Isaac Chuang “Quantum Digital Signatures” (2001).

[50] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf, “Quantum Fingerprinting” Physical Review Letters 87 (2001).

[51] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran, “Evaluating analytic gradients on quantum hardware” Phys. Rev. A 99, 032331 (2019).

[52] J.C. Spall “Implementation of the simultaneous perturbation algorithm for stochastic optimization” IEEE Transactions on Aerospace and Electronic Systems 34, 817–823 (1998).

[53] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, “Quantum circuit learning” Physical Review A 98 (2018).

[54] Robert M. Parrish, Edward G. Hohenstein, Peter L. McMahon, and Todd J. Martinez, “Hybrid Quantum/​Classical Derivative Theory: Analytical Gradients and Excited-State Dynamics for the Multistate Contracted Variational Quantum Eigensolver” (2019).

[55] Gavin E. Crooks “Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition” (2019).

[56] Andrea Mari, Thomas R. Bromley, and Nathan Killoran, “Estimating the gradient and higher-order derivatives on quantum hardware” Physical Review A 103 (2021).

[57] Leonardo Banchiand Gavin E. Crooks “Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule” Quantum 5, 386 (2021).

[58] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles, “Cost function dependent barren plateaus in shallow parametrized quantum circuits” Nature Communications 12 (2021).

[59] Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven, “Barren plateaus in quantum neural network training landscapes” Nature Communications 9 (2018).

[60] Tobias Haugand M. S. Kim “Optimal training of variational quantum algorithms without barren plateaus” (2021).

[61] Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, and Marcello Benedetti, “An initialization strategy for addressing barren plateaus in parametrized quantum circuits” Quantum 3, 214 (2019).

[62] Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J. Coles, “Variational Quantum Linear Solver” (2020).

[63] Héctor Abraham et al. “Qiskit: An Open-source Framework for Quantum Computing” (2019).

[64] J. Demmel “On condition numbers and the distance to the nearest ill-posed problem” Numerische Mathematik 51, 251–289 (1987).

[65] Guifré Vidal “Efficient Simulation of One-Dimensional Quantum Many-Body Systems” Physical Review Letters 93, 040502 (2004).

[66] A. J. Daley, C. Kollath, U. Schollwock, and G. Vidal, “Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces” Journal of Statistical Mechanics-Theory and Experiment P04005 (2004).

[67] Steven R. Whiteand Adrian E. Feiguin “Real-Time Evolution Using the Density Matrix Renormalization Group” Physical Review Letters 93, 076401 (2004).

[68] Giuseppe Carleoand Matthias Troyer “Solving the quantum many-body problem with artificial neural networks” Science 355, 602–606 (2017).

[69] Markus Schmittand Markus Heyl “Quantum Many-Body Dynamics in Two Dimensions with Artificial Neural Networks” Physical Review Letters 125, 100503 (2020) Publisher: American Physical Society.

[70] Stefano Barison “Github repository” (2021).

Cited by

[1] Yong-Xin Yao, Niladri Gomes, Feng Zhang, Cai-Zhuang Wang, Kai-Ming Ho, Thomas Iadecola, and Peter P. Orth, “Adaptive Variational Quantum Dynamics Simulations”, PRX Quantum 2 3, 030307 (2021).

[2] Jonathan Wei Zhong Lau, Tobias Haug, Leong Chuan Kwek, and Kishor Bharti, “NISQ Algorithm for Hamiltonian Simulation via Truncated Taylor Series”, arXiv:2103.05500.

[3] Michael R. Geller, Zoë Holmes, Patrick J. Coles, and Andrew Sornborger, “Experimental Quantum Learning of a Spectral Decomposition”, arXiv:2104.03295.

[4] Kian Hwee Lim, Tobias Haug, Leong Chuan Kwek, and Kishor Bharti, “Fast-Forwarding with NISQ Processors without Feedback Loop”, arXiv:2104.01931.

[5] Tobias Haug and M. S. Kim, “Optimal training of variational quantum algorithms without barren plateaus”, arXiv:2104.14543.

[6] Julien Gacon, Christa Zoufal, Giuseppe Carleo, and Stefan Woerner, “Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information”, arXiv:2103.09232.

[7] Yongdan Yang, Bing-Nan Lu, and Ying Li, “Accelerated quantum Monte Carlo with mitigated error on noisy quantum computer”, arXiv:2106.09880.

[8] Paolo P. Mazza, Dominik Zietlow, Federico Carollo, Sabine Andergassen, Georg Martius, and Igor Lesanovsky, “Machine learning time-local generators of open quantum dynamics”, Physical Review Research 3 2, 023084 (2021).

[9] Refik Mansuroglu, Samuel Wilkinson, Ludwig Nützel, and Michael J. Hartmann, “Classical Variational Optimization of Gate Sequences for Time Evolution of Large Quantum Systems”, arXiv:2106.03680.

[10] Michael R. Geller, Andrew Arrasmith, Zoë Holmes, Bin Yan, Patrick J. Coles, and Andrew Sornborger, “Quantum simulation of operator spreading in the chaotic Ising model”, arXiv:2106.16170.

[11] Lucas Slattery, Benjamin Villalonga, and Bryan K. Clark, “Unitary Block Optimization for Variational Quantum Algorithms”, arXiv:2102.08403.

[12] Rouven Koch and Jose L. Lado, “Neural network enhanced hybrid quantum many-body dynamical distributions”, arXiv:2105.03129.

[13] Kishor Bharti, Tobias Haug, Vlatko Vedral, and Leong-Chuan Kwek, “NISQ Algorithm for Semidefinite Programming”, arXiv:2106.03891.

The above citations are from SAO/NASA ADS (last updated successfully 2021-07-28 10:06:09). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2021-07-28 10:06:07: Could not fetch cited-by data for 10.22331/q-2021-07-28-512 from Crossref. This is normal if the DOI was registered recently.

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Causal and compositional structure of unitary transformations



Robin Lorenz1,2 and Jonathan Barrett1

1Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK
2Cambridge Quantum Computing Ltd, 17 Beaumont Street, Oxford OX1 2NA, UK

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The causal structure of a unitary transformation is the set of relations of possible influence between any input subsystem and any output subsystem. We study whether such causal structure can be understood in terms of compositional structure of the unitary. Given a quantum circuit with no path from input system $A$ to output system $B$, system $A$ cannot influence system $B$. Conversely, given a unitary $U$ with a no-influence relation from input $A$ to output $B$, it follows from [B. Schumacher and M. D. Westmoreland, Quantum Information Processing 4 no. 1, (Feb, 2005)] that there exists a circuit decomposition of $U$ with no path from $A$ to $B$. However, as we argue, there are unitaries for which there does not exist a circuit decomposition that makes all causal constraints evident $textit{simultaneously}$. To address this, we introduce a new formalism of `extended circuit diagrams’, which goes beyond what is expressible with quantum circuits, with the core new feature being the ability to represent direct sum structures in addition to sequential and tensor product composition. A $textit{causally faithful}$ extended circuit decomposition, representing a unitary $U$, is then one for which there is a path from an input $A$ to an output $B$ if and only if there actually is influence from $A$ to $B$ in $U$. We derive causally faithful extended circuit decompositions for a large class of unitaries, where in each case, the decomposition is implied by the unitary’s respective causal structure. We hypothesize that every finite-dimensional unitary transformation has a causally faithful extended circuit decomposition.

Whenever one is able to draw an intuitive picture that succinctly captures the essential features of a complicated whole, then not only does it usually help communication, arguably, this also is a sign of having achieved good conceptual understanding. Also for quantum theory there is a rich history of developing and employing diagrammatic reasoning, where quantum circuit diagrams capture aspects of how quantum systems interact with each other and evolve over time. Circuit diagrams are the basic ingredient to one of the main paradigms of quantum computation, and have also proven useful in research ranging from the foundations of quantum theory to the design of algorithms and efficient quantum compilers. One particular area of research of foundational, as well as applied importance is causality. Unitary transformations that play a fundamental role in the quantum formalism, describe the evolution of a set of systems and have a clear notion of causal structure: which of the systems can causally influence which other systems.

Combining all these lines of thought, this paper asks: can one understand the causal structure of a unitary transformation in compositional terms, i.e. through an intuitive diagram, where its components are connected up in such a way that lays bare where and how causal influence goes? We first show that this is generally not possible with ordinary circuit diagrams. We then derive new kinds of decompositions for many causal structures, as well as introduce a new graphical language of ‘extended circuit diagrams’ to visualise them. This novel perspective to study causal structure and its ramifications is of conceptual significance, as well as likely to facilitate progress with other open problems, which it in fact already has in the study of indefinite causal order, a much debated area of quantum physics.

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► References

[1] G. Chiribella, G. M. D’Ariano, and P. Perinotti. Transforming quantum operations: Quantum supermaps. EPL (Europhysics Letters), 83 (3): 30004, jul 2008. 10.1209/​0295-5075/​83/​30004.

[2] Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti. Theoretical framework for quantum networks. Phys. Rev. A, 80: 022339, Aug 2009. 10.1103/​PhysRevA.80.022339.

[3] Giulio Chiribella, Giacomo Mauro D’Ariano, Paolo Perinotti, and Benoit Valiron. Quantum computations without definite causal structure. Phys. Rev. A, 88: 022318, Aug 2013. 10.1103/​PhysRevA.88.022318.

[4] Rafael Chaves, Lukas Luft, and David Gross. Causal structures from entropic information: geometry and novel scenarios. New Journal of Physics, 16 (4): 043001, apr 2014. 10.1088/​1367-2630/​16/​4/​043001.

[5] Christopher J Wood and Robert W Spekkens. The lesson of causal discovery algorithms for quantum correlations: causal explanations of bell-inequality violations require fine-tuning. New Journal of Physics, 17 (3): 033002, mar 2015. 10.1088/​1367-2630/​17/​3/​033002.

[6] Tobias Fritz. Beyond bell’s theorem ii: Scenarios with arbitrary causal structure. Communications in Mathematical Physics, 341 (2): 391–434, 2016. 10.1007/​s00220-015-2495-5.

[7] Rafael Chaves. Polynomial bell inequalities. Phys. Rev. Lett., 116: 010402, Jan 2016. 10.1103/​PhysRevLett.116.010402.

[8] Marc-Olivier Renou, Elisa Bäumer, Sadra Boreiri, Nicolas Brunner, Nicolas Gisin, and Salman Beigi. Genuine quantum nonlocality in the triangle network. Phys. Rev. Lett., 123: 140401, Sep 2019. 10.1103/​PhysRevLett.123.140401.

[9] Robert R Tucci. Quantum bayesian nets. International Journal of Modern Physics B, 9 (03): 295–337, 1995. 10.1142/​S0217979295000148.

[10] Robert R. Tucci. Factorization of quantum density matrices according to bayesian and markov networks. arXiv preprint quant-ph/​0701201. 2007.

[11] M. S. Leifer and Robert W. Spekkens. Towards a formulation of quantum theory as a causally neutral theory of bayesian inference. Phys. Rev. A, 88: 052130, Nov 2013. 10.1103/​PhysRevA.88.052130.

[12] Rafael Chaves, Christian Majenz, and David Gross. Information–theoretic implications of quantum causal structures. Nature communications, 6 (1): 1–8, 2015. 10.1038/​ncomms6766.

[13] Fabio Costa and Sally Shrapnel. Quantum causal modelling. New Journal of Physics, 18 (6): 063032, jun 2016. 10.1088/​1367-2630/​18/​6/​063032.

[14] John-Mark A. Allen, Jonathan Barrett, Dominic C. Horsman, Ciarán M. Lee, and Robert W. Spekkens. Quantum common causes and quantum causal models. Phys. Rev. X, 7: 031021, Jul 2017. 10.1103/​PhysRevX.7.031021.

[15] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov. Quantum causal models. arXiv preprint quant-ph/​1906.10726. 2019.

[16] Lucien Hardy. Probability theories with dynamic causal structure: a new framework for quantum gravity. arXiv preprint gr-qc/​0509120. 2005. https:/​/​​abs/​gr-qc/​0509120.

[17] Giulio Chiribella. Perfect discrimination of no-signalling channels via quantum superposition of causal structures. Phys. Rev. A, 86: 040301, Oct 2012. 10.1103/​PhysRevA.86.040301.

[18] Ognyan Oreshkov, Fabio Costa, and Caslav Brukner. Quantum correlations with no causal order. Nature Commun., 3: 1092, 2012. 10.1038/​ncomms2076.

[19] Philippe Allard Guérin, Adrien Feix, Mateus Araújo, and Časlav Brukner. Exponential communication complexity advantage from quantum superposition of the direction of communication. Phys. Rev. Lett., 117: 100502, Sep 2016. 10.1103/​PhysRevLett.117.100502.

[20] Caslav Brukner. Quantum causality. Nat Phys, 10 (4): 259–263, April 2014. 10.1038/​nphys2930.

[21] Christopher Portmann, Christian Matt, Ueli Maurer, Renato Renner, and Björn Tackmann. Causal boxes: Quantum information-processing systems closed under composition. IEEE Transactions on Information Theory, 63 (5): 3277–3305, 2017. 10.1109/​TIT.2017.2676805.

[22] Ciarán M. Lee and Matty J. Hoban. Towards device-independent information processing on general quantum networks. Phys. Rev. Lett., 120: 020504, Jan 2018. 10.1103/​PhysRevLett.120.020504.

[23] David Beckman, Daniel Gottesman, M. A. Nielsen, and John Preskill. Causal and localizable quantum operations. Phys. Rev., A64: 052309, 2001. 10.1103/​PhysRevA.64.052309.

[24] Benjamin Schumacher and Michael D. Westmoreland. Locality and information transfer in quantum operations. Quantum Information Processing, 4 (1): 13–34, Feb 2005. ISSN 1573-1332. 10.1007/​s11128-004-3193-y.

[25] Pablo Arrighi, Vincent Nesme, and Reinhard Werner. Unitarity plus causality implies localizability. Journal of Computer and System Sciences, 77 (2): 372–378, 2011a. 10.1016/​j.jcss.2010.05.004.

[26] T Eggeling, D Schlingemann, and R. F Werner. Semicausal operations are semilocalizable. Europhysics Letters (EPL), 57 (6): 782–788, mar 2002. 10.1209/​epl/​i2002-00579-4.

[27] M. Piani, M. Horodecki, P. Horodecki, and R. Horodecki. Properties of quantum nonsignaling boxes. Phys. Rev. A, 74: 012305, Jul 2006. 10.1103/​PhysRevA.74.012305.

[28] Benjamin Schumacher and Michael D Westmoreland. Isolation and information flow in quantum dynamics. Foundations of Physics, 42 (7): 926–931, 2012. 10.1007/​s10701-012-9651-y.

[29] Cédric Bény. Causal structure of the entanglement renormalization ansatz. New Journal of Physics, 15 (2): 023020, feb 2013. 10.1088/​1367-2630/​15/​2/​023020.

[30] B. Schumacher and R. F. Werner. Reversible quantum cellular automata. arXiv preprint quant-ph/​0405174. 2004.

[31] Pablo Arrighi, Vincent Nesme, and Reinhard Werner. One-dimensional quantum cellular automata over finite, unbounded configurations. In International Conference on Language and Automata Theory and Applications, pages 64–75. Springer, 2008. 10.1007/​978-3-540-88282-4_8.

[32] Pablo Arrighi, Renan Fargetton, Vincent Nesme, and Eric Thierry. Applying causality principles to the axiomatization of probabilistic cellular automata. In Conference on Computability in Europe, pages 1–10. Springer, 2011b. 10.1007/​978-3-642-21875-0_1.

[33] Asif Shakeel and Peter J Love. When is a quantum cellular automaton (qca) a quantum lattice gas automaton (qlga)? Journal of Mathematical Physics, 54 (9): 092203, 2013. 10.1063/​1.4821640.

[34] Terence C. Farrelly and Anthony J. Short. Causal fermions in discrete space-time. Phys. Rev. A, 89: 012302, Jan 2014. 10.1103/​PhysRevA.89.012302.

[35] Giacomo Mauro D’Ariano and Paolo Perinotti. Derivation of the dirac equation from principles of information processing. Phys. Rev. A, 90: 062106, Dec 2014. 10.1103/​PhysRevA.90.062106.

[36] Alessandro Bisio, Giacomo Mauro D’Ariano, and Paolo Perinotti. Special relativity in a discrete quantum universe. Phys. Rev. A, 94: 042120, Oct 2016. 10.1103/​PhysRevA.94.042120.

[37] Pablo Arrighi and Simon Martiel. Quantum causal graph dynamics. Physical Review D, 96 (2): 024026, 2017. 10.1103/​PhysRevD.96.024026.

[38] Pablo Arrighi, Cédric Bény, and Terry Farrelly. A quantum cellular automaton for one-dimensional qed. Quantum Information Processing, 19 (3): 1–28, 2020. 10.1007/​s11128-019-2555-4.

[39] Paolo Perinotti. Cellular automata in operational probabilistic theories. Quantum, 4: 294, July 2020. ISSN 2521-327X. 10.22331/​q-2020-07-09-294.

[40] Samson Abramsky and Bob Coecke. A categorical semantics of quantum protocols. 2004. 10.1109/​LICS.2004.1319636.

[41] Samson Abramsky and Ross Duncan. A categorical quantum logic. Mathematical Structures in Computer Science, 16 (3): 469–489, 2006. 10.1017/​S0960129506005275.

[42] Bob Coecke and Aleks Kissinger. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning. Cambridge University Press, 2017. 10.1017/​9781316219317.

[43] Chris Heunen and Jamie Vicary. Categories for Quantum Theory: an introduction. Oxford University Press, USA, 2019. 10.1093/​oso/​9780198739623.001.0001.

[44] Bob Coecke and Raymond Lal. Causal categories: relativistically interacting processes. Foundations of Physics, 43 (4): 458–501, 2013. 10.1007/​s10701-012-9646-8.

[45] Aleks Kissinger, Matty Hoban, and Bob Coecke. Equivalence of relativistic causal structure and process terminality. https:/​/​​abs/​1708.04118. 2017.

[46] A. Jamiołkowski. Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics, 3 (4): 275 – 278, 1972. ISSN 0034-4877. https:/​/​​10.1016/​0034-4877(72)90011-0.

[47] Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Applications, 10 (3): 285 – 290, 1975. ISSN 0024-3795. https:/​/​​10.1016/​0024-3795(75)90075-0.

[48] Sandu Popescu, Master’s thesis, unpublished.

[49] H. Buhrman and S. Massar. Causality and tsirelson’s bounds. Phys. Rev. A, 72: 052103, Nov 2005. 10.1103/​PhysRevA.72.052103.

[50] Robert Spekkens. Private communication.

[51] Yakir Aharonov and Lev Vaidman. The Two-State Vector Formalism: An Updated Review, pages 399–447. Springer Berlin Heidelberg, Berlin, Heidelberg, 2008. ISBN 978-3-540-73473-4. 10.1007/​978-3-540-73473-4_13.

[52] Yakir Aharonov, Sandu Popescu, Jeff Tollaksen, and Lev Vaidman. Multiple-time states and multiple-time measurements in quantum mechanics. Phys. Rev. A, 79: 052110, May 2009. 10.1103/​PhysRevA.79.052110.

[53] Ralph Silva, Yelena Guryanova, Nicolas Brunner, Noah Linden, Anthony J. Short, and Sandu Popescu. Pre- and postselected quantum states: Density matrices, tomography, and kraus operators. Phys. Rev. A, 89: 012121, Jan 2014. 10.1103/​PhysRevA.89.012121.

[54] Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen. Each instant of time a new universe. Springer, Milano, Milano, 2014. ISBN 978-88-470-5216-1. 10.1007/​978-88-470-5217-8_3.

[55] Ralph Silva, Yelena Guryanova, Anthony J Short, Paul Skrzypczyk, Nicolas Brunner, and Sandu Popescu. Connecting processes with indefinite causal order and multi-time quantum states. New Journal of Physics, 19 (10): 103022, oct 2017. 10.1088/​1367-2630/​aa84fe.

[56] Mateus Araújo, Cyril Branciard, Fabio Costa, Adrien Feix, Christina Giarmatzi, and Časlav Brukner. Witnessing causal nonseparability. New Journal of Physics, 17 (10): 102001, oct 2015. 10.1088/​1367-2630/​17/​10/​102001.

[57] Ognyan Oreshkov and Nicolas J Cerf. Operational quantum theory without predefined time. New Journal of Physics, 18 (7): 073037, jul 2016. 10.1088/​1367-2630/​18/​7/​073037.

[58] Ognyan Oreshkov and Christina Giarmatzi. Causal and causally separable processes. New Journal of Physics, 18 (9): 093020, sep 2016. 10.1088/​1367-2630/​18/​9/​093020.

[59] Felix A. Pollock, César Rodríguez-Rosario, Thomas Frauenheim, Mauro Paternostro, and Kavan Modi. Non-markovian quantum processes: Complete framework and efficient characterization. Phys. Rev. A, 97: 012127, Jan 2018a. 10.1103/​PhysRevA.97.012127.

[60] Felix A. Pollock, César Rodríguez-Rosario, Thomas Frauenheim, Mauro Paternostro, and Kavan Modi. Operational markov condition for quantum processes. Phys. Rev. Lett., 120: 040405, Jan 2018b. 10.1103/​PhysRevLett.120.040405.

[61] Augustin Vanrietvelde, Hlér Kristjánsson, and Jonathan Barrett. Routed quantum circuits. https:/​/​​abs/​2011.08120. 2020.

[62] David J Reutter and Jamie Vicary. Shaded tangles for the design and verification of quantum circuits. Proceedings of the Royal Society A, 475 (2224): 20180338, 2019a. 10.1098/​rspa.2018.0338.

[63] David J Reutter and Jamie Vicary. Biunitary constructions in quantum information. Higher Structures, 3, 2019b.

[64] Jamie Vicary. Higher quantum theory. arXiv preprint arXiv:1207.4563. 2012.

[65] Ross Duncan. Generalised Proof-Nets for Compact Categories with Biproducts, pages 70–134. Cambridge University Press, 2009. Preprint available at http:/​/​​abs/​0903.5154.

Cited by

[1] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, “Quantum Causal Models”, arXiv:1906.10726.

[2] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, “Cyclic quantum causal models”, Nature Communications 12, 885 (2021).

[3] Wataru Yokojima, Marco Túlio Quintino, Akihito Soeda, and Mio Murao, “Consequences of preserving reversibility in quantum superchannels”, arXiv:2003.05682.

[4] Augustin Vanrietvelde, Hlér Kristjánsson, and Jonathan Barrett, “Routed quantum circuits”, arXiv:2011.08120.

[5] Paolo Perinotti, “Causal influence in operational probabilistic theories”, arXiv:2012.15213.

The above citations are from SAO/NASA ADS (last updated successfully 2021-07-28 09:46:13). The list may be incomplete as not all publishers provide suitable and complete citation data.

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A Soil-Science Revolution Upends Plans to Fight Climate Change



The hope was that the soil might save us. With civilization continuing to pump ever-increasing amounts of carbon dioxide into the atmosphere, perhaps plants — nature’s carbon scrubbers — might be able to package up some of that excess carbon and bury it underground for centuries or longer.

That hope has fueled increasingly ambitious climate change–mitigation plans. Researchers at the Salk Institute, for example, hope to bioengineer plants whose roots will churn out huge amounts of a carbon-rich, cork-like substance called suberin. Even after the plant dies, the thinking goes, the carbon in the suberin should stay buried for centuries. This Harnessing Plants Initiative is perhaps the brightest star in a crowded firmament of climate change solutions based on the brown stuff beneath our feet.

Such plans depend critically on the existence of large, stable, carbon-rich molecules that can last hundreds or thousands of years underground. Such molecules, collectively called humus, have long been a keystone of soil science; major agricultural practices and sophisticated climate models are built on them.

But over the past 10 years or so, soil science has undergone a quiet revolution, akin to what would happen if, in physics, relativity or quantum mechanics were overthrown. Except in this case, almost nobody has heard about it — including many who hope soils can rescue the climate. “There are a lot of people who are interested in sequestration who haven’t caught up yet,” said Margaret Torn, a soil scientist at Lawrence Berkeley National Laboratory.

A new generation of soil studies powered by modern microscopes and imaging technologies has revealed that whatever humus is, it is not the long-lasting substance scientists believed it to be. Soil researchers have concluded that even the largest, most complex molecules can be quickly devoured by soil’s abundant and voracious microbes. The magic molecule you can just stick in the soil and expect to stay there may not exist.

“I have The Nature and Properties of Soils in front of me — the standard textbook,” said Gregg Sanford, a soil researcher at the University of Wisconsin, Madison. “The theory of soil organic carbon accumulation that’s in that textbook has been proven mostly false … and we’re still teaching it.”

The consequences go far beyond carbon sequestration strategies. Major climate models such as those produced by the Intergovernmental Panel on Climate Change are based on this outdated understanding of soil. Several recent studies indicate that those models are underestimating the total amount of carbon that will be released from soil in a warming climate. In addition, computer models that predict the greenhouse gas impacts of farming practices — predictions that are being used in carbon markets — are probably overly optimistic about soil’s ability to trap and hold on to carbon.

It may still be possible to store carbon underground long term.  Indeed, radioactive dating measurements suggest that some amount of carbon can stay in the soil for centuries. But until soil scientists build a new paradigm to replace the old — a process now underway — no one will fully understand why.

The Death of Humus

Soil doesn’t give up its secrets easily. Its constituents are tiny, varied and outrageously numerous. At a bare minimum, it consists of minerals, decaying organic matter, air, water, and enormously complex ecosystems of microorganisms. One teaspoon of healthy soil contains more bacteria, fungi and other microbes than there are humans on Earth.

The German biologist Franz Karl Achard was an early pioneer in making sense of the chaos. In a seminal 1786 study, he used alkalis to extract molecules made of long carbon chains from peat soils. Over the centuries, scientists came to believe that such long chains, collectively called humus, constituted a large pool of soil carbon that resists decomposition and pretty much just sits there. A smaller fraction consisting of shorter molecules was thought to feed microbes, which respired carbon dioxide to the atmosphere.

This view was occasionally challenged, but by the mid-20th century, the humus paradigm was “the only game in town,” said Johannes Lehmann, a soil scientist at Cornell University. Farmers were instructed to adopt practices that were supposed to build humus. Indeed, the existence of humus is probably one of the few soil science facts that many non-scientists could recite.

What helped break humus’s hold on soil science was physics. In the second half of the 20th century, powerful new microscopes and techniques such as nuclear magnetic resonance and X-ray spectroscopy allowed soil scientists for the first time to peer directly into soil and see what was there, rather than pull things out and then look at them.

What they found — or, more specifically, what they didn’t find — was shocking: there were few or no long “recalcitrant” carbon molecules — the kind that don’t break down. Almost everything seemed to be small and, in principle, digestible.

“We don’t see any molecules in soil that are so recalcitrant that they can’t be broken down,” said Jennifer Pett-Ridge, a soil scientist at Lawrence Livermore National Laboratory. “Microbes will learn to break anything down — even really nasty chemicals.”

Lehmann, whose studies using advanced microscopy and spectroscopy were among the first to reveal the absence of humus, has become the concept’s debunker-in-chief. A 2015 Nature paper he co-authored states that “the available evidence does not support the formation of large-molecular-size and persistent ‘humic substances’ in soils.” In 2019, he gave a talk with a slide containing a mock death announcement for “our friend, the concept of Humus.”

Over the past decade or so, most soil scientists have come to accept this view. Yes, soil is enormously varied. And it contains a lot of carbon. But there’s no carbon in soil that can’t, in principle, be broken down by microorganisms and released into the atmosphere. The latest edition of The Nature and Properties of Soils, published in 2016, cites Lehmann’s 2015 paper and acknowledges that “our understanding of the nature and genesis of soil humus has advanced greatly since the turn of the century, requiring that some long-accepted concepts be revised or abandoned.”

Old ideas, however, can be very recalcitrant. Few outside the field of soil science have heard of humus’s demise.

Buried Promises

At the same time that soil scientists were rediscovering what exactly soil is, climate researchers were revealing that increasing amounts of carbon dioxide in the atmosphere were rapidly warming the climate, with potentially catastrophic consequences.

Thoughts soon turned to using soil as a giant carbon sink. Soils contain enormous amounts of carbon — more carbon than in Earth’s atmosphere and all its vegetation combined. And while certain practices such as plowing can stir up that carbon — farming, over human history, has released an estimated 133 billion metric tons of carbon into the atmosphere — soils can also take up carbon, as plants die and their roots decompose.

Scientists began to suggest that we might be able to coax large volumes of atmospheric carbon back into the soil to dampen or even reverse the damage of climate change.

In practice, this has proved difficult. An early idea to increase carbon stores — planting crops without tilling the soil — has mostly fallen flat. When farmers skipped the tilling and instead drilled seeds into the ground, carbon stores grew in upper soil layers, but they disappeared from lower layers. Most experts now believe that the practice redistributes carbon within the soil rather than increases it, though it can improve other factors such as water quality and soil health.

Efforts like the Harnessing Plants Initiative represent something like soil carbon sequestration 2.0: a more direct intervention to essentially jam a bunch of carbon into the ground.

The initiative emerged when two plant geneticists at the Salk Institute, Joanne Chory and Wolfgang Busch, came up with an idea: Create plants whose roots produce an excess of carbon-rich molecules. By their calculations, if grown widely, such plants might sequester up to 20% of the excess carbon dioxide that humans add to the atmosphere every year.

The researchers zeroed in on a complex, cork-like molecule called suberin, which is produced by many plant roots. Studies from the 1990s and 2000s had hinted that suberin and similar molecules could resist decomposition in soil.

With flashy marketing, the Harnessing Plants Initiative gained attention. An initial round of fundraising in 2019 brought in over $35 million. Last year, the multibillionaire Jeff Bezos contributed $30 million from his “Earth Fund.”

But as the project gained momentum, it attracted doubters. One group of researchers noted in 2016 that no one had actually observed the suberin decomposition process. When those authors did the relevant experiment, they found that much of the suberin decayed quickly.

In 2019, Chory described the project at a TED conference. Asmeret Asefaw Berhe, a soil scientist at the University of California, Merced, who spoke at the same conference, pointed out to Chory that according to modern soil science, suberin, like any carbon-containing compound, should break down in soil. (Berhe, who has been nominated to lead the U.S. Department of Energy’s Office of Science, declined an interview request.)

Around the same time, Hanna Poffenbarger, a soil researcher at the University of Kentucky, made a similar comment after hearing Busch speak at a workshop. “You should really get some soil scientists on board, because the assumption that we can breed for more recalcitrant roots — that may not be valid,” Poffenbarger recalls telling Busch.

Questions about the project surfaced publicly earlier this year, when Jonathan Sanderman, a soil scientist at the Woodwell Climate Research Center in Woods Hole, Massachusetts, tweeted, “I thought the soil biogeochem community had moved on from the idea that there is a magical recalcitrant plant compound. Am I missing some important new literature on suberin?” Another soil scientist responded, “Nope, the literature suggests that suberin will be broken down just like every other organic plant component. I’ve never understood why the @salkinstitute has based their Harnessing Plant Initiative on this premise.”

Busch, in an interview, acknowledged that “there is no unbreakable biomolecule.” But, citing published papers on suberin’s resistance to decomposition, he said, “We are still very optimistic when it comes to suberin.”

He also noted a second initiative Salk researchers are pursuing in parallel to enhancing suberin. They are trying to design plants with longer roots that could deposit carbon deeper in soil. Independent experts such as Sanderman agree that carbon tends to stick around longer in deeper soil layers, putting that solution on potentially firmer conceptual ground.

Chory and Busch have also launched collaborations with Berhe and Poffenbarger, respectively. Poffenbarger, for example, will analyze how soil samples containing suberin-rich plant roots change under different environmental conditions. But even those studies won’t answer questions about how long suberin sticks around, Poffenbarger said — important if the goal is to keep carbon out of the atmosphere long enough to make a dent in global warming.

Beyond the Salk project, momentum and money are flowing toward other climate projects that would rely on long-term carbon sequestration and storage in soils. In an April speech to Congress, for example, President Biden suggested paying farmers to plant cover crops, which are grown not for harvest but to nurture the soil in between plantings of cash crops. Evidence suggests that when cover crop roots break down, some of their carbon stays in the soil — although as with suberin, how long it lasts is an open question.

Not Enough Bugs in the Code

Recalcitrant carbon may also be warping climate prediction.

In the 1960s, scientists began writing large, complex computer programs to predict the global climate’s future. Because soil both takes up and releases carbon dioxide, climate models attempted to take into account soil’s interactions with the atmosphere. But the global climate is fantastically complex, and to enable the programs to run on the machines of the time, simplifications were necessary. For soil, scientists made a big one: They ignored microbes in the soil entirely. Instead, they basically divided soil carbon into short-term and long-term pools, in accordance with the humus paradigm.

More recent generations of models, including ones that the Intergovernmental Panel on Climate Change uses for its widely read reports, are essentially palimpsests built on earlier ones, said Torn. They still assume soil carbon exists in long-term and short-term pools. As a consequence, these models may be overestimating how much carbon will stick around in soils and underestimating how much carbon dioxide they will emit.

Last summer, a study published in Nature examined how much carbon dioxide was released when researchers artificially warmed the soil in a Panamanian rainforest to mimic the long-term effects of climate change. They found that the warmed soil released 55% more carbon than nearby unwarmed areas — a much larger release than predicted by most climate models. The researchers think that microbes in the soil grow more active at the warmer temperatures, leading to the increase.

The study was especially disheartening because most of the world’s soil carbon is in the tropics and the northern boreal zone. Despite this, leading soil models are calibrated to results of soil studies in temperate countries such as the U.S. and Europe, where most studies have historically been done. “We’re doing pretty bad in high latitudes and the tropics,” said Lehmann.

Even temperate climate models need improvement. Torn and colleagues reported earlier this year that, contrary to predictions, deep soil layers in a California forest released roughly a third of their carbon when warmed for five years.

Ultimately, Torn said, models need to represent soil as something closer to what it actually is: a complex, three-dimensional environment governed by a hyper-diverse community of carbon-gobbling bacteria, fungi and other microscopic beings. But even smaller steps would be welcome. Just adding microbes as a single class would be major progress for most models, she said.

Fertile Ground

If the humus paradigm is coming to an end, the question becomes: What will replace it?

One important and long-overlooked factor appears to be the three-dimensional structure of the soil environment. Scientists describe soil as a world unto itself, with the equivalent of continents, oceans and mountain ranges. This complex microgeography determines where microbes such as bacteria and fungi can go and where they can’t; what food they can gain access to and what is off limits.

A soil bacterium “may be only 10 microns away from a big chunk of organic matter that I’m sure they would love to degrade, but it’s on the other side of a cluster of minerals,” said Pett-Ridge. “It’s literally as if it’s on the other side of the planet.”

Another related, and poorly understood, ingredient in a new soil paradigm is the fate of carbon within the soil. Researchers now believe that almost all organic material that enters soil will get digested by microbes. “Now it’s really clear that soil organic matter is just this loose assemblage of plant matter in varying degrees of degradation,” said Sanderman. Some will then be respired into the atmosphere as carbon dioxide. What remains could be eaten by another microbe — and a third, and so on. Or it could bind to a bit of clay or get trapped inside a soil aggregate: a porous clump of particles that, from a microbe’s point of view, could be as large as a city and as impenetrable as a fortress. Studies of carbon isotopes have shown that a lot of carbon can stick around in soil for centuries or even longer. If humus isn’t doing the stabilizing, perhaps minerals and aggregates are.

Before soil science settles on a new theory, there will doubtless be more surprises. One may have been delivered recently by a group of researchers at Princeton University who constructed a simplified artificial soil using microfluidic devices — essentially, tiny plastic channels for moving around bits of fluid and cells. The researchers found that carbon they put inside an aggregate made of bits of clay was protected from bacteria. But when they added a digestive enzyme, the carbon was freed from the aggregate and quickly gobbled up. “To our surprise, no one had drawn this connection between enzymes, bacteria and trapped carbon,” said Howard Stone, an engineer who led the study.

Lehmann is pushing to replace the old dichotomy of stable and unstable carbon with a “soil continuum model” of carbon in progressive stages of decomposition. But this model and others like it are far from complete, and at this point, more conceptual than mathematically predictive.

Researchers agree that soil science is in the midst of a classic paradigm shift. What nobody knows is exactly where the field will land — what will be written in the next edition of the textbook. “We’re going through a conceptual revolution,” said Mark Bradford, a soil scientist at Yale University. “We haven’t really got a new cathedral yet. We have a whole bunch of churches that have popped up.”

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