Yao.jl: Extensible, Efficient Framework for Quantum Algorithm Design

Xiu-Zhe Luo1,2,3,4, Jin-Guo Liu1, Pan Zhang2, and Lei Wang1,5

1Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
2Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
3Department of Physics and Astronomy, University of Waterloo, Waterloo N2L 3G1, Canada
4Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We introduce $\texttt{Yao}$, an extensible, efficient open-source framework for quantum algorithm design. $\texttt{Yao}$ features generic and differentiable programming of quantum circuits. It achieves state-of-the-art performance in simulating small to intermediate-sized quantum circuits that are relevant to near-term applications. We introduce the design principles and critical techniques behind $\texttt{Yao}$. These include the quantum block intermediate representation of quantum circuits, a builtin automatic differentiation engine optimized for reversible computing, and batched quantum registers with GPU acceleration. The extensibility and efficiency of $\texttt{Yao}$ help boost innovation in quantum algorithm design.

► BibTeX data

► References

[1] John Preskill. Quantum computing in the nisq era and beyond. Quantum, 2: 79, 2018. 10.22331/​q-2018-08-06-79.

[2] 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: 4213, 2014a. 10.1038/​ncomms5213.

[3] Dave Wecker, Matthew B Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Physical Review A, 92 (4): 042303, 2015. 10.1103/​physreva.92.042303.

[4] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18 (2): 023023, 2016. 10.1088/​1367-2630/​18/​2/​023023.

[5] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028, 2014.

[6] Edward Farhi and Hartmut Neven. Classification with Quantum Neural Networks on Near Term Processors. arXiv e-prints, art. arXiv:1802.06002, February 2018.

[7] Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, and Keisuke Fujii. Quantum circuit learning. Physical Review A, 98 (3): 032309, 2018. 10.1103/​physreva.98.032309.

[8] Marcello Benedetti, Delfina Garcia-Pintos, Oscar Perdomo, Vicente Leyton-Ortega, Yunseong Nam, and Alejandro Perdomo-Ortiz. A generative modeling approach for benchmarking and training shallow quantum circuits. npj Quantum Information, 5 (1), May 2019a. ISSN 2056-6387. 10.1038/​s41534-019-0157-8. URL http:/​/​dx.doi.org/​10.1038/​s41534-019-0157-8.

[9] Jin-Guo Liu and Lei Wang. Differentiable learning of quantum circuit born machines. Physical Review A, 98 (6): 062324, 2018. 10.1103/​physreva.98.062324.

[10] Peter JJ O’Malley et al. Scalable quantum simulation of molecular energies. Physical Review X, 6 (3): 031007, 2016. 10.21236/​ada387360.

[11] 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 (7671): 242, 2017a. 10.1038/​nature23879. URL https:/​/​www.nature.com/​articles/​nature23879.

[12] 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 (7747): 209, 2019. 10.1038/​s41586-019-0980-2.

[13] Daiwei Zhu et al. Training of quantum circuits on a hybrid quantum computer. Science Advances, 5 (10): eaaw9918, 2019. 10.1126/​sciadv.aaw9918.

[14] G. Pagano, A. Bapat, P. Becker, K. S. Collins, A. De, P. W. Hess, H. B. Kaplan, A. Kyprianidis, W. L. Tan, C Baldwin, L T Brady, A Deshpande, F Liu, S Jordan, A V Gorshkov, and C Monroe. Quantum Approximate Optimization with a Trapped-Ion Quantum Simulator. 2019. URL http:/​/​arxiv.org/​abs/​1906.02700.

[15] Vicente Leyton-Ortega, Alejandro Perdomo-Ortiz, and Oscar Perdomo. Robust implementation of generative modeling with parametrized quantum circuits. arXiv preprint arXiv:1901.08047, 2019.

[16] Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. Nat. Commun., 9 (1): 4812, 2018. ISSN 2041-1723. 10.1038/​s41467-018-07090-4. URL https:/​/​doi.org/​10.1038/​s41467-018-07090-4.

[17] Tim Besard, Christophe Foket, and Bjorn De Sutter. Effective extensible programming: Unleashing julia on gpus. CoRR, abs/​1712.03112, 2017. 10.1109/​tpds.2018.2872064. URL http:/​/​arxiv.org/​abs/​1712.03112.

[18] Guillermo García-Pérez, Matteo A. C. Rossi, and Sabrina Maniscalco. Ibm q experience as a versatile experimental testbed for simulating open quantum systems, 2019. URL https:/​/​arxiv.org/​abs/​1906.07099. 10.1038/​s41534-019-0235-y.

[19] Differentiable Programming. https:/​/​en.wikipedia.org/​wiki/​Differentiable_programming, a.

[20] Karpathy, Andrej. Software 2.0. https:/​/​medium.com/​@karpathy/​software-2-0-a64152b37c35.

[21] Tianqi Chen et al. Mxnet: A flexible and efficient machine learning library for heterogeneous distributed systems. arXiv preprint arXiv:1512.01274, 2015.

[22] Martín Abadi et al. Tensorflow: A system for large-scale machine learning. In 12th $\{$USENIX$\}$ Symposium on Operating Systems Design and Implementation ($\{$OSDI$\}$ 16), pages 265–283, 2016.

[23] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, high-performance deep learning library. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d' Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems 32, pages 8024–8035. Curran Associates, Inc., 2019. https:/​/​arxiv.org/​abs/​1912.01703.

[24] Dougal Maclaurin, David Duvenaud, and Ryan P Adams. Autograd: Effortless gradients in numpy. In ICML 2015 AutoML Workshop, volume 238, 2015.

[25] Michael Innes, Elliot Saba, Keno Fischer, Dhairya Gandhi, Marco Concetto Rudilosso, Neethu Mariya Joy, Tejan Karmali, Avik Pal, and Viral Shah. Fashionable modelling with flux. CoRR, abs/​1811.01457, 2018. URL http:/​/​arxiv.org/​abs/​1811.01457.

[26] Mike Innes, Alan Edelman, Keno Fischer, Chris Rackauckus, Elliot Saba, Viral B Shah, and Will Tebbutt. Zygote: A differentiable programming system to bridge machine learning and scientific computing. arXiv preprint arXiv:1907.07587, 2019.

[27] C. H. Bennett. Logical reversibility of computation. IBM Journal of Research and Development, 17 (6): 525–532, Nov 1973. ISSN 0018-8646. 10.1147/​rd.176.0525.

[28] Jin-Guo Liu, Yi-Hong Zhang, Yuan Wan, and Lei Wang. Variational quantum eigensolver with fewer qubits. Phys. Rev. Research, 1: 023025, Sep 2019a. 10.1103/​PhysRevResearch.1.023025. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.1.023025.

[29] Alexander S Green, Peter LeFanu Lumsdaine, Neil J Ross, Peter Selinger, and Benoı̂t Valiron. Quipper: a scalable quantum programming language. In ACM SIGPLAN Notices, volume 48, pages 333–342. ACM, 2013. 10.1145/​2491956.2462177.

[30] Damian S Steiger, Thomas Häner, and Matthias Troyer. Projectq: an open source software framework for quantum computing. arXiv preprint arXiv:1612.08091, 2016. 10.22331/​q-2018-01-31-49.

[31] Krysta Svore, Martin Roetteler, Alan Geller, Matthias Troyer, John Azariah, Christopher Granade, Bettina Heim, Vadym Kliuchnikov, Mariia Mykhailova, and Andres Paz. Q#: Enabling scalable quantum computing and development with a high-level dsl. Proceedings of the Real World Domain Specific Languages Workshop 2018 on - RWDSL2018, 2018. 10.1145/​3183895.3183901. URL http:/​/​dx.doi.org/​10.1145/​3183895.3183901.

[32] Cirq: A Python framework for creating, editing, and invoking noisy intermediate scale quantum (NISQ) circuits. https:/​/​github.com/​quantumlib/​Cirq.

[33] qulacs: Variational Quantum Circuit Simulator for Quantum Computation Research. https:/​/​github.com/​qulacs/​qulacs.

[34] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, and Nathan Killoran. Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968, 2018.

[35] Héctor Abraham et al. Qiskit: An open-source framework for quantum computing, 2019.

[36] Tyson Jones, Anna Brown, Ian Bush, and Simon C. Benjamin. Quest and high performance simulation of quantum computers. Scientific Reports, 9 (1), Jul 2019. ISSN 2045-2322. 10.1038/​s41598-019-47174-9. URL http:/​/​dx.doi.org/​10.1038/​s41598-019-47174-9.

[37] Mark Fingerhuth, Tomáš Babej, and Peter Wittek. Open source software in quantum computing. PloS one, 13 (12): e0208561, 2018. 10.1371/​journal.pone.0208561.

[38] Ryan LaRose. Overview and Comparison of Gate Level Quantum Software Platforms. Quantum, 3: 130, March 2019. ISSN 2521-327X. 10.22331/​q-2019-03-25-130. URL https:/​/​doi.org/​10.22331/​q-2019-03-25-130.

[39] Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini. Parameterized quantum circuits as machine learning models. Quantum Science and Technology, 4 (4): 043001, nov 2019b. 10.1088/​2058-9565/​ab4eb5. URL https:/​/​doi.org/​10.1088.

[40] J. Bezanson. “Why is Julia fast? Can it be faster?” 2015, JuliaCon India. https:/​/​www.youtube.com/​watch?v=xUP3cSKb8sI.

[41] Jeff Bezanson, Stefan Karpinski, Viral B Shah, and Alan Edelman. Julia: A fast dynamic language for technical computing. arXiv preprint arXiv:1209.5145, 2012.

[42] Jarrod R. McClean et al. Openfermion: The electronic structure package for quantum computers, jun 2020. URL https:/​/​doi.org/​10.1088/​2058-9565/​ab8ebc.

[43] D Coppersmith. An approximate fourier transform useful in quantum computing. Technical report, Technical report, IBM Research Division, 1994. https:/​/​arxiv.org/​abs/​quant-ph/​0201067.

[44] Artur Ekert and Richard Jozsa. Quantum computation and shor's factoring algorithm. Reviews of Modern Physics, 68 (3): 733, 1996. 10.1103/​RevModPhys.68.733.

[45] Richard Jozsa. Quantum algorithms and the fourier transform. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454 (1969): 323–337, 1998. 10.1098/​rspa.1998.0163.

[46] Peter J Karalekas, Nikolas A Tezak, Eric C Peterson, Colm A Ryan, Marcus P da Silva, and Robert S Smith. A quantum-classical cloud platform optimized for variational hybrid algorithms. Quantum Science and Technology, 5 (2): 024003, apr 2020. 10.1088/​2058-9565/​ab7559. URL https:/​/​doi.org/​10.1088.

[47] Krylovkit.jl: Krylov methods for linear problems, eigenvalues, singular values and matrix functions. https:/​/​github.com/​Jutho/​KrylovKit.jl.

[48] Christopher Rackauckas and Qing Nie. Differentialequations.jl – a performant and feature-rich ecosystem for solving differential equations in julia. The Journal of Open Research Software, 5 (1), 2017. 10.5334/​jors.151. URL https:/​/​app.dimensions.ai/​details/​publication/​pub.1085583166 and http:/​/​openresearchsoftware.metajnl.com/​articles/​10.5334/​jors.151/​galley/​245/​download/​. Exported from https:/​/​app.dimensions.ai on 2019/​05/​05.

[49] Atilim Gunes Baydin, Barak A Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind. Automatic differentiation in machine learning: a survey. Journal of machine learning research, 18 (153), 2018. https:/​/​arxiv.org/​abs/​1502.05767.

[50] Andreas Griewank and Andrea Walther. Evaluating Derivatives. Society for Industrial and Applied Mathematics, jan 2008. 10.1137/​1.9780898717761. URL https:/​/​doi.org/​10.1137.

[51] Aidan N Gomez, Mengye Ren, Raquel Urtasun, and Roger B Grosse. The reversible residual network: Backpropagation without storing activations. In Advances in neural information processing systems, pages 2214–2224, 2017.

[52] Tian Qi Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. In Advances in neural information processing systems, pages 6571–6583, 2018a.

[53] Akira Hirose. Complex-valued neural networks: theories and applications, volume 5. World Scientific, 2003. 10.1142/​5345.

[54] Mike Giles. An extended collection of matrix derivative results for forward and reverse mode algorithmic differentiation. Technical report, 2008. URL https:/​/​people.maths.ox.ac.uk/​gilesm/​files/​NA-08-01.pdf.

[55] Jin-Guo Liu, Liang Mao, Pan Zhang, and Lei Wang. Solving quantum statistical mechanics with variational autoregressive networks and quantum circuits. 2019b. URL http:/​/​arxiv.org/​abs/​1912.11381.

[56] 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. Nat. Commun., 5: 4213, 2014b. 10.1038/​ncomms5213. URL https:/​/​www.nature.com/​articles/​ncomms5213.

[57] 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 (7671): 242, 2017b. 10.1038/​nature23879.

[58] Gavin E Crooks. Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition. URL https:/​/​arxiv.org/​abs/​1905.13311.

[59] Jun Li, Xiaodong Yang, Xinhua Peng, and Chang-Pu Sun. Hybrid quantum-classical approach to quantum optimal control. Phys. Rev. Lett., 118: 150503, Apr 2017. 10.1103/​physrevlett.118.150503. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.118.150503.

[60] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Phys. Rev. A, 99 (3): 032331, 2019. ISSN 24699934. 10.1103/​PhysRevA.99.032331.

[61] Ken M Nakanishi, Keisuke Fujii, and Synge Todo. Sequential minimal optimization for quantum-classical hybrid algorithms. 10.21236/​ada212800. URL https:/​/​arxiv.org/​abs/​1903.12166.

[62] Shakir Mohamed, Mihaela Rosca, Michael Figurnov, and Andriy Mnih. Monte carlo gradient estimation in machine learning. URL https:/​/​arxiv.org/​abs/​1906.10652.

[63] Chun-Liang Li, Wei-Cheng Chang, Yu Cheng, Yiming Yang, and Barnabás Póczos. MMD GAN: Towards Deeper Understanding of Moment Matching Network. URL http:/​/​arxiv.org/​abs/​1705.08584.

[64] Arthur Gretton, Karsten M Borgwardt, Malte J Rasch, Bernhard Schölkopf, and Alexander Smola. A kernel two-sample test. Journal of Machine Learning Research, 13 (Mar): 723–773, 2012. URL http:/​/​www.jmlr.org/​papers/​v13/​gretton12a.html.

[65] Michael A Nielsen and Isaac L Chuang. Quantum Computation and Quantum Information. Cambridge university press, 2010. 10.1017/​cbo9780511976667.016.

[66] William James Huggins, Piyush Patil, Bradley Mitchell, K Birgitta Whaley, and Miles Stoudenmire. Towards quantum machine learning with tensor networks. Quantum Science and Technology, 2018. 10.1088/​2058-9565/​aaea94.

[67] Frederica Darema, David A George, V Alan Norton, and Gregory F Pfister. A single-program-multiple-data computational model for epex/​fortran. Parallel Computing, 7 (1): 11–24, 1988. 10.1016/​0167-8191(88)90094-4.

[68] Statically sized arrays for Julia. https:/​/​github.com/​JuliaArrays/​StaticArrays.jl.

[69] A luxury sparse matrix package for julia. https:/​/​github.com/​QuantumBFS/​LuxurySparse.jl.

[70] Thomas Häner and Damian S Steiger. 0.5 petabyte simulation of a 45-qubit quantum circuit. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, page 33. ACM, 2017. 10.1145/​3126908.3126947.

[71] Igor L Markov and Yaoyun Shi. Simulating quantum computation by contracting tensor networks. SIAM Journal on Computing, 38 (3): 963–981, 2008. 10.1137/​050644756.

[72] Edwin Pednault, John A Gunnels, Giacomo Nannicini, Lior Horesh, Thomas Magerlein, Edgar Solomonik, and Robert Wisnieff. Breaking the 49-qubit barrier in the simulation of quantum circuits. arXiv preprint arXiv:1710.05867, 2017.

[73] Fang Zhang et al. Alibaba cloud quantum development kit: Large-scale classical simulation of quantum circuits. arXiv preprint arXiv:1907.11217, 2019.

[74] Pyquest-cffi: A python interface to the quest quantum simulator (cffi based). https:/​/​github.com/​HQSquantumsimulations/​PyQuEST-cffi.

[75] PennyLane is a cross-platform Python library for quantum machine learning, automatic differentiation, and optimization of hybrid quantum-classical computations. https:/​/​github.com/​XanaduAI/​pennylane, a.

[76] Review of PennyLane benchmark. https:/​/​github.com/​Roger-luo/​quantum-benchmarks/​pull/​7, b.

[77] Aer is a high performance simulator for quantum circuits that includes noise models. https:/​/​github.com/​Qiskit/​qiskit-aer, a.

[78] Terra provides the foundations for Qiskit. It allows the user to write quantum circuits easily, and takes care of the constraints of real hardware. https:/​/​github.com/​Qiskit/​qiskit-terra, b.

[79] py.test fixture for benchmarking code. https:/​/​github.com/​ionelmc/​pytest-benchmark.

[80] Jiahao Chen and Jarrett Revels. Robust benchmarking in noisy environments. arXiv preprint arXiv:1608.04295, 2016.

[81] Benchmarking Quantum Circuit Emulators For Your Daily Research Usage. https:/​/​github.com/​Roger-luo/​quantum-benchmarks.

[82] Frank Arute et al. Quantum supremacy using a programmable superconducting processor. Nature, 574 (7779): 505–510, 2019. 10.1038/​s41586-019-1666-5.

[83] CuYao.jl: CUDA extension for Yao.jl. https:/​/​github.com/​QuantumBFS/​CuYao.jl.

[84] Jinfeng Zeng, Yufeng Wu, Jin-Guo Liu, Lei Wang, and Jiangping Hu. Learning and inference on generative adversarial quantum circuits. Physical Review A, 99 (5): 052306, 2019. 10.1103/​physreva.99.052306.

[85] Weishi Wang, Jin-Guo Liu, and Lei Wang. A variational quantum state compression algorithm. to appear.

[86] Vivek V. Shende, Igor L. Markov, and Stephen S. Bullock. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A, 69: 062321, Jun 2004. 10.1103/​PhysRevA.69.062321. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.69.062321.

[87] Michael Innes. Don't unroll adjoint: Differentiating ssa-form programs. CoRR, abs/​1810.07951, 2018. URL http:/​/​arxiv.org/​abs/​1810.07951.

[88] Andrew W Cross, Lev S Bishop, John A Smolin, and Jay M Gambetta. Open quantum assembly language. arXiv preprint arXiv:1707.03429, 2017.

[89] Xiang Fu et al. eqasm: An executable quantum instruction set architecture. In 2019 IEEE International Symposium on High Performance Computer Architecture (HPCA), pages 224–237. IEEE, 2019. 10.1109/​hpca.2019.00040.

[90] Robert S Smith, Michael J Curtis, and William J Zeng. A practical quantum instruction set architecture. arXiv preprint arXiv:1608.03355, 2016.

[91] RBNF: A DSL for modern parsing. https:/​/​github.com/​thautwarm/​RBNF.jl.

[92] Bidirectional transformation between Yao Quantum Block IR and QASM. https:/​/​github.com/​QuantumBFS/​YaoQASM.jl, a.

[93] YaoLang: The next DSL for Yao and quantum programs. https:/​/​github.com/​QuantumBFS/​YaoLang.jl, b.

[94] ZXCalculus.jl: An implementation of ZX-calculus in Julia. https:/​/​github.com/​QuantumBFS/​ZXCalculus.jl, c.

[95] Aleks Kissinger and John van de Wetering. PyZX: Large Scale Automated Diagrammatic Reasoning. In Bob Coecke and Matthew Leifer, editors, Proceedings 16th International Conference on Quantum Physics and Logic, Chapman University, Orange, CA, USA., 10-14 June 2019, volume 318 of Electronic Proceedings in Theoretical Computer Science, pages 229–241. Open Publishing Association, 2020. 10.4204/​EPTCS.318.14.

[96] Raban Iten, David Sutter, and Stefan Woerner. Efficient template matching in quantum circuits. arXiv preprint arXiv:1909.05270, 2019.

[97] Dmitri Maslov, Gerhard W Dueck, D Michael Miller, and Camille Negrevergne. Quantum circuit simplification and level compaction. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 27 (3): 436–444, 2008. 10.1109/​tcad.2007.911334.

[98] Aleks Kissinger and John van de Wetering. Reducing T-count with the ZX-calculus. arXiv preprint arXiv:1903.10477, 2019. 10.1103/​PhysRevA.102.022406.

[99] Piotr Gawron, Dariusz Kurzyk, and Łukasz Pawela. Quantuminformation.jl—a julia package for numerical computation in quantum information theory. PLOS ONE, 13 (12): e0209358, Dec 2018. ISSN 1932-6203. 10.1371/​journal.pone.0209358. URL http:/​/​dx.doi.org/​10.1371/​journal.pone.0209358.

[100] Sergio Boixo, Sergei V Isakov, Vadim N Smelyanskiy, and Hartmut Neven. Simulation of low-depth quantum circuits as complex undirected graphical models. arXiv preprint arXiv:1712.05384, 2017.

[101] Jianxin Chen, Fang Zhang, Mingcheng Chen, Cupjin Huang, Michael Newman, and Yaoyun Shi. Classical simulation of intermediate-size quantum circuits. arXiv preprint arXiv:1805.01450, 2018b.

[102] Chu Guo, Yong Liu, Min Xiong, Shichuan Xue, Xiang Fu, Anqi Huang, Xiaogang Qiang, Ping Xu, Junhua Liu, Shenggen Zheng, He-Liang Huang, Mingtang Deng, Dario Poletti, Wan-Su Bao, and Junjie Wu. General-purpose quantum circuit simulator with projected entangled-pair states and the quantum supremacy frontier. Phys. Rev. Lett., 123: 190501, Nov 2019. 10.1103/​PhysRevLett.123.190501. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.123.190501.

[103] Feng Pan, Pengfei Zhou, Sujie Li, and Pan Zhang. Contracting arbitrary tensor networks: general approximate algorithm and applications in graphical models and quantum circuit simulations. Phys. Rev. Lett., 2020. 10.1103/​PhysRevLett.125.060503.

[104] Edwin Stoudenmire and David J Schwab. Supervised learning with tensor networks. pages 4799–4807, 2016. URL http:/​/​papers.nips.cc/​paper/​6211-supervised-learning-with-tensor-networks.pdf.

[105] Zhao-Yu Han, Jun Wang, Heng Fan, Lei Wang, and Pan Zhang. Unsupervised generative modeling using matrix product states. Phys. Rev. X, 8: 031012, Jul 2018. 10.1103/​PhysRevX.8.031012. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.8.031012.

[106] Song Cheng, Lei Wang, Tao Xiang, and Pan Zhang. Tree tensor networks for generative modeling. Phys. Rev. B, 99: 155131, Apr 2019. 10.1103/​PhysRevB.99.155131. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.99.155131.

[107] Ivan Glasser, Ryan Sweke, Nicola Pancotti, Jens Eisert, and Ignacio Cirac. Expressive power of tensor-network factorizations for probabilistic modeling. In Advances in Neural Information Processing Systems, pages 1496–1508, 2019.

[108] Tai-Danae Bradley, E M Stoudenmire, and John Terilla. Modeling sequences with quantum states: a look under the hood. Machine Learning: Science and Technology, 1 (3): 035008, jul 2020. 10.1088/​2632-2153/​ab8731. URL https:/​/​doi.org/​10.1088.

[109] YaoTensorNetwork: Dump a quantum circuit in Yao to a tensor network graph model. https:/​/​github.com/​QuantumBFS/​YaoTensorNetwork.jl, d.

[110] Sergio Boixo, Sergei V Isakov, Vadim N Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, Michael J Bremner, John M Martinis, and Hartmut Neven. Characterizing quantum supremacy in near-term devices. Nature Physics, 14 (6): 595, 2018. 10.1038/​s41567-018-0124-x.

[111] Miriam Backens. The ZX-calculus is complete for stabilizer quantum mechanics. New Journal of Physics, 16 (9): 093021, sep 2014. 10.1088/​1367-2630/​16/​9/​093021. URL https:/​/​doi.org/​10.1088.

[112] Multi-language suite for high-performance solvers of differential equations. https:/​/​github.com/​JuliaDiffEq/​DifferentialEquations.jl, b.

[113] General Permutation Matrix. https:/​/​en.wikipedia.org/​wiki/​Generalized_permutation_matrix.

[114] Thomas Häner, Damian S Steiger, Mikhail Smelyanskiy, and Matthias Troyer. High performance emulation of quantum circuits. In SC'16: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, pages 866–874. IEEE, 2016. 10.1109/​sc.2016.73.

[115] Ryan LaRose, Arkin Tikku, Étude O’Neel-Judy, Lukasz Cincio, and Patrick J. Coles. Variational quantum state diagonalization. npj Quantum Information, 5 (1), Jun 2019. ISSN 2056-6387. 10.1038/​s41534-019-0167-6. URL http:/​/​dx.doi.org/​10.1038/​s41534-019-0167-6.

[116] Cristina Cirstoiu, Zoe Holmes, Joseph Iosue, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger. Variational fast forwarding for quantum simulation beyond the coherence time, 2019. 10.1038/​s41534-020-00302-0.

[117] Lukasz Cincio, Yiğit Subaşı, Andrew T Sornborger, and Patrick J Coles. Learning the quantum algorithm for state overlap. New Journal of Physics, 20 (11): 113022, Nov 2018. ISSN 1367-2630. 10.1088/​1367-2630/​aae94a. URL http:/​/​dx.doi.org/​10.1088/​1367-2630/​aae94a.

[118] Xiaoguang Wang, Zhe Sun, and Z. D. Wang. Operator fidelity susceptibility: An indicator of quantum criticality. Physical Review A, 79 (1), Jan 2009. ISSN 1094-1622. 10.1103/​physreva.79.012105. URL http:/​/​dx.doi.org/​10.1103/​PhysRevA.79.012105.

[119] Richard H Byrd, Peihuang Lu, Jorge Nocedal, and Ciyou Zhu. A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing, 16 (5): 1190–1208, 1995. 10.2172/​204262. URL https:/​/​doi.org/​10.1137/​0916069.

[120] Patrick Kofod Mogensen and Asbjørn Nilsen Riseth. Optim: A mathematical optimization package for Julia. Journal of Open Source Software, 3 (24): 615, 2018. 10.21105/​joss.00615.

Cited by

[1] Feng Pan, Pengfei Zhou, Sujie Li, and Pan Zhang, "Contracting Arbitrary Tensor Networks: General Approximate Algorithm and Applications in Graphical Models and Quantum Circuit Simulations", Physical Review Letters 125 6, 060503 (2020).

[2] Sirui Lu, Lu-Ming Duan, and Dong-Ling Deng, "Quantum adversarial machine learning", Physical Review Research 2 3, 033212 (2020).

[3] Jin-Guo Liu, Liang Mao, Pan Zhang, and Lei Wang, "Solving Quantum Statistical Mechanics with Variational Autoregressive Networks and Quantum Circuits", arXiv:1912.11381.

[4] Tatiana A. Bespalova and Oleksandr Kyriienko, "Hamiltonian operator approximation for energy measurement and ground state preparation", arXiv:2009.03351.

[5] Tong Liu, Jin-Guo Liu, and Heng Fan, "Probabilistic Nonunitary Gate in Imaginary Time Evolution", arXiv:2006.09726.

[6] Stavros Efthymiou, Sergi Ramos-Calderer, Carlos Bravo-Prieto, Adrián Pérez-Salinas, Diego García-Martín, Artur Garcia-Saez, José Ignacio Latorre, and Stefano Carrazza, "Qibo: a framework for quantum simulation with hardware acceleration", arXiv:2009.01845.

[7] Jin-Guo Liu, Lei Wang, and Pan Zhang, "Tropical Tensor Network for Ground States of Spin Glasses", arXiv:2008.06888.

[8] Jin-Guo Liu and Taine Zhao, "Differentiate Everything with a Reversible Domain-Specific Language", arXiv:2003.04617.

[9] Carsten Bauer, "Fast and stable determinant quantum Monte Carlo", arXiv:2003.05286.

[10] The Quingo Development Team, "Quingo: A Programming Framework for Heterogeneous Quantum-Classical Computing with NISQ Features", arXiv:2009.01686.

[11] Vincent Paul Su, "Variational Preparation of the Sachdev-Ye-Kitaev Thermofield Double", arXiv:2009.04488.

[12] Andrea Mari, Thomas R. Bromley, and Nathan Killoran, "Estimating the gradient and higher-order derivatives on quantum hardware", arXiv:2008.06517.

The above citations are from SAO/NASA ADS (last updated successfully 2020-10-23 05:19:51). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2020-10-23 05:19:49).