# Quantum-assisted quantum compiling

Sumeet Khatri1,2, Ryan LaRose1,3, Alexander Poremba1,4, Lukasz Cincio1, Andrew T. Sornborger5, and Patrick J. Coles1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM USA.
2Hearne Institute for Theoretical Physics and Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA USA.
3Department of Computational Mathematics, Science, and Engineering and Department of Physics and Astronomy, Michigan State University, East Lansing, MI USA.
4Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA USA.
5Information Sciences, Los Alamos National Laboratory, Los Alamos, NM USA.

### Abstract

Compiling quantum algorithms for near-term quantum computers (accounting for connectivity and native gate alphabets) is a major challenge that has received significant attention both by industry and academia. Avoiding the exponential overhead of classical simulation of quantum dynamics will allow compilation of larger algorithms, and a strategy for this is to evaluate an algorithm's cost on a quantum computer. To this end, we propose a variational hybrid quantum-classical algorithm called quantum-assisted quantum compiling (QAQC). In QAQC, we use the overlap between a target unitary $U$ and a trainable unitary $V$ as the cost function to be evaluated on the quantum computer. More precisely, to ensure that QAQC scales well with problem size, our cost involves not only the global overlap ${\rm Tr}(V^†U)$ but also the local overlaps with respect to individual qubits. We introduce novel short-depth quantum circuits to quantify the terms in our cost function, and we prove that our cost cannot be efficiently approximated with a classical algorithm under reasonable complexity assumptions. We present both gradient-free and gradient-based approaches to minimizing this cost. As a demonstration of QAQC, we compile various one-qubit gates on IBM's and Rigetti's quantum computers into their respective native gate alphabets. Furthermore, we successfully simulate QAQC up to a problem size of 9 qubits, and these simulations highlight both the scalability of our cost function as well as the noise resilience of QAQC. Future applications of QAQC include algorithm depth compression, black-box compiling, noise mitigation, and benchmarking.

Ordinary computers require a compiler that converts one's code into a machine-level language. Quantum computers require a compiler as well. However, a new challenge for such "quantum compilers" is that they should be optimal, i.e., they should return a machine-level program that has as few operations as possible. This optimality is crucial for current noisy quantum devices, where longer programs accumulate more errors while shorter programs avoid errors. In this work, we introduce an algorithm for optimal quantum compiling. The key feature that allows for optimality is that we propose to use quantum computers themselves to assist in the compiling process. Hence, our algorithm is called quantum-assisted quantum compiling (QAQC, pronounced "Quack").

The idea is that one needs to quantify the distance between the original program and the compiled program, with the goal of trying to minimize this distance. We prove that this distance calculation cannot be done efficiently on a classical computer. On the other hand, we provide an efficient quantum circuit for computing it.

In addition to shortening the length of one's quantum program, QAQC can be used to learn algorithms that compensate for a given quantum computer's noise and also to benchmark the noise processes occurring on a quantum computer. We successfully implement QAQC for small programs using currently available quantum computers from IBM and Rigetti, and we use simulators to explore the compilation of larger programs. Overall, QAQC appears to be a promising tool for mitigating errors in the era of noisy intermediate-scale quantum computers.

### ► References

[1] P. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journal on Computing 26, 1484 (1997).
https:/​/​doi.org/​10.1137/​S0097539795293172

[2] E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028 (2014).
arXiv:1411.4028

[3] R. P. Feynman, Simulating physics with computers, International Journal of Theoretical Physics 21, 467 (1982).
https:/​/​doi.org/​10.1007/​BF02650179

[4] J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
https:/​/​doi.org/​10.22331/​q-2018-08-06-79

[5] J. Preskill, Quantum computing and the entanglement frontier, arXiv:1203.5813 (2012).
arXiv:1203.5813

[6] C. Neill, P. Roushan, K. Kechedzhi, S. Boixo, S. V. Isakov, V. Smelyanskiy, et al., A blueprint for demonstrating quantum supremacy with superconducting qubits, Science 360, 195 (2018).
https:/​/​doi.org/​10.1126/​science.aao4309

[7] D. Venturelli, M. Do, E. Rieffel, and J. Frank, Compiling quantum circuits to realistic hardware architectures using temporal planners, Quantum Science and Technology 3, 025004 (2018).
https:/​/​doi.org/​10.1088/​2058-9565/​aaa331

[8] K. E. C. Booth, M. Do, J. C. Beck, E. Rieffel, D. Venturelli, and J. Frank, Comparing and integrating constraint programming and temporal planning for quantum circuit compilation, arXiv:1803.06775 (2018).
arXiv:1803.06775

[9] L. Cincio, Y. Subaşi, A. T. Sornborger, and P. J. Coles, Learning the quantum algorithm for state overlap, New Journal of Physics 20, 113022 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aae94a

[10] D. Maslov, G. W. Dueck, D. M. Miller, and C. Negrevergne, Quantum circuit simplification and level compaction, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 27, 436 (2008).

[11] A. G. Fowler, Constructing arbitrary Steane code single logical qubit fault-tolerant gates, Quantum Information and Computation 11, 867 (2011).
http:/​/​dl.acm.org/​citation.cfm?id=2230936.2230946

[12] J. Booth Jr, Quantum compiler optimizations, arXiv:1206.3348 (2012).
arXiv:1206.3348

[13] Y. Nam, N. J. Ross, Y. Su, A. M. Childs, and D. Maslov, Automated optimization of large quantum circuits with continuous parameters, npj Quantum Information 4, 23 (2018).
https:/​/​doi.org/​10.1038/​s41534-018-0072-4

[14] F. T. Chong, D. Franklin, and M. Martonosi, Programming languages and compiler design for realistic quantum hardware, Nature 549, 180 (2017).
https:/​/​doi.org/​10.1038/​nature23459

[15] L. E. Heyfron and E. T. Campbell, An efficient quantum compiler that reduces T count, Quantum Science and Technology 4, 015004 (2018).

[16] T. Häner, D. S. Steiger, K. Svore, and M. Troyer, A software methodology for compiling quantum programs, Quantum Science and Technology 3, 020501 (2018).
https:/​/​doi.org/​10.1088/​2058-9565/​aaa5cc

[17] A. Oddi and R. Rasconi, in International Conference on the Integration of Constraint Programming, Artificial Intelligence, and Operations Research (Springer, 2018) pp. 446–461.

[18] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'Brien, A variational eigenvalue solver on a photonic quantum processor, Nature Communications 5, 4213 (2014).
https:/​/​doi.org/​10.1038/​ncomms5213

[19] P. D. Johnson, J. Romero, J. Olson, Y. Cao, and A. Aspuru-Guzik, QVECTOR: an algorithm for device-tailored quantum error correction, arXiv:1711.02249 (2017).
arXiv:1711.02249

[20] M. Benedetti, D. Garcia-Pintos, O. Perdomo, V. Leyton-Ortega, Y. Nam, and A. Perdomo-Ortiz, A generative modeling approach for benchmarking and training shallow quantum circuits, arXiv:1801.07686 (2018a).
arXiv:1801.07686

[21] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Physical Review A 98, 032309 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.032309

[22] G. Verdon, J. Pye, and M. Broughton, A Universal Training Algorithm for Quantum Deep Learning, arXiv:1806.09729 (2018).
arXiv:1806.09729

[23] J. Romero, J. P. Olson, and A. Aspuru-Guzik, Quantum autoencoders for efficient compression of quantum data, Quantum Science and Technology 2, 045001 (2017).
https:/​/​doi.org/​10.1088/​2058-9565/​aa8072

[24] J. Romero, J. P. Olson, and A. Aspuru-Guzik, Quantum autoencoders for short depth quantum circuit synthesis, GitHub article (2018).
https:/​/​github.com/​zapatacomputing/​cusp_cirq_demo/​blob/​master/​cusp_protocol.pdf

[25] B. Dive, A. Pitchford, F. Mintert, and D. Burgarth, In situ upgrade of quantum simulators to universal computers, Quantum 2, 80 (2018).
https:/​/​doi.org/​10.22331/​q-2018-08-08-80

[26] E. Knill and R. Laflamme, Power of one bit of quantum information, Physical Review Letters 81, 5672 (1998).
https:/​/​doi.org/​10.1103/​PhysRevLett.81.5672

[27] K. Fujii, H. Kobayashi, T. Morimae, H. Nishimura, S. Tamate, and S. Tani, Impossibility of Classically Simulating One-Clean-Qubit Model with Multiplicative Error, Physical Review Letters 120, 200502 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.200502

[28] B. Rosgen and J. Watrous, in 20th Annual IEEE Conference on Computational Complexity (CCC'05) (2005) pp. 344–354.
https:/​/​doi.org/​10.1109/​CCC.2005.21

[29] R. S. Smith, M. J. Curtis, and W. J. Zeng, A practical quantum instruction set architecture, arXiv:1608.03355 (2016).
arXiv:1608.03355

[30] A. W. Cross, L. S. Bishop, J. A. Smolin, and J. M. Gambetta, Open Quantum Assembly Language, arXiv:1707.03429 (2017).
arXiv:1707.03429

[31] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
https:/​/​doi.org/​10.1017/​CBO9780511976667

[32] A. Kitaev, Quantum computations: algorithms and error correction, Russian Mathematical Surveys 52, 1191 (1997).
https:/​/​doi.org/​10.1070/​RM1997v052n06ABEH002155

[33] C. M. Dawson and M. A. Nielsen, The Solovay-Kitaev algorithm, Quantum Information and Compututation 6, 81 (2006).
http:/​/​dl.acm.org/​citation.cfm?id=2011679.2011685

[34] T. T. Pham, R. Van Meter, and C. Horsman, Optimization of the Solovay-Kitaev algorithm, Physical Review A 87, 052332 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.052332

[35] V. Kliuchnikov, D. Maslov, and M. Mosca, Asymptotically optimal approximation of single qubit unitaries by Clifford and T circuits using a constant number of ancillary qubits, Physical Review Letters 110, 190502 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.110.190502

[36] V. Kliuchnikov, A. Bocharov, and K. M. Svore, Asymptotically optimal topological quantum compiling, Physical Review Letters 112, 140504 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.140504

[37] Y. Zhiyenbayev, V. M. Akulin, and A. Mandilara, Quantum compiling with diffusive sets of gates, Physical Review A 98, 012325 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.012325

[38] M. Horodecki, P. Horodecki, and R. Horodecki, General teleportation channel, singlet fraction, and quasidistillation, Physical Review A 60, 1888 (1999).
https:/​/​doi.org/​10.1103/​PhysRevA.60.1888

[39] M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Physics Letters A 303, 249 (2002).
https:/​/​doi.org/​10.1016/​S0375-9601(02)01272-0

[40] A. Gepp and P. Stocks, A review of procedures to evolve quantum algorithms, Genetic Programming and Evolvable Machines 10, 181 (2009).
https:/​/​doi.org/​10.1007/​s10710-009-9080-7

[41] M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and monte carlo simulations, Physics Letters A 146, 319 (1990).
https:/​/​doi.org/​10.1016/​0375-9601(90)90962-N

[42] T. Jones and S. C. Benjamin, Quantum compilation and circuit optimisation via energy dissipation, arXiv:1811.03147 (2018).
arXiv:1811.03147

[43] J. C. Garcia-Escartin and P. Chamorro-Posada, Swap test and Hong-Ou-Mandel effect are equivalent, Physical Review A 87, 052330 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.052330

[44] P. W. Shor and S. P. Jordan, Estimating jones polynomials is a complete problem for one clean qubit, Quantum Information & Computation 8, 681 (2008).
http:/​/​www.rintonpress.com/​xxqic8/​qic-8-89/​0681-0714.pdf

[45] IBM Q 5 Tenerife backend specification, (2018a).
https:/​/​github.com/​QISKit/​qiskit-backend-information/​tree/​master/​backends/​tenerife/​V1

[46] IBM Q 16 Rueschlikon backend specification, (2018b).
https:/​/​github.com/​Qiskit/​qiskit-backend-information/​tree/​master/​backends/​rueschlikon/​V1

[47] Rigetti 8Q-Agave specification v.2.0.0.dev0, (2018).
http:/​/​docs.rigetti.com/​en/​latest/​qpu.html

[48] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature Communications 9, 4812 (2018).
https:/​/​doi.org/​10.1038/​s41467-018-07090-4

[49] A. G. R. Day, M. Bukov, P. Weinberg, P. Mehta, and D. Sels, Glassy phase of optimal quantum control, Physical Review Letters 122, 020601 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.020601

[50] X. Glorot and Y. Bengio, in In Proceedings of the International Conference on Artificial Intelligence and Statistics (2010) pp. 249–256.
http:/​/​proceedings.mlr.press/​v9/​glorot10a/​glorot10a.pdf?hc_location=ufi

[51] M. Benedetti, D. Garcia-Pintos, O. Perdomo, V. Leyton-Ortega, Y. Nam, and A. Perdomo-Ortiz, A generative modeling approach for benchmarking and training shallow quantum circuits, arXiv:1801.07686 (2018b).
arXiv:1801.07686

[52] R. LaRose, A. Tikku, É. O'Neel-Judy, L. Cincio, and P. J. Coles, Variational quantum state diagonalization, arXiv:1810.10506 (2018).
arXiv:1810.10506

[53] A. Kandala, K. Temme, A. D. Corcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Extending the computational reach of a noisy superconducting quantum processor, Nature 567, 491 (2018).
https:/​/​doi.org/​10.1038/​s41586-019-1040-7

[54] Scikit-optimize, (2018a).
https:/​/​github.com/​scikit-optimize/​scikit-optimize

[55] J. Močkus, in Optimization Techniques IFIP Technical Conference Novosibirsk, July 1–7, 1974 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1975) pp. 400–404.
https:/​/​doi.org/​10.1007/​978-3-662-38527-2_55

[56] M. A. Osborne, R. Garnett, and S. J. Roberts, in 3rd International Conference on Learning and Intelligent Optimization (LION3) 2009 (2009).
https:/​/​www.cse.wustl.edu/​~garnett/​files/​papers/​osborne_et_al_lion_2009.pdf

[57] P. Rebentrost, M. Schuld, L. Wossnig, F. Petruccione, and S. Lloyd, Quantum gradient descent and Newton's method for constrained polynomial optimization, arXiv:1612.01789 (2016).
arXiv:1612.01789

[58] I. Kerenidis and A. Prakash, Quantum gradient descent for linear systems and least squares, arXiv:1704.04992 (2017).
arXiv:1704.04992

[59] A. Gilyén, S. Arunachalam, and N. Wiebe, Optimizing quantum optimization algorithms via faster quantum gradient computation, in Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1425–1444.
https:/​/​doi.org/​10.1137/​1.9781611975482.87

[60] P. B. M. Sousa and R. V. Ramos, Universal quantum circuit for $n$-qubit quantum gate: A programmable quantum gate, Quantum Information and Computation 7, 228 (2007).
http:/​/​dl.acm.org/​citation.cfm?id=2011717.2011721

[61] F. Vatan and C. Williams, Optimal quantum circuits for general two-qubit gates, Physical Review A 69, 032315 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.69.032315

[62] Scipy optimization and root finding, (2018b).
https:/​/​docs.scipy.org/​doc/​scipy/​reference/​optimize.html

[63] X.-Q. Zhou, T. C. Ralph, P. Kalasuwan, M. Zhang, A. Peruzzo, B. P. Lanyon, and J. L. O'Brien, Adding control to arbitrary unknown quantum operations, Nature Communications 2, 413 (2011).
https:/​/​doi.org/​10.1038/​ncomms1392

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[2] Tyler Volkoff and Patrick J Coles, "Large gradients via correlation in random parameterized quantum circuits", Quantum Science and Technology 6 2, 025008 (2021).

[3] M. Fanizza, M. Rosati, M. Skotiniotis, J. Calsamiglia, and V. Giovannetti, "Beyond the Swap Test: Optimal Estimation of Quantum State Overlap", Physical Review Letters 124 6, 060503 (2020).

[4] Ryan Shaffer, Eli Megidish, Joseph Broz, Wei-Ting Chen, and Hartmut Häffner, "Practical verification protocols for analog quantum simulators", npj Quantum Information 7 1, 46 (2021).

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[8] Alexander J. Buser, Tanmoy Bhattacharya, Lukasz Cincio, and Rajan Gupta, "State preparation and measurement in a quantum simulation of the O(3) sigma model", Physical Review D 102 11, 114514 (2020).

[9] Jacques Carolan, Masoud Mohseni, Jonathan P. Olson, Mihika Prabhu, Changchen Chen, Darius Bunandar, Murphy Yuezhen Niu, Nicholas C. Harris, Franco N. C. Wong, Michael Hochberg, Seth Lloyd, and Dirk Englund, "Variational quantum unsampling on a quantum photonic processor", Nature Physics 16 3, 322 (2020).

[10] Tariq M. Khan and Antonio Robles-Kelly, "Machine Learning: Quantum vs Classical", IEEE Access 8, 219275 (2020).

[11] Carlos Bravo-Prieto, Josep Lumbreras-Zarapico, Luca Tagliacozzo, and José I. Latorre, "Scaling of variational quantum circuit depth for condensed matter systems", Quantum 4, 272 (2020).

[12] Tyler J. Volkoff, "Efficient Trainability of Linear Optical Modules in Quantum Optical Neural Networks", Journal of Russian Laser Research (2021).

[14] B Jaderberg, A Agarwal, K Leonhardt, M Kiffner, and D Jaksch, "Minimum hardware requirements for hybrid quantum–classical DMFT", Quantum Science and Technology 5 3, 034015 (2020).

[15] Kunal Sharma, Sumeet Khatri, M Cerezo, and Patrick J Coles, "Noise resilience of variational quantum compiling", New Journal of Physics 22 4, 043006 (2020).

[16] Alicia B. Magann, Christian Arenz, Matthew D. Grace, Tak-San Ho, Robert L. Kosut, Jarrod R. McClean, Herschel A. Rabitz, and Mohan Sarovar, "From Pulses to Circuits and Back Again: A Quantum Optimal Control Perspective on Variational Quantum Algorithms", PRX Quantum 2 1, 010101 (2021).

[17] Lukas Burgholzer, Richard Kueng, and Robert Wille, Proceedings of the 26th Asia and South Pacific Design Automation Conference 767 (2021) ISBN:9781450379991.

[18] Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini, "Parameterized quantum circuits as machine learning models", Quantum Science and Technology 4 4, 043001 (2019).

[19] Ryan LaRose and Brian Coyle, "Robust data encodings for quantum classifiers", Physical Review A 102 3, 032420 (2020).

[20] Cheng Xue, Zhao-Yun Chen, Yu-Chun Wu, and Guo-Ping Guo, "Effects of Quantum Noise on Quantum Approximate Optimization Algorithm ", Chinese Physics Letters 38 3, 030302 (2021).

[21] Lingling Lao and Carmen G. Almudever, "Fault-tolerant quantum error correction on near-term quantum processors using flag and bridge qubits", Physical Review A 101 3, 032333 (2020).

[22] Brian Coyle, Daniel Mills, Vincent Danos, and Elham Kashefi, "The Born supremacy: quantum advantage and training of an Ising Born machine", npj Quantum Information 6 1, 60 (2020).

[23] I. Meyerov, A. Liniov, M. Ivanchenko, and S. Denisov, "Modeling Complex Quantum Dynamics: Evolution of Numerical Algorithms in the HPC Context", Lobachevskii Journal of Mathematics 41 8, 1509 (2020).

[24] Shavindra P. Premaratne and A. Y. Matsuura, 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) 278 (2020) ISBN:978-1-7281-8969-7.

[25] 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 1, 1791 (2021).

[26] Shi-Ning Sun, Mario Motta, Ruslan N. Tazhigulov, Adrian T.K. Tan, Garnet Kin-Lic Chan, and Austin J. Minnich, "Quantum Computation of Finite-Temperature Static and Dynamical Properties of Spin Systems Using Quantum Imaginary Time Evolution", PRX Quantum 2 1, 010317 (2021).

[27] Margarite L. LaBorde, Allee C. Rogers, and Jonathan P. Dowling, "Finding broken gates in quantum circuits: exploiting hybrid machine learning", Quantum Information Processing 19 8, 230 (2020).

[28] Daniel Mills, Seyon Sivarajah, Travis L. Scholten, and Ross Duncan, "Application-Motivated, Holistic Benchmarking of a Full Quantum Computing Stack", Quantum 5, 415 (2021).

[29] Xiaosi Xu, Simon C. Benjamin, and Xiao Yuan, "Variational Circuit Compiler for Quantum Error Correction", Physical Review Applied 15 3, 034068 (2021).

[30] Benjamin Weder, Johanna Barzen, Frank Leymann, and Marie Salm, "Automated Quantum Hardware Selection for Quantum Workflows", Electronics 10 8, 984 (2021).

[31] A. Roggero and A. Baroni, "Short-depth circuits for efficient expectation-value estimation", Physical Review A 101 2, 022328 (2020).

[32] Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles, "An Adaptive Optimizer for Measurement-Frugal Variational Algorithms", Quantum 4, 263 (2020).

[33] 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 1, 82 (2020).

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[48] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran, "Evaluating analytic gradients on quantum hardware", Physical Review A 99 3, 032331 (2019).

[49] 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 (2018).

[50] Juan Miguel Arrazola, Thomas R. Bromley, Josh Izaac, Casey R. Myers, Kamil Brádler, and Nathan Killoran, "Machine learning method for state preparation and gate synthesis on photonic quantum computers", Quantum Science and Technology 4 2, 024004 (2019).

[51] Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik, "Quantum Chemistry in the Age of Quantum Computing", arXiv:1812.09976.

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[54] Matteo G. Pozzi, Steven J. Herbert, Akash Sengupta, and Robert D. Mullins, "Using Reinforcement Learning to Perform Qubit Routing in Quantum Compilers", arXiv:2007.15957.

[55] Ed Younis, Koushik Sen, Katherine Yelick, and Costin Iancu, "QFAST: Quantum Synthesis Using a Hierarchical Continuous Circuit Space", arXiv:2003.04462.

[56] Mark M. Wilde, "Coherent Quantum Channel Discrimination", arXiv:2001.02668.

[57] Marc Grau Davis, Ethan Smith, Ana Tudor, Koushik Sen, Irfan Siddiqi, and Costin Iancu, "Heuristics for Quantum Compiling with a Continuous Gate Set", arXiv:1912.02727.

[58] Ed Younis, Koushik Sen, Katherine Yelick, and Costin Iancu, "QFAST: Conflating Search and Numerical Optimization for Scalable Quantum Circuit Synthesis", arXiv:2103.07093.

The above citations are from Crossref's cited-by service (last updated successfully 2021-05-06 16:31:13) and SAO/NASA ADS (last updated successfully 2021-05-06 16:31:14). The list may be incomplete as not all publishers provide suitable and complete citation data.