Robust quantum compilation and circuit optimisation via energy minimisation

Tyson Jones and Simon C. Benjamin

Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK

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Abstract

We explore a method for automatically recompiling a quantum circuit $\mathcal{A}$ into a target circuit $\mathcal{B}$, with the goal that both circuits have the same action on a specific input i.e. $\mathcal{B}{\mid{in}\rangle}=\mathcal{A}{\mid{in}\rangle}$. This is of particular relevance to hybrid, NISQ-era algorithms for dynamical simulation or eigensolving. The user initially specifies $\mathcal{B}$ as a blank template: a layout of parameterised unitary gates configured to the identity. The compilation then proceeds using quantum hardware to perform an isomorphic energy-minimisation task, and an optional gate elimination phase to compress the circuit. If $\mathcal{B}$ is insufficient for perfect recompilation then the method will result in an approximate solution. We optimise using imaginary time evolution, and a recent extension of quantum natural gradient for noisy settings. We successfully recompile a $7$-qubit circuit involving $186$ gates of multiple types into an alternative form with a different topology, far fewer two-qubit gates, and a smaller family of gate types. Moreover we verify that the process is $robust$, finding that per-gate noise of up to $1\%$ can still yield near-perfect recompilation. We test the scaling of our algorithm on up to $20$ qubits, recompiling into circuits with up to $400$ parameterized gates, and incorporate a custom adaptive timestep technique. We note that a classical simulation of the process can be useful to optimise circuits for today's prototypes, and more generally the method may enable `blind' compilation i.e. harnessing a device whose response to control parameters is deterministic but unknown.

The code and resources used to generate our results are openly available online [1] [2]. A simple Mathematica demonstration of our algorithm can be found at questlink.qtechtheory.org.

To run a program, a computer needs to understand how to perform each operation written in the code. But there are possibly thousands of programming languages which contain extremely many basic operations. How can one computer run them all? It can do so using compilers: Software written in a language the computer understands which can translate programs into instructions the computer knows how to carry out. Compilers can often also optimise the program, by reducing the number of basic operations yet still obtain the same final result. You’re probably viewing this webpage with a device running tens of compilers at once!

But quantum computers run a different kind of program. Quantum programs are written and understood in a very different way to the programs running on your home computer. They therefore need very different methods to compile them, that is shorten them or re-express them using a different set of operations. Many existing quantum compilers use clever maths to find alternate sequences of operations which have an identical effect on the quantum state. However, they run on classical computers, and require the result of these operations on the quantum state are precisely known and are storable in the computer.

But on noisy, near-future quantum computers, we often don’t know precisely what our elementary operations are doing to the quantum state. Sometimes we wish re-express our programs using a new set of operations which cannot precisely recreate the effect on the quantum state, but can bring us pretty close. How can we approximately translate our program?

In this work, we introduce a new method for performing approximate compilation to optimise the programs written for near-future quantum computers. It recasts the problem of compiling into one of finding the lowest energy state of a quantum system, for which recent developments have shown that near-future quantum computers can be quite good at. Our algorithm can run on real quantum hardware, or be simulated on a classical machine, in order to shorten quantum programs or translate them into different families of instructions. We explore one fascinating application of our recompilation method – to prolong the accuracy of physical simulation circuits by recompiling them on the fly! In line with our open science philosophy, we share all our code to simulate our algorithm online, and have done our best to make it readable, runnable and extendable.

► BibTeX data

► References

[1] github.com/​QTechTheory/​DissipativeRecompiler.
https:/​/​github.com/​QTechTheory/​DissipativeRecompiler

[2] github.com/​QTechTheory/​RecompilerSqueezeScaling.
https:/​/​github.com/​QTechTheory/​RecompilerSqueezeScaling

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

[4] S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.022316

[5] Ying Li, A magic state's fidelity can be superior to the operations that created it, New J. Phys. 17, 023037 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​2/​023037

[6] E. T. Campbell and M. Howard, Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost, Phys. Rev. A 95, 022316 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.022316

[7] David Gosset, Vadym Kliuchnikov, Michele Mosca, and Vincent Russo, An algorithm for the T-count, Quantum Information & Computation 14, 1261 (2014).
https:/​/​dl.acm.org/​doi/​10.5555/​2685179.2685180

[8] Neil J. Ross and Peter Selinger, Optimal ancilla-free Clifford+$T$ approximation of z-rotations, Quantum Information & Computation 16, 0901 (2016).
http:/​/​www.scopus.com/​inward/​record.url?eid=2-s2.0-84978986918&partnerID=MN8TOARS

[9] Matthew Amy and Michele Mosca, T-count optimization and Reed-Muller codes, IEEE Transactions on Information Theory 65, 8 (2019).
https:/​/​doi.org/​10.1109/​TIT.2019.2906374

[10] Luke E. Heyfron and Earl T. Campbell, An Efficient Quantum Compiler that reduces T count, Qantum Sci. Technol. 4, 015004 (2019).
https:/​/​iopscience.iop.org/​article/​10.1088/​2058-9565/​aad604

[11] Yunseong Nam, Neil J. Ross, Yuan Su, Andrew M. Childs and Dmitri 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

[12] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O'Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, Superconducting quantum circuits at the surface code threshold for fault tolerance, Nature 508, 500 (2014).
https:/​/​doi.org/​10.1038/​nature13171

[13] T. P. Harty, D. T. C. Allcock, C. J. Ballance, L. Guidoni, H. A. Janacek, N. M. Linke, D. N. Stacey, and D. M. Lucas, High-fidelity preparation, gates, memory and readout of a trapped-ion quantum bit, Phys. Rev. Lett. 113, 220501 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.220501

[14] C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, High-fidelity quantum logic gates using trapped-ion hyperfine qubits, Phys. Rev. Lett. 117, 060504 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.117.060504

[15] J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, High-fidelity universal gate set for $^9$Be$^+$ ion qubits, Phys. Rev. Lett. 117, 060505 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.117.060505

[16] Naomi H. Nickerson, Joseph F. Fitzsimons and Simon C. Benjamin, Freely Scalable Quantum Technologies Using Cells of 5-to-50 Qubits with Very Lossy and Noisy Photonic Links, Phys. Rev. X 4, 041041 (2014).
https:/​/​doi.org/​10.1103/​PhysRevX.4.041041

[17] Kentaro Heya, Yasunari Suzuki, Yasunobu Nakamura, and Keisuke Fujii, Variational Quantum Gate Optimization, arxiv:1810.12745 (2018).
arXiv:1810.12745

[18] Harper R. Grimsley, Sophia E. Economou, Edwin Barnes and Nicholas J. Mayhall, An adaptive variational algorithm for exact molecular simulations on a quantum computer, Nature Comms. 10 3007 (2019).
https:/​/​doi.org/​10.1038/​s41467-019-10988-2

[19] Arthur G. Rattew, Shaohan Hu, Marco Pistoia, Richard Chen and Steve Wood, A Domain-agnostic, Noise-resistant, Hardware-efficient Evolutionary Variational Quantum Eigensolver, arXiv:1910.09694 (2019).
arXiv:1910.09694

[20] M. Bilkis, M. Cerezo, Guillaume Verdon, Patrick J. Coles, Lukasz Cincio, A semi-agnostic ansatz with variable structure for quantum machine learning, arXiv:2103.06712 (2021).
arXiv:2103.06712

[21] P. Wocjan, D. Janzing, and T. Beth, Two QCMA-complete problems, Quant. Inf. & Comp., 3, 6 (2003).
https:/​/​dl.acm.org/​doi/​10.5555/​2011556.2011563

[22] For a recent review describing hybrid variational techniques in chemistry, see e.g. Sam McArdle, Suguru Endo, Alan Aspuru-Guzik, Simon Benjamin and Xiao Yuan, Quantum computational chemistry, Rev. Mod. Phys. (2019).
https:/​/​doi.org/​10.1103/​RevModPhys.92.015003

[23] Sumeet Khatri, Ryan LaRose, Alexander Poremba1, Lukasz Cincio1, Andrew T. Sornborger, and Patrick J. Coles, Quantum-assisted quantum compiling, Quantum 3, 140 (2019).
https:/​/​doi.org/​10.22331/​q-2019-05-13-140

[24] 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, Nat. Phys. (2020).
https:/​/​doi.org/​10.1038/​s41567-019-0747-6

[25] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, Patrick J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3(9), pp.625-644 (2021).
https:/​/​doi.org/​10.1038/​s42254-021-00348-9

[26] Suguru Endo, Zhenyu Cai, Simon C. Benjamin, Xiao Yuan, Hybrid quantum-classical algorithms and quantum error mitigation, J. Phys. Soc. Jpn., 90, 032001 (2021).
https:/​/​doi.org/​10.7566/​JPSJ.90.032001

[27] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C. Benjamin, Theory of variational quantum simulation, Quantum 3, 191 (2019).
https:/​/​doi.org/​10.22331/​q-2019-10-07-191

[28] Sam McArdle, Tyson Jones, Suguru Endo, Ying Li, Simon Benjamin and Xiao Yuan, Variational ansatz-based quantum simulation of imaginary time evolution, npj Quantum Information 5, 75 (2019).
https:/​/​doi.org/​10.1038/​s41534-019-0187-2

[29] Bálint Koczor, Simon C. Benjamin, Quantum natural gradient generalised to non-unitary circuits, arXiv:1912.08660v2 (2020).
arXiv:1912.08660

[30] James Stokes, Josh Izaac, Nathan Killoran, Giuseppe Carleo, Quantum Natural Gradient, Quantum 4, 269 (2020).
https:/​/​doi.org/​10.22331/​q-2020-05-25-269

[31] Y. Li and S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization, Phys. Rev. X 7, 021050 (2017).
https:/​/​doi.org/​10.1103/​PhysRevX.7.021050

[32] Tyson Jones & Suguru Endo, Sam McArdle, Xiao Yuan and Simon Benjamin, Variational quantum algorithms for discovering Hamiltonian spectra, Phys. Rev. A 99, 062304 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.062304

[33] Barnaby van Straaten, Bálint Koczor, Measurement cost of metric-aware variational quantum algorithms, PRX Quantum 2, 030324 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.030324

[34] Lukasz Cincio, Yigit Subasi, Andrew T Sornborger and Patrick J Coles, Learning the quantum algorithm for state overlap, New J. Phys. 20 113022 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aae94a

[35] Sang Min Lee, Jinhyoung Lee and Jeongho Bang, Learning unknown pure quantum states, Phys. Rev. A 98, 052302 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.052302

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

[37] Ming-Cheng Chen, Ming Gong, Xiaosi Xu, Xiao Yuan, Jian-Wen Wang, Can Wang, Chong Ying, Jin Lin, Yu Xu, Yulin Wu, Shiyu Wang, Hui Deng, Futian Liang, Cheng-Zhi Peng, Simon C. Benjamin, Xiaobo Zhu, Chao-Yang Lu, and Jian-Wei Pan, Demonstration of Adiabatic Variational Quantum Computing with a Superconducting Quantum Coprocessor, Phys. Rev. Lett., 125, 180501 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.180501

[38] Grippo, Luigi, Francesco Lampariello, and Stephano Lucidi, A nonmonotone line search technique for Newton's method. SIAM Journal on Numerical Analysis 23, 4 (1986).
https:/​/​doi.org/​10.1137/​0723046

[39] Hale F. Trotter, On the product of semi-groups of operators, Proceedings of the American Mathematical Society 10, 4 (1959).
https:/​/​doi.org/​10.2307/​2033649

[40] Seth Lloyd, Universal quantum simulators, Science 1073-1078 (1996).
https:/​/​doi.org/​10.1126/​science.273.5278.1073

[41] Kunal Sharma, Sumeet Khatri, M Cerezo, Patrick J Coles, Noise resilience of variational quantum compiling, New J. Phys. 22 043006 (2020).
https:/​/​doi.org/​10.1088/​1367-2630/​ab784c

[42] Kristan Temme, Sergey Bravyi and Jay M. Gambetta, Error Mitigation for Short-Depth Quantum Circuits, Phys. Rev. Lett. 119, 180509 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.180509

[43] Abhinav Kandala, Kristan Temme, Antonio D. Córcoles, Antonio Mezzacapo, Jerry M. Chow & Jay M. Gambetta Error mitigation extends the computational reach of a noisy quantum processor, Nature 567, 491 (2019).
https:/​/​doi.org/​10.1038/​s41586-019-1040-7

[44] Suguru Endo, Simon Benjamin and Ying Li, Practical Quantum Error Mitigation for Near-Future Applications, Phys. Rev. X, 8,031027 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.031027

[45] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio and Patrick J. Coles, Cost function dependent barren plateaus in shallow parametrized quantum circuits, Nature Comms. 12, 1791 (2021).
https:/​/​doi.org/​10.1038/​s41467-021-21728-w

[46] Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio and Patrick J. Coles, An Adaptive Optimizer for Measurement-Frugal Variational Algorithms, Quantum 4, 263 (2020).
https:/​/​doi.org/​10.22331/​q-2020-05-11-263

[47] Andrew Arrasmith, Lukasz Cincio, Rolando D. Somma and Patrick J. Coles, Operator Sampling for Shot-frugal Optimization in Variational Algorithms, arXiv:2004.06252 (2020).
arXiv:2004.06252

[48] Jian Ma, Xiaoguang Wang, C. P. Sun, Franco Nori, Quantum spin squeezing, Physics Reports 509, 2-3 (2011).
https:/​/​doi.org/​10.1016/​j.physrep.2011.08.003

[49] Duger Ulam-Orgikh and Masahiro Kitagawa, Spin squeezing and decoherence limit in Ramsey spectroscopy, Phys. Rev. A 64, 052106 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.64.052106

[50] Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, and Marcello Benedetti, An initialization strategy for addressing barren plateaus in parametrized quantum circuits, Quantum 3, 214 (2019).
https:/​/​doi.org/​10.22331/​q-2019-12-09-214

[51] K. Temme, T. J. Osborne, K. G. Vollbrecht, D. Poulin and F. Verstraete, Quantum Metropolis sampling, Nature 471, 87 (2011).
https:/​/​doi.org/​10.1038/​nature09770

[52] Man-Hong Yung and Alán Aspuru-Guzik, A quantum-quantum Metropolis algorithm, PNAS 109, 754 (2012).
https:/​/​doi.org/​10.1073/​pnas.1111758109

[53] Mohammad H. Amin, Evgeny Andriyash, Jason Rolfe, Bohdan Kulchytskyy and Roger Melko, Quantum Boltzmann Machine Phys. Rev. X 8, 021050 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.021050

[54] Fernando G.S.L. Brandao and Krysta Svore, Quantum Speed-ups for Semidefinite Programming, Proceedings FOCS 2017 (2017) and arXiv:1609.05537 (2016).
arXiv:1609.05537

[55] Joran van Apeldoorn, András Gilyén, Improvements in Quantum SDP-Solving with Applications, arXiv:1804.05058 (2018).
https:/​/​doi.org/​10.4230/​LIPIcs.ICALP.2019.99
arXiv:1804.05058

[56] Dave Wecker, Matthew B. Hastings, Nathan Wiebe, Bryan K. Clark, Chetan Nayak and Matthias Troyer, Solving strongly correlated electron models on a quantum computer, Phys. Rev. A 92, 062318 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.062318

[57] Lukasz Cincio, Kenneth Rudinger, Mohan Sarovar and Patrick J. Coles, Machine Learning of Noise-Resilient Quantum Circuits, PRX Quantum 2 010324 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.010324

[58] Artur K. Ekert, Carolina Moura Alves, Daniel K. L. Oi, Michal Horodecki, Pawel Horodecki, and L. C. Kwek, Direct estimations of linear and nonlinear functionals of a quantum state, Phys. Rev. Lett. 88, 217901 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.88.217901

[59] Andrew M. Childs, Aaron Ostrander and Yuan Su, Faster quantum simulation by randomization Quantum 3, 182 (2019).
https:/​/​doi.org/​10.22331/​q-2019-09-02-182

[60] Earl Campbell, A random compiler for fast Hamiltonian simulation, Phys. Rev. Lett. 123, 070503 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.070503

[61] Tyson Jones, Anna Brown, Ian Bush and Simon Benjamin, QuEST and High Performance Simulation of Quantum Computers, Scientific Reports 9, 10736 (2019).
https:/​/​doi.org/​10.1038/​s41598-019-47174-9

[62] Tyson Jones, and Simon C Benjamin, QuESTlink - Mathematica embiggened by a hardware-optimised quantum emulator, Quantum Sci. Technol. 5 034012 (2020).
https:/​/​doi.org/​10.1088/​2058-9565/​ab8506

[63] Contributors and GSL Project, GSL - GNU Scientific Library - GNU Project - Free Software, The GNU Operating System (2010).
https:/​/​www.gnu.org/​software/​gsl/​

[64] Gene Golub and Charles Van Loan, Matrix computations, The Mathematical Gazette 83, 498 (1999).
https:/​/​doi.org/​10.1137/​1028073

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[2] Enrico Fontana, M. Cerezo, Andrew Arrasmith, Ivan Rungger, and Patrick J. Coles, "Non-trivial symmetries in quantum landscapes and their resilience to quantum noise", Quantum 6, 804 (2022).

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[6] Brian Coyle, Mina Doosti, Elham Kashefi, and Niraj Kumar, "Progress toward practical quantum cryptanalysis by variational quantum cloning", Physical Review A 105 4, 042604 (2022).

[7] Changsu Cao, Jiaqi Hu, Wengang Zhang, Xusheng Xu, Dechin Chen, Fan Yu, Jun Li, Han-Shi Hu, Dingshun Lv, and Man-Hong Yung, "Progress toward larger molecular simulation on a quantum computer: Simulating a system with up to 28 qubits accelerated by point-group symmetry", Physical Review A 105 6, 062452 (2022).

[8] 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, 1791 (2021).

[9] 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", arXiv:2012.09265.

[10] Suguru Endo, Zhenyu Cai, Simon C. Benjamin, and Xiao Yuan, "Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation", Journal of the Physical Society of Japan 90 3, 032001 (2021).

[11] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles, "Noise-induced barren plateaus in variational quantum algorithms", Nature Communications 12, 6961 (2021).

[12] Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J. Coles, "Variational Quantum Linear Solver", arXiv:1909.05820.

[13] Bobak Toussi Kiani, Giacomo De Palma, Milad Marvian, Zi-Wen Liu, and Seth Lloyd, "Learning quantum data with the quantum Earth Mover's distance", arXiv:2101.03037, Quantum Science and Technology 7 4, 045002 (2021).

[14] Xiaosi Xu, Jinzhao Sun, Suguru Endo, Ying Li, Simon C. Benjamin, and Xiao Yuan, "Variational algorithms for linear algebra", Science Bulletin 66 21, 2181 (2021).

[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] Samson Wang, Piotr Czarnik, Andrew Arrasmith, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?", arXiv:2109.01051.

[17] Matthias C. Caro, Hsin-Yuan Huang, Nicholas Ezzell, Joe Gibbs, Andrew T. Sornborger, Lukasz Cincio, Patrick J. Coles, and Zoë Holmes, "Out-of-distribution generalization for learning quantum dynamics", arXiv:2204.10268.

[18] Bálint Koczor, Suguru Endo, Tyson Jones, Yuichiro Matsuzaki, and Simon C. Benjamin, "Variational-state quantum metrology", New Journal of Physics 22 8, 083038 (2020).

[19] Andrew Arrasmith, Lukasz Cincio, Rolando D. Somma, and Patrick J. Coles, "Operator Sampling for Shot-frugal Optimization in Variational Algorithms", arXiv:2004.06252.

[20] Pei Zeng, Jinzhao Sun, and Xiao Yuan, "Universal quantum algorithmic cooling on a quantum computer", arXiv:2109.15304.

[21] Sam McArdle and David P. Tew, "Improving the accuracy of quantum computational chemistry using the transcorrelated method", arXiv:2006.11181.

[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, 60 (2020).

[23] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo, "Quantum Natural Gradient", arXiv:1909.02108.

[24] Cheng Xue, Zhao-Yun Chen, Yu-Chun Wu, and Guo-Ping Guo, "Effects of Quantum Noise on Quantum Approximate Optimization Algorithm", arXiv:1909.02196, Chinese Physics Letters 38 3, 030302 (2019).

[25] Xin Wang, Zhixin Song, and Youle Wang, "Variational Quantum Singular Value Decomposition", arXiv:2006.02336.

[26] Hayata Yamasaki, Kosuke Fukui, Yuki Takeuchi, Seiichiro Tani, and Masato Koashi, "Polylog-overhead highly fault-tolerant measurement-based quantum computation: all-Gaussian implementation with Gottesman-Kitaev-Preskill code", arXiv:2006.05416.

[27] Lukasz Cincio, Kenneth Rudinger, Mohan Sarovar, and Patrick J. Coles, "Machine learning of noise-resilient quantum circuits", arXiv:2007.01210.

[28] Mohammad Pirhooshyaran and Tamas Terlaky, "Quantum Circuit Design Search", arXiv:2012.04046.

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

[30] Zhan Yu, Xuanqiang Zhao, Benchi Zhao, and Xin Wang, "Optimal quantum dataset for learning a unitary transformation", arXiv:2203.00546.

[31] Sumeet Khatri, Ryan LaRose, Alexander Poremba, Lukasz Cincio, Andrew T. Sornborger, and Patrick J. Coles, "Quantum-assisted quantum compiling", arXiv:1807.00800.

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[34] Michael R. Geller, Zoë Holmes, Patrick J. Coles, and Andrew Sornborger, "Experimental quantum learning of a spectral decomposition", Physical Review Research 3 3, 033200 (2021).

[35] Nikita A. Nemkov, Evgeniy O. Kiktenko, Ilia A. Luchnikov, and Aleksey K. Fedorov, "Efficient variational synthesis of quantum circuits with coherent multi-start optimization", arXiv:2205.01121.

[36] Ryan LaRose, Arkin Tikku, Étude O'Neel-Judy, Lukasz Cincio, and Patrick J. Coles, "Variational Quantum State Diagonalization", arXiv:1810.10506.

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[38] Zhimin He, Lvzhou Li, Shenggen Zheng, Yongyao Li, and Haozhen Situ, "Variational quantum compiling with double Q-learning", New Journal of Physics 23 3, 033002 (2021).

[39] Nathan Fitzpatrick, Harriet Apel, and David Muñoz Ramo, "Evaluating low-depth quantum algorithms for time evolution on fermion-boson systems", arXiv:2106.03985.

[40] Zhimin He, Chuangtao Chen, Lvzhou Li, Shenggen Zheng, and Haozhen Situ, "Quantum Architecture Search with Meta-learning", arXiv:2106.06248.

[41] Brian Coyle, "Machine learning applications for noisy intermediate-scale quantum computers", arXiv:2205.09414.

[42] Sam McArdle, Alex Mayorov, Xiao Shan, Simon Benjamin, and Xiao Yuan, "Digital quantum simulation of molecular vibrations", arXiv:1811.04069.

[43] Zhimin He, Junjian Su, Chuangtao Chen, Minghua Pan, and Haozhen Situ, "Search space pruning for quantum architecture search", European Physical Journal Plus 137 4, 491 (2022).

[44] Prakash Murali, Lingling Lao, Margaret Martonosi, and Dan Browne, "Designing calibration and expressivity-efficient instruction sets for quantum computing", arXiv:2106.15490.

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