TFermion: A non-Clifford gate cost assessment library of quantum phase estimation algorithms for quantum chemistry

Pablo A. M. Casares1, Roberto Campos1,2, and M. A. Martin-Delgado1,3

1Departamento de Física Teórica, Universidad Complutense de Madrid.
2Quasar Science Resources, SL.
3CCS-Center for Computational Simulation, Universidad Politécnica de Madrid.

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Quantum Phase Estimation is one of the most useful quantum computing algorithms for quantum chemistry and as such, significant effort has been devoted to designing efficient implementations. In this article, we introduce TFermion, a library designed to estimate the T-gate cost of such algorithms, for an arbitrary molecule. As examples of usage, we estimate the T-gate cost of a few simple molecules and compare the same Taylorization algorithms using Gaussian and plane-wave basis.

Presentation of TFermion in the APS March meeting 2022:

Chemistry and material science are often thought of as the killer application of quantum computing. Specifically, quantum phase estimation is a cornerstone quantum algorithm that can be used to study the energy of quantum systems, and thus estimate many of their chemical properties. On the other hand, implementing this algorithm depends on a few crucial choices, including how the system is represented and how it is made to evolve. These decisions will ultimately be reflected in the total time required to execute the algorithm, a key consideration if we want it to be useful. TFermion is a software library that estimates the number of the most costly logical gates in quantum phase estimation, thus allowing for comparing the cost of different options, and evaluating their practicality.

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[1] Daniel S Abrams and Seth Lloyd. Simulation of many-body fermi systems on a universal quantum computer. Physical Review Letters, 79 (13): 2586, 1997. https:/​/​​10.1103/​PhysRevLett.79.2586.

[2] Alán Aspuru-Guzik, Anthony D Dutoi, Peter J Love, and Martin Head-Gordon. Simulated quantum computation of molecular energies. Science, 309 (5741): 1704–1707, 2005. https:/​/​​10.1126/​science.1113479.

[3] Ryan Babbush, Dominic W Berry, Ian D Kivlichan, Annie Y Wei, Peter J Love, and Alán Aspuru-Guzik. Exponentially more precise quantum simulation of fermions in second quantization. New Journal of Physics, 18 (3): 033032, 2016. https:/​/​​10.1088/​1367-2630/​18/​3/​033032.

[4] Ryan Babbush, Dominic W Berry, Yuval R Sanders, Ian D Kivlichan, Artur Scherer, Annie Y Wei, Peter J Love, and Alán Aspuru-Guzik. Exponentially more precise quantum simulation of fermions in the configuration interaction representation. Quantum Science and Technology, 3 (1): 015006, 2017. https:/​/​​10.1088/​2058-9565/​aa9463.

[5] Ryan Babbush, Craig Gidney, Dominic W Berry, Nathan Wiebe, Jarrod R McClean, Alexandru Paler, Austin Fowler, and Hartmut Neven. Encoding electronic spectra in quantum circuits with linear t complexity. Physical Review X, 8 (4): 041015, 2018a. https:/​/​​10.1103/​physrevx.8.041015.

[6] Ryan Babbush, Nathan Wiebe, Jarrod R McClean, James McClain, Hartmut Neven, and Garnet Kin-Lic Chan. Low-depth quantum simulation of materials. Physical Review X, 8 (1): 011044, 2018b. https:/​/​​10.1103/​physrevx.8.011044.

[7] Ryan Babbush, Dominic W Berry, Jarrod R McClean, and Hartmut Neven. Quantum simulation of chemistry with sublinear scaling in basis size. npj Quantum Information, 5 (1): 1–7, 2019. https:/​/​​10.1038/​s41534-019-0199-y.

[8] Adriano Barenco, Charles H Bennett, Richard Cleve, David P DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A Smolin, and Harald Weinfurter. Elementary gates for quantum computation. Physical Review A, 52 (5): 3457, 1995. https:/​/​​10.1103/​PhysRevA.52.3457.

[9] Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Somma. Simulating hamiltonian dynamics with a truncated taylor series. Physical Review Letters, 114 (9): 090502, 2015. https:/​/​​10.1103/​physrevlett.114.090502.

[10] Dominic W Berry, Mária Kieferová, Artur Scherer, Yuval R Sanders, Guang Hao Low, Nathan Wiebe, Craig Gidney, and Ryan Babbush. Improved techniques for preparing eigenstates of fermionic hamiltonians. npj Quantum Information, 4 (1): 1–7, 2018. https:/​/​​10.1038/​s41534-018-0071-5.

[11] Dominic W Berry, Craig Gidney, Mario Motta, Jarrod R McClean, and Ryan Babbush. Qubitization of arbitrary basis quantum chemistry leveraging sparsity and low rank factorization. Quantum, 3: 208, 2019. https:/​/​​10.22331/​q-2019-12-02-208.

[12] Evan E Bolton, Yanli Wang, Paul A Thiessen, and Stephen H Bryant. Pubchem: integrated platform of small molecules and biological activities. In Annual Reports in Computational Chemistry, volume 4, pages 217–241. Elsevier, 2008. https:/​/​​10.1016/​s1574-1400(08)00012-1.

[13] Hector Bombin and Miguel Angel Martin-Delgado. Topological computation without braiding. Physical Review Letters, 98 (16): 160502, 2007. https:/​/​​10.1103/​physrevlett.98.160502.

[14] Earl Campbell. Shorter gate sequences for quantum computing by mixing unitaries. Physical Review A, 95 (4): 042306, 2017. https:/​/​​10.1103/​physreva.95.042306.

[15] Earl Campbell. Random compiler for fast hamiltonian simulation. Physical Review Letters, 123 (7): 070503, 2019. https:/​/​​10.1103/​PhysRevLett.123.070503.

[16] Earl Campbell. Early fault-tolerant simulations of the hubbard model. Quantum Science and Technology, 7 (1): 015007, 2021. https:/​/​​10.1088/​2058-9565/​ac3110.

[17] Yudong Cao, Jonathan Romero, Jonathan P Olson, Matthias Degroote, Peter D Johnson, Mária Kieferová, Ian D Kivlichan, Tim Menke, Borja Peropadre, Nicolas PD Sawaya, et al. Quantum chemistry in the age of quantum computing. Chemical Reviews, 119 (19): 10856–10915, 2019. https:/​/​​10.1021/​acs.chemrev.8b00803.

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

[19] Richard Cleve, Artur Ekert, Chiara Macchiavello, and Michele Mosca. Quantum algorithms revisited. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454 (1969): 339–354, 1998. https:/​/​​10.1098/​rspa.1998.0164.

[20] Steven A Cuccaro, Thomas G Draper, Samuel A Kutin, and David Petrie Moulton. A new quantum ripple-carry addition circuit. arXiv preprint quant-ph/​0410184, 2004. https:/​/​​10.48550/​arXiv.quant-ph/​0410184.

[21] Alain Delgado, Pablo Antonio Moreno Casares, Roberto dos Reis, Modjtaba Shokrian Zini, Roberto Campos, Norge Cruz-Hernández, Arne-Christian Voigt, Angus Lowe, Soran Jahangiri, Miguel Angel Martin-Delgado, Jonathan E. Mueller, and Juan Miguel Arrazola. How to simulate key properties of lithium-ion batteries with a fault-tolerant quantum computer. arXiv preprint arXiv:2204.11890, 2022. 10.48550/​ARXIV.2204.11890. URL https:/​/​​abs/​2204.11890.

[22] Vincent E Elfving, Benno W Broer, Mark Webber, Jacob Gavartin, Mathew D Halls, K Patrick Lorton, and A Bochevarov. How will quantum computers provide an industrially relevant computational advantage in quantum chemistry? arXiv preprint arXiv:2009.12472, 2020. https:/​/​​10.48550/​arXiv.2009.12472.

[23] Andrew J Ferris. Fourier transform for fermionic systems and the spectral tensor network. Physical Review Letters, 113 (1): 010401, 2014. https:/​/​​10.1103/​physrevlett.113.010401.

[24] Richard P Feynman. Simulating physics with computers. In Feynman and computation, pages 133–153. CRC Press, 2018. https:/​/​​10.1201/​9780429500459-11.

[25] Alberto Galindo and Miguel Angel Martin-Delgado. Information and computation: Classical and quantum aspects. Reviews of Modern Physics, 74 (2): 347, 2002. https:/​/​​10.1103/​revmodphys.74.347.

[26] Yimin Ge, Jordi Tura, and J Ignacio Cirac. Faster ground state preparation and high-precision ground energy estimation with fewer qubits. Journal of Mathematical Physics, 60 (2): 022202, 2019. https:/​/​​10.1063/​1.5027484.

[27] Craig Gidney. Halving the cost of quantum addition. Quantum, 2: 74, 2018. https:/​/​​10.22331/​q-2018-06-18-74.

[28] Joshua J Goings, Alec White, Joonho Lee, Christofer S Tautermann, Matthias Degroote, Craig Gidney, Toru Shiozaki, Ryan Babbush, and Nicholas C Rubin. Reliably assessing the electronic structure of cytochrome p450 on today's classical computers and tomorrow's quantum computers. arXiv preprint arXiv:2202.01244, 2022. https:/​/​​10.48550/​arXiv.2202.01244.

[29] Harper R. Grimsley, S. Economou, Edwin Barnes, and Nicholas J. Mayhall. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nature Communications, 10, 2019. https:/​/​​10.1038/​s41467-019-10988-2.

[30] Matthew B. Hastings, Dave Wecker, Bela Bauer, and Matthias Troyer. Improving quantum algorithms for quantum chemistry. Quantum Information and Computation, 15 (1–2): 1–21, jan 2015. ISSN 1533-7146. https:/​/​​10.26421/​qic15.1-2-1.

[31] Frank Jensen. Atomic orbital basis sets. Wiley Interdisciplinary Reviews: Computational Molecular Science, 3 (3): 273–295, 2013. https:/​/​​10.1002/​wcms.1123.

[32] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. Chow, and J. Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549: 242–246, 2017. https:/​/​​10.1038/​nature23879.

[33] Julia Kempe, Alexei Kitaev, and Oded Regev. The complexity of the local hamiltonian problem. SIAM Journal on Computing, 35 (5): 1070–1097, 2006. https:/​/​​10.1137/​s0097539704445226.

[34] Mária Kieferová, Artur Scherer, and Dominic W Berry. Simulating the dynamics of time-dependent hamiltonians with a truncated dyson series. Physical Review A, 99 (4): 042314, 2019. https:/​/​​10.1103/​physreva.99.042314.

[35] Isaac H Kim, Ye-Hua Liu, Sam Pallister, William Pol, Sam Roberts, and Eunseok Lee. Fault-tolerant resource estimate for quantum chemical simulations: Case study on li-ion battery electrolyte molecules. Physical Review Research, 4 (2): 023019, 2022. https:/​/​​10.1103/​physrevresearch.4.023019.

[36] Ian D Kivlichan, Craig Gidney, Dominic W Berry, Nathan Wiebe, Jarrod R McClean, Wei Sun, Zhang Jiang, Nicholas C Rubin, Austin Fowler, Alán Aspuru-Guzik, et al. Improved fault-tolerant quantum simulation of condensed-phase correlated electrons via trotterization. Quantum, 4: 296, 2020. https:/​/​​10.22331/​q-2020-07-16-296.

[37] Jorge Kohanoff. Electronic structure calculations for solids and molecules: theory and computational methods. Cambridge university press, 2006. https:/​/​​10.1017/​CBO9780511755613.

[38] Emiel Koridon, Saad Yalouz, Bruno Senjean, Francesco Buda, Thomas E O'Brien, and Lucas Visscher. Orbital transformations to reduce the 1-norm of the electronic structure hamiltonian for quantum computing applications. Physical Review Research, 3 (3): 033127, 2021. https:/​/​​10.1103/​physrevresearch.3.033127.

[39] Joonho Lee, Dominic W Berry, Craig Gidney, William J Huggins, Jarrod R McClean, Nathan Wiebe, and Ryan Babbush. Even more efficient quantum computations of chemistry through tensor hypercontraction. PRX Quantum, 2 (3): 030305, 2021. https:/​/​​10.1103/​prxquantum.2.030305.

[40] Zhendong Li, Junhao Li, Nikesh S Dattani, CJ Umrigar, and Garnet Kin-Lic Chan. The electronic complexity of the ground-state of the femo cofactor of nitrogenase as relevant to quantum simulations. The Journal of Chemical Physics, 150 (2): 024302, 2019. https:/​/​​10.1063/​1.5063376.

[41] Lin Lin and Yu Tong. Near-optimal ground state preparation. Quantum, 4: 372, 2020. https:/​/​​10.22331/​q-2020-12-14-372.

[42] Seth Lloyd. Universal quantum simulators. Science, pages 1073–1078, 1996. https:/​/​​10.1126/​science.273.5278.1073.

[43] Guang Hao Low and Isaac L Chuang. Optimal hamiltonian simulation by quantum signal processing. Physical Review Letters, 118 (1): 010501, 2017. https:/​/​​10.1103/​physrevlett.118.010501.

[44] Guang Hao Low and Isaac L Chuang. Hamiltonian simulation by qubitization. Quantum, 3: 163, 2019. https:/​/​​10.22331/​q-2019-07-12-163.

[45] Guang Hao Low and Nathan Wiebe. Hamiltonian simulation in the interaction picture. arXiv preprint arXiv:1805.00675, 2018. https:/​/​​10.48550/​arXiv.1805.00675.

[46] Guang Hao Low, Vadym Kliuchnikov, and Luke Schaeffer. Trading t-gates for dirty qubits in state preparation and unitary synthesis. arXiv preprint arXiv:1812.00954, 2018. https:/​/​​10.48550/​arXiv.1812.00954.

[47] Sam McArdle, Tyson Jones, Suguru Endo, Y. Li, S. Benjamin, and Xiao Yuan. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Information, 5: 1–6, 2018. https:/​/​​10.1038/​s41534-019-0187-2.

[48] Sam McArdle, Earl Campbell, and Yuan Su. Exploiting fermion number in factorized decompositions of the electronic structure hamiltonian. Physical Review A, 105 (1): 012403, 2022. https:/​/​​10.1103/​physreva.105.012403.

[49] Jarrod R McClean, Nicholas C Rubin, Kevin J Sung, Ian D Kivlichan, Xavier Bonet-Monroig, Yudong Cao, Chengyu Dai, E Schuyler Fried, Craig Gidney, Brendan Gimby, et al. Openfermion: the electronic structure package for quantum computers. Quantum Science and Technology, 5 (3): 034014, 2020. https:/​/​​10.1088/​2058-9565/​ab8ebc.

[50] Mario Motta, Erika Ye, Jarrod R McClean, Zhendong Li, Austin J Minnich, Ryan Babbush, and Garnet Kin Chan. Low rank representations for quantum simulation of electronic structure. npj Quantum Information, 7 (1): 1–7, 2021. https:/​/​​10.1038/​s41534-021-00416-z.

[51] Felix Motzoi, Michael P Kaicher, and Frank K Wilhelm. Linear and logarithmic time compositions of quantum many-body operators. Physical Review Letters, 119 (16): 160503, 2017. https:/​/​​10.1103/​physrevlett.119.160503.

[52] Edgard Muñoz-Coreas and Himanshu Thapliyal. T-count optimized design of quantum integer multiplication. arXiv preprint arXiv:1706.05113, 2017. https:/​/​​10.48550/​arXiv.1706.05113.

[53] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010. 10.1017/​CBO9780511976667.

[54] Alberto Peruzzo, Jarrod R 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, 2014. https:/​/​​10.1038/​ncomms5213.

[55] David Poulin, Matthew B Hastings, Dave Wecker, Nathan Wiebe, Andrew C Doherty, and Matthias Troyer. The trotter step size required for accurate quantum simulation of quantum chemistry. arXiv preprint arXiv:1406.4920, 2014. https:/​/​​10.26421/​qic15.5-6-1.

[56] David Poulin, Alexei Kitaev, Damian S Steiger, Matthew B Hastings, and Matthias Troyer. Quantum algorithm for spectral measurement with a lower gate count. Physical Review Letters, 121 (1): 010501, 2018. https:/​/​​10.1103/​physrevlett.121.010501.

[57] Abhishek Rajput, Alessandro Roggero, and Nathan Wiebe. Hybridized methods for quantum simulation in the interaction picture. arXiv preprint arXiv:2109.03308, 2021. https:/​/​​10.48550/​arXiv.2109.03308.

[58] Markus Reiher, Nathan Wiebe, Krysta M Svore, Dave Wecker, and Matthias Troyer. Elucidating reaction mechanisms on quantum computers. Proceedings of the National Academy of Sciences, 114 (29): 7555–7560, 2017. https:/​/​​10.1073/​pnas.1619152114.

[59] Elvira R Sayfutyarova, Qiming Sun, Garnet Kin-Lic Chan, and Gerald Knizia. Automated construction of molecular active spaces from atomic valence orbitals. Journal of Chemical Theory and Computation, 13 (9): 4063–4078, 2017. https:/​/​​10.1021/​acs.jctc.7b00128.s001.

[60] Peter Selinger. Efficient clifford+t approximation of single-qubit operators. Quantum Info. Comput., 15 (1–2): 159–180, jan 2015. ISSN 1533-7146. https:/​/​​10.26421/​qic15.1-2-10.

[61] Vivek V Shende, Stephen S Bullock, and Igor L Markov. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 25 (6): 1000–1010, 2006. https:/​/​​10.1109/​tcad.2005.855930.

[62] Yuan Su, Dominic W Berry, Nathan Wiebe, Nicholas C Rubin, and Ryan Babbush. Fault-tolerant quantum simulations of chemistry in first quantization. PRX Quantum, 2 (4): 040332, 2021a. https:/​/​​10.1103/​prxquantum.2.040332.

[63] Yuan Su, Hsin-Yuan Huang, and Earl T Campbell. Nearly tight trotterization of interacting electrons. Quantum, 5: 495, 2021b. https:/​/​​10.22331/​q-2021-07-05-495.

[64] Qiming Sun, Timothy C Berkelbach, Nick S Blunt, George H Booth, Sheng Guo, Zhendong Li, Junzi Liu, James D McClain, Elvira R Sayfutyarova, Sandeep Sharma, et al. Pyscf: the python-based simulations of chemistry framework. Wiley Interdisciplinary Reviews: Computational Molecular Science, 8 (1): e1340, 2018. https:/​/​​10.1002/​wcms.1340.

[65] Masuo Suzuki. Fractal decomposition of exponential operators with applications to many-body theories and monte carlo simulations. Physics Letters A, 146 (6): 319–323, 1990. https:/​/​​10.1016/​0375-9601(90)90962-n.

[66] Masuo Suzuki. General theory of fractal path integrals with applications to many-body theories and statistical physics. Journal of Mathematical Physics, 32 (2): 400–407, 1991. https:/​/​​10.1063/​1.529425.

[67] Himanshu Thapliyal, TSS Varun, Edgard Munoz-Coreas, Keith A Britt, and Travis S Humble. Quantum circuit designs of integer division optimizing t-count and t-depth. In 2017 IEEE International Symposium on Nanoelectronic and Information Systems (iNIS), pages 123–128. IEEE, 2017. https:/​/​​10.1109/​inis.2017.34.

[68] Jack E Volder. The cordic trigonometric computing technique. IRE Transactions on electronic computers, (3): 330–334, 1959. https:/​/​​10.1109/​tec.1959.5222693.

[69] Vera von Burg, Guang Hao Low, Thomas Häner, Damian S Steiger, Markus Reiher, Martin Roetteler, and Matthias Troyer. Quantum computing enhanced computational catalysis. Physical Review Research, 3 (3): 033055, 2021. https:/​/​​10.1103/​physrevresearch.3.033055.

[70] Kianna Wan, Mario Berta, and Earl Campbell. A randomized quantum algorithm for statistical phase estimation. arXiv preprint arXiv:2110.12071, 2021. https:/​/​​10.48550/​arXiv.2110.12071.

[71] Dave Wecker, Matthew B Hastings, Nathan Wiebe, Bryan K Clark, Chetan Nayak, and Matthias Troyer. Solving strongly correlated electron models on a quantum computer. Physical Review A, 92 (6): 062318, 2015. https:/​/​​10.1103/​physreva.92.062318.

[72] Steven R White. Hybrid grid/​basis set discretizations of the schrödinger equation. The Journal of Chemical Physics, 147 (24): 244102, 2017. https:/​/​​10.1063/​1.5007066.

[73] Steven R White and E Miles Stoudenmire. Multisliced gausslet basis sets for electronic structure. Physical Review B, 99 (8): 081110, 2019. https:/​/​​10.1103/​PhysRevB.99.081110.

[74] James D Whitfield, Jacob Biamonte, and Alán Aspuru-Guzik. Simulation of electronic structure hamiltonians using quantum computers. Molecular Physics, 109 (5): 735–750, 2011. https:/​/​​10.1080/​00268976.2011.552441.

[75] Nathan Wiebe and Chris Granade. Efficient bayesian phase estimation. Physical Review Letters, 117 (1): 010503, 2016. https:/​/​​10.1103/​physrevlett.117.010503.

[76] Ruizhe Zhang, Guoming Wang, and Peter Johnson. Computing Ground State Properties with Early Fault-Tolerant Quantum Computers. Quantum, 6: 761, July 2022. ISSN 2521-327X. 10.22331/​q-2022-07-11-761. URL https:/​/​​10.22331/​q-2022-07-11-761.

Cited by

[1] Xiantao Li, "Some error analysis for the quantum phase estimation algorithms", Journal of Physics A Mathematical General 55 32, 325303 (2022).

[2] Alain Delgado, Pablo A. M. Casares, Roberto dos Reis, Modjtaba Shokrian Zini, Roberto Campos, Norge Cruz-Hernández, Arne-Christian Voigt, Angus Lowe, Soran Jahangiri, M. A. Martin-Delgado, Jonathan E. Mueller, and Juan Miguel Arrazola, "How to simulate key properties of lithium-ion batteries with a fault-tolerant quantum computer", arXiv:2204.11890.

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