1Quantinuum, 13-15 Hills Road, CB2 1NL, Cambridge, United Kingdom
2Yusuf Hamied Department of Chemistry, University of Cambridge, Cambridge, United Kingdom
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
Subspace diagonalisation methods have appeared recently as promising means to access the ground state and some excited states of molecular Hamiltonians by classically diagonalising small matrices, whose elements can be efficiently obtained by a quantum computer. The recently proposed Variational Quantum Phase Estimation (VQPE) algorithm uses a basis of real time-evolved states, for which the energy eigenvalues can be obtained directly from the unitary matrix $U=e^{-iH{Delta}t}$, which can be computed with cost linear in the number of states used. In this paper, we report a circuit-based implementation of VQPE for arbitrary molecular systems and assess its performance and costs for the $H_2$, $H_3^+$ and $H_6$ molecules. We also propose using Variational Fast Forwarding (VFF) to decrease to quantum depth of time-evolution circuits for use in VQPE. We show that the approximation provides a good basis for Hamiltonian diagonalisation even when its fidelity to the true time evolved states is low. In the high fidelity case, we show that the approximate unitary U can be diagonalised instead, preserving the linear cost of exact VQPE.
Featured image: Controlled variationally fast forwarded compilation of the time evolution operator to be used in calculating the matrix elements of the sub space matrix to be diagonalised to calculate the spectrum of the Hamiltonian.
Popular summary
This work is based on the Variational Quantum Phase Estimation (VQPE) algorithm, which uses the time evolution operator to generate basis states, which have a series of mathematically convenient properties. Among these, the eigenfunctions can be computed from the matrix of the time-evolution operator itself, which has a linear number of distinct elements for a uniform time grid. Nevertheless, conventional approaches to expressing the time-evolution operator on a quantum device, such as Trotterised time-evolution, lead to intractably deep quantum circuits for chemistry Hamiltonians.
We combine this method with the Variational Fast Forwarding (VFF) approach, which generates a constant-circuit-dept approximation to the time evolution operator. We show that the method converges well even when the VFF approximation is not extremely accurate. When it is, it can take advantage of the same cost-reduction properties as the original VQPE algorithm, making the algorithm much more amenable to NISQ hardware.
► BibTeX data
► References
[1] John Preskill. “Quantum Computing in the NISQ era and beyond”. Quantum 2, 79 (2018).
https://doi.org/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”. Nat. Commun. 5, 4213 (2014).
https://doi.org/10.1038/ncomms5213
[3] P. J. J. O’Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, P. V. Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, and J. M. Martinis. “Scalable quantum simulation of molecular energies”. Phys. Rev. X 6, 031007 (2016).
https://doi.org/10.1103/PhysRevX.6.031007
[4] Cornelius Hempel, Christine Maier, Jonathan Romero, Jarrod McClean, Thomas Monz, Heng Shen, Petar Jurcevic, Ben P. Lanyon, Peter Love, Ryan Babbush, Alán Aspuru-Guzik, Rainer Blatt, and Christian F. Roos. “Quantum chemistry calculations on a trapped-ion quantum simulator”. Phys. Rev. X 8, 031022 (2018).
https://doi.org/10.1103/PhysRevX.8.031022
[5] Sam McArdle, Tyson Jones, Suguru Endo, Ying Li, Simon C. Benjamin, and Xiao Yuan. “Variational ansatz-based quantum simulation of imaginary time evolution”. npj Quantum Info. 5, 75 (2019).
https://doi.org/10.1038/s41534-019-0187-2
[6] Robert M. Parrish and Peter L. McMahon. “Quantum filter diagonalization: Quantum eigendecomposition without full quantum phase estimation” (2019). arXiv:1909.08925.
arXiv:1909.08925
[7] A Yu Kitaev. “Quantum measurements and the abelian stabilizer problem” (1995). arXiv:quant-ph/9511026.
arXiv:quant-ph/9511026
[8] Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, and Martin Head-Gordon. “Chemistry: Simulated quantum computation of molecular energies”. Science 309, 1704–1707 (2005).
https://doi.org/10.1126/science.1113479
[9] Katherine Klymko, Carlos Mejuto-Zaera, Stephen J. Cotton, Filip Wudarski, Miroslav Urbanek, Diptarka Hait, Martin Head-Gordon, K. Birgitta Whaley, Jonathan Moussa, Nathan Wiebe, Wibe A. de Jong, and Norm M. Tubman. “Real-time evolution for ultracompact hamiltonian eigenstates on quantum hardware”. PRX Quantum 3, 020323 (2022).
https://doi.org/10.1103/PRXQuantum.3.020323
[10] Jarrod R. McClean, Mollie E. Kimchi-Schwartz, Jonathan Carter, and Wibe A. de Jong. “Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states”. Phys. Rev. A 95, 042308 (2017).
https://doi.org/10.1103/PhysRevA.95.042308
[11] William J Huggins, Joonho Lee, Unpil Baek, Bryan O’Gorman, and K Birgitta Whaley. “A non-orthogonal variational quantum eigensolver”. New J. Phys. 22 (2020). arXiv:1909.09114.
https://doi.org/10.1088/1367-2630/ab867b
arXiv:1909.09114
[12] Mario Motta, Chong Sun, Adrian T. K. Tan, Matthew J. O’Rourke, Erika Ye, Austin J. Minnich, Fernando G. S. L. Brandão, and Garnet Kin-Lic Chan. “Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution”. Nat. Phys. 16, 231 (2020).
https://doi.org/10.1038/s41567-019-0704-4
[13] Nicholas H. Stair, Renke Huang, and Francesco A. Evangelista. “A multireference quantum krylov algorithm for strongly correlated electrons”. J. Chem. Theory Comput. 16, 2236–2245 (2020).
https://doi.org/10.1021/acs.jctc.9b01125
[14] Cristian L. Cortes and Stephen K. Gray. “Quantum krylov subspace algorithms for ground- and excited-state energy estimation”. Phys. Rev. A 105, 022417 (2022).
https://doi.org/10.1103/PhysRevA.105.022417
[15] G.H. Golub and C.F. Van Loan. “Matrix computations”. The North Oxford Academic paperback. North Oxford Academic. (1983).
https://doi.org/10.56021/9781421407944
[16] Cristina Cı̂rstoiu, Zoë Holmes, Joseph Iosue, Lukasz Cincio, Patrick J Coles, and Andrew Sornborger. “Variational fast forwarding for quantum simulation beyond the coherence time”. npj Quantum Inf. 6, 82 (2020).
https://doi.org/10.1038/s41534-020-00302-0
[17] Joe Gibbs, Kaitlin Gili, Zoë Holmes, Benjamin Commeau, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger. “Long-time simulations with high fidelity on quantum hardware” (2021). arXiv:2102.04313.
arXiv:2102.04313
[18] A. Krylov. “De la résolution numérique de l’équation servant à déterminer dans des questions de mécanique appliquée les fréquences de petites oscillations des systèmes matériels.”. Bull. Acad. Sci. URSS 1931, 491–539 (1931).
[19] P. Jordan and E. Wigner. “Über das Paulische Äquivalenzverbot”. Z. Phys. 47, 631–651 (1928).
https://doi.org/10.1007/BF01331938
[20] Sergey B. Bravyi and Alexei Yu Kitaev. “Fermionic Quantum Computation”. Ann. Phys. 298, 210–226 (2002).
https://doi.org/10.1006/aphy.2002.6254
[21] Alexander Cowtan, Silas Dilkes, Ross Duncan, Will Simmons, and Seyon Sivarajah. “Phase gadget synthesis for shallow circuits”. EPTCS 318, 213–228 (2020).
https://doi.org/10.4204/EPTCS.318.13
[22] Hans Hon Sang Chan, David Muñoz Ramo, and Nathan Fitzpatrick. “Simulating non-unitary dynamics using quantum signal processing with unitary block encoding” (2023). arXiv:2303.06161.
arXiv:2303.06161
[23] Bryan T. Gard, Linghua Zhu, George S. Barron, Nicholas J. Mayhall, Sophia E. Economou, and Edwin Barnes. “Efficient symmetry-preserving state preparation circuits for the variational quantum eigensolver algorithm”. npj Quantum Inf. 6, 10 (2020).
https://doi.org/10.1038/s41534-019-0240-1
[24] Kyle Poland, Kerstin Beer, and Tobias J. Osborne. “No free lunch for quantum machine learning” (2020).
[25] Qiskit contributors. “Qiskit: An open-source framework for quantum computing” (2023).
[26] Andrew Tranter, Cono Di Paola, David Zsolt Manrique, David Muñoz Ramo, Duncan Gowland, Evgeny Plekhanov, Gabriel Greene-Diniz, Georgia Christopoulou, Georgia Prokopiou, Harry Keen, Iakov Polyak, Irfan Khan, Jerzy Pilipczuk, Josh Kirsopp, Kentaro Yamamoto, Maria Tudorovskaya, Michal Krompiec, Michelle Sze, and Nathan Fitzpatrick. “InQuanto: Quantum Computational Chemistry” (2022). Version 2.
[27] D C Liu and J Nocedal. “On the limited memory bfgs method for large scale optimization”. Math. Programm. 45, 503–528 (1989).
https://doi.org/10.1007/BF01589116
[28] Kaoru Mizuta, Yuya O. Nakagawa, Kosuke Mitarai, and Keisuke Fujii. “Local variational quantum compilation of large-scale hamiltonian dynamics”. PRX Quantum 3, 040302 (2022). url: https://doi.org/10.1103/PRXQuantum.3.040302.
https://doi.org/10.1103/PRXQuantum.3.040302
[29] Norbert M. Linke, Dmitri Maslov, Martin Roetteler, Shantanu Debnath, Caroline Figgatt, Kevin A. Landsman, Kenneth Wright, and Christopher Monroe. “Experimental comparison of two quantum computing architectures”. PNAS 114, 3305–3310 (2017).
https://doi.org/10.1073/pnas.1618020114
[30] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. “Theory of trotter error with commutator scaling”. Phys. Rev. X 11, 011020 (2021).
https://doi.org/10.1103/PhysRevX.11.011020
[31] Yosi Atia and Dorit Aharonov. “Fast-forwarding of hamiltonians and exponentially precise measurements”. Nat. Commun. 8, 1572 (2017).
https://doi.org/10.1038/s41467-017-01637-7
[32] Kentaro Yamamoto, Samuel Duffield, Yuta Kikuchi, and David Muñoz Ramo. “Demonstrating bayesian quantum phase estimation with quantum error detection” (2023). arXiv:2306.16608.
https://doi.org/10.1103/PhysRevResearch.6.013221
arXiv:2306.16608
[33] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin. “Fast quantum gates for neutral atoms”. Phys. Rev. Lett. 85, 2208–2211 (2000).
https://doi.org/10.1103/PhysRevLett.85.2208
[34] Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. “Quantum computation by adiabatic evolution” (2000). arXiv:quant-ph/0001106.
arXiv:quant-ph/0001106
[35] Edward Farhi, Jeffrey Goldstone, Sam Gutmann, Joshua Lapan, Andrew Lundgren, and Daniel Preda. “A quantum adiabatic evolution algorithm applied to random instances of an np-complete problem”. Science 292, 472–475 (2001).
https://doi.org/10.1126/science.1057726
Cited by
[1] Francois Jamet, Connor Lenihan, Lachlan P. Lindoy, Abhishek Agarwal, Enrico Fontana, Baptiste Anselme Martin, and Ivan Rungger, “Anderson impurity solver integrating tensor network methods with quantum computing”, arXiv:2304.06587, (2023).
The above citations are from SAO/NASA ADS (last updated successfully 2024-03-24 12:04:32). 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 2024-03-24 12:04:31).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.
- SEO Powered Content & PR Distribution. Get Amplified Today.
- PlatoData.Network Vertical Generative Ai. Empower Yourself. Access Here.
- PlatoAiStream. Web3 Intelligence. Knowledge Amplified. Access Here.
- PlatoESG. Carbon, CleanTech, Energy, Environment, Solar, Waste Management. Access Here.
- PlatoHealth. Biotech and Clinical Trials Intelligence. Access Here.
- Source: https://quantum-journal.org/papers/q-2024-03-13-1278/
