Quantum Adiabatic Perturbation Theory Overview
Explore Quantum Adiabatic Perturbation Theory in this comprehensive overview by Nicholas Cariello, covering motivation, adiabatic analysis, implementation on a quantum computer, and conclusions. Supported by the U.S. Department of Energy and facilitated by the Facility for Rare Isotope Beams.
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Quantum Adiabatic Perturbation Theory Nicholas Cariello Graduate Research Assistant cariello@frib.msu.edu 30 May 2025 This material is based upon work supported by the U.S. Department of Energy under Award Number DE-SC0023658, as well as being supported by the Office of Science, Office of Nuclear Physics and used resources of the Facility for Rare Isotope Beams (FRIB) Operations, which is a DOE Office of Science User Facility under Award Number DE-SC0023633.
Overview Motivation Adiabatic Analysis Implementation on a Quantum Computer Conclusion Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 2
Motivation Elhatisari, S., Bovermann, L., Ma, YZ. et al. Wavefunction matching for solving quantum many-body problems. Nature630, 59 63 (2024). Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 3
Motivation Rodeo Algorithm Quantum Adiabatic Perturbation Theory Wavefunction Matching Prepare eigenstate of ?0 ? = ?0+ ??1 ?????????? ? Elhatisari, S., Bovermann, L., Ma, YZ. et al. Wavefunction matching for solving quantum many-body problems. Nature630, 59 63 (2024). Choi, Kenneth, et al. "Rodeo algorithm for quantum computing." Physical Review Letters 127.4 (2021): 040505. Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 4
Adiabatic Theorem A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian s spectrum. - Born and Fock (1928) ? ? = ?0+ ???? ?1 Reduced Time ? =? Interpolation Function k - smoothness ? M. Born and V. A. Fock (1928). "Beweis des Adiabatensatzes". Zeitschrift f r Physik A. 51 (3 4): 165 180. Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 5
Adiabatic Approximation ? 1 = ?0+ ??1 ? ? = ?0+ ???? ?1 ? 0 = ?0 ???? = ??????|??? Instantaneous Eigensystem: ?? ? ? ? ? ? ? = ? ? |? ? ?? ??(? ) ??? = 0 M. Born and V. A. Fock (1928). "Beweis des Adiabatensatzes". Zeitschrift f r Physik A. 51 (3 4): 165 180. Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 6
Perturbative Expansion of the Adiabatic Term ???? = ??????|??? ?? ? ? ?? ??(? ) ??? = 0 Looking at the ?? -order perturbative term at the end of the evolution: ? 1 ?? ? ??? ??????1 1 (0) ??? ? ?(1) + ? = e ???? ? 1 ?? ?=0?? 1 ? ? ! ?=0 Lower order terms that diverge in ? ?? -order term of interest Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 7
Non-Adiabatic Term ???? ??????? ?? = ? + ? Probability amplitudes that the final system transitioned into a different state: ? ??? ? |??? ??? ??(?) ??? 1 ??? (??? ??(? )) ??? 0 ?? ?1 = ?? 0 1 Successive integration by parts yields powers of ? in the denominator ?????? ? ? ? ?2+? ? Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 8
Implementation Dyson Series ? 1 = ?0+ ??1 ? ? = ?0+ ???? ?1 ? 0 = ?0 Solve the time-dependent Schr dinger equation perturbatively: 1 1 ?2 (0) ??1? ?1 ?2?2 ?? = 1 ??? ??1??2? ?2? ?1 + |?? 0 0 0 ? ? = ?+???0???? ?1? ???0? Raimes, S. (1972) Many-Electron Theory. North-Holland, Amsterdam. Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 9
Circuit Schematic Find the first-order correction to the measurement of some operator ? 1 0 Namely, 2 ?? ? ?? Do this by creating a superposition of the 0th order and 1st order Dyson series terms. Measuring ?? on the ancilla removes divergences. This measurement yields the imaginary part of a 0th order transition amplitude that we know must be fully real. Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 10
General Circuit Schematic j controlled insertions i anti-controlled insertions ? ? Finds the transition element 2 ?? ? ?? The controlled and anti-controlled ?1 insertions can happen in any order Need to subtract out divergences related to lower-order energy corrections in post-processing Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 11
Conclusions Method to find perturbative corrections to a wavefunction using a quantum computer Able to recursively find higher order corrections When run in tandem with other methods, this can be a powerful tool to solve large, difficult to simulate Hamiltonians Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 12
Acknowledgements Advisors Dean Lee Morten Hjorth-Jensen Group Members Danny Jammooa Patrick Cook Gabriel Given This material is based upon work supported by the U.S. Department of Energy under Award Number DE-SC0023658, as well as being supported by the Office of Science, Office of Nuclear Physics and used resources of the Facility for Rare Isotope Beams (FRIB) Operations, which is a DOE Office of Science User Facility under Award Number DE-SC0023633. Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 13
Thank You Facility for Rare Isotope Beams U.S. Department of Energy Office of Science | Michigan State University 640 South Shaw Lane East Lansing, MI 48824, USA frib.msu.edu N. Cariello, 30 May 2025, Slide 14
