Quantum Computation as Geometry
Open accessAbstract
Quantum computers hold great promise for solving interesting computational problems, but it remains a challenge to find efficient quantum circuits that can perform these complicated tasks. Here we show that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry. By recasting the problem of finding quantum circuits as a geometric problem, we open up the possibility of using the mathematical techniques of Riemannian geometry to suggest new quantum algorithms or to prove limitations on the power of quantum computers.
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A Riemannian metric on M can be introduced that penalizes the forbidden subspace B, so that as the penalty grows, length minimization forces the Hamiltonian to remain in A.
“Nevertheless, it is possible to derive the geodesic equation for these shortest-distance curves by using a clever trick [35,39]: one introduces a Riemannian metric on M that applies a penalty to the forbidden subspace B, so that in the limit when the penalty is large, minimizing the path length also forces HðtÞ to stay within A.”
From Quantum Brachistochrone Curves as Geodesics: Obtaining Accurate Minimum-Time Protocols for the Control of Quantum Systems (2015) · cites this paper as [35]
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<i>Quantum Computation and Quantum Information</i>
Nielsen, M. A., Chuang, I., & Grover, L. K. (2002) · American Journal of Physics · doi:10.1119/1.1463744 · Cited by 21,895 · see citing works
Self-citationQuantum Computation and Quantum Information: 10th Anniversary Edition
Nielsen, M. A., & Chuang, I. L. (2012) · doi:10.1017/cbo9780511976667 · Cited by 8,436 · see citing works
Self-citationSakurai, J. J., & Napolitano, J. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108587280 · Cited by 3,672 · see citing works
Sakurai, J. J., & Napolitano, J. (2017) · Cambridge University Press eBooks · doi:10.1017/9781108499996 · Cited by 1,653 · see citing works
Jurdjevic, V. (1996) · Cambridge University Press eBooks · doi:10.1017/cbo9780511530036 · Cited by 1,155 · see citing works
Topological Methods in Hydrodynamics
Arnold, V. I., & Khesin, B. (1998) · Applied mathematical sciences · doi:10.1007/b97593 · Cited by 1,115 · see citing works
Topological Methods in Hydrodynamics
Arnold, V. I., & Khesin, B. (1992) · Annual Review of Fluid Mechanics · doi:10.1146/annurev.fl.24.010192.001045 · Cited by 1,063 · see citing works
A Tour of Subriemannian Geometries, Their Geodesics and Applications
Montgomery, R. (2006) · Mathematical surveys and monographs · doi:10.1090/surv/091 · Cited by 947 · see citing works
A Panoramic View of Riemannian Geometry
Berger, M. (2003) · Springer eBooks · doi:10.1007/978-3-642-18245-7 · Cited by 781 · see citing works
Time optimal control in spin systems
Khaneja, N., Brockett, R. W., & Glaser, S. J. (2001) · Physical Review A · doi:10.1103/physreva.63.032308 · Cited by 755 · see citing works
Khaneja, N., Glaser, S. J., & Brockett, R. W. (2002) · Physical Review A · doi:10.1103/physreva.65.032301 · Cited by 245 · see citing works
Cartan decomposition of SU(2n) and control of spin systems
Khaneja, N., & Glaser, S. J. (2001) · Chemical Physics · doi:10.1016/s0301-0104(01)00318-4 · Cited by 232 · see citing works
A geometric approach to quantum circuit lower boundsPreprint
Nielsen, M. A. (2005) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0502070 · Cited by 49 · see citing works
Self-citationKhaneja, N., & Glaser, S. J. (2000) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0010100 · Cited by 31 · see citing works
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Research works printed but not linked: 4
- [1]1. P. W. Shor, Proceedings, 35th Annual Symposium on Fundamentals of Computer Science (IEEE Press, Los Alamitos, 1994).
- [4]4. N. Khaneja, R. Brockett, S. J. Glaser, Phys. Rev. A 63 , 032308 (2001). arXiv:quantph/0006114.
- [6]6. N. Khaneja, S. J. Glaser, R. Brockett, Phys. Rev. A 65 , 032301 (2002). arXiv:quantph/0106099.
- [11]11. J. Milnor, Morse theory (Princeton University Press, 1969).
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Faster than Hermitian quantum mechanics
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Geng, J., Wu, Y., Wang, X., Xu, K., Shi, F., Xie, Y., Rong, X., & Du, J. (2016) · Physical Review Letters · doi:10.1103/physrevlett.117.170501 · Cited by 68 · see citing works
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- Authors
- Nielsen, M. A., Dowling, M. R., Gu, M., & Doherty, A. C.
- Affiliations
- The University of Queensland (AU)
- Corresponding author
- Nielsen, Michael A.
- Venue
- Science
- Published
- 2006-02-24
- Volume
- 311
- Issue
- 5764
- Pages
- 1133-1135
- ISSN
- 0036-8075 (ISSN-L)
- 1095-9203
- OpenAlex
- W1967883933
- Cited by
- 539
- References
- 14
- Citations by year
- 2026: 38 · 2025: 60 · 2024: 65 · 2023: 53 · 2022: 56 · 2021: 64 · 2020: 49 · 2019: 41 · 2018: 20 · 2017: 14 · 2016: 14 · 2015: 5 · 2014: 5 · 2013: 8 · 2012: 4
- FWCI
- 9.89
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Science (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- quantum circuit optimization, quantum computing, Riemannian geometry, curved geometries, shortest path, quantum algorithms
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