Built on

M. Berger, “Variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive”, Ann. Scuola Norm. Sup. Pisa, 15, 179 (1961)
1961
Earlier work this paper cites.
John Milnor, “Curvatures of Left Invariant Metrics on Lie Groups”, Advances in Mathematics 21, 293-329 (1976)
1976
Earlier work this paper cites.
T. Sakai, “Cut loci of Berger’s spheres”, Hokkaido Mathematical Journal Vol. 10, p. 143-155 (1981)
1981
Earlier work this paper cites.
K. D. Mulmuley and M. Sohoni. Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems. SIAM J. Comput. 31(2), 496-526 (2001)
2001
Earlier work this paper cites.
Navin Khaneja, Roger Brockett, Steffen Glaser, “Time Optimal Control in Spin Systems” Phys. Rev. A 63
2001
Earlier work this paper cites.
A. Y. Kitaev, “Quantum computations: algorithms and error correction”, Russ. Math. Surv., 52 (6) 1191-1249 (1997); C. M. Dawson & M. A. Nielsen, “The Solovay-Kitaev algorithm”, arXiv:quantph/0505030 (2005)
2005
Earlier work this paper cites.
M. A. Nielsen, M. Dowling, M. Gu, and A. C. Doherty, “Quantum Computation as Geometry”, Science 311, 1133 (2006), arXiv:quant-ph/0603161
2006
Earlier work this paper cites.
M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, “Optimal control, geometry, and quantum computing”, Phys. Rev. A 73, 062323 (2006), arXiv:quant-ph/0603160
2006
Earlier work this paper cites.
U. Boscain & F. Rossi, “Invariant Carnot-Caratheodory metrics on S 3 S^{3} , SO(3), SL(2) and lens spaces”, SIAM J. Control Optim. 47, no. 4, 1851-1878 (2008)
2008
Earlier work this paper cites.
Bin Li, Zu-Huan Yu, Shao-Ming Fei, “Geometry of Quantum Computation with Qutrits”, Scientific Reports 3
Original
2013
Earlier work this paper cites.

Similar

Then

S. Chapman, M. P. Heller, H. Marrochio and F. Pastawski, “Toward a Definition of Complexity for Quantum Field Theory States,” Phys. Rev. Lett. 120
Original
2018
Later among the works it cites.
W. Cottrell and M. Montero, “Complexity is simple!,” JHEP 1802
Original
2018
Later among the works it cites.
M. Guo, J. Hernandez, R. C. Myers and S. M. Ruan, “Circuit Complexity for Coherent States,” JHEP 1810
Original
2018
Later among the works it cites.
L. Hackl and R. C. Myers, “Circuit complexity for free fermions,” JHEP 1807
Original
2018
Later among the works it cites.
R. Q. Yang, “Complexity for quantum field theory states and applications to thermofield double states,” Phys. Rev. D 97

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2019

The Complexity Geometry of a Single Qubit

Brown, Adam R., Susskind, Leonard

Understand

The computational complexity of a quantum state quantifies how hard it is to make.

`Complexity geometry', first proposed by Nielsen, is an approach to defining computational complexity using the tools of differential geometry.
Here we demonstrate many of the attractive features of complexity geometry using the example of a single qubit, which turns out to be rich enough to be illustrative but simple enough to be illuminating.

Built on

M. Berger, “Variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive”, Ann. Scuola Norm. Sup. Pisa, 15, 179 (1961)
1961
Earlier work this paper cites.
John Milnor, “Curvatures of Left Invariant Metrics on Lie Groups”, Advances in Mathematics 21, 293-329 (1976)
1976
Earlier work this paper cites.
T. Sakai, “Cut loci of Berger’s spheres”, Hokkaido Mathematical Journal Vol. 10, p. 143-155 (1981)
1981
Earlier work this paper cites.
K. D. Mulmuley and M. Sohoni. Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems. SIAM J. Comput. 31(2), 496-526 (2001)
2001
Earlier work this paper cites.
Navin Khaneja, Roger Brockett, Steffen Glaser, “Time Optimal Control in Spin Systems” Phys. Rev. A 63
2001
Earlier work this paper cites.
A. Y. Kitaev, “Quantum computations: algorithms and error correction”, Russ. Math. Surv., 52 (6) 1191-1249 (1997); C. M. Dawson & M. A. Nielsen, “The Solovay-Kitaev algorithm”, arXiv:quantph/0505030 (2005)
2005
Earlier work this paper cites.
M. A. Nielsen, M. Dowling, M. Gu, and A. C. Doherty, “Quantum Computation as Geometry”, Science 311, 1133 (2006), arXiv:quant-ph/0603161
2006
Earlier work this paper cites.
M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, “Optimal control, geometry, and quantum computing”, Phys. Rev. A 73, 062323 (2006), arXiv:quant-ph/0603160
2006
Earlier work this paper cites.
U. Boscain & F. Rossi, “Invariant Carnot-Caratheodory metrics on S 3 S^{3} , SO(3), SL(2) and lens spaces”, SIAM J. Control Optim. 47, no. 4, 1851-1878 (2008)
2008
Earlier work this paper cites.
Bin Li, Zu-Huan Yu, Shao-Ming Fei, “Geometry of Quantum Computation with Qutrits”, Scientific Reports 3
Original
2013
Earlier work this paper cites.

Similar

Then

S. Chapman, M. P. Heller, H. Marrochio and F. Pastawski, “Toward a Definition of Complexity for Quantum Field Theory States,” Phys. Rev. Lett. 120
Original
2018
Later among the works it cites.
W. Cottrell and M. Montero, “Complexity is simple!,” JHEP 1802
Original
2018
Later among the works it cites.
M. Guo, J. Hernandez, R. C. Myers and S. M. Ruan, “Circuit Complexity for Coherent States,” JHEP 1810
Original
2018
Later among the works it cites.
L. Hackl and R. C. Myers, “Circuit complexity for free fermions,” JHEP 1807
Original
2018
Later among the works it cites.
R. Q. Yang, “Complexity for quantum field theory states and applications to thermofield double states,” Phys. Rev. D 97

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao, “Complexity, action, and black holes,” Phys. Rev. D 93

Original

2016

Cited alongside, same era.

A. R. Brown, L. Susskind and Y. Zhao, “Quantum Complexity and Negative Curvature,” Phys. Rev. D 95

Original

2017

Cited alongside, same era.

R. Jefferson and R. C. Myers, “Circuit complexity in quantum field theory,” JHEP 1710

Original

2017

Cited alongside, same era.

A. R. Brown and L. Susskind, “Second law of quantum complexity,” Phys. Rev. D 97

Original

2018

Cited alongside, same era.

A. V. Podobryaev, “Diameter of the Berger Sphere”, Math Notes 103: 846 (2018)

2018

Cited alongside, same era.

Michael A. Nielsen, “A geometric approach to quantum circuit lower bounds,” arXiv:quant-ph/0502070

Cited in the paper.

Mark R. Dowling and Michael A. Nielsen, “The geometry of quantum computation” arXiv:quant-ph/0701004

Cited in the paper.

Mile Gu, Andrew Doherty & Michael Nielsen “Quantum control via geometry: An explicit example”, Physical Review A, 78

Original

Cited in the paper.

Original

2018

Later among the works it cites.

A. Bhattacharyya, A. Shekar and A. Sinha, “Circuit complexity in interacting QFTs and RG flows,” JHEP 1810

Original

2018

Later among the works it cites.

R. Khan, C. Krishnan and S. Sharma, “Circuit Complexity in Fermionic Field Theory,” Phys. Rev. D 98

Original

2018

Later among the works it cites.

J. Diamond, “Guns, Germs, and Steel”, Norton (1997); J. B. Pavlik & A. T. Young, “Did Technology Transfer More Rapidly East-West than North-South?”, ssrn:3141510 (2019)

2019

Closest in time.

H. W. Lin, “Cayley graphs and complexity geometry,” JHEP 1902

Original

2019

Closest in time.