Abstract
We consider the circuit complexity of free bosons and free fermions in 1+1 dimensions. Motivated by the results of [1, 2, 3] who found different behavior in the complexity of free bosons and fermions, in any dimension, we consider the 1+1 dimensional case where, thanks to the bosonisation equivalence of the Hilbert spaces, we can consider the same state from both the bosonic and the fermionic perspectives. This allows us to study the dependence of the complexity on the choice of the set of gates, which explains the discrepancy. We study the effect in two classes of states: i) bosonic-coherent / fermionic- gaussian states; ii) states that are both bosonic- and fermionic-gaussian. We consider the complexity relative to the ground state. In the first class, the different complexities can be related to each other by introducing a mode-dependent cost function in one of the descriptions. The differences in the second class are more important, in terms of the structure of UV divergencies and the overall behavior of the complexity.
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References
R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].
L. Hackl and R.C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [arXiv:1803.10638] [INSPIRE].
R. Khan, C. Krishnan and S. Sharma, Circuit complexity in fermionic field theory, Phys. Rev. D 98 (2018) 126001 [arXiv:1801.07620] [INSPIRE].
L. Susskind, Computational complexity and black hole horizons, Fortsch. Phys. 64 (2016) 24 [Addendum ibid. 64 (2016) 44] [arXiv:1403.5695] [INSPIRE].
D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
L. Susskind and Y. Zhao, Switchbacks and the bridge to nowhere, arXiv:1408.2823 [INSPIRE].
L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].
L. Susskind, Three lectures on complexity and black holes, arXiv:1810.11563 [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic complexity equals bulk action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part I, JHEP 06 (2018) 046 [arXiv:1804.07410] [INSPIRE].
S. Chapman, H. Marrochio and R.C. Myers, Holographic complexity in Vaidya spacetimes. Part II, JHEP 06 (2018) 114 [arXiv:1805.07262] [INSPIRE].
D. Carmi, R.C. Myers and P. Rath, Comments on holographic complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].
L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].
S. Chapman, D. Ge and G. Policastro, Holographic complexity for defects distinguishes action from volume, JHEP 05 (2019) 049 [arXiv:1811.12549] [INSPIRE].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Toward a definition of complexity for quantum field theory states, Phys. Rev. Lett. 120 (2018) 121602 [arXiv:1707.08582] [INSPIRE].
M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, quant-ph/0701004.
P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT, JHEP 11 (2017) 097 [arXiv:1706.07056] [INSPIRE].
A. Bhattacharyya, P. Caputa, S.R. Das, N. Kundu, M. Miyaji and T. Takayanagi, Path-integral complexity for perturbed CFTs, JHEP 07 (2018) 086 [arXiv:1804.01999] [INSPIRE].
D.A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 04 (2017) 121 [arXiv:1610.04903] [INSPIRE].
S. Chapman et al., Complexity and entanglement for thermofield double states, SciPost Phys. 6 (2019) 034 [arXiv:1810.05151] [INSPIRE].
A. Bhattacharyya, A. Shekar and A. Sinha, Circuit complexity in interacting QFTs and RG flows, JHEP 10 (2018) 140 [arXiv:1808.03105] [INSPIRE].
A. Belin, A. Lewkowycz and G. Sárosi, Complexity and the bulk volume, a New York time story, JHEP 03 (2019) 044 [arXiv:1811.03097] [INSPIRE].
R.-Q. Yang, Y.-S. An, C. Niu, C.-Y. Zhang and K.-Y. Kim, Principles and symmetries of complexity in quantum field theory, Eur. Phys. J. C 79 (2019) 109 [arXiv:1803.01797] [INSPIRE].
R.-Q. Yang, Y.-S. An, C. Niu, C.-Y. Zhang and K.-Y. Kim, More on complexity of operators in quantum field theory, JHEP 03 (2019) 161 [arXiv:1809.06678] [INSPIRE].
T. Ali, A. Bhattacharyya, S. Shajidul Haque, E.H. Kim and N. Moynihan, Post-quench evolution of distance and uncertainty in a topological system: complexity, entanglement and revivals, arXiv:1811.05985 [INSPIRE].
J. Watrous, Quantum computational complexity, arXiv:0804.3401.
R. Cleve, An introduction to quantum complexity theory, quant-ph/9906111.
J. Cotler, M.R. Mohammadi Mozaffar, A. Mollabashi and A. Naseh, Renormalization group circuits for weakly interacting continuum field theories, Fortsch. Phys. 67 (2019) 1900038 [arXiv:1806.02831] [INSPIRE].
J. von Delft and H. Schoeller, Bosonization for beginners: refermionization for experts, Annalen Phys. 7 (1998) 225 [cond-mat/9805275] [INSPIRE].
M. Guo, J. Hernandez, R.C. Myers and S.-M. Ruan, Circuit complexity for coherent states, JHEP 10 (2018) 011 [arXiv:1807.07677] [INSPIRE].
R.-Q. Yang, Complexity for quantum field theory states and applications to thermofield double states, Phys. Rev. D 97 (2018) 066004 [arXiv:1709.00921] [INSPIRE].
I. Bengtsson and K. Zyczkowski, Geometry of quantum states: an introduction to quantum entanglement, Cambridge University Press, New York, NY, U.S.A. (2006).
F.M. Paula, T.R. de Oliveira and M.S. Sarandy, Geometric quantum discord through the Schatten 1-norm, Phys. Rev. A 87 (2013) 064101.
A. Bernamonti, F. Galli, J. Hernandez, R.C. Myers, S.-M. Ruan and J. Simón, First law of holographic complexity, Phys. Rev. Lett. 123 (2019) 081601 [arXiv:1903.04511] [INSPIRE].
T. Giamarchi, Quantum physics in one dimension, Oxford University Press, Oxford, U.K. (2003).
A.O. Gogolin, A.A. Nersesian and A.M. Tsvelik, Bosonization and strongly correlated systems, Cambridge University Press, Cambridge, U.K. (2004) [INSPIRE].
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Ge, D., Policastro, G. Circuit complexity and 2D bosonisation. J. High Energ. Phys. 2019, 276 (2019). https://doi.org/10.1007/JHEP10(2019)276
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DOI: https://doi.org/10.1007/JHEP10(2019)276