We study the relationship between quantum chaos and pseudorandomness by developing probes of unitary design. A natural probe of randomness is the “frame poten-tial,” which is minimized by unitary k-designs and measures the 2-norm distance between the Haar random unitary ensemble and another ensemble. A natural probe of quantum chaos is out-of-time-order (OTO) four-point correlation functions. We show that the norm squared of a generalization of out-of-time-order 2k-point correlators is proportional to the kth frame potential, providing a quantitative connection between chaos and pseudorandomness. Additionally, we prove that these 2k-point correlators for Pauli operators completely determine the k-fold channel of an ensemble of unitary operators. Finally, we use a counting argument to obtain a lower bound on the quantum circuit complexity in terms of the frame potential. This provides a direct link between chaos, complexity, and randomness.
F. Dupuis, M. Berta, J. Wullschleger and R. Renner, One-shot decoupling, Comm. Math. Phys. 328 (2014) 251 [arXiv:1012.6044].
W. Brown and O. Fawzi, Decoupling with random quantum circuits, Comm. Math. Phys. 340 (2015) 867.
. Chamon, A. Hamma and E.R. Mucciolo, Emergent irreversibility and entanglement spectrum statistics, Phys. Rev. Lett. 112 (2014) 240501 [arXiv:1310.2702].
A. Larkin and Y. Ovchinnikov, Quasiclassical method in the theory of superconductivity, JETP 28 (1969) 1200.
A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, talk given at the Fundamental Physics Prize Symposium, November 10 (2014).
D.P. DiVincenzo, D.W. Leung and B.M. Terhal, Quantum data hiding, IEEE Trans. Inf. Theory 48 (2002) 580.
J. Emerson, E. Livine and S. Lloyd, Convergence conditions for random quantum circuits, Phys. Rev. A 72 (2005) 060302.
A. Ambainis and J. Emerson, Quantum t-designs: t-wise independence in the quantum world, in the proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity (CCC’07), June 13-16, Washington, U.S.A. (2007).
D. Gross, K. Audenaert and J. Eisert, Evenly distributed unitaries: on the structure of unitary designs, J. Math. Phys. 48 (2007) 052104 [quant-ph/0611002]
C. Dankert, R. Cleve, J. Emerson and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. A 80 (2009) 012304.
J. Emerson et al., Pseudo-random unitary operators for quantum information processing, Science 302 (2003) 2098.
W.G. Brown and L. Viola, Convergence rates for arbitrary statistical moments of random quantum circuits, Phys. Rev. Lett. 104 (2010) 250501.
A.W. Harrow and R.A. Low, Random quantum circuits are approximate 2-designs, Comm. Math. Phys. 291 (2009) 257 [arXiv:0802.1919].
E. Knill et al., Randomized benchmarking of quantum gates, Phys. Rev. A 77 (2008) 012307.
F.G.S.L. Brandao, A.W. Harrow and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, arXiv:1208.0692.
R. Kueng and D. Gross, Qubit stabilizer states are complex projective 3-designs, arXiv:1510.02767.
Z. Webb, The Clifford group forms a unitary 3-design, arXiv:1510.02769.
R.A. Low., Pseudo-randomness and learning in quantum computation, arXiv:1006.5227.
A.J. Scott, Optimizing quantum process tomography with unitary 2-designs, J. Phys. A 41 (2008) 055308.
H. Zhu, Multiqubit clifford groups are unitary 3-designs, arXiv:1510.02619.
B. Collins and I. Nechita, Random matrix techniques in quantum information theory, J. Math. Phys. 57 (2016) 015215 [arXiv:1509.04689].
A. Kitaev, Ph/CS 219C: quantum computation, course taught at Caltech, California, U.S.A. (2016).
Y. Gu, Moments of random matrices and weingarten functions, M.Sc. thesis, Queen’s University, Ontario, Canada (2013).
J. Watrous, Theory of quantum information, lecture notes (2015)
B. Collins, Moments and cumulants of polynomial random variables on unitarygroups, the Itzykson-Zuber integral, and free probability, Int. Math. Res. Not. 2003 (2003) 953.
A. Roy and A.J. Scott, Unitary designs and codes, Des. Codes Cryptogr. 53 (2009) 13.
A. Kitaev, A simple model of quantum holography, talks given at KITP, April 7 and May 27 Santa Barbara, U.S.A. (2015).
M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, quant-ph/0701004.
S. Chapman, H. Marrochio and R.C. Myers, Complexity of formation in holography, to appear.
J. Maldacena, Spacetime from entanglement, talk give at KITP, August 20, Santa Barbara, U.S.A. (2013).
A. Almheiri, X. Dong, and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041].
M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge, U.K. (2000).
M. Mozrzymas et al., Local random quantum circuits are approximate polynomial-designs: numerical results, J. Phys. A 46 (2013) 305301 [arXiv:1212.2556].
A. Brown, Wormholes and complexity, talk given at the Perimeter Institute for Theoretical Physics, August 21, Waterloo, Canada (2015).
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ArXiv ePrint: 1610.04903
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Roberts, D.A., Yoshida, B. Chaos and complexity by design. J. High Energ. Phys. 2017, 121 (2017). https://doi.org/10.1007/JHEP04(2017)121
- AdS-CFT Correspondence
- Gauge-gravity correspondence
- Random Systems
- Holography and condensed matter physics (AdS/CMT)