Geometry of quantum complexity
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Geometry of quantum complexity. / Auzzi, Roberto; Baiguera, Stefano; De Luca, G. Bruno; Legramandi, Andrea; Nardelli, Giuseppe; Zenoni, Nicolo.
I: Physical Review D, Bind 103, Nr. 10, 106021, 19.05.2021.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Geometry of quantum complexity
AU - Auzzi, Roberto
AU - Baiguera, Stefano
AU - De Luca, G. Bruno
AU - Legramandi, Andrea
AU - Nardelli, Giuseppe
AU - Zenoni, Nicolo
PY - 2021/5/19
Y1 - 2021/5/19
N2 - Computational complexity is a quantum information concept that recently has found applications in the holographic understanding of the black hole interior. We consider quantum computational complexity for n qubits using Nielsen's geometrical approach. In the definition of complexity there is a big amount of arbitrariness due to the choice of the penalty factors, which parametrizes the cost of the elementary computational gates. In order to reproduce desired features in holography, such as ergodicity and exponential maximal complexity for large number of qubits n, negative curvatures are required. With the simplest choice of penalties, this is achieved at the price of singular sectional curvatures in the large n limit. We investigate a choice of penalties in which we can obtain negative curvatures in a smooth way. We also analyze the relation between operator and state complexities, framing the discussion with the language of Riemannian submersions. This provides a direct relation between geodesics and curvatures in the unitaries and the states spaces, which we also exploit to give a closed-form expression for the metric on the states in terms of the one for the operators. Finally, we study conjugate points for a large number of qubits in the unitary space and we provide a strong indication that maximal complexity scales exponentially with the number of qubits in a certain regime of the penalties space.
AB - Computational complexity is a quantum information concept that recently has found applications in the holographic understanding of the black hole interior. We consider quantum computational complexity for n qubits using Nielsen's geometrical approach. In the definition of complexity there is a big amount of arbitrariness due to the choice of the penalty factors, which parametrizes the cost of the elementary computational gates. In order to reproduce desired features in holography, such as ergodicity and exponential maximal complexity for large number of qubits n, negative curvatures are required. With the simplest choice of penalties, this is achieved at the price of singular sectional curvatures in the large n limit. We investigate a choice of penalties in which we can obtain negative curvatures in a smooth way. We also analyze the relation between operator and state complexities, framing the discussion with the language of Riemannian submersions. This provides a direct relation between geodesics and curvatures in the unitaries and the states spaces, which we also exploit to give a closed-form expression for the metric on the states in terms of the one for the operators. Finally, we study conjugate points for a large number of qubits in the unitary space and we provide a strong indication that maximal complexity scales exponentially with the number of qubits in a certain regime of the penalties space.
U2 - 10.1103/PhysRevD.103.106021
DO - 10.1103/PhysRevD.103.106021
M3 - Journal article
VL - 103
JO - Physical Review D
JF - Physical Review D
SN - 2470-0010
IS - 10
M1 - 106021
ER -
ID: 272651129