Complexity of operators generated by quantum mechanical Hamiltonians
Abstract
We propose how to compute the complexity of operators generated by Hamiltonians in quantum field theory (QFT) and quantum mechanics (QM). The Hamiltonians in QFT/QM and quantum circuit have a few essential differences, for which we introduce new principles and methods for complexity. We show that the complexity geometry corresponding to one-dimensional quadratic Hamiltonians is equivalent to AdS3 spacetime. Here, the requirement that the complexity is nonnegative corresponds to the fact that the Hamiltonian is lower bounded and the speed of a particle is not superluminal. Our proposal proves the complexity of the operator generated by a free Hamiltonian is zero, as expected. By studying a non-relativistic particle in compact Riemannian manifolds we find the complexity is given by the global geometric property of the space. In particular, we show that in low energy limit the critical spacetime dimension to ensure the `nonnegative' complexity is the 3+1 dimension.
- Publication:
-
Journal of High Energy Physics
- Pub Date:
- March 2019
- DOI:
- 10.1007/JHEP03(2019)010
- arXiv:
- arXiv:1810.09405
- Bibcode:
- 2019JHEP...03..010Y
- Keywords:
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- Gauge-gravity correspondence;
- Holography and condensed matter physics (AdS/CMT);
- High Energy Physics - Theory;
- Quantum Physics
- E-Print:
- A revised version and gives more detailed examples on why "complexity'' in quantum mechanics and quantum field theory should be bi-invariant