Kavli Affiliate: Matthew P. A. Fisher
| First 5 Authors: Yaodong Li, Romain Vasseur, Matthew P. A. Fisher, Andreas W. W. Ludwig,
| Summary:
We introduce an exact mapping of Clifford (stabilizer) random tensor networks
(RTNs) and monitored quantum circuits, onto a statistical mechanics model. With
Haar unitaries, the fundamental degrees of freedom (‘spins’) are permutations
because all operators commuting with the action of the unitaries on a tensor
product arise from permutations of the tensor factors (‘Schur-Weyl duality’).
For unitaries restricted to the smaller Clifford group, the set of commuting
operators, the ‘commutant’, forming the new ‘spin’ degrees of freedom, will be
larger. We use the recent full characterization of this commutant by Gross et
al., Comm. Math. Phys. 385, 1325 (2021), to construct the Clifford statistical
mechanics models for on-site Hilbert space dimensions which are powers of a
prime number $p$. We show that the Boltzmann weights are invariant under a
symmetry group involving orthogonal matrices with entries in the finite number
field ${bf F}_p$. This implies that the symmetry group, and consequently all
universal properties of entanglement transitions in Clifford circuits and RTNs
will in general depend on, and only on the prime $p$. We show that Clifford
monitored circuits with on-site Hilbert space dimension $d=p^M$ are described
by percolation in the limits $d to infty$ at (a) $p=$ fixed but $Mto
infty$, and at (b) $M= 1$ but $p to infty$. In the limit (a) we calculate
the effective central charge, and in the limit (b) we derive the following
universal minimal cut entanglement entropy $S_A =(sqrt{3}/pi)ln p ln L_A$
for $d=p$ large at the transition. We verify those predictions numerically, and
present extensive numerical results for critical exponents at the transition in
monitored Clifford circuits for prime number on-site Hilbert space dimension
$d=p$ for a variety of different values of $p$, and find that they approach
percolation values at large $p$.
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