Abstract
Motivated by recent work showing that a quantum error correcting code can be generated by hybrid dynamics of unitaries and measurements, we study the long time behavior of such systems. We demonstrate that even in the ``mixed'' phase, a maximally mixed initial density matrix is purified on a time scale equal to the Hilbert space dimension (i.e., exponential in system size), albeit with noisy dynamics at intermediate times which we connect to Dyson Brownian motion. In contrast, we show that free fermion systems—i.e., ones where the unitaries are generated by quadratic Hamiltonians and the measurements are of fermion bilinears—purify in a time quadratic in the system size. In particular, a volume law phase for the entanglement entropy cannot be sustained in a free fermion system.
Highlights
It has been argued that a low-dimensional quantum system which mixes unitary evolution by local circuits with local measurements can act as a quantum memory [1,2,3,4,5,6,7]
We persist in using the term many-body; in particular, one may hope that sufficiently deep quantum circuits for a tensor product Hilbert space can be well-approximated by our Haar random measurements [8,9,10]
In the free fermion case, the Hilbert space is a Fock space of fermions, and measurements are only allowed to be of fermion bilinears
Summary
The terms of order δ3 are bounded by O(poly(log(N ))/N 3/2) with probability 1 − 1/N α for any constant α and so we may neglect them when estimating to order 1/N. We formalize this into the following ‘non-perturbative’. For a nearly pure state, tr ρ2 = 1 − , we show in Eq (59) in Appendix B that the noise is upper bounded by 4 2/N
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