7,724 research outputs found
Circuit Complexity and 2D Bosonisation
We consider the circuit complexity of free bosons, or equivalently free
fermions, in 1+1 dimensions. Motivated by the results of [1] and [2, 3] who
found different behavior in the complexity of free bosons and fermions, in any
dimension, we consider the 1+1 dimensional case where, thanks to the
bosonisation equivalence, we can consider the same state from both the bosonic
and the fermionic perspectives. In this way the discrepancy can be attributed
to a different choice of the set of gates allowed in the circuit. We study the
effect in two classes of states: i) bosonic-coherent / fermionic-gaussian
states; ii) states that are both bosonic- and fermionic-gaussian. We consider
the complexity relative to the ground state. In the first class, the different
results can be reconciled admitting a mode-dependent cost function in one of
the descriptions. The differences in the second class are more important, in
terms of the cutoff-dependence and the overall behavior of the complexity.Comment: Fix typos and add reference
Boolean Circuit Complexity of Regular Languages
In this paper we define a new descriptional complexity measure for
Deterministic Finite Automata, BC-complexity, as an alternative to the state
complexity. We prove that for two DFAs with the same number of states
BC-complexity can differ exponentially. In some cases minimization of DFA can
lead to an exponential increase in BC-complexity, on the other hand
BC-complexity of DFAs with a large state space which are obtained by some
standard constructions (determinization of NFA, language operations), is
reasonably small. But our main result is the analogue of the "Shannon effect"
for finite automata: almost all DFAs with a fixed number of states have
BC-complexity that is close to the maximum.Comment: In Proceedings AFL 2014, arXiv:1405.527
Circuit complexity in interacting QFTs and RG flows
We consider circuit complexity in certain interacting scalar quantum field
theories, mainly focusing on the theory. We work out the circuit
complexity for evolving from a nearly Gaussian unentangled reference state to
the entangled ground state of the theory. Our approach uses Nielsen's geometric
method, which translates into working out the geodesic equation arising from a
certain cost functional. We present a general method, making use of integral
transforms, to do the required lattice sums analytically and give explicit
expressions for the cases. Our method enables a study of circuit
complexity in the epsilon expansion for the Wilson-Fisher fixed point. We find
that with increasing dimensionality the circuit depth increases in the presence
of the interaction eventually causing the perturbative calculation to
breakdown. We discuss how circuit complexity relates with the renormalization
group.Comment: 50 pages, 2 figures; references updated; version to appear in JHE
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