arXiv subjects
Publications and source records attributed to M. H. Freedman.
Quantum information is about the entanglement of states. To this starting point we add parameters whereby a single state becomes a non-vanishing section of a bundle. We consider through examples the possible entanglement patterns of sections.
We construct families of cell complexes that generalize expander graphs. These families are called non-$k$-hyperfinite, generalizing the idea of a non-hyperfinite (NH) family of graphs. Roughly speaking, such a complex has the property that one cannot remove a small fraction of points and be left with an object that looks $k-1$-dimensional at large scales. We then consider certain quantum systems on these complexes. A future goal is to construct a family of Hamiltonians such that every low energy state has topological order as part of an attempt to prove the quantum PCP conjecture. This goal is approached by constructing a toric code Hamiltonian with the property that every low energy state without vertex defects has topological order, a property that would not hold for any local system in any lattice $Z^d$ or indeed on any 1-hyperfinite complex. Further, such NH complexes find application in quantum coding theory. The hypergraph product codes[1] of Tillich and Zémor are generalized using NH complexes.
We consider the adiabatic quantum algorithm for systems with "no sign problem", such as the transverse field Ising mode, and analyze the equilibration time for quantum Monte Carlo (QMC) on these systems. We ask: if the spectral gap is only inverse polynomially small, will equilibration methods based on slowly changing the Hamiltonian parameters in the QMC simulation succeed in a polynomial time? We show that this is not true, by constructing counter-examples. Some examples are Hamiltonians where the space of configurations where the wavefunction has non-negligible amplitude has a nontrivial fundamental group, causing the space of trajectories in imaginary time to break into disconnected components, with only negligible probability outside these components. For the simplest example we give with an abelian fundamental group, QMC does not equilibrate but still solves the optimization problem. More severe effects leading to failure to solve the optimization can occur when the fundamental group is a free group on two generators. Other examples where QMC fails have a trivial fundamental group, but still use ideas from topology relating group presentations to simplicial complexes. We define gadgets to realize these Hamiltonians as the effective low-energy dynamics of a transverse field Ising model. We present some analytic results on equilibration times which may be of some independent interest in the theory of equilibration of Markov chains. Conversely, we show that a small spectral gap implies slow equilibration at low temperature for some initial conditions and for a natural choice of local QMC updates.
In a topological quantum computer, universal quantum computation is performed by dragging quasiparticle excitations of certain two dimensional systems around each other to form braids of their world lines in 2+1 dimensional space-time. In this paper we show that any such quantum computation that can be done by braiding $n$ identical quasiparticles can also be done by moving a single quasiparticle around n-1 other identical quasiparticles whose positions remain fixed.