304 research outputs found
Fourier transform of fermionic systems and the spectral tensor network
Leveraging the decomposability of the fast Fourier transform, I propose a new
class of tensor network that is efficiently contractible and able to represent
many-body systems with local entanglement that is greater than the area law.
Translationally invariant systems of free fermions in arbitrary dimensions as
well as 1D systems solved by the Jordan-Wigner transformation are shown to be
exactly represented in this class. Further, it is proposed that these tensor
networks be used as generic structures to variationally describe more
complicated systems, such as interacting fermions. This class shares some
similarities with Evenbly & Vidal's branching MERA, but with some important
differences and greatly reduced computational demands.Comment: Accepted in Phys. Rev. Lett. 9 pages, 13 figures. This version is
reorganized to be more pedagogical and includes a new derivation of the FFT
decomposition, as well as extra details on the contraction scheme in the
Appendi
Multimode analysis of non-classical correlations in double well Bose-Einstein condensates
The observation of non-classical correlations arising in interacting two to
size weakly coupled Bose-Einstein condensates was recently reported by Esteve
et al. [Nature 455, 1216 (2008)]. In order to observe fluctuations below the
standard quantum limit, they utilized adiabatic passage to reduce the thermal
noise to below that of thermal equilibrium at the minimum realizable
temperature. We present a theoretical analysis that takes into account the
spatial degrees of freedom of the system, allowing us to calculate the expected
correlations at finite temperature in the system, and to verify the hypothesis
of adiabatic passage by comparing the dynamics to the idealized model.Comment: 12 pages, 7 figure
Variational Monte Carlo with the Multi-Scale Entanglement Renormalization Ansatz
Monte Carlo sampling techniques have been proposed as a strategy to reduce
the computational cost of contractions in tensor network approaches to solving
many-body systems. Here we put forward a variational Monte Carlo approach for
the multi-scale entanglement renormalization ansatz (MERA), which is a unitary
tensor network. Two major adjustments are required compared to previous
proposals with non-unitary tensor networks. First, instead of sampling over
configurations of the original lattice, made of L sites, we sample over
configurations of an effective lattice, which is made of just log(L) sites.
Second, the optimization of unitary tensors must account for their unitary
character while being robust to statistical noise, which we accomplish with a
modified steepest descent method within the set of unitary tensors. We
demonstrate the performance of the variational Monte Carlo MERA approach in the
relatively simple context of a finite quantum spin chain at criticality, and
discuss future, more challenging applications, including two dimensional
systems.Comment: 11 pages, 12 figures, a variety of minor clarifications and
correction
Tensor Networks and Quantum Error Correction
We establish several relations between quantum error correction (QEC) and
tensor network (TN) methods of quantum many-body physics. We exhibit
correspondences between well-known families of QEC codes and TNs, and
demonstrate a formal equivalence between decoding a QEC code and contracting a
TN. We build on this equivalence to propose a new family of quantum codes and
decoding algorithms that generalize and improve upon quantum polar codes and
successive cancellation decoding in a natural way.Comment: Accepted in Phys. Rev. Lett. 8 pages, 9 figure
Perfect Sampling with Unitary Tensor Networks
Tensor network states are powerful variational ans\"atze for many-body ground
states of quantum lattice models. The use of Monte Carlo sampling techniques in
tensor network approaches significantly reduces the cost of tensor
contractions, potentially leading to a substantial increase in computational
efficiency. Previous proposals are based on a Markov chain Monte Carlo scheme
generated by locally updating configurations and, as such, must deal with
equilibration and autocorrelation times, which result in a reduction of
efficiency. Here we propose a perfect sampling scheme, with vanishing
equilibration and autocorrelation times, for unitary tensor networks -- namely
tensor networks based on efficiently contractible, unitary quantum circuits,
such as unitary versions of the matrix product state (MPS) and tree tensor
network (TTN), and the multi-scale entanglement renormalization ansatz (MERA).
Configurations are directly sampled according to their probabilities in the
wavefunction, without resorting to a Markov chain process. We also describe a
partial sampling scheme that can result in a dramatic (basis-dependent)
reduction of sampling error.Comment: 11 pages, 9 figures, renamed partial sampling to incomplete sampling
for clarity, extra references, plus a variety of minor change
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