5 research outputs found
Matrix Product Representation of Locality Preserving Unitaries
The matrix product representation provides a useful formalism to study not
only entangled states, but also entangled operators in one dimension. In this
paper, we focus on unitary transformations and show that matrix product
operators that are unitary provides a necessary and sufficient representation
of 1D unitaries that preserve locality. That is, we show that matrix product
operators that are unitary are guaranteed to preserve locality by mapping local
operators to local operators while at the same time all locality preserving
unitaries can be represented in a matrix product way. Moreover, we show that
the matrix product representation gives a straight-forward way to extract the
GNVW index defined in Ref.\cite{Gross2012} for classifying 1D locality
preserving unitaries. The key to our discussion is a set of `fixed point'
conditions which characterize the form of the matrix product unitary operators
after blocking sites. Finally, we show that if the unitary condition is relaxed
and only required for certain system sizes, the matrix product operator
formalism allows more possibilities than locality preserving unitaries. In
particular, we give an example of a simple matrix product operator which is
unitary only for odd system sizes, does not preserve locality and carries a
`fractional' index as compared to their locality preserving counterparts.Comment: 14 page
Boson condensation and instability in the tensor network representation of string-net states
The tensor network representation of many-body quantum states, given by local
tensors, provides a promising numerical tool for the study of strongly
correlated topological phases in two dimension. However, tensor network
representations may be vulnerable to instabilities caused by small
perturbations of the local tensor, especially when the local tensor is not
injective. For example, the topological order in tensor network representations
of the toric code ground state has been shown to be unstable under certain
small variations of the local tensor, if these small variations do not obey a
local symmetry of the tensor. In this paper, we ask the questions of
whether other types of topological orders suffer from similar kinds of
instability and if so, what is the underlying physical mechanism and whether we
can protect the order by enforcing certain symmetries on the tensor. We answer
these questions by showing that the tensor network representation of all
string-net models are indeed unstable, but the matrix product operator (MPO)
symmetries of the local tensor can help to protect the order. We find that,
`stand-alone' variations that break the MPO symmetries lead to instability
because they induce the condensation of bosonic quasi-particles and destroy the
topological order in the system. Therefore, such variations must be forbidden
for the encoded topological order to be reliably extracted from the local
tensor. On the other hand, if a tensor network based variational algorithm is
used to simulate the phase transition due to boson condensation, then such
variation directions must be allowed in order to access the continuous phase
transition process correctly.Comment: 44 pages, 85 figures, comments welcom
Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains
Quantum error correction was invented to allow for fault-tolerant quantum computation. Systems with topological order turned out to give a natural physical realization of quantum error correcting codes (QECC) in their ground spaces. More recently, in the context of the anti-de Sitter/conformal field theory correspondence, it has been argued that eigenstates of CFTs with a holographic dual should also form QECCs. These two examples raise the question of how generally eigenstates of many-body models form quantum codes. In this Letter we establish new connections between quantum chaos and translation invariance in many-body spin systems, on one hand, and approximate quantum error correcting codes (AQECC), on the other hand. We first observe that quantum chaotic systems obeying the eigenstate thermalization hypothesis have eigenstates forming approximate quantum error-correcting codes. Then we show that AQECC can be obtained probabilistically from translation-invariant energy eigenstates of every translation-invariant spin chain, including integrable models. Applying this result to 1D classical systems, we describe a method for using local symmetries to construct parent Hamiltonians that embed these codes into the low-energy subspace of gapless 1D quantum spin chains. As explicit examples we obtain local AQECC in the ground space of the 1D ferromagnetic Heisenberg model and the Motzkin spin chain model with periodic boundary conditions, thereby yielding nonstabilizer codes in the ground space and low energy subspace of physically plausible 1D gapless models