27 research outputs found
Tensor Network Methods for Invariant Theory
Invariant theory is concerned with functions that do not change under the
action of a given group. Here we communicate an approach based on tensor
networks to represent polynomial local unitary invariants of quantum states.
This graphical approach provides an alternative to the polynomial equations
that describe invariants, which often contain a large number of terms with
coefficients raised to high powers. This approach also enables one to use known
methods from tensor network theory (such as the matrix product state
factorization) when studying polynomial invariants. As our main example, we
consider invariants of matrix product states. We generate a family of tensor
contractions resulting in a complete set of local unitary invariants that can
be used to express the R\'enyi entropies. We find that the graphical approach
to representing invariants can provide structural insight into the invariants
being contracted, as well as an alternative, and sometimes much simpler, means
to study polynomial invariants of quantum states. In addition, many tensor
network methods, such as matrix product states, contain excellent tools that
can be applied in the study of invariants.Comment: 21 page
Quantum Circuits for General Multiqubit Gates
We consider a generic elementary gate sequence which is needed to implement a
general quantum gate acting on n qubits -- a unitary transformation with 4^n
degrees of freedom. For synthesizing the gate sequence, a method based on the
so-called cosine-sine matrix decomposition is presented. The result is optimal
in the number of elementary one-qubit gates, 4^n, and scales more favorably
than the previously reported decompositions requiring 4^n-2^n+1 controlled NOT
gates.Comment: 4 pages, 3 figure
Community Detection in Quantum Complex Networks
Determining community structure is a central topic in the study of complex
networks, be it technological, social, biological or chemical, in static or
interacting systems. In this paper, we extend the concept of community
detection from classical to quantum systems---a crucial missing component of a
theory of complex networks based on quantum mechanics. We demonstrate that
certain quantum mechanical effects cannot be captured using current classical
complex network tools and provide new methods that overcome these problems. Our
approaches are based on defining closeness measures between nodes, and then
maximizing modularity with hierarchical clustering. Our closeness functions are
based on quantum transport probability and state fidelity, two important
quantities in quantum information theory. To illustrate the effectiveness of
our approach in detecting community structure in quantum systems, we provide
several examples, including a naturally occurring light-harvesting complex,
LHCII. The prediction of our simplest algorithm, semiclassical in nature,
mostly agrees with a proposed partitioning for the LHCII found in quantum
chemistry literature, whereas our fully quantum treatment of the problem
uncovers a new, consistent, and appropriately quantum community structure.Comment: 16 pages, 4 figures, 1 tabl