Dr. Axel Dahlberg, QuTech, Delft: "Transforming graphs states using
single-qubit operations"
Tuesday, August 7, Haus Room, 36-428
4:00-5:00PM
Abstract:
Graph states are ubiquitous in quantum information with diverse
applications ranging from quantum network protocols to measurement based
quantum computing. Here we consider the question whether one graph (source)
state can be transformed into another graph (target) state, using a
specific set of quantum operations (LC+LPM+CC): single-qubit Clifford
operations (LC), single-qubit Pauli measurements (LPM) and classical
communication (CC) between sites holding the individual qubits. We first
show that deciding whether a graph state |G> can be transformed into
another graph state |G'> using LC+LPM+CC is NP-Complete, even if |G'> is
restricted to be a GHZ-state. However, we also provide efficient algorithms
for two situations of practical interest:
1. |G> has Schmidt-rank width one and |G'> is a GHZ-state. The Schmidt-rank
width is an entanglement measure of quantum states, meaning this algorithm
is efficient if the original state has little entanglement. Our algorithm
has runtime O(|V(G')||V(G)|^3), and is also efficient in practice even on
small instances as further showcased by a freely available software
implementation.
2. |G> is in a certain class of states with unbounded Schmidt-rank width,
and |G'> is a GHZ-state of a constant size. Here the runtime is
O(poly(|V(G)|)), showing that more efficient algorithms can in principle be
found even for states holding a large amount of entanglement, as long as
the output state has constant size.
Our results make use of the insight that deciding whether a graph state |G>
can be transformed to another graph state |G'> is equivalent to a known
decision problem in graph theory, namely the problem of deciding whether a
graph G' is a vertex-minor of a graph G. Many of the technical tools
developed to obtain our results may be of independent interest.
--
Dirk R. Englund
Associate Professor of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
77 Massachusetts Avenue, Room 36-591
englund(a)mit.edu; (617) 324-7014;
http://qplab.mit.edu
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