2 research outputs found
On the Satisfiability of Quantum Circuits of Small Treewidth
It has been known for almost three decades that many -hard
optimization problems can be solved in polynomial time when restricted to
structures of constant treewidth. In this work we provide the first extension
of such results to the quantum setting. We show that given a quantum circuit
with uninitialized inputs, gates, and treewidth ,
one can compute in time a classical
assignment that maximizes the acceptance probability of up
to a additive factor. In particular, our algorithm runs in polynomial
time if is constant and . For unrestricted values
of , this problem is known to be complete for the complexity class
, a quantum generalization of MA. In contrast, we show that the
same problem is -complete if even when is
constant.
On the other hand, we show that given a -input quantum circuit of
treewidth , and a constant , it is
-complete to determine whether there exists a quantum state
such that the acceptance
probability of is greater than , or whether
for every such state , the acceptance probability of
is less than . As a consequence, under the
widely believed assumption that , we have that
quantum witnesses are strictly more powerful than classical witnesses with
respect to Merlin-Arthur protocols in which the verifier is a quantum circuit
of logarithmic treewidth.Comment: 30 Pages. A preliminary version of this paper appeared at the 10th
International Computer Science Symposium in Russia (CSR 2015). This version
has been submitted to a journal and is currently under revie
Carving-width and contraction trees for tensor networks
We study the problem of finding contraction orderings on tensor networks for
physical simulations using a syncretic abstract data type, the
, and explain its connection to temporal and spatial
measures of tensor contraction computational complexity (nodes express time;
arcs express space). We have implemented the Ratcatcher of Seymour and Thomas
for determining the carving-width of planar networks, in order to offer
experimental evidence that this measure of spatial complexity makes a generally
effective heuristic for limiting their total contraction time.Comment: 16 pages, 7 figures, 1 tabl