588 research outputs found
Counting single-qubit Clifford equivalent graph states is #P-Complete
Graph states, which include for example Bell states, GHZ states and cluster
states, form a well-known class of quantum states with applications ranging
from quantum networks to error-correction. Deciding whether two graph states
are equivalent up to single-qubit Clifford operations is known to be decidable
in polynomial time and have been studied both in the context of producing
certain required states in a quantum network but also in relation to stabilizer
codes. The reason for the latter this is that single-qubit Clifford equivalent
graph states exactly corresponds to equivalent stabilizer codes. We here
consider the computational complexity of, given a graph state |G>, counting the
number of graph states, single-qubit Clifford equivalent to |G>. We show that
this problem is #P-Complete. To prove our main result we make use of the notion
of isotropic systems in graph theory. We review the definition of isotropic
systems and point out their strong relation to graph states. We believe that
these isotropic systems can be useful beyond the results presented in this
paper.Comment: 10 pages, no figure
Counting single-qubit Clifford equivalent graph states is #â-complete
Graph states, which include for example Bell states, GHZ states and cluster states, form a well-known class of quantum states with applications ranging from quantum networks to error-correction. Deciding whether two graph states are equivalent up to single-qubit Clifford operations is known to be decidable in polynomial time and have been studied both in the context of producing certain required states in a quantum network but also in relation to stabilizer codes. The reason for the latter this is that single-qubit Clifford equivalent graph states exactly corresponds to equivalent stabilizer codes. We here consider the computational complexity of, given a graph state |G>, counting the number of graph states, single-qubit Clifford equivalent to |G>. We show that this problem is #P-Complete. To prove our main result we make use of the notion of isotropic systems in graph theory. We review the definition of isotropic systems and point out their strong relation to graph states. We believe that these isotropic systems can be useful beyond the results presented in this paper
Some Ulam's reconstruction problems for quantum states
Provided a complete set of putative -body reductions of a multipartite
quantum state, can one determine if a joint state exists? We derive necessary
conditions for this to be true. In contrast to what is known as the quantum
marginal problem, we consider a setting where the labeling of the subsystems is
unknown. The problem can be seen in analogy to Ulam's reconstruction conjecture
in graph theory. The conjecture - still unsolved - claims that every graph on
at least three vertices can uniquely be reconstructed from the set of its
vertex-deleted subgraphs. When considering quantum states, we demonstrate that
the non-existence of joint states can, in some cases, already be inferred from
a set of marginals having the size of just more than half of the parties. We
apply these methods to graph states, where many constraints can be evaluated by
knowing the number of stabilizer elements of certain weights that appear in the
reductions. This perspective links with constraints that were derived in the
context of quantum error-correcting codes and polynomial invariants. Some of
these constraints can be interpreted as monogamy-like relations that limit the
correlations arising from quantum states. Lastly, we provide an answer to
Ulam's reconstruction problem for generic quantum states.Comment: 22 pages, 3 figures, v2: significantly revised final versio
Optical generation of matter qubit graph states
We present a scheme for rapidly entangling matter qubits in order to create
graph states for one-way quantum computing. The qubits can be simple 3-level
systems in separate cavities. Coupling involves only local fields and a static
(unswitched) linear optics network. Fusion of graph state sections occurs with,
in principle, zero probability of damaging the nascent graph state. We avoid
the finite thresholds of other schemes by operating on two entangled pairs, so
that each generates exactly one photon. We do not require the relatively slow
single qubit local flips to be applied during the growth phase: growth of the
graph state can then become a purely optical process. The scheme naturally
generates graph states with vertices of high degree and so is easily able to
construct minimal graph states, with consequent resource savings. The most
efficient approach will be to create new graph state edges even as qubits
elsewhere are measured, in a `just in time' approach. An error analysis
indicates that the scheme is relatively robust against imperfections in the
apparatus.Comment: 10 pages in 2 column format, includes 4 figures. Problems with
figures resolve
Quantum Commuting Circuits and Complexity of Ising Partition Functions
Instantaneous quantum polynomial-time (IQP) computation is a class of quantum
computation consisting only of commuting two-qubit gates and is not universal
in the sense of standard quantum computation. Nevertheless, it has been shown
that if there is a classical algorithm that can simulate IQP efficiently, the
polynomial hierarchy (PH) collapses at the third level, which is highly
implausible. However, the origin of the classical intractability is still less
understood. Here we establish a relationship between IQP and computational
complexity of the partition functions of Ising models. We apply the established
relationship in two opposite directions. One direction is to find subclasses of
IQP that are classically efficiently simulatable in the strong sense, by using
exact solvability of certain types of Ising models. Another direction is
applying quantum computational complexity of IQP to investigate (im)possibility
of efficient classical approximations of Ising models with imaginary coupling
constants. Specifically, we show that there is no fully polynomial randomized
approximation scheme (FPRAS) for Ising models with almost all imaginary
coupling constants even on a planar graph of a bounded degree, unless the PH
collapses at the third level. Furthermore, we also show a multiplicative
approximation of such a class of Ising partition functions is at least as hard
as a multiplicative approximation for the output distribution of an arbitrary
quantum circuit.Comment: 36 pages, 5 figure
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