3,403 research outputs found
On weakly symmetric graphs of order twice a prime
A graph is weakly symmetric if its automorphism group is both vertex-transitive and edge-transitive. In 1971, Chao characterized all weakly symmetric graphs of prime order and showed that such graphs are also transitive on directed edges. In this paper we determine all weakly symmetric graphs of order twice a prime and show that these graphs too are directed-edge transitive. © 1987
Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count
We present an infinite family of protocols to distill magic states for
-gates that has a low space overhead and uses an asymptotic number of input
magic states to achieve a given target error that is conjectured to be optimal.
The space overhead, defined as the ratio between the physical qubits to the
number of output magic states, is asymptotically constant, while both the
number of input magic states used per output state and the -gate depth of
the circuit scale linearly in the logarithm of the target error (up to
). Unlike other distillation protocols, this protocol
achieves this performance without concatenation and the input magic states are
injected at various steps in the circuit rather than all at the start of the
circuit. The protocol can be modified to distill magic states for other gates
at the third level of the Clifford hierarchy, with the same asymptotic
performance. The protocol relies on the construction of weakly self-dual CSS
codes with many logical qubits and large distance, allowing us to implement
control-SWAPs on multiple qubits. We call this code the "inner code". The
control-SWAPs are then used to measure properties of the magic state and detect
errors, using another code that we call the "outer code". Alternatively, we use
weakly-self dual CSS codes which implement controlled Hadamards for the inner
code, reducing circuit depth. We present several specific small examples of
this protocol.Comment: 39 pages, (v2) renamed "odd" and "even" weakly self-dual CSS codes of
(v1) to "normal" and "hyperbolic" codes, respectively. (v3) published in
Quantu
Dynamical properties of profinite actions
We study profinite actions of residually finite groups in terms of weak
containment. We show that two strongly ergodic profinite actions of a group are
weakly equivalent if and only if they are isomorphic. This allows us to
construct continuum many pairwise weakly inequivalent free actions of a large
class of groups, including free groups and linear groups with property (T). We
also prove that for chains of subgroups of finite index, Lubotzky's property
() is inherited when taking the intersection with a fixed subgroup of
finite index. That this is not true for families of subgroups in general leads
to answering the question of Lubotzky and Zuk, whether for families of
subgroups, property () is inherited to the lattice of subgroups generated
by the family. On the other hand, we show that for families of normal subgroups
of finite index, the above intersection property does hold. In fact, one can
give explicite estimates on how the spectral gap changes when passing to the
intersection. Our results also have an interesting graph theoretical
consequence that does not use the language of groups. Namely, we show that an
expander covering tower of finite regular graphs is either bipartite or stays
bounded away from being bipartite in the normalized edge distance.Comment: Corrections made based on the referee's comment
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