. The logical depth of our encoding and decoding operations is constant and depends only on the channel in question. For channels described by an arbitrary discrete group

G

, i.e.~with a discrete number,

\lvert G\rvert

, of possible noise operators, perfect transmission at a rate

m/(m+r)

is achieved with an overhead that scales at most as

\mathcal{O}(d^r)

where the number of auxiliary qudits,

r

, depends solely on the group in question. Moreover, this overhead is independent of the number of transmitted qudits,

m

. For certain groups, e.g.~cyclic groups, we find that the overhead scales only linearly with the number of group elements

|G|

DOI: 10.26421/qic13.3-4-9

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m

. The logical depth of our encoding and decoding operations is constant and depends only on the channel in question. For channels described by an arbitrary discrete group

G

, i.e.~with a discrete number,

\lvert G\rvert

, of possible noise operators, perfect transmission at a rate

m/(m+r)

is achieved with an overhead that scales at most as

\mathcal{O}(d^r)

where the number of auxiliary qudits,

r

, depends solely on the group in question. Moreover, this overhead is independent of the number of transmitted qudits,

m

. For certain groups, e.g.~cyclic groups, we find that the overhead scales only linearly with the number of group elements

|G|

\mathcal{O}(d^r)

where the number of auxiliary qudits, r, depends solely on the group in question. Moreover, this overhead is independent of the number of transmitted qudits, m. For certain groups, e.g. cyclic groups, we find that the overhead scales only linearly with the number of group elements |G|.

n-1

classical bits are distributed by using

2n

qubits in our protocols, the transmission rate is close to that of noiseless BB84 protocol. Furthermore, when considering the cost of transmitting classical bits, the efficiencies of these protocols are even higher than that of BB84 protocol.

C^*

-algebra embedding. Our noise model for the transmission is a phase damping channel which admits a noiseless or decoherence-free subspace or subsystem. More precisely, the transmission channel is obtained from convex combinations of a set of lowest rank yes/no measurements that leave a component of the messenger state unchanged. The objective of our encoding is to distribute quantum information optimally across the noise-susceptible component of the transmission when the noiseless component is not large enough to contain all the quantum information to be transmitted. We derive simple geometric conditions for optimal encoding and construct examples.

T

and every

\epsilon >0

there exists a threshold

p(\epsilon,T)

for the gate error probability below which rates larger than$C-\epsilon

are fault-tolerantly achievable with vanishing overall communication error, where

C$denotes the usual capacity. Our results are not only relevant in communication over large distances, but also on-chip, where distant parts of a quantum computer might need to communicate under higher levels of noise than affecting the local gates.

Decoherence-Insensitive Quantum Communication by Optimal

C^{\ast }

-Encoding