# Decoding quantum errors with subspace expansions. (arXiv:1903.05786v1 [quant-ph])

With the rapid developments in quantum hardware comes a push towards the
first practical applications on these devices. While fully fault-tolerant
quantum computers may still be years away, one may ask if there exist
intermediate forms of error correction or mitigation that might enable
practical applications before then. In this work, we consider the idea of
post-processing error decoders using existing quantum codes, which are capable
of mitigating errors on encoded logical qubits using classical post-processing
with no complicated syndrome measurements or additional qubits beyond those
used for the logical qubits. This greatly simplifies the experimental
exploration of quantum codes on near-term devices, removing the need for
locality of syndromes or fast feed-forward, allowing one to study performance
aspects of codes on real devices. We provide a general construction equipped
with a simple stochastic sampling scheme that does not depend explicitly on a
number of terms that we extend to approximate projectors within a subspace.
This theory then allows one to generalize to the correction of some logical
errors in the code space, correction of some physical unencoded Hamiltonians
without engineered symmetries, and corrections derived from approximate
symmetries. In this work, we develop the theory of the method and demonstrate
it on a simple example with the perfect $[[5,1,3]]$ code, which exhibits a
pseudo-threshold of $p \approx 0.50$ under a single qubit depolarizing channel
applied to all qubits. We also provide a demonstration under the application of
a logical operation and performance on an unencoded hydrogen molecule, which
exhibits a significant improvement over the entire range of possible errors
incurred under a depolarizing channel.