\chapter*{Abstract}

% Current state of the art

\Ac{qec} protects fragile quantum states against decoherence by
encoding logical information into a larger number of physical qubits.
To obtain parity information on an encoded state without disturbing it, a
syndrome extraction is performed.
Because the syndrome extraction circuitry is itself implemented on
noisy quantum hardware, practical \ac{qec} must be fault-tolerant,
accounting for errors introduced by the correction procedure itself.
Fault tolerance considerations and the syndrome extraction circuit
are captured by \acp{dem}, which provide a unified framework for passing
this information to the decoder.

Accounting for fault tolerance substantially inflates the
decoding problem.
At the same time, \ac{qec} imposes strict latency constraints due to
the backlog problem, where syndrome data accumulates faster than it
can be decoded.
Together, these factors pose a serious challenge for practical decoders.
Sliding-window decoding addresses this challenge by exploiting the
repeated structure of the syndrome extraction circuitry, partitioning
the check matrix of the \ac{dem} into overlapping windows that can be
decoded sequentially.
Therefore, decoding can begin as soon as the syndrome components
associated with the first window have been measured.

% Our work: Identify research gap

In this thesis, we perform a review of the existing literature on
sliding-window decoding and draw an analogy to windowed
decoding of classical spatially-coupled low-density parity-check
(\acs{sc}-\acs{ldpc}) codes.
We recognize that in contrast to the latter, existing realizations
of sliding-window decoding for \ac{qec} discard the soft information
produced inside one window before moving to the subsequent window.

% Our work: Warm-start

% TODO: Quantify improvement. Also for conclusion
To take this information into account, we propose warm-start
sliding-window decoding, in which the \ac{bp} messages on the edges
crossing into the overlap region of the previous window are reused to
initialize the corresponding messages of the next window.
The warm start is formulated first for standard \ac{bp} and then extended to
\ac{bp} with guided decimation (\acs{bpgd}).
For both standard \ac{bp} and \ac{bpgd} decoding, the warm-start
initialization provides a consistent improvement across all examined
parameter settings.
We attribute this to an effective increase in \ac{bp} iterations on
variable nodes in the overlap regions: each such VN is processed by
multiple consecutive windows, and warm-starting lets these
invocations accumulate iterations rather than restart from scratch.
Crucially, the warm-start modification incurs no additional
computational cost relative to cold-start decoding, as it differs
only in the initialization of the \ac{bp} messages.