Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes C2⊆C1, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs (C1,C2)
Nearby in the stack
whose resulting CSS codes realize a target logical diagonal gate via transversal physical
Z
-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit
Z
-rotations and multi-qubit controlled-
Z
rotations via transversal physical
Z
-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an
[[n′,k′]]
CSS code
Q′
and a target logical
Z
-rotation (single-qubit or multi-controlled)
UL
, and extends
Q′
by systematically appending
n′′
physical qubits to obtain an
[[n,k]]
CSS code
Q
with
n=n′+n′′
and
k=k′
. The target logical gate
UL
is realized in
Q
by applying a well-chosen physical transversal
Z
-rotation to the
n′′
appended physical qubits. The CSS code
Q
may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code
Q′
to obtain a CSS code
Q
that supports fault-tolerant implementations of multiple desired logical
Z
-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows.
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