  A quantum cellular automaton (QCA) or a causal unitary is by definition an
automorphism of local operator algebra, by which local operators are mapped to
local operators. Quantum circuits of small depth, local Hamiltonian evolutions
for short time, and translations (shifts) are examples. A Clifford QCA is one
that maps any Pauli operator to a finite tensor product of Pauli operators.
Here, we obtain a complete table of groups $\mathfrak C(\mathsf d,p)$ of
translation invariant Clifford QCA in any spatial dimension $\mathsf d \ge 0$
modulo Clifford quantum circuits and shifts over prime $p$-dimensional qudits,
where the circuits and shifts are allowed to obey only coarser translation
invariance. The group $\mathfrak C(\mathsf d,p)$ is nonzero only for $\mathsf d
= 2k+3$ if $p=2$ and $\mathsf d = 4k+3$ if $p$ is odd where~$k \ge 0$ is any
integer, in which case $\mathfrak C(\mathsf d,p) \cong \widetilde{\mathfrak
W}(\mathbb F_p)$, the classical Witt group of nonsingular quadratic forms over
the finite field $\mathbb F_p$. It is well known that $\widetilde{\mathfrak
W}(\mathbb F_2) \cong \mathbb Z/2\mathbb Z$, $\widetilde{\mathfrak W}(\mathbb
F_p) \cong \mathbb Z/4\mathbb Z$ if $p = 3 \bmod 4$, and $\widetilde{\mathfrak
W}(\mathbb F_p)\cong \mathbb Z/2\mathbb Z \oplus \mathbb Z/2\mathbb Z$ if $p =
1 \bmod 4$. The classification is achieved by a dimensional descent, which is a
reduction of Laurent extension theorems for algebraic $L$-groups of surgery
theory in topology.
