Topological Phases of Unitary Dynamics: Classification in Clifford Category

A quantum cellular automaton (QCA) or a causal unitary is by definition an automorphism of a local operator algebra, by which local operators are mapped to nearby 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 \({\mathfrak {C}}({\textsf{d}},p)\) of translation invariant Clifford QCA in any spatial dimension \({\textsf{d}}\ge 0\) 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 \({\mathfrak {C}}({\textsf{d}},p)\) is nonzero only for \({\textsf{d}}= 2k+3\) if \(p=2\) and \({\textsf{d}}= 4k+3\) if p is odd where \(k \ge 0\) is any integer, in which case \({\mathfrak {C}}({\textsf{d}},p) \cong {\widetilde{\mathfrak {W}}}({\mathbb {F}}_p)\), 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.