32 problems

Conjecture that $(-1)^{\int w_2^4}$ is a nontrivial QCA. The response $Y=(-1)^{\int w_2^4}$ is a nontrivial quantum cellular automaton.

Fidkowski's conjecture. $Y$ is a nontrivial QCA if the invertible phase it entangles has a nontrivial partition function on some orientable manifold.

Let $n,m\geq0$ satisfy $d+1=2n+3m$ with $d\geq3$, and let $A_{d-1}\in Z^{d-1}(M_{d+1};\mathbb Z_2)$ and $B_2\in Z^2(M_{d+1};\mathbb Z_2)$ be cocycles. Consider the QCA family whose…

Let $S(X)=\frac{\sigma(X)}{8}\pmod{1}$ be the invertibly normalized response of the seven-dimensional generalized-semion QCA, where $\sigma(X)$ is the signature of a closed oriente…

Let $\mathbf Q(\mathrm{pt})$ be the spectrum of quantum cellular automata governing lattice anomalies, and let $\mathbf{4Vect}^{\times}$ be the spectrum of invertible objects and m…

Let $\mathcal W$ denote the Witt group of $(2+1)$-dimensional topological quantum field theories, identified in the source with the Witt group of nondegenerate braided fusion categ…

An invertible stabilizer code is an invertible lattice stabilizer phase, and a quantum cellular automaton (QCA) is considered modulo finite-depth quantum circuits and lattice trans…

Let $\mathbf{TS}^*$ be a proposed categorical spectrum of topological stabilizer codes, with $\mathbf{TS}^n$ its degree-$n$ component, and let $\mathbf Q^*(\mathrm{pt})$ denote the…

Let $k$ be a field, let $\mathbf Q(\mathbb Z^3)$ be the QCA space in dimension three, and let $\mathrm{Witt}_k$ denote the Witt group of braided fusion $k$-linear categories. The Q…

For each spatial dimension $d$, let $\mathcal Q^*(\mathbb Z^d)$ be the group of quantum cellular automata and let $\mathcal C^*(\mathbb Z^d)$ be its normal subgroup of quantum circ…

For each spatial dimension $d$, let $\mathcal I(\mathbb Z^d)$ be the abelian group of invertible gapped phases, and let $\mathrm{IP}(\mathbb Z^d)$ denote a proposed space-level con…

Consider the invertible phases, including invertible stabilizer phases and quantum cellular automata (QCA), and their classification by a generalized cohomology theory. Kitaev's co…

Let $(\operatorname{QCA}_d)_{d\in\mathbb{Z}}$ and $(\operatorname{inv}_d)_{d\in\mathbb{Z}}$ be the proposed spectra of quantum cellular automata and invertible phases, respectively…

Let $\Pi(\operatorname{QCA}_{d+1})$ be the higher category whose objects are $(d+1)$-dimensional quantum cellular automata and whose morphisms are blends and higher morphisms as de…

Let $G$ be a discrete group, let $\mathrm{B}G$ be its classifying space, and let $\mathrm{QCA}_{d+1}$ denote the proposed $(d+1)$-dimensional quantum-cellular-automaton spectrum. W…

Let $Q_{d+1}^{(0)}$ denote the group of quantum cellular automaton classes that can preserve an on-site LIOM algebra, and let $d\geq1$ be the spatial dimension parameter. On-site-p…

Let $G$ be a finite group, and let $\mathcal{U}$ and $\mathcal{W}$ be $G$-representations. They are blend-equivalent if they are related by the lattice blending relation, and conju…

Completeness conjecture. If two $G$-representations have the same total index, then they belong to the same equivalence class.

The analysis concerns renormalising cellular automata on $\mathbb Z^s$ through a Kadanoff-like coarse-graining procedure. In the special case $s=1$ with qubit cells, the coarse-gra…

Kitaev's no-go conjecture. No commuting strictly local Hamiltonian on a two-dimensional lattice with locally finite-dimensional degrees of freedom can realize a chiral topological…

Classical Witt-group conjecture. The group of such QCAs is isomorphic to the classical Witt group.

2-primary QCA group conjecture. The QCAs presented in this work generate a group isomorphic to

Chiral-semion QCA nontriviality conjecture. The QCA $\alpha_1$ is nontrivial, because otherwise there would exist a standalone two-dimensional commuting projector Hamiltonian reali…

Generator conjecture. The QCA $b1_1$ generates the $a95_8$ subgroup, and $b1_1b1_2^{-1}$ generates the $a95_2$ subgroup.

Categorical Witt-group classification conjecture. This subgroup is improper; equivalently, the group of three-dimensional QCAs is isomorphic to the categorical Witt group.