# Topological phases of unitary dynamics: Classification in Clifford category

@inproceedings{Haah2022TopologicalPO, title={Topological phases of unitary dynamics: Classification in Clifford category}, author={Jeongwan Haah}, year={2022} }

. 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 C ( d , p ) of translation invariant Cliﬀord QCA in any spatial…

## References

SHOWING 1-10 OF 38 REFERENCES

Clifford quantum cellular automata: Trivial group in 2D and Witt group in 3D

- MathematicsJournal of Mathematical Physics
- 2021

We study locality preserving automorphisms of operator algebras on lattices (QCA), specializing in those that are translation invariant and map every prime $p$-dimensional Pauli matrix to a tensor…

Nontrivial Quantum Cellular Automata in Higher Dimensions

- Mathematics
- 2018

We construct a three-dimensional quantum cellular automaton (QCA), an automorphism of the local operator algebra on a lattice of qubits, which disentangles the ground state of the Walker-Wang three…

Classification of Quantum Cellular Automata

- Computer Science
- 2020

In two dimensions, it is shown that an extension of this index theory (including torsion) fully classifies quantum cellular automata, at least in the absence of fermionic degrees of freedom.

Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices

- Mathematics
- 2018

We prove that on any two-dimensional lattice of qudits of a prime dimension, every translation invariant Pauli stabilizer group with local generators and with code distance being the linear system…

The Group Structure of Quantum Cellular Automata

- MathematicsCommunications in Mathematical Physics
- 2022

We consider the group structure of quantum cellular automata (QCA) modulo circuits and show that it is abelian even without assuming the presence of ancillas, at least for most reasonable choices of…

A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory

- MathematicsAnnales Henri Poincaré
- 2022

Unitary dynamics with a strict causal cone (or “light cone”) have been studied extensively, under the name of quantum cellular automata (QCAs). In particular, QCAs in one dimension have been…

Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies

- Physics
- 2021

We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four…

Commuting Pauli Hamiltonians as Maps between Free Modules

- Mathematics
- 2013

We study unfrustrated spin Hamiltonians that consist of commuting tensor products of Pauli matrices. Assuming translation-invariance, a family of Hamiltonians that belong to the same phase of matter…

Reversible quantum cellular automata

- Computer Science
- 2004

The main structure theorem asserts that any quantum cellular automaton is structurally reversible, i.e., that it can be obtained by applying two blockwise unitary operations in a generalized Margolus partitioning scheme.

Classifying quantum phases using Matrix Product States and PEPS

- Mathematics
- 2010

We give a classification of gapped quantum phases of one-dimensional systems in the framework of Matrix Product States (MPS) and their associated parent Hamiltonians, for systems with unique as well…