Classification of Matrix-Product Unitaries with Symmetries

Abstract

We prove that matrix-product unitaries with on-site unitary symmetries are completely classified by the (chiral) index and the cohomology class of the symmetry group G, provided that we can add trivial and symmetric ancillas with arbitrary on-site representations of G. If the representations in both system and ancillas are fixed to be the same, we can define symmetry-protected indices (SPIs) which quantify the imbalance in the transport associated to each group element and greatly refines the classification. These SPIs are stable against disorder and measurable in interferometric experiments. Our results lead to a systematic construction of two-dimensional Floquet symmetry-protected topological phases beyond the standard classification, and thus shed new light on understanding nonequilibrium phases of quantum matter.

A review of Quantum Cellular Automata

“…This works similarly to the classification of matrix product states in the presence of symmetries [107]. The main result of [108] is that one-dimensional matrix product unitaries are completely classified by (i) the QCA index and (ii) the cohomology class of the representation of the symmetry group acting on the bond indices, as conjectured in [109]. The classification considers two matrix product unitaries U 0 and U 1 to be equivalent if, possibly by adding ancillas transforming under arbitrary representations of the symmetry group, there is a continuous path of matrix product unitaries U t joining U 0 and U 1 .…”

“…x U = n σ n x , so U is clearly not locality preserving. These MPUs can be further classified in the presence of on-site unitary symmetries [108]. This works similarly to the classification of matrix product states in the presence of symmetries [107].…”

“…(We will give an example showing why this is true below.) The motivation for this decomposition of QCAs in terms of tensor networks is that it is useful for characterizing symmetries of QCAs [105,108], and also topological phases of Floquet models [25]. Note that these results only apply to finite lattices.…”

Quantum Cellular Automata, Tensor Networks, and Area Laws

“…So far, complete solutions have been obtained in one [7] and two spatial dimensions [8][9][10][11]. Additionally, in the first case, QCA have been identified [12][13][14][15][16][17][18] with matrix product operators, a 1D version of tensor networks (TN), which satisfy an extra condition named simpleness [14] (this has been recently extended to fermionic systems [19,20]). This identification connects QCA with TN, a very active area of research in many-body physics and quantum information.…”

“…The identification of QCA with PEPU allows one to use the established techniques based on TN for numerical simulations of their action [34,35]. This also gives us a very natural framework to investigate the classification of (symmetry-protected) topological (SPT) phases for QCA [18] in higher dimensions, with possible implications for the classification of Floquet SPT phases [13,[36][37][38]. Additionally, QCA inherit the holographic principle of PEPS [39], which can also be used for their classification.…”