We study phases of one-dimensional matrix-product states (MPS) when transformations are restricted to symmetric local circuits supplemented with symmetric measurements and feedforward (G-CMF). Building on the framework introduced in Gunn et al., Phys. Rev. B 111, 115110 (2025), we extend the analysis to all finite nilpotent groups for which we obtain a complete classification of G-CMF phases. We construct explicit symmetry-respecting protocols that map any symmetry-protected topological (SPT) or non-normal (GHZ-type) MPS to the trivial phase-and vice versa-with success probability approaching one in the thermodynamic limit. The key technical ingredient is a finite hierarchical structure of irreducible representations of nilpotent groups, which enables successive rounds of symmetric measurements to systematically reduce non-abelian components to abelian ones. Our results demonstrate that allowing symmetric measurements and feedforward fundamentally simplifies the phase structure of 1D systems with nilpotent symmetries: all SPT and non-normal MPS phases collapse into a single asymptotically equivalent phase under G-CMF transformations.
We study phases of one-dimensional matrix-product states (MPS) when transformations are restricted to symmetric local circuits supplemented with symmetric measurements and feedforward (G-CMF). Building on the framework introduced in Gunn et al., Phys. Rev. B 111, 115110 (2025), we extend the analysis to all finite nilpotent groups for which we obtain a complete classification of G-CMF phases. We construct explicit symmetry-respecting protocols that map any symmetry-protected topological (SPT) or non-normal (GHZ-type) MPS to the trivial phase-and vice versa-with success probability approaching one in the thermodynamic limit. The key technical ingredient is a finite hierarchical structure of irreducible representations of nilpotent groups, which enables successive rounds of symmetric measurements to systematically reduce non-abelian components to abelian ones. Our results demonstrate that allowing symmetric measurements and feedforward fundamentally simplifies the phase structure of 1D systems with nilpotent symmetries: all SPT and non-normal MPS phases collapse into a single asymptotically equivalent phase under G-CMF transformations.
Two matrix product states (MPS) are in the same phase in the presence of symmetries if they can be transformed into one another via symmetric short-depth circuits. We consider how symmetry-preserving measurements with feedforward alter the phase classification of MPS in the presence of global on-site symmetries. We demonstrate that, for all finite Abelian symmetries, any two symmetric MPS belong to the same phase. We give an explicit protocol that achieves a transformation between any two phases and that uses only a depth-two symmetric circuit, a single round of symmetric measurements, and a constant number of auxiliary systems per site. In the case of non-Abelian symmetries, symmetry protection prevents one from deterministically transforming symmetry-protected topological (SPT) states to product states directly via measurements, thereby complicating the analysis. Nonetheless, we provide protocols that allow for asymptotically deterministic transformations between the trivial phase, SPT phases, and GHZ phases of some non-Abelian nilpotent groups.
Abstract Classifying and understanding phases of matter is an important task in condensed matter physics. The class of ‘conventional’ symmetry-broken phases is well understood in terms of Landau’s theory. A paradigmatic example is the Z2-symmetric Ising model with symmetric (paramagnetic) and symmetry-broken (ferromagnet) phases, which can be distinguished by measuring the magnetization as a local order parameter. Topological phases of matter, however, are less understood and no complete classification is known. These lecture notes discuss schemes that provide an understanding and characterization of certain topological phases of matter in models with local Hamiltonians. The main focus is on symmetry-protected topological (SPT) phases in one-dimensional bosonic systems. On the basis of the entanglement properties of such systems, a motivation is given for the matrix product state (MPS) representation of ground states. Using the MPS framework, it is demonstrated how SPT phases can be classified using projective representations of the symmetries.
We classify one-dimensional symmetry-protected topological (SPT) phases protected by dipole symmetries. A dipole symmetry comprises two sets of symmetry generators: charge and dipole operators, which together form a nontrivial algebra with translations. Using matrix product states (MPS), we show that for a G dipole symmetry with G a finite Abelian group, the one-dimensional dipolar SPTs are classified by the group ${H}^{2}[G\ifmmode\times\else\texttimes\fi{}G,U(1)]/{H}^{2}{[G,U(1)]}^{2}$. Because of the symmetry algebra, the MPS tensors exhibit an unusual property, prohibiting the fractionalization of charge operators at the edges. For each phase in the classification, we explicitly construct a stabilizer Hamiltonian to realize the SPT phase and derive the response field theories by coupling the dipole symmetry to background tensor gauge fields. These field theories generalize the Dijkgraaf-Witten theories to twisted finite tensor gauge theories.
We classify the different ways in which matrix product states (MPSs) can stay invariant under the action of matrix product operator (MPO) symmetries. This is achieved through a local characterization of how the MPSs, that generate a ground space, remain invariant under a global MPO symmetry. This characterization yields a set of quantities satisfying the coupled pentagon equations, associated with a module category over the fusion category that describes the MPO symmetry. Equivalence classes of these quantities provide complete invariants for an MPO symmetry protected phase: they are robust under continuous deformations of the MPS tensor, and two phases with the same equivalence class can be connected by a symmetric gapped path. Our techniques match and extend the known renormalization fixed point classifications and facilitate the numerical study of these systems. For MPO symmetries described by a group, we recover the symmetry protected topological order classification for unique and degenerate ground states. Moreover, we study the interplay between time reversal symmetry and an MPO symmetry and we also provide examples of our classification, together with explicit constructions based on groups. Finally, we elaborate on the connection between our setup and gapped boundaries of two-dimensional topological systems, where MPO symmetries also play a key role.
Adaptive quantum circuits, which combine local unitary gates, midcircuit measurements, and feedforward operations, have recently emerged as a promising avenue for efficient state preparation, particularly on near-term quantum devices limited to shallow-depth circuits. Matrix product states (MPS) comprise a significant class of many-body entangled states, efficiently describing the ground states of one-dimensional gapped local Hamiltonians and finding applications in a number of recent quantum algorithms. Recently, it has been shown that the Affleck-Kennedy-Lieb-Tasaki state—a paradigmatic example of an MPS—can be exactly prepared with an adaptive quantum circuit of constant depth, an impossible feat with local unitary gates alone due to its nonzero correlation length [Smith , PRX Quantum 4, 020315 (2023)]. In this work, we broaden the scope of this approach and demonstrate that a diverse class of MPS can be exactly prepared using constant-depth adaptive quantum circuits, outperforming theoretically optimal preparation with unitary circuits. We show that this class includes short- and long-ranged entangled MPS, symmetry-protected topological (SPT) and symmetry-broken states, MPS with finite Abelian, non-Abelian, and continuous symmetries, resource states for MBQC, and families of states with tunable correlation length. Moreover, we illustrate the utility of our framework for designing constant-depth sampling protocols, such as for random MPS or for generating MPS in a particular SPT phase. We present sufficient conditions for particular MPS to be preparable in constant time, with global on-site symmetry playing a pivotal role. Altogether, this work demonstrates the immense promise of adaptive quantum circuits for efficiently preparing many-body entangled states and provides explicit algorithms that outperform known protocols to prepare an essential class of states. Published by the American Physical Society 2024
Matrix product states (MPS) provide a powerful framework for characterizing one-dimensional symmetry-protected topological (SPT) phases of matter and for formulating Lieb-Schultz-Mattis (LSM)-type constraints. Here we generalize the MPS formalism to translationally invariant systems with general modulated symmetries. We show that the standard symmetry "push-through" condition for conventional global symmetry must be revised to account for symmetry modulation, and we derive the appropriate generalized condition. Using this generalized push-through structure, we classify one-dimensional SPT phases with modulated symmetries and formulate LSM-type constraints within the same MPS-based framework.