Phases of matrix-product states with symmetries and measurements: Finite nilpotent groups

TL;DR

The paper shows that for finite nilpotent symmetry groups G, symmetric CMF transformations consisting of G-symmetric measurements and feedforward drive all MPS phases—both SPT and non-normal (GHZ-type)—to the trivial phase in the thermodynamic limit. This is achieved via a hierarchical irrep structure that yields a finite sequence of measurement rounds, each projecting onto equivalence classes [lpha]_m and reducing the effective symmetry channel from G_m to G_{m-1}. By constructing explicit GHZ- and SPT-representative states and detailing resource bounds, the authors prove asymptotically deterministic trivialization with polylogarithmic circuit depth, thereby collapsing the phase diagram for these non-abelian nilpotent groups under CMF. The results illuminate how symmetry-respecting measurements empower operational phase transformations and raise open questions about extending to solvable but non-nilpotent or non-finite symmetry groups. The work has implications for the controllability of quantum phases in 1D tensor-network settings and informs future explorations of phase classification under measurement-based operations.

Abstract

We classify phases of one-dimensional matrix-product states (MPS) under symmetric circuits augmented with symmetric measurements and feedforward. Building on the framework introduced in Gunn et al., Phys. Rev. B 111, 115110 (2025), we extend the analysis from abelian and class-2 nilpotent groups to all finite nilpotent groups. For any such symmetry group $G$, we construct explicit protocols composed of $G$-symmetric circuits and measurements with feedforward that transform symmetry-protected topological (SPT) states into the trivial phase and vice versa using a finite number of measurement rounds determined by the nilpotency class of $G$. Although these transformations are approximate, we prove that their success probability converges to unity in the thermodynamic limit, establishing asymptotically deterministic equivalence. Consequently, all SPT phases protected by finite nilpotent groups collapse to a single phase once symmetric measurements and feedforward are allowed. We further show that the same holds for non-normal MPS with long-range correlations, including GHZ-type states. The central technical ingredient is a hierarchical structure of irreducible representations of nilpotent groups, which enables a recursive reduction of non-abelian components to abelian ones. Our results demonstrate that symmetric measurements lead to a complete collapse of both symmetry-protected and non-normal MPS phases for all finite nilpotent symmetry groups.

Figures (5)

• Figure 1: (a) We consider transformations between states belonging to different phases. Starting from an initial state (black dot), for example in the trivial phase, we first transform the state into a phase representative (black circle). This transformation can be implemented by a local symmetric circuit of depth $O(\mathrm{polylog}(N))$chen2010localhastings2005quasiadiabaticnachtergaele2019quasicoser2019classification. In the second step, as we will show in detail below, the phase representative is transformed into representatives of other phases using CMF transformations, which include symmetric measurements. In the final step, we again apply a local symmetric circuit of depth $O(\mathrm{polylog}(N))$ to obtain the desired target state. By constructing explicit protocols that transform between phase representatives, we thereby demonstrate a trivialization of the phase diagram. (b) Circuit representation of a CMF transformation: Blue boxes represent local symmetric unitary circuits corresponding to transformations within phases, i.e., the solid lines in (a). Red boxes represent CMF consisting of multiple rounds of symmetric measurements followed by short-depth symmetric quantum circuits which depend on the measurement outcomes. For class-$M$ nilpotent groups our protocols consist of $M$ rounds of measurements and circuits. These circuits consist of sequences of nearest-neighbor SWAP gates. While some of our protocols make use of auxiliary systems, other achieve the desired transformation without auxiliary systems.
• Figure 2: An example of the SPT protocol for the non-abelian class-3 nilpotent group $D_{16}$, transforming an SPT state to the trivial phase. (a) We depict the initial non-trivial state, $\ket{\rm SPT}$, corresponding to a chain of Bell pairs between nearest neighbors. This state is symmetric under $U_g^{\otimes N}$, with the effective symmetry defined in Eq. \ref{['eq:Ug']}. (b) In the first round of the protocol, we project onto the equivalence classes in $\mathrm{Irr}_3/{\sim}$, with projectors represented by blue boxes. Due to the isomorphism in Eq. \ref{['eq:groupIso']} we have that $\mathrm{Irr}_3/{\sim} \cong G_2/G_3\cong \mathbb{Z}_2$. Therefore, measurement outcomes can be labeled by elements of $\mathbb{Z}_2$. Note, that the total product of outcomes is equal to $0$, the identity element. Next, we partition the output into minimal connected substrings that multiply to the identity. Here, the longest minimal connected substring that multiplies to the identity is of length $4$. Therefore, we need $L^{(1)}\ge 16$ to ensure the protocol does not fail in this step. In a second step we use a SWAP circuit to permute these minimal connected substrings to get substrings of length at most $|G_2|/|G_3|=2$ that multiply to the identity. This is to ensure that subsequent measurements act on only $O(1)$ sites. By inspection, we see that the second round will measure $N^{(2)}=10$ supersites. (c) In the second round of the protocol, we then project onto the equivalence classes in $\mathrm{Irr}_{2}/{\sim}$, with projectors represented by green boxes. Outcomes are labeled by $\mathrm{Irr}_{2}/{\sim}\cong G_1/G_2 \cong \mathds{Z}_2$. As before, we use a circuit to permute the outcome. Then, substrings have support on subspace of 1D irreps. Finally, in (d), we project onto 1D subspaces, represented by orange boxes. As these measurements are rank-1, the resulting state is a symmetric pure product state with local sites of size $O(|G|)$.
• Figure 3: Example of Part 1 of the protocol transforming a state in the trivial phase to the GHZ state for the class-$4$ nilpotent group $D_{32}$---the dihedral group with thirty-two elements. For this group we have that the first three levels of the hierarchy are equipped with the group structure $\mathrm{Irr}_m\cong G_{m-1}/G_m\cong \mathbb{Z}_2$ for $m=4,3,2$, i.e., the outcomes in each round can be labeled by $0,1\in\mathbb{Z}_2$. The protocol begins with the state in (a)---local copies of the fiducial state of the GHZ tensor, where the middle qudits, indicated by red lines, represent physical sites that remain untouched until the end of the protocol. Clearly the state is separable between sites and therefore in the trivial phase. After using a nearest-neighbor SWAP circuit to move each right auxiliary qudit to its right neighbor, the first round begins by performing measurements similar as in the SPT case. Note that, unlike the SPT case, the measurement outcomes are not guaranteed to multiply to the identity. Therefore, in each round, we have the possibility of a "remainder" parity. For example, in (b), the product of all measurement outcomes yields $1$ instead of $0$. As a result, there is a remainder at the end. The sites are permuted so that the remainder is moved to the end of the string. The protocol then proceeds to the next round, (c), leaving the remainder unmeasured until Part 2 of the protocol. Although measurement outcomes are not guaranteed to multiply to the identity, they may do so, as in (c), in which case there is no remainder. This is represented by $\emptyset$ in the string $y$ which denotes the remainder parity. For the above sequence of outputs, $y=((1),\emptyset,(1))$. This concludes Part 1 of the protocol, for Part 2 see Fig. \ref{['fig:GHZProtocolPart2']}.
• Figure 4: Example of Part 2 of the protocol, which establishes the parity constraint. At the end of Part 1 we can describe the remainder by $\boldsymbol{y}=((1),\emptyset,(1))$; see Fig. \ref{['fig:GHZProtocolPart1']}(d). (a) We append auxiliary systems in the state $\ket{\phi_{00}}$ and perform the measurement $P^{(2),M}_{[\alpha]}$. The measurement outcome can belong to levels $2,3$ or $4$ of the hierarchy. Let us assume we obtained an outcome belonging to $\mathrm{Irr}_{2}/{\sim}$ (green box). The remainder updates to $\boldsymbol{y}=((1),(1),(1))$, where none of the $y_i$ contain a subset whose elements multiply to the identity. Therefore, in (b), we rearrange the sites and append another auxiliary system in the state $\ket{\phi_{00}}$. We measure again, obtaining the updated remainder $\boldsymbol{y} = ((1),(1),(1,1))$. Now $y_3$ multiplies to the identity, so in (c) we perform another measurement, yielding $\boldsymbol{y} = ((1),(1,1),\emptyset)$. Next, $y_2$ multiplies to the identity, and in (d) we measure once more, obtaining $\boldsymbol{y} = ((1,1),\emptyset,\emptyset)$. At this point, the total parity of the state is the identity, and we can proceed to Part 3.
• Figure 5: Example of projective measurements on the first substring of sites in round $3$ for a class-3 Nilpotent group (assuming no permutation circuits are required). Let us discuss this image as an aid to understanding the notation introduced. Looking at this image, we can read off the measurement history of the protocol. Starting from the top of the image, we see the final projective measurement, corresponding to Eq. \ref{['eq:AppendixAbelianBasisRegRep']}. Moving down, there were two projective measurements (green boxes) in round two. Thus, these two measurements must have totaled to identity; i.e., $N_1^{(2)}=2$ and $x^{(2)}_{11}\circ\dots\circ x^{(2)}_{1N^{(2)}_1}=[1]$. Looking under the first round-two-measurement, $x^{(2)}_{11}$, we see three round one measurements (blue boxes). Thus, these measurements must have totaled to identity; i.e., $N^{(1)}_{11}=3$ and $x_{111}^{(1)}\circ\dots\circ x_{11N^{(1)}_{11}}^{(1)}=[1]$. Note, this is just the first substring ($i=1$); the state continues to the right.

Theorems & Definitions (5)

• Lemma 1
• proof
• proof
• Lemma 1
• proof