Exact strong zero modes in quantum circuits and spin chains with non-diagonal boundary conditions

Clear explanations

Technically strong result in the context of integrability

Important question of strong zero modes

It feels like more "incremental" than a fundamental new results or new techniques (but this is said without reducing the technical challenges involved and interest of the results themselves in the integrability community).

In this paper, exact strong zero modes are constructed explicitly for the XXZ chains and related quantum circuits. Such zero modes have been studied actively recently, and in particular constructed for these models (e.g. refs [5,20]), however this was for certain boundary conditions that preserve the bulk $U(1)$ symmetry of the time evolution operator. Here, this is generalised to the most general boundary conditions that break it while preserving a $\mathbb Z_2$ symmetry (as is required by the general definition of strong zero mode).

The paper is clearly written, even though, as is usual in these studies, the expressions of strong zero modes (as operators acting on the quantum chain or circuit) can be quite complicated. The results are also interesting, generalising previous results to an important class, with essentially arbitrary boundary conditions. The results appear to be correct, with good numerical checks.

Without implying any reduction in the interest, technical strength and accuracy of the results, I would still characterise them more as “incremental” than as game-changing within the area of study, so I would suggest publication in Scipost Physics Core, once the requested changes below are addressed.

• P2 after eq 3: $U(1) \otimes Z_2$: isn’t this a semi-direct product instead of a tensor product?

• Eq 4: you mean for $\vec n= \vec h_1$?

• Eq 7: It would be clearer to write the form of $V_{j,j+1}$ more generally.

• Eq 8 and line after: I think it is the notation $\boldsymbol \ell$, not $\vec\ell$, that is used throughout.

• P 5: “The real parameters $\ell$ and $r$ characterizing the quantum circuit in Eq. (8) are obtained by an appropriate choice of …” what does this mean - are all values of $\xi^{(R)}$ etc allowed, or only some values? In the latter case, what are the constraints?

• Eq 11: probably a typo, $\tilde{\Delta \tau}$ should be $\tilde\Delta\tau$?

• From eq 11 it looks like there are constraints on $\tau$, e.g. it cannot be near to $\pi$ as otherwise, for many choices of $\tilde\Delta$, the quantity $\sin(\delta)$ would not lie within $[-1,1]$.

• In find that the description from eq 9 to 14 could be clarified: it would clarify if all variables ($\tau,\,\Delta,\,\eta,\,\xi^{(\cdot)},\,s^{(\cdot)},\,\varphi^{(\cdot)}$) were given their exact range of possible values, along with the constraints written.

• Eq 30: $\tilde{\mathcal A}$, etc only appeared up to then within a box in a graphical representation, this equation I suppose means equality of these box quantities but it is a bit strangely presented here. Same thing for $D$ in paragraph above eq 32.

• Eq 31: $\tilde{\mathcal A}$ is a 4-index quantity, so diagonalisation has to be explained. Supposedly in terms of groupings into 2 bi-indices. It would be good to clarify how these graphical symbols are interpreted as matrices.

• I think, what the letter, non-graphical symbol represents is indeed the corresponding matrix, please clarify.

• In don’t find the arguments in eqs 41-43 very convincing. Everything hinges on the expected decay of $C^{\mathcal O’}(t)$ as $t\to\infty$. But how is this explained? It would be better perhaps to use projection, something like von Neumann’s mean ergodic theorem (as applied to unitary operators): $\lim_{T\to\infty} T^{-1}\int_0^T dt U_t = P_1$ (or similar for sums over discrete values of time) for any unitary $U$ on any Hilbert space, where $P_1$ is the projection onto its eigenvalue-1 eigenspace. Such an argument leads to the following. In the Hilbert space induced by the Hilbert-Schmidt inner product (induced by Cauchy completion), the assumption is that the eigenvalue-1 eigenspace of the evolution operator is composed of $\mathbb \Psi$ along with operators have have zero-overlap with operators sitting around site 0. That is the main assumption.

• P 6: “if (18) is fulfilled then we have…” Is this general statement proven somewhere? Below it seems like the statement eq 19 is shown for the MPO representation, but I’m not sure the general statement “if … then …” is actually shown.

• P6 also, eqs 21-23 are these shown in the calculations that follow? Please clarify the logical organisation, what are the statements, which ones are shown and where, and in what level of generality.

Accept in alternative Journal (see Report)