SciPost Submission Page
Brick Wall Quantum Circuits with Global Fermionic Symmetry
by Pietro Richelli, Kareljan Schoutens, Alberto Zorzato
Submission summary
| Authors (as registered SciPost users): | Alberto Zorzato |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2402.18440v2 (pdf) |
| Date submitted: | 2024-03-05 16:48 |
| Submitted by: | Zorzato, Alberto |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We study brick wall quantum circuits enjoying a global fermionic symmetry. The constituent 2-qubit gate, and its fermionic symmetry, derive from a 2-particle scattering matrix in integrable, supersymmetric quantum field theory in 1+1 dimensions. Our 2-qubit gate, as a function of three free parameters, is of so-called free fermionic or matchgate form, allowing us to derive the spectral structure of both the brick wall unitary $U_F$ and its, non-trivial, hamiltonian limit $H_{\gamma}$ in closed form. We find that the fermionic symmetry pins $H_{\gamma}$ to a surface of critical points, whereas breaking that symmetry leads to non-trivial topological phases. We briefly explore quench dynamics and entanglement build up for this class of circuits.
Current status:
Reports on this Submission
Strengths
1- clear and detailed study of the spectrum and (absence of) topological features
Weaknesses
1- the physical motivation for such circuits with fermionic symmetry is not clearly explained
2- the study of the spectrum of the circuit dynamics does not bring about a huge amount of novelty with respect to that of the Hamiltonian limit
3- analysis of entanglement based on numerics for very small system sizes (10 sites at most), hence no strong conclusions can be reached
Report
This paper conducts an in-depth study of a new sort of brickwork quantum circuits, characterized by a fermionic symmetry inspired from previous studies in quantum field theory.
From the statistical mechanics point of view those are equivalent to special regimes of previously known eight vertex model, however the fermionic symmetry brings about new features : first, it introduces a graded tensor product structure; second, and more importantly, imposing the fermionic symmetry imposes stronger restrictions which pin the model to a gapless phase. This is clearly seen in the Hamiltonian limit, connecting to the well-known Kitaev chain, but the authors further extend their study to the circit geometry.
This a potentially interesting paper, however the physical motivation for the study of such circuits should be better explained in my opinion.
Why, from the condensed matter point of view, study such circuits ? If my understanding is correct that the difference between these and circuits constructed directly from the eight vertex model (that is, without the graded tensor product structure) boils down to a question of boundary conditions, then are there important physical differences between the two settings ? And should we expect to see one more naturally than the other in experiments or physical materials ?
I would appreciate if the authors could elaborate more on these issues, as well as on the more specific points addressed in the "Weaknesses" and "Requested changes" sections.
Besides this, this is a well-written and technically solid paper, which may be suitable for publication in SciPost.
Requested changes
1- Section 1 : could the authors give more physical motivation for the study of such circuits ? (see Report section)
2-Section 1.3: when mentioning statistical mechanics, the name "eight vertex model" is never used. Would seem relevant though.
3- At the end of Sec. 2.4, it would be useful to see the expression of the fermionic symmetry generators $Q^l$, $Q^r$ in terms of the fermion creation/annihilation operators
4- Before eq. (3.13) and eq. (4.9) : notation $|- \rangle$ should be explained
5- p. 15: one or several references would be appreciated when discussing the class BDI
Recommendation
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