{"id":"b3a77779-c7bf-4f0d-b064-b0cc42475c2a","arxiv_id":"2505.08888","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"Simulations show that storing qubits in Majorana edge modes and executing gates with vortex-core Majorana zero modes realizes Z, X, and approximate Hadamard gates in a 2D topological superconductor.","lead":"This paper uses computer simulations to show that Majorana edge modes and vortex-bound Majorana zero modes in a two-dimensional superconductor can work together to perform quantum logic gates. It proposes a layout where the edge modes store quantum information while movable vortices execute the gate operations, potentially simplifying how topological quantum computers could be built.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (6)'s phase interpolation may create phase slips; simulated vortex motion might not represent braiding.","rationale":"The paper provides a substantial numerical demonstration: full many-body dynamics, transition probabilities near unity for √Z, Z, X, and 3-qubit √X gates, and geometric phase differences matching Ising anyon statistics. These are real independent evidence if the time-dependent Hamiltonian faithfully represents vortex braiding. The weakest link is the phenomenological construction of the moving vortex in Appendix A. The magnitude profile is admittedly non-self-consistent but supported for robustness by Ref. [55]; the phase interpolation in Eq. (6), however, is not examined. Linearly interpolating the phase between endpoint vortex configurations is a topological operation that does not generally correspond to a single moving vortex; it can induce phase slips where the winding number around a fixed loop changes discontinuously, potentially closing the topological gap and creating non-adiabatic excitations. The paper's diagnostics, final transition probabilities and geometric phases, are computed within the same interpolation and cannot reveal such phase slips if they occur away from the relevant wavefunction overlap. A direct computation of plaquette vorticity, or a re-run using Eq. (5) evaluated at the continuously moving core position, would settle whether the interpolation is benign. This concern is specific and testable, but it does not overturn the reader's CONDITIONAL verdict: the paper should be accepted only after the vortex motion is shown to be topologically consistent with a single moving vortex throughout the braid.","tokens_in":11068,"tokens_out":15465,"duration_ms":155023,"concrete_test":"Compute the gauge-invariant plaquette vorticity v_p(t) = (1/2π) Σ_δ arg[Δ_r(t) Δ_{r+δ}^*(t)] along the vortex paths of Figs. 2 and 3, and locate plaquettes with v_p = ±1 as a function of time. If a vortex position jumps, or a plaquette's vorticity changes without |Δ| vanishing there, Eq. (6) introduces phase slips. A stronger check: rerun the Fig. 2 √Z gate with φ(r,t) recomputed directly from Eq. (5) using the moving R_i(t) of Eq. (3) instead of the interpolation in Eq. (6), and compare p_E_{1,1}(t_f) and the geometric phase difference Δφ; a material difference would show the interpolation is load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires the time-dependent Hamiltonian to describe vortices moving continuously along the paths in Figs. 2–4. Appendix A models this by interpolating the superconducting phase as φ(r,t) = φ(r,t0) + [φ(r,t0+tV)−φ(r,t0)] s(t,t0,tV) (Eq. 6), with endpoint phases from Eq. (5). Linear interpolation of endpoint phases is not equivalent to translating a vortex: for a vortex moving from R0 to R1, the interpolated phase is (1−s)φ0 + sφ1, and the gauge-invariant winding number around a loop enclosing R0 changes from 1 to 0 at some intermediate s — a phase slip, a topological event not accompanied by |Δ| vanishing on the loop. Thus the instantaneous Hamiltonian may leave the single-vortex sector, closing the gap and generating non-adiabatic excitations. The stated adiabaticity condition (tV ≫ ℏ/Δt) and the final transition probabilities are computed within this interpolation and cannot detect phase slips occurring away from MEM/MZM wavefunction overlap. The Appendix A caveat, with Ref. [55], addresses only the magnitude profile |Δr(t)|, not the phase interpolation. If phase slips occur, the reported p = 1 and p = 0.999 do not establish braiding of physical vortices.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes using a two-dimensional topological superconductor containing topological and trivial domains as a platform for topological quantum computing. Qubit states are stored in delocalized Majorana edge modes (MEMs), and vortices carrying vortex-core Majorana zero modes (MZMs) are moved through domain walls to execute gates. The authors perform full many-body time evolution and report near-unit transition probabilities for sqrt(Z), Z, and X gates, an approximately uniform 8-state superposition for a 3-qubit sqrt(X)/Hadamard-type gate, and geometric phase differences consistent with Ising anyon statistics. They also visualize the processes through non-equilibrium local density of states and Majorana world lines.","tokens_in":11377,"tokens_out":6078,"duration_ms":63280,"significance":"If the central claim holds, the proposed MEM-as-memory and MZM-as-gate architecture is a genuinely new and potentially scalable way to use 2D topological superconductors for quantum information processing. The paper's strengths are its direct many-body time evolution rather than fitting to expected outcomes, the use of a gauge-invariant geometric phase as a consistency check, and the concrete visualization of the gate processes. These methods give the results a concreteness that is often missing in braiding proposals. However, the physical interpretation of the engineered vortex motion and the strength of the claims ('fault-tolerant', 'topologically protected', 'Hadamard gate') need to be brought into line with what is actually demonstrated before the paper can be accepted.","major_comments":[{"comment":"The time dependence of the superconducting phase in Eq. (6) is not equivalent to translating a vortex. For a vortex initially at R_i(t0) moving to R_i(t0)+a, the endpoint fields phi(r,t0) and phi(r,t0+tV) have branch cuts in different locations, and the interpolation phi(r,t) = phi(r,t0) + [phi(r,t0+tV)-phi(r,t0)] s(t,t0,tV) changes the winding number around a loop that encloses only the initial core from 1 to 0 at some intermediate s while |Delta_r(t)| stays nonzero on that loop. Such a change requires phase singularities (phase slips) not accompanied by a vanishing of |Delta|, so the instantaneous Hamiltonian can leave the single-vortex sector and close the gap. Because all reported gate probabilities and geometric phases are computed within this interpolation, the central claim that physical vortices have been braided is not yet fully supported. The adiabaticity discussion and the robustness caveat in Appendix A concern the magnitude profile only, not this phase-interpolation issue. Please either implement a phase profile that rigidly translates the vortex (for example phi(r - R_i(t)) with a consistent branch choice) or provide a numerical check that the local winding number of the order parameter is conserved throughout the trajectory.","section":"Appendix A, Eq. (6)"},{"comment":"The terms 'fault-tolerant' and 'topologically protected' are stronger than what the simulations demonstrate. The calculations use a fixed disorder-free Hamiltonian, a phenomenological vortex trajectory, and a single quasiparticle broadening Gamma = 0.01 t_e; there is no analysis of quasiparticle poisoning, unwanted vortex-vortex interactions, disorder, initialization or measurement errors, or a systematic study of how errors scale with system size. A single gate fidelity of p = 0.999 above a quantum error correction threshold does not by itself establish fault tolerance. Please either provide the missing error analysis or rephrase the claims to say that the architecture is a promising platform for fault-tolerant topological quantum computing rather than that fault tolerance has been achieved.","section":"Abstract and Discussion"},{"comment":"The abstract claims the successful simulation of Z-, X-, and Hadamard gates, but the Hadamard gate is not directly simulated in this paper. The text around Fig. 4 presents an N=3 qubit sqrt(X)-gate and says that this operation 'can be utilized as a Hadamard gate', citing Ref. [44]; no time-dependent transition probabilities or final-state fidelities for a Hadamard operation are shown here. If the Hadamard gate is one of the central claimed results, please provide the explicit simulation or, alternatively, revise the abstract and the claims to say that a sqrt(X)/Hadamard-type gate is realized.","section":"Abstract; Fig. 4 and surrounding text"},{"comment":"There is an apparent inconsistency in the geometric phase criterion. The paper states that Ising anyon statistics give Delta_phi as an odd integer multiple of pi/2, yet the Z-gate is reported with Delta_phi = -13 pi, which is an even integer multiple of pi/2 (though it is an odd multiple of pi, as expected for a double exchange). Please state the criterion separately for single exchange (sqrt(Z)) and double exchange (Z), or explain explicitly why -13 pi is consistent with the stated odd-multiple-of-pi/2 condition.","section":"Fig. 2(k) and Appendix B"}],"minor_comments":[{"comment":"As written, the smoothed Heaviside function s(t,t0,tV) = sin^2((t-t0)/tV) does not reach 1 at t = t0 + tV; it reaches sin^2(1) which is about 0.708. Presumably the intended form is sin^2(pi(t-t0)/(2 tV)) or an equivalent expression with the correct boundary values.","section":"Appendix A, Eq. (2)"},{"comment":"The Data Availability statement contains the placeholder text '(insert link to Zenodo depository)'; the actual Zenodo link should be provided.","section":"Data Availability"},{"comment":"The statement that p = 0.999 is 'above the threshold for quantum error correction' cites both Ref. [32] and Ref. [53]; Ref. [32] is about controlled vortex manipulation and does not appear to contain a threshold result, so the threshold citation should be to Ref. [53] alone or to the appropriate error-correction literature.","section":"Section Results, X-gate paragraph"},{"comment":"The caption states that panels (a),(b) use a 24x24 system while panel (c) uses a 40x40 system, but the text does not discuss finite-size effects or why the system size changes mid-protocol; a brief explanation would help the reader.","section":"Fig. 4 caption"}],"recommendation":"major_revision","confidential_remarks":"The main technical risk is the phase-interpolation issue in Eq. (6): if the authors can switch to a rigidly translating phase profile and still reproduce the reported transition probabilities, or can convincingly rule out phase slips, the paper would be a solid contribution. The paper relies heavily on the authors' prior numerical methods (Refs. [42,43]), but the MEM+MZM architecture itself appears new relative to that prior work. I do not see a circularity problem in the central derivation; the geometric phase comparison to Ising statistics is a genuine consistency check rather than a fit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious numerical proposal from a group that has built this machinery over several papers, and the specific architecture—store qubits in delocalized Majorana edge modes, execute gates by dragging vortices through domain walls, and transfer MEM states into vortex-core MZM states—is not in the cited literature as far as I can tell. The full many-body time evolution is a real step beyond static spectral checks, and the numbers are consistent: p near 1 for Z and sqrt(Z), 0.999 for X, a flat ~1/8 distribution for the three-qubit gate, and geometric phase differences in odd multiples of pi/2 matching Ising anyon statistics. The mixed-encoded qubit transfer is the most original piece and could be practically useful.\n\nWhere I would push back. The abstract advertises a Hadamard gate, but the paper simulates a 3-qubit sqrt(X) and relies on a previous paper to call it Hadamard. That is an overstatement. Also 'fault-tolerant' and 'topologically protected' are asserted without any error budget, decoherence analysis, or disorder study; this is an ideal-simulation claim, not a fault-tolerance proof. The data availability line is a placeholder ('insert link to Zenodo'), and no code is included.\n\nThe bigger issue is whether the time-dependent order parameter in Eqs. (4)-(6) really describes moving vortices. The stress-test note is correct: Eq. (6) linearly interpolates the phase between endpoint configurations. For a loop enclosing the old vortex position, the interpolated winding is (1-s) times an integer, which is not the phase of a translating vortex unless branch cuts are handled in a very specific way. That can create artificial phase slips and non-adiabatic excitations that the simulation may not detect if they occur away from the MEM/MZM wavefunction overlap. The Appendix A caveat addresses only the magnitude profile, not the phase interpolation. This is a load-bearing soft spot for the claim that the simulated process is braiding of physical vortices.\n\nWho this is for: researchers working on 2D magnet-superconductor hybrids and on dynamical simulations of Majorana systems. The architecture is a reasonable extension of the group's prior work, and the numerics deserve serious engagement. I would send it to a referee, but not with the abstract as written—the Hadamard and fault-tolerance claims need recalibration, and the phase-interpolation worry needs either a convincing response or a revised simulation scheme.","headline":"A serious numerical proposal for MEM-based memory with vortex-core MZM gates, but the abstract overclaims a Hadamard gate and fault tolerance, and the phase interpolation in Eq. (6) may be a genuine load-bearing problem.","tokens_in":11879,"tokens_out":3893,"would_cite":false,"duration_ms":43179,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Dragging vortices through domain walls turns Majorana edge modes into a quantum memory that executes fault-tolerant gates.","keywords":["Majorana edge modes","Majorana zero modes","topological quantum computing","vortex braiding","Ising anyon statistics","two-dimensional topological superconductor","quantum memory","non-equilibrium Green's functions"],"falsifier":"A self-consistent time-dependent simulation of a vortex dragged across a domain wall at the same model parameters, using the actual self-consistent order parameter instead of the smoothed-Heaviside profile, should reproduce $p^E_{1,1}(t_f)=1$ for the $\\sqrt{Z}$ and $Z$ gates and $p^E_{10,01}(t_f)=0.999$ for the $X$ gate; any substantial departure, or a geometric phase difference $\\Delta\\phi$ that is not an odd multiple of $\\pi/2$, would falsify the claim.","tokens_in":10890,"feed_emoji":"🌀","tokens_out":10867,"duration_ms":95654,"temperature":0.7,"pith_summary":"This paper argues that a two-dimensional topological superconductor containing both topological and trivial domains can serve as a working platform for fault-tolerant quantum computing. In the proposed architecture, qubit states are stored in delocalized Majorana edge modes (MEMs) along domain walls, while magnetic vortices are dragged through those walls: each vortex picks up an edge mode as a vortex-core Majorana zero mode (MZM), braids it, and returns it, executing $Z$, $X$, and Hadamard gates. The authors compute the full many-body dynamics and report success probabilities of $p^E_{1,1}(t_f)=1$ for the $\\sqrt{Z}$ and $Z$ gates, $p^E_{10,01}(t_f)=0.999$ for the $X$ gate, and approximately $0.125=2^{-3}$ for each of the eight outputs of a three-qubit $\\sqrt{X}$/Hadamard gate, with geometric phase differences matching Ising anyon statistics. The paper claims this gives a scalable architecture in which adding one trivial domain bubble containing two vortices adds one qubit.","feed_headline":"Vortex moves turn Majorana edge modes into quantum gates","feed_subtitle":"Full many-body simulations show Z, X, and Hadamard gates on Majorana qubits with near-perfect fidelity.","key_machinery":"The load-bearing object is the time-dependent superconducting order parameter $\\Delta_{\\mathbf r}(t)=|\\Delta_{\\mathbf r}(t)|e^{i\\phi(\\mathbf r,t)}$, with Majorana edge modes being the one-dimensional Majorana states living on domain walls and Majorana zero modes their point-like vortex-core counterparts. The order parameter's magnitude is suppressed to zero inside a radius $R_V$ around each moving vortex core via a $\\sin^2$ profile, and its phase is interpolated between configurations by a smoothed Heaviside function $s(t,t_0,t_V)$. Driving this profile slowly enough ($t_V$ much larger than $\\hbar/\\Delta_t$, where $\\Delta_t$ is the topological gap) adiabatically converts a delocalized MEM into an MZM without populating Caroli-de Gennes–Matricon states. The full many-body dynamics are computed from a time-resolved Green's function, and the Ising anyon statistics are read off from the gauge-invariant geometric phase difference $\\Delta\\phi$ between even- and odd-parity states.","core_discovery":"On the paper's own terms, the central discovery is that a delocalized Majorana edge mode can be adiabatically transferred into a vortex-core Majorana zero mode and back, and that repeated transfer and braiding implements topologically protected quantum gates. The authors demonstrate this for a $\\sqrt{Z}$ gate (geometric phase difference $\\Delta\\phi=-9\\pi/2$), a $Z$ gate ($\\Delta\\phi=-13\\pi$), an $X$ gate with odd-parity transition probability $p^E_{10,01}(t_f)=0.999$, and a three-qubit $\\sqrt{X}$ gate that acts as a Hadamard gate with all eight even-parity output probabilities near $0.125$. Along the way, a qubit state initialized in the MEM occupation basis evolves adiabatically into a mixed-encoded state shared between vortex-core MZMs and their MEM partners, so the platform can define qubits using both types of Majorana states and convert between them.","pith_inferences":["Beyond the paper: the same adiabatic transfer could be paired with measurement-only braiding protocols, where parity measurements replace some vortex world-line crossings and reduce the number of slow vortex moves needed per gate.","A concrete testable extension is to repeat the simulation with a self-consistently computed moving-vortex order parameter; the zero-energy spectral weight should follow the same domain-wall-to-core path if the smoothed-Heaviside model is faithful.","The architecture suggests a memory hierarchy in which only the vortex array needs dynamical control while the edge-mode network stays static, which could simplify experimental control in large multi-bubble arrays.","Chaining these gates into multi-step quantum algorithms is a natural next step given the demonstrated three-qubit superposition, and the paper's companion work already begins that program."],"forward_implications":["Qubit states can be stored in MEMs along domain walls, so the edge modes act as a quantum memory that holds the information while the vortices are out in the trivial region.","Vortex-core MZMs, braided by dragging vortices with an STM tip, magnetic force microscopy, or SQUID-based manipulation, execute the gate operations on those stored states.","The adiabatic MEM-to-MZM transfer provides a new route to initialize the many-body states of vortex-core MZMs.","The architecture scales: each extra trivial domain bubble containing two vortices adds one qubit, as demonstrated by the three-qubit $\\sqrt{X}$/Hadamard gate with eight outputs at roughly $0.125$.","The reported success probabilities of at least $0.999$ lie above the two-dimensional fault-tolerance threshold, supporting the claim that these gates are topologically protected."],"supporting_citations":[{"why":"Establishes that 2D topological superconductors host Majorana edge modes on domain boundaries, the memory element used in the architecture.","marker":"[1–7]"},{"why":"Proposes that disorder in FeSe1−xTex yields coexisting topological and trivial domains, the geometry the gate scheme exploits.","marker":"[40, 41]"},{"why":"Supplies the many-body simulation approach that avoids an exponential Hilbert space, enabling full dynamics of the braiding process.","marker":"[42]"},{"why":"Provides the gate-simulation framework and the gauge-invariant geometric-phase functional used to extract the Ising anyon statistics.","marker":"[43]"},{"why":"Shows the sqrt(X) gate can act as a Hadamard gate, a step the three-qubit demonstration relies on.","marker":"[44]"},{"why":"Defines the Chern-number C = -1 topological phase of a ferromagnetic Shiba lattice, the model phase used in the simulations.","marker":"[47]"},{"why":"Shows vortex-core Majorana zero modes are robust to the spatial profile of the superconducting order parameter, justifying the phenomenological profile used for moving vortices.","marker":"[55]"},{"why":"Supplies the two-dimensional fault-tolerance threshold against which the reported gate success probabilities are measured.","marker":"[53]"}],"fun_headline_variants":["Majorana edge modes become quantum memory for gate operations","Vortex cores pair with edge modes for topological qubit gates","Majorana edge modes store qubits for fault-tolerant computing","Edge-mode memory enables Z, X, Hadamard gates on Majorana qubits","Delocalized Majorana modes act as memory for quantum gates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The scheme rests on the assumption that a vortex dragged through a domain wall moves adiabatically: the smoothed time-dependent order-parameter profile must transfer the Majorana state into the vortex core without creating ordinary excited states, so the reported probabilities (1, 0.999, and 0.125 per output) are not artifacts of the chosen profile and interpolation time $t_V$.","fun_headline_variants_meta":{"raw":{"variants":["Majorana edge modes become quantum memory for gate operations","Vortex cores pair with edge modes for topological qubit gates","Majorana edge modes store qubits for fault-tolerant computing","Edge-mode memory enables Z, X, Hadamard gates on Majorana qubits","Delocalized Majorana modes act as memory for quantum gates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000226,"raw_usage":{"total_tokens":1407,"prompt_tokens":823,"completion_tokens":584,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":439,"completion_tokens_details":{"reasoning_tokens":496}},"tokens_in":439,"tokens_out":584,"duration_ms":5683,"temperature":1.0,"reasoning_tokens":496,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:45:25.241362+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A self-consistent time-dependent simulation of a vortex dragged across a domain wall at the same model parameters, using the actual self-consistent order parameter instead of the smoothed-Heaviside profile, should reproduce $p^E_{1,1}(t_f)=1$ for the $\\sqrt{Z}$ and $Z$ gates and $p^E_{10,01}(t_f)=0.999$ for the $X$ gate; any substantial departure, or a geometric phase difference $\\Delta\\phi$ that is not an odd multiple of $\\pi/2$, would falsify the claim.","supporting_citations":[{"cited_title":"Mascot, T","cited_arxiv_id":null,"evidence_quote":"Supplies the many-body simulation approach that avoids an exponential Hilbert space, enabling full dynamics of the braiding process."},{"cited_title":"Bedow, E","cited_arxiv_id":null,"evidence_quote":"Provides the gate-simulation framework and the gauge-invariant geometric-phase functional used to extract the Ising anyon statistics."},{"cited_title":"Rachel, E","cited_arxiv_id":null,"evidence_quote":"Defines the Chern-number C = -1 topological phase of a ferromagnetic Shiba lattice, the model phase used in the simulations."},{"cited_title":"Nagai, H","cited_arxiv_id":null,"evidence_quote":"Shows vortex-core Majorana zero modes are robust to the spatial profile of the superconducting order parameter, justifying the phenomenological profile used for moving vortices."}],"review_version":1}