{"id":"1641cf2e-952e-4a3e-b022-e26090f2ad33","arxiv_id":"2501.04091","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For unitary-invariant random gate distributions, operator spreading follows drift-diffusion with distribution-dependent butterfly velocity and diffusion constant, plus finite-time and domain-wall corrections.","lead":"This paper studies how quantum operators spread in random quantum circuits when the gates are drawn from a broad class of rotationally symmetric random distributions, not just the fully random Haar distribution. It shows the spreading still obeys a drift-diffusion law, with modified butterfly velocity and diffusion constant, plus finite-time and finite-width corrections that vanish in the Haar limit.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The general-ensemble claim rests on the uncontrolled truncation (41), validated only along the Poisson-kernel curve; a non-Poisson gate-distribution test is needed to support v_B and D for arbitrary unitary-invariant ensembles.","rationale":"The reader's weakest_assumption identifies exactly the truncation of the n-point density hierarchy, Eq. (41), as the load-bearing uncontrolled step. I agree: the paper's general results for arbitrary unitary-invariant gate distributions rely on this truncation, and the evidence for its validity is restricted to the Poisson kernel family. This is genuinely the single most load-bearing concern because every general statement about v_B, D, and the drift-diffusion limit flows through the closed hierarchy obtained by that truncation. The exact closure of the projected binary process (lumpability of the Pauli-string Markov chain under the binary partition) is not in question, so no hidden issue undermines the framework before the truncation. The Poisson-kernel calculations are convincing on their own: the analytical results reproduce Haar and trivial limits, the truncation converges rapidly in n, and the direct simulation of the exact binary process agrees. The concern is therefore about the extrapolation from the Poisson-kernel curve to the full allowed parameter space, not about internal consistency. A targeted simulation at a non-Poisson point, together with a direct test of the factorized ansatz (41), would settle whether the general claim holds. Since the conditional verdict already flags this exact issue and the Poisson-kernel results remain intact, no change to the verdict is warranted.","tokens_in":28170,"tokens_out":14966,"duration_ms":144652,"concrete_test":"Simulate the exact binary Markov chain (33) for a gate distribution at a point in the allowed parameter space of Fig. 2 far from the Poisson-kernel curve—e.g., U0 fixed to a permutation or Clifford matrix with Haar-random conjugation, giving a specific (Re A1, A2+A3). Measure the front velocity and diffusion constant from the late-time spreading of the right-propagating density ¯ρ(0)(Δx;t) via the slope and broadening of the front, with N≥2×10^5 realizations. Compare with the analytical v_B^(n), D^(n) for n=0,2,4. If any n=4 prediction differs from the simulation by more than 5% of the Haar value, the truncation (41) fails for general ensembles.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim that operator spreading for any unitary-invariant gate distribution is governed by drift-diffusion with computed v_B and D depends on the truncation in Eq. (41): the (n+1)-point density is replaced by its maximally random factorized form beyond a finite distance n_DW. This is an uncontrolled structural assumption about the operator front. It is verified only for the Poisson kernel ensemble, i.e., along a one-parameter curve in the allowed (Re A1, A2+A3) parameter space of Fig. 2. The numerical simulations in Figs. 4 and 5 simulate the same binary Markov process that the approximation is meant to solve, so they validate the Poisson-kernel case but do not constrain other unitary-invariant distributions. If some allowed ensemble (for example, a fixed unitary conjugated by Haar randomness) has a much larger domain-wall width, the truncated hierarchy would yield incorrect v_B and D, invalidating the general claim. The closure of the projected binary process itself is exact by lumpability, so the truncation is the sole uncontrolled approximation in the derivation of the general drift-diffusion coefficients.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies operator spreading in one-dimensional random unitary circuits whose two-qudit gates are drawn from any unitary-invariant distribution. It shows that the ensemble-averaged Pauli-string weights satisfy a closed Markov process, which can be projected exactly onto a binary (identity/non-identity) stochastic growth model. The evolution of right-propagating n-point densities is closed by a maximal-randomness truncation, Eq. (41), leading to drift-diffusion equations with a butterfly velocity v_B and diffusion constant D. For the Poisson-kernel ensemble, the paper derives explicit formulas for the relevant moments, for v_B and D, for the binary-relaxation time τ_b, and for the domain-wall correction to the OTOC. The analytical Poisson-kernel results are compared with numerical simulations of the projected binary stochastic process, and the paper also treats the Dyson Brownian-motion ensemble in an appendix.","tokens_in":28354,"tokens_out":13364,"duration_ms":124852,"significance":"If the general-ensemble claim holds, the paper extends the hydrodynamic description of operator spreading from Haar-random circuits to the full unitary-invariant class, with quantitative predictions that depend on the gate ensemble through Re A1 and A2+A3. The paper is unusually careful in several places: the Markovian reduction and the binary projection are exact, the Poisson-kernel parameters are not fitted, and the results reproduce the Haar and trivial limits correctly. The explicit Poisson-kernel formulas, including the finite τ_b and finite domain-wall width n_DW, are new and are supported by convergence-in-order checks and by simulations of the same stochastic process. The main limitation is that the truncation (41) is an uncontrolled approximation for the general unitary-invariant class, so the breadth of the central claim currently outpaces the evidence.","major_comments":[{"comment":"The general-ensemble calculation of v_B and D rests on replacing the (n+1)-point density by its maximally random factorized form in Eq. (41). This is an uncontrolled closure except in the Haar limit and the q→∞ limit, and it is verified only for the Poisson-kernel ensemble in Figs. 4–6 and, without numerical simulation, for the Brownian-motion ensemble in App. E. Because the simulations in Figs. 4 and 5 run the same projected binary Markov process (33) that the truncation is meant to solve, they do not constrain other unitary-invariant distributions. I ask the authors to test Eq. (41) at least one additional point in the allowed (Re A1, A2+A3) region of Fig. 2, for example a fixed two-qudit gate U0 conjugated by Haar-random V, and to compare direct simulations of the binary process with v_B^(n) and D^(n) for n=0,2,4. If no such test is provided, the claims in Secs. III and VI about the entire unitary-invariant class should be explicitly restricted to ensembles for which convergence of the truncation has been demonstrated.","section":"Sec. III D, Eq. (41)"},{"comment":"The statement that λ_W<1 for every non-trivial unitary-invariant distribution is asserted without proof, even though the allowed parameter region in Fig. 2 is only sampled numerically. Since λ_W<1 is what guarantees the finite relaxation time τ_b, this should be justified, either by an analytic bound using the unitarity relations (17) or by a dense numerical scan of the allowed moment region. Without such a justification, the generality of the finite-τ_b claim remains an assumption.","section":"Sec. V A, Eq. (70)"}],"minor_comments":[{"comment":"There are typos: “Pauli strong” should be “Pauli string” in the abstract, and “out-of-time-ordrered” should be “out-of-time-ordered” in Sec. I.","section":"Abstract and Sec. I"},{"comment":"The degeneracy formula for even q is garbled: “Ni = q2/ − 8” should be a proper expression such as (q^4 − 8)/4 or the intended polynomial; please correct it.","section":"Sec. V A"},{"comment":"The Poisson-kernel parameter α is introduced as a complex number in Eqs. (6)–(7), but all final results depend only on |α|; the authors should state explicitly that the phase can be absorbed by a redefinition of the gate or that it is otherwise irrelevant.","section":"Sec. IV and App. F"},{"comment":"Inline notation such as “|α|2q4” is ambiguous: it can be read as |α|^2 q^4 or as |α|^{2q^4}. Using explicit braces or separate factors would avoid this confusion.","section":"App. F, Eqs. (F3)–(F10)"},{"comment":"The Brownian-motion ensemble results are presented without a numerical check; a sentence explaining that they follow the same convergence checks as the Poisson-kernel results, or a small figure, would strengthen the appendix.","section":"App. E"}],"recommendation":"major_revision","confidential_remarks":"The Poisson-kernel analysis is solid and the paper is honest about its main approximation. The central issue is scope: the abstract and conclusions claim results for all unitary-invariant gate distributions, but the only controlled numerical evidence for the truncation (41) is along the Poisson-kernel curve. A non-Poisson test in the (Re A1, A2+A3) plane would resolve whether this is a genuine general result or a property of the specific ensembles studied. If the test is not added, the paper would still be valuable if restricted to the ensembles explicitly verified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper does something real: it takes the Nahum–von Keyserlingk operator-spreading machinery for Haar-random circuits and extends it to any unitary-invariant two-qudit gate distribution. The new pieces are the Markovian reduction to binary strings, the n-point density hierarchy, and the explicit Poisson-kernel results: v_B and D as functions of alpha, the finite time tau_b to binary weights, and the finite domain-wall width. The math is careful; the Poisson-kernel formulas reproduce the Haar limit at alpha=0 and the trivial limit at alpha=1, and the stochastic-process simulations match the analytical curves well.\n\nThe soft spot is exactly the one the authors admit in the conclusion: the hierarchy is closed by replacing the (n+1)-point density with its maximally random approximation, Eq. (41). That is an uncontrolled approximation for a general unitary-invariant ensemble. It is validated numerically only along the Poisson-kernel one-parameter curve. The general claim that v_B and D are given by the drift-diffusion data for arbitrary unitary-invariant distributions is therefore a conjecture supported by one example, not a theorem. The stress-test note is right on this point. What saves the paper is that the paper itself says so, and the Poisson-kernel test is a genuine non-trivial family interpolating all the way from trivial to Haar.\n\nMinor: the degeneracy formulas in Eq. (70) have clear typos that should be fixed before publication.\n\nWho should read this: people working on operator spreading, random circuits, and scrambling, especially those who need non-Haar gates. The citation pattern is standard and appropriate. It is a serious, well-executed paper that deserves a proper referee. My recommendation is to engage, and conditional acceptance with a request for a sharper justification of the truncation (or at least one non-Poisson test) is the right posture.","headline":"Solid extension of Haar-random-circuit operator spreading to unitary-invariant gates; the Poisson-kernel results are reliable, but the general-ensemble claim rests on an unproven truncation.","tokens_in":28876,"tokens_out":2592,"would_cite":true,"duration_ms":24464,"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":"Operator spreading in random unitary circuits remains drift-diffusive for general unitary-invariant gate distributions, with gate-dependent butterfly velocity and diffusion constant; for the Poisson kernel the two coefficients are derived…","keywords":["random unitary circuits","operator spreading","butterfly velocity","unitary-invariant gate distribution","Poisson kernel","Pauli strings","out-of-time-ordered correlator","drift-diffusion equation"],"falsifier":"Run the binary stochastic growth process (33) for a unitary-invariant gate ensemble whose moments place $(\\mathrm{Re}\\,A_1,\\,A_2+A_3)$ at the boundary of the allowed region in Fig. 2 and check whether the extracted $v_{\\rm B}^{(n)}$ and $\\mathcal{D}^{(n)}$ converge with truncation order $n$; failure of convergence would show that the maximally-random ansatz (41) does not hold at finite distance.","tokens_in":27913,"feed_emoji":"⚛️","tokens_out":7807,"duration_ms":68555,"temperature":0.7,"pith_summary":"When a local quantum operator evolves in a random unitary circuit, it spreads into a sum of many nonlocal operator strings. This paper shows that the statistics of that spreading is governed by a classical random walk for a very broad class of gate distributions: any distribution unchanged by unitary rotations. As a result, the speed and diffusion of the operator front are now properties of the gate ensemble, not fixed numbers, and the paper derives them explicitly for the Poisson kernel ensemble, which interpolates between a trivial circuit and a maximally random one. The same hydrodynamic description also governs the out-of-time-ordered correlator that is commonly used to probe scrambling.","feed_headline":"Operator fronts obey drift-diffusion for any unitary-invariant ensemble","feed_subtitle":"Gate-dependent speed and diffusion characterize the operator front; explicit Poisson kernel formulas included.","key_machinery":"The key machinery is the mapping of the Pauli-string evolution to a classical stochastic growth model on projected binary strings, whose pair transition probabilities factorize and depend only on four moments of the gate distribution, $R_{1;1}$, $R_{2;2}$, $R_{1,1;1,1}$, and $R_{1,1;2}$. The hierarchy of right-propagating $n$-point densities is closed by the maximally-random truncation ansatz (41), and a two-component spinor rewrite converts the growth process into a drift-diffusion equation whose coefficients are given in terms of the transition matrix by Eqs. (50) and (51).","core_discovery":"The central claim is that for any two-qudit gate distribution satisfying the unitary-invariance condition $P(U)=P(VUV^\\dagger)$, the ensemble-averaged Pauli-string weights $\\rho_p(t)=\\langle|\\gamma_p(t)|^2\\rangle$ evolve under a closed Markov chain, and in the long-time limit the right-moving front of Pauli strings obeys a drift-diffusion equation with butterfly velocity $v_{\\rm B}$ and diffusion constant $\\mathcal{D}$ that depend on the gate distribution. For the Poisson kernel ensemble, the paper obtains explicit formulas for $v_{\\rm B}$ and $\\mathcal{D}$ (Eqs. (58) and (59)) that reduce to the Haar values at $\\alpha=0$, vanish in the trivial limit $\\alpha\\to1$, and in the large-$q$ limit give $v_{\\rm B}=(1-|\\alpha|^4)/(1+|\\alpha|^4)$ and $\\mathcal{D}=4|\\alpha|^4(1-|\\alpha|^4)/(1+|\\alpha|^4)^2$. Two qualitative differences from the Haar case are established: a finite time $\\tau_{\\rm b}$ elapses before Pauli-string weights acquire a binary form, and the operator front has a finite domain-wall width $n_{\\rm DW}$ separating a random-matrix-like bulk from a trivial region.","pith_inferences":["If the unitary-invariant result holds generally, the scrambling hydrodynamics of a random circuit is parametrized by the moments of the gate distribution rather than by the qudit dimension alone; the Poisson kernel is only the maximum-entropy representative of that family.","The dependence of $\\tau_{\\rm b}$ and $n_{\\rm DW}$ on the gate ensemble suggests a practical diagnostic: the subleading OTOC correction (Fig. 8) could be used experimentally to distinguish Haar-like scrambling from structured unitary-invariant scrambling even when the main front profile looks identical.","The agreement between the Poisson-kernel and Brownian-motion ensembles at large $q$ (after identifying $|\\alpha|^2=\\mathrm{e}^{-\\lambda}$) raises the possibility that the leading front parameters depend only on the first moment $\\langle U\\rangle$ in that limit, a property the paper does not prove."],"forward_implications":["For any unitary-invariant gate distribution, the operator front in a random circuit is asymptotically a drift-diffusion front, so measuring the OTOC profile determines the gate-dependent $v_{\\rm B}$ and $\\mathcal{D}$ directly.","For the Poisson kernel at fixed $|\\alpha|>0$ in the large-$q$ limit, the front remains diffusive with $\\mathcal{D}>0$ and $v_{\\rm B}<1$, in contrast to the Haar limit where $v_{\\rm B}\\to1$ and $\\mathcal{D}\\to0$.","The finite time $\\tau_{\\rm b}$ to reach a binary Pauli-string distribution and the finite width $n_{\\rm DW}$ of the domain wall grow only when the circuit approaches the trivial limit $|\\alpha|\\to1$.","The long-time OTOC is well approximated by a complementary error function profile, with a subleading correction $\\delta$ that becomes independent of position and time at late times.","The same truncation scheme, applied to the Dyson Brownian-motion interpolating ensemble, yields closed-form drift-diffusion coefficients that agree with the Poisson-kernel results in the large-$q$ limit."],"supporting_citations":[{"why":"Supplies the Haar-circuit mapping to a stochastic growth model and the baseline values $v_{\\rm B}=(q^2-1)/(q^2+1)$ and $\\mathcal{D}=4q^2/(q^2+1)^2$ that this paper generalizes.","marker":"[4]"},{"why":"Provides the expression for the OTOC in terms of right-propagating densities and the long-time complementary error function profile, which this paper extends with the domain-wall correction $\\delta$.","marker":"[5]"},{"why":"The diagrammatic method for integration over the unitary group is used to derive the pair transition probability (26) from the four moments of the gate distribution.","marker":"[24]"},{"why":"Supplies the parametrization $U=(\\alpha\\mathbf{1}-U_0)(\\mathbf{1}-\\alpha^*U_0)^{-1}$ of Poisson-kernel distributed matrices, which is used to compute the moments for the explicit results.","marker":"[16]"},{"why":"Identifies the Poisson kernel as the maximum-information-entropy distribution with fixed ensemble average $\\langle U\\rangle=\\alpha\\mathbf{1}$, motivating the interpolation between the trivial and Haar limits.","marker":"[17]"},{"why":"Defines operator spreading and the out-of-time-ordered correlator used throughout the paper as the observable characterizing scrambling.","marker":"[12]"}],"fun_headline_variants":["Universal drift-diffusion for any unitary-invariant gate ensemble","Beyond Haar: operator fronts get a finite boundary layer","Unitary-invariant circuits: universal diffusion, finite-time front width"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, at distances more than a few sites behind the operator front, the Pauli-string distribution is already maximally random, so the hierarchy of $n$-point densities can be truncated at finite $n$; this is verified numerically for the Poisson kernel but not proven for every unitary-invariant ensemble.","fun_headline_variants_meta":{"raw":{"variants":["Universal drift-diffusion for any unitary-invariant gate ensemble","Beyond Haar: operator fronts get a finite boundary layer","Unitary-invariant circuits: universal diffusion, finite-time front width"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002358,"raw_usage":{"total_tokens":9136,"prompt_tokens":1050,"completion_tokens":8086,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":666,"completion_tokens_details":{"reasoning_tokens":8031}},"tokens_in":666,"tokens_out":8086,"duration_ms":56041,"temperature":1.0,"reasoning_tokens":8031,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:40:53.845052+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the binary stochastic growth process (33) for a unitary-invariant gate ensemble whose moments place $(\\mathrm{Re}\\,A_1,\\,A_2+A_3)$ at the boundary of the allowed region in Fig. 2 and check whether the extracted $v_{\\rm B}^{(n)}$ and $\\mathcal{D}^{(n)}$ converge with truncation order $n$; failure of convergence would show that the maximally-random ansatz (41) does not hold at finite distance.","supporting_citations":[{"cited_title":"Nahum, J","cited_arxiv_id":null,"evidence_quote":"Supplies the Haar-circuit mapping to a stochastic growth model and the baseline values $v_{\\rm B}=(q^2-1)/(q^2+1)$ and $\\mathcal{D}=4q^2/(q^2+1)^2$ that this paper generalizes."},{"cited_title":"Following Ref","cited_arxiv_id":null,"evidence_quote":"The diagrammatic method for integration over the unitary group is used to derive the pair transition probability (26) from the four moments of the gate distribution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the parametrization $U=(\\alpha\\mathbf{1}-U_0)(\\mathbf{1}-\\alpha^*U_0)^{-1}$ of Poisson-kernel distributed matrices, which is used to compute the moments for the explicit results."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the Poisson kernel as the maximum-information-entropy distribution with fixed ensemble average $\\langle U\\rangle=\\alpha\\mathbf{1}$, motivating the interpolation between the trivial and Haar limits."}],"review_version":1}