{"id":"b6a21c0a-d0df-40e0-b236-85d9b2111255","arxiv_id":"2608.06459","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The single-particle Green's function in a noisy U(1) random circuit has KPZ scaling: its spatial center wanders as t^{2/3} and its log-intensity fluctuations grow as t^{1/3}, with a weak-noise crossover time of order gamma^{-3/2}.","lead":"In a one-dimensional random quantum circuit with particle loss, a single flipped spin's signal is predicted to spread with universal wiggly fluctuations: its center wanders like the 2/3 power of time. The paper derives this KPZ-like spreading from mappings to waves and polymers in random media, and finds the same behavior at strong and weak noise.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Weak-noise claim for genuine single-sample Green's functions rests on the untested source-manifold ansatz; numerics validate only the phase-annealed norm N.","rationale":"The reader's weakest assumption is the source-manifold ansatz, and that is exactly the most load-bearing vulnerability of the central claim. The strong-noise mapping is exact in the gamma >> 1 regime and the moderate-noise numerics are suggestive, but the weak-noise statement for genuine G is essential to the abstract's claim and is not directly tested. The paper itself concedes that verifying weak-noise DWRM for genuine single-sample Green's functions is outstanding. The phase-annealed norm N is a proxy; it removes the quenched phase disorder that distinguishes DWRM from DPRM, and the replica continuation in the Supplementary Material is a theoretical extrapolation rather than a numerical check. A direct numerical test of G at weak noise, pushed past the predicted crossover scale gamma^{-3/2}, is feasible in principle because gamma ~ 0.02 gives a crossover of only a few hundred steps, and the paper already reaches t ~ 10^4 steps for N at those parameters. If such a simulation reproduces the predicted exponents, the central claim is substantially strengthened; if it does not, the weak-noise half of the claim would need to be revised. Since the reader's conditional verdict already reflects this limitation, no change in verdict is needed, but the proposed test would be the decisive next step.","tokens_in":23812,"tokens_out":13570,"duration_ms":121652,"concrete_test":"Run single-copy TEBD simulations of the noisy U(1)-conserving circuit at gamma = 0.02, 0.04, and 0.06 on systems of length L >= 200, with N_samples ~ 1500, and measure Var[-log sum_x |G(x,t)|^2] and <x(t)>^2 for genuine G (not the phase-annealed norm) up to t ~ 10 gamma^{-3/2}. Collapse the data against tau = gamma^{3/2} t and check for t^{2/3} free-energy variance and t^{4/3} wandering growth. If G exhibits the predicted DWRM scaling with the same crossover time, the concern is resolved; if G remains diffusive or annealed on accessible timescales, the source-manifold ansatz is falsified for quenched local correlators.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that G(x,t) is governed by directed waves at weak noise is built on Eq. (5), which the paper explicitly labels 'our principal assumption' and carries over from Ref. [39]. This source-manifold ansatz projects the q=1 sector onto a single sigma+ coherence moving in a symmetric-exclusion fluid with the same hopping disorder. It is not derived from the microscopic circuit. The numerical evidence at weak noise (Fig. 2) tests only the phase-annealed norm N, where phase averaging makes the coherence path classical (Eq. (7)); it does not test quenched interference in genuine G. The replica treatment in Supplementary Material II extends the ansatz to arbitrary replica number q, but it inherits the same projection and adds further approximations (point-absorber reduction at all q, Feynman–polaron replacement on collision-free segments). The paper's Discussion explicitly states that verifying the weak-noise DWRM scaling for genuine single-sample Green's functions remains an outstanding task. Therefore, if the source-manifold ansatz fails at weak noise—for example, if multi-coherence sectors such as sigma+ sigma+ sigma- contribute beyond the O(gamma log(1/gamma)) point-absorber estimate—then the directed-polymer/DWRM mapping for G does not follow, and the headline claim for genuine local correlators is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the single-particle Green's function G(x,t)=<sigma^-_x(0)sigma^+_0(t)> in one-dimensional, particle-number-conserving random unitary circuits with local noise. For a fixed circuit realization, it argues that G is governed, in both the strong- and weak-noise regimes, by directed waves in a random medium, leading to KPZ scaling: the center of the normalized distribution |G|^2 wanders as t^{2/3}, and sample-to-sample fluctuations of -log sum_x |G|^2 grow as t^{1/3}, with a weak-noise crossover time of order gamma^{-3/2}. In the strong-noise limit, the paper derives an exact mapping to a discrete non-Hermitian Schr\\\"odinger equation with random complex potential and confirms the predicted scaling by tensor-network simulations at moderate noise. In the weak-noise regime, it introduces a projected 'source-manifold' ansatz for the q=1 operator sector, derives a DPRM description for the phase-annealed norm N=E_phi|G|^2 via a macroscopic fluctuation theory, and tests the resulting scaling by numerically exact two-copy simulations. A replica treatment in the Supplementary Material extends the weak-noise mapping to balanced moments of the genuine Green's function, yielding a DWRM description. The paper explicitly states that direct verification of the weak-noise DWRM scaling for genuine single-sample Green's functions remains an outstanding task.","tokens_in":24101,"tokens_out":6090,"duration_ms":56113,"significance":"If the weak-noise claim for genuine single-sample Green's functions holds, the paper identifies a new strong-disorder universality class for a phase-sensitive observable in a diffusive random quantum circuit, contrasting with the standard hydrodynamics of conserved densities. The strong-noise mapping is exact in the gamma-to-infinity limit and is a clean, self-contained contribution. The weak-noise MFT derivation is transparent and yields a specific, parameter-free crossover scale gamma^{-3/2}; the numerics for the phase-annealed norm N provide a direct benchmark of the projected-dynamics ansatz for that observable. However, the central advertised claim for genuine G at weak noise rests on an ansatz that is explicitly labeled as the paper's principal assumption and is carried over from the author's own Ref. [39]; it is not derived from the microscopic circuit, and the numerical evidence at weak noise tests only N, not the interference in G.","major_comments":[{"comment":"The weak-noise treatment of the genuine single-particle Green's function is built on the source-manifold ansatz, which the manuscript explicitly labels 'our principal assumption' and which is carried over from Ref. [39]. This ansatz projects the q=1 sector onto a single local coherence plus a diagonal SEP fluid, and it is not derived from the microscopic circuit. Because Eq. (5) is the starting point for all subsequent weak-noise results, including the DPRM description of N and the DWRM description of G, the paper's abstract claim that G is governed by directed waves 'in both the strong- and weak-noise limits' is not yet established for genuine G. The paper's own Discussion confirms this: verifying the weak-noise DWRM scaling for genuine single-sample Green's functions is listed as an outstanding task. The revision should either provide numerical evidence for genuine G at weak noise, derive the ansatz from the circuit, or substantially temper the abstract and introduction to present the weak-noise claim for G as a conjecture.","section":"Main text, 'Effective dynamics at weak noise', Eq. (5)"},{"comment":"The replica extension from the phase-annealed norm N to balanced moments of the genuine Green's function inherits the source-manifold ansatz and adds further uncontrolled approximations: the point-absorber reduction at all replica numbers and the Feynman-polaron replacement on collision-free segments. The first sentence of SM II states that the derivation assumes the projected operator dynamics ansatz. The point-absorber reduction is justified only by the estimate N_QSS^u ~ gamma log(1/gamma) in the End Matter and SM II, which is itself computed within the ansatz. If multi-coherence sectors contribute beyond this estimate, the DWRM mapping for G does not follow. The claim in Eq. (S65) that G_F matches all moments of G upon coarse-graining is therefore conditional on these approximations and should be stated as such in the main text.","section":"Supplementary Material II, Eq. (S14) and Eq. (S65)"},{"comment":"The numerical confirmation at weak noise applies only to the phase-annealed norm N, not to genuine G. Fig. 2 simulates the full two-copy transfer-matrix dynamics for N, which is exact for N but is insensitive to the quenched interference that distinguishes directed waves from directed polymers. The claim that the numerics 'provide a direct benchmark of both the projected-dynamics ansatz and the weak-noise scaling' is therefore only valid for the phase-annealed observable. Since the abstract's headline prediction concerns G, the distinction between N and G should be made prominent throughout, and any statement that the KPZ predictions for G are 'confirmed' should be limited to the strong-noise regime and to the phase-annealed proxy at weak noise.","section":"Main text, 'Effective dynamics at weak noise' and Fig. 2"}],"minor_comments":[{"comment":"The phrase 'amcentral problem' appears to be a typo for 'a central problem'.","section":"Introduction, first paragraph"},{"comment":"The sentence 'We call the resulting low-entropy region avoid' appears to contain a typo; it should likely read 'we call the resulting low-entropy region a void.'","section":"Main text, 'MFT formulation'"},{"comment":"The figures would benefit from explicit error bars and a description of the fitting windows used to extract the t^{2/3} and tau^{2/3} guides; the collapse in Fig. 2 uses a free reference time t0, and the sensitivity of the collapse to t0 should be discussed.","section":"Fig. 1 and Fig. 2"},{"comment":"The notation in Eq. (S38) is somewhat compressed; defining P^{(0)}_{I_i}[X_a] and the Feynman-polaron action S_{F,I} explicitly before use would improve readability.","section":"Supplementary Material II, Eq. (S38)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest about its limitations, and the strong-noise part is sound and valuable. The main issue is that the weak-noise claim for genuine G, which is advertised in the abstract and title, is not supported by the numerical evidence or a derivation that is independent of an untested ansatz. A revision that clearly scopes the weak-noise results to the phase-annealed norm N and labels the genuine-G DWRM claim as conditional would be acceptable; the paper should not be rejected because the strong-noise result and the MFT framework are significant contributions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Ewan,\n\nRead the McCulloch paper. The useful news: the strong-noise limit is genuinely solved in a way I had not seen before. The projected transfer matrix on the single-coherence sector is written down exactly, coarse-grains to a non-Hermitian Schrödinger equation with a random potential, and the numerics at gamma=2/3 show the KPZ wandering and t^{1/3} free-energy fluctuations. That part is a real contribution, and the paper correctly discloses that Ref. [60] anticipated KPZ scaling in a related noiseless setting. The citation pattern is clean.\n\nThe weak-noise story is where I have reservations. The phase-annealed norm N is mapped to a directed polymer with a crossover time gamma^{-3/2}, and the data collapse for gamma=0.02–0.06 is convincing at face value. But the whole construction passes through Eq. (5), the source-manifold/projected operator-dynamics ansatz, which is imported from Ref. [39] and not derived. The paper is admirably explicit about this: it calls the ansatz \"our principal assumption\" and its Discussion says direct verification for genuine single-sample Green's functions is an outstanding task. The numerics at weak noise test only N, where phase averaging makes the coherence path classical; they do not test quenched interference in G. The replica treatment in the Supplementary Material extends the ansatz but inherits the same projection, so it is not independent evidence.\n\nThe stress-test note is not overstated. If the source-manifold ansatz fails—for example, if multi-coherence sectors contribute beyond the O(gamma log(1/gamma)) estimate—then the DPRM/DWRM description for genuine G does not follow. That is an unproven load-bearing step, but it is not a contradiction. The paper is honest, the KPZ exponents are not fitted, and the strong-noise result stands independently.\n\nI would send this to peer review. The right referee outcome is conditional acceptance: require either a direct numerical check of G at weak noise on accessible timescales or a sharper derivation of the q=1 projection, and ask for error bars and for code/data. The paper is for people working on operator hydrodynamics, rare-event physics in random circuits, and KPZ universality in quantum dynamics; they will get real value from the strong-noise mapping and the explicit crossover prediction even if the weak-noise extension is still open.\n\nYours,","headline":"The strong-noise directed-wave mapping is solid and the weak-noise crossover prediction is genuinely new, but the headline claim for quenched Green's functions rests on an unproven ansatz—still worth refereeing.","tokens_in":24600,"tokens_out":2758,"would_cite":true,"duration_ms":25889,"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":"For fixed spacetime disorder, the single-particle Green's function in a one-dimensional random unitary circuit is governed by a directed wave in a random medium, with KPZ wandering $t^{2/3}$ and free-energy fluctuations $t^{1/3}$.","keywords":["Kardar-Parisi-Zhang universality","directed polymer in a random medium","directed wave in a random medium","random unitary circuits","single-particle Green's function","operator hydrodynamics","macroscopic fluctuation theory","quantum thermalization"],"falsifier":"A decisive calculation is to evolve one fixed weak-noise circuit for times $t\\gg\\gamma^{-3/2}$, then measure across samples the wandering variance $\\mathrm{Var}\\langle x\\rangle$ and the free-energy variance $\\mathrm{Var}[-\\log\\sum_x|G|^2]$; if these do not grow as $t^{4/3}$ and $t^{2/3}$ respectively, the central claim is falsified.","tokens_in":23606,"feed_emoji":"⚫️","tokens_out":8248,"duration_ms":66212,"temperature":0.7,"pith_summary":"This paper argues that a basic phase-sensitive observable—the single-particle Green's function of a particle-number-conserving random quantum circuit coupled to a bath—does not simply diffuse or decay exponentially in a fixed realization of the disorder. Instead, for fixed spacetime disorder, the Green's function is governed by a strong-disorder fixed point: a directed wave (or, after phase averaging, a directed polymer) moving in a random medium. The paper predicts that the center of the normalized distribution $p(x,t)\\propto |G(x,t)|^2$ wanders on a length scale $O(t^{2/3})$, and that sample-to-sample fluctuations of $-\\log \\sum_x |G(x,t)|^2$ grow as $t^{1/3}$. At weak noise there is a crossover to this fixed point at a parametrically long time $O(\\gamma^{-3/2})$. If right, this means that intrinsically quantum, phase-sensitive correlators in an otherwise diffusive system are controlled by a different universality class than classical conserved densities.","feed_headline":"Local correlators wander with t^{2/3}, not diffuse","feed_subtitle":"In noisy random circuits the center of a one-particle correlator wanders on a t^{2/3} scale, not diffusively.","key_machinery":"The central objects are the projected single-coherence subspace and, at weak noise, the source-manifold (projected operator-dynamics) ansatz. In strong noise the dynamics closes on states $|x\\rangle=\\sigma_x^+$; each half-layer gives a random $2\\times 2$ matrix, leading to a discrete directed wave in a random medium and the continuum equation above. In weak noise, the ansatz represents $G$ as a path sum over a coherence trajectory $X$ and an exclusion-process fluid history $n$: $G\\simeq\\sum_X\\sum_n P_U[n|X]A_U[X|n]$. Phase averaging imposes local occupation agreement and reduces the norm to $\\mathcal{N}\\simeq\\sum_X P_{\\rm coh}[X]\\,Z_{\\rm 2-copy}[X]$. The workhorse is then the point-absorber / macroscopic fluctuation theory (MFT) action for the $r$-symbol density, whose saddle gives the void size, and the Feynman\\textendash polaron treatment of the internal coherence motion, which yields the coarse-grained directed-polymer action. These ingredients convert microscopic gate randomness into an $O(1)$ coarse-grained random potential with the KPZ/DPRM scaling.","core_discovery":"The central claim, stated on the paper's own terms, is that for a fixed circuit realization and fixed noise rate $\\gamma$, the infinite-temperature Green's function $G_\\gamma(x,t)=\\langle \\sigma^-_x(0)\\sigma^+_0(t)\\rangle$ is described at long times by a directed wave in a random medium (DWRM). In the strong-noise limit this is derived exactly: large $\\gamma$ projects the operator dynamics onto a single moving coherence $\\sigma^+_x$, whose transfer matrix is a contractive non-Hermitian Schr\\\"odinger evolution with random complex hoppings and a random complex potential, coarse-graining to $\\partial_t G = D\\partial_x^2 G - \\Gamma G + \\eta(x,t)G$. The KPZ exponents follow from the DWRM fixed point: the center wanders as $t^{2/3}$ and the free energy $F=-\\log\\sum_x |G|^2$ has $t^{1/3}$ fluctuations. At weak noise, the paper extends the projected operator-dynamics ('source-manifold') ansatz of Ref. [39] to a coherence dressed by a diagonal exclusion-process fluid with quenched random hopping rates; phase averaging turns the norm $\\mathcal{N}=\\mathbb{E}_\\phi |G|^2$ into a sub-stochastic sum over coherence paths with a two-copy exclusion weight, whose stationary saddle is a slow 'void' of size $\\xi\\sim\\sqrt{D_0/\\gamma}$. Restoring hopping disorder makes the void coordinate a directed polymer in a random medium with $O(1)$ random potential on the rescaled spacetime scale $\\xi^2/D_{\\rm void}\\sim \\gamma^{-3/2}$, giving the same KPZ exponents. Tensor-network simulations of individual circuits at $\\gamma=2/3$, and of the full two-copy transfer matrix at weak noise, show the predicted $t^{2/3}$ and $t^{1/3}$ scalings.","pith_inferences":["A consequence the author leaves implicit is that the same strong-disorder fixed point should govern other charge-carrying phase-sensitive correlators, such as fermionic or bosonic single-particle Green's functions in diffusive random circuits, wherever a similar single-coherence subspace can be projected out.","A direct testable extension is to measure the center wandering of a local spin coherence in a cold-atom quantum simulator; the predicted $t^{2/3}$ wandering of the spin-correlation cloud should be visible once the bath coupling is fixed and the disorder realization is held.","The weak-noise analysis explicitly treats the phase-annealed norm; whether genuine single-sample Green's functions at weak noise show the same DWRM behavior is still a prediction rather than a demonstrated fact, so a reader should treat that part as the open edge of the paper's claim."],"forward_implications":["If the claim holds, the single-particle Green's function in a fixed circuit is not self-averaging at the diffusive scale: it shows KPZ wandering, so phase-sensitive local probes see strong-disorder fluctuations even where densities diffuse.","The weak-noise crossover time $\\gamma^{-3/2}$ is parametrically larger than the void-formation time $\\gamma^{-1}$, so experiments or simulations at small $\\gamma$ must wait beyond that scale to see the KPZ regime.","The mapping predicts $\\mathrm{Var}[F]\\sim t^{2/3}$ and wandering $\\langle x\\rangle^2\\sim t^{4/3}$ simultaneously, giving a sharp numerical signature in tensor-network or analog-simulation data.","The result implies that annealing over circuit phases, as in earlier ensemble studies, misses the typical single-sample behavior; phase-quenched disorder is relevant for this observable."],"supporting_citations":[{"why":"Supplies the projected operator-dynamics (source-manifold) ansatz that the weak-noise analysis extends to fixed spacetime disorder, and the Feynman\\textendash polaron treatment of the internal coherence motion.","marker":"[39]"},{"why":"Defines the directed-wave-in-random-medium universality class and the KPZ exponents $t^{2/3}$, $t^{1/3}$ used for the strong-noise Green's function.","marker":"[40–45]"},{"why":"Provides the Kardar\\textendash Parisi\\textendash Zhang equation and its review, the source of the predicted wandering and free-energy scaling.","marker":"[46,47]"},{"why":"Gives the directed polymer in a random medium (DPRM) theory used for the phase-annealed norm at weak noise.","marker":"[48–53]"},{"why":"Supplies the macroscopic fluctuation theory action used for the point-absorber problem and the large-deviation analysis of the void.","marker":"[65–69]"},{"why":"Establishes the symmetric-exclusion-process hydrodynamics for the diagonal operator sector in which the coherence moves.","marker":"[35,36]"},{"why":"Provides the time-evolving block decimation algorithm used to simulate individual circuits and two-copy dynamics numerically.","marker":"[71–74]"}],"fun_headline_variants":["KPZ superdiffusion replaces diffusion in random circuits","t^{2/3} wandering: KPZ signature in noisy quantum circuits","Random quantum circuits show superdiffusive correlator wander","Noisy circuits: correlator center scales as t^{2/3}, not t^{1/2}","Directed wave picture yields KPZ exponents in random circuits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the paper's 'projected operator-dynamics' ansatz in the weak-noise regime: after projecting onto operators with one local coherence, the surrounding diagonal operators evolve as a symmetric exclusion process with the same random hopping rates as the gates. If that representation fails, the directed-polymer mapping and the KPZ exponents do not follow.","fun_headline_variants_meta":{"raw":{"variants":["KPZ superdiffusion replaces diffusion in random circuits","t^{2/3} wandering: KPZ signature in noisy quantum circuits","Random quantum circuits show superdiffusive correlator wander","Noisy circuits: correlator center scales as t^{2/3}, not t^{1/2}","Directed wave picture yields KPZ exponents in random circuits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000372,"raw_usage":{"total_tokens":2118,"prompt_tokens":1202,"completion_tokens":916,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":818,"completion_tokens_details":{"reasoning_tokens":823}},"tokens_in":818,"tokens_out":916,"duration_ms":7430,"temperature":1.0,"reasoning_tokens":823,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:33:27.417494+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive calculation is to evolve one fixed weak-noise circuit for times $t\\gg\\gamma^{-3/2}$, then measure across samples the wandering variance $\\mathrm{Var}\\langle x\\rangle$ and the free-energy variance $\\mathrm{Var}[-\\log\\sum_x|G|^2]$; if these do not grow as $t^{4/3}$ and $t^{2/3}$ respectively, the central claim is falsified.","supporting_citations":[],"review_version":2}