R88 G02 — conditional near-linear preparation from the R87 parent-gap candidate
Status: current-agent mathematical candidate; original G02 remains open pending full dependency audit. No independent review has been performed.

Precise conditional statement.
Let H(g)=-1/4 sum_<i,j> ZiZj-g sum_i Xi on the L by L torus, L>=3, N=L^2, beta_min<=beta<beta_c=2 log(1+sqrt(2)), and delta=beta_c-beta. Suppose the EXACT doubled parents of R87 have a unique thermofield-double ground state and gap at least c_gamma min(1,delta)^q0, uniformly along (i) beta_s=s beta,g=0 and (ii) beta fixed,g_s=sg, whenever |g|<=c_gap delta^p. In R87 the proposed q0 is 4 C_*, with fixed p determined by its quantitative stability argument. These are hypotheses here, not newly certified facts.

Then, after decreasing the fixed field constant, the actual Gibbs mixture can be prepared within trace distance epsilon by a uniform finite-universal-gate circuit, with gates, classical bit operations and ancillary space bounded by
 C N max(1,delta^-(28q0+1)) [1+log(N/epsilon)]^A poly(b),
where A is an absolute fixed exponent and b is input bit length. Arbitrary gate connectivity is allowed, as in the frozen problem. No dense local matrix oracle, qRAM, Gibbs input or postselection is supplied.

Proof and resource chain.
1. At g=0 the parent is sum_i(P_i+h_i(beta)); at beta=0 each summand equals I-|Phi+><Phi+| on its doubled site. Thus the first input is prepared with O(N) gates. At fixed beta put t=beta/2, A0=-H(0), V=sum X, tau_s(O)=exp(s A0)O exp(-s A0), and K_g=exp(t(A0+gV))exp(-t A0). The annihilators are K_g P_i K_g^-1 and K_g h_i K_g^-1, and their Gram terms are summed separately. Grouping the annihilators would change the parent and is not done.

2. The interaction-picture ODE U_I'=g tau_-s(V)U_I gives an ordered nested-commutator expansion of each annihilator. Its order-m integrand is tau_t(ad_B(u1)...ad_B(um)(tau_-t(A_i))), with t>=u1>=...>=um>=0 and B(u)=tau_-u(V). The latest time u1 is the OUTERMOST commutator; this follows directly from C'=g[B,C]. The symbolic Pauli construction and bounds are in work-01. Before the final classical conjugation the physical support is at most 13+4m. Only at most 5(13+4r) star centers can contribute at step r. Commuting Ising conjugations give an explicit binary Pauli expansion, not a dense matrix.

3. An LCU majorant is alpha_m<=C_A 2^72 exp(36t) q^m (13/4)_m/m!, q=40t exp(10t)2^20 |g|. It and its fixed polynomial m-weighted versions are summable for q<=1/8. Taking, for example, an additional |g|<=2^-48 restriction is sufficient because t<=1. All these are fixed constants, however impractical numerically. This merely shrinks the positive constant sought in G02.

An order register can be prepared with probabilities proportional to the computed majorants for 0<=m<=M using O(M) controlled rotations; its normalization stays bounded. At each step a uniform rank in 0,...,5(13+4r)-1 can be prepared by binary prefix counts and scalar rotations, to the chosen precision. This avoids an uncharged factor 2 per branching step. Alternatively that factor can be charged and the field constant shrunk again. Padding unused centers contributes zero. Sorting m uniform grid times implements the ordered simplex average with the t^m/m! factor already paid. The grid size enters only through its number of bits. Pauli support construction does not depend on the time values, including branches whose coefficient vanishes at zero time. Coefficient flags implement top-left amplitudes coherently. No successful measurement outcome is assumed.

4. Truncate at M=O(log(N/(epsilon gamma))), where gamma is a conservative lower gap bound after truncation. Defining Q^M as the sum of the separate truncated Gram terms gives full norm error <=C N(M+1)^2 2^-M. Its unique ground state remains gapped and is O(epsilon)-close to the exact endpoint by ordinary eigenvector perturbation. The truncated parent need not be frustration-free and its ground energy need not be known. Its anchored terms and their first two path derivatives have constant norm and block-encoding normalization, radius r0=O(M), and uniform reversible coefficient circuits. Derivatives in g_s use polynomials m g^m s^(m-1), including s=0; there is no division by g_s. The beta path has only constant-size terms. The paths agree at their join.

5. Use the elementary approximate spectral-flow filter
 F_sigma(t)=sign(t) erfc(sigma|t|/sqrt(2))/2,
 whose transform under exp(i omega t) is i(1-exp(-omega^2/(2sigma^2)))/omega.
For A_sigma=int F_sigma(t) exp(itQ^M) Q^M' exp(-itQ^M)dt and U'=i A_sigma U, the excited-ground derivative error is at most ||Q^M'|| gamma^-1 exp(-gamma^2/(2sigma^2)). Integrating that error uses unitary variation of constants, with no exponential N loss. Choose ell=O(1+log(N/(epsilon gamma))), sigma proportional to gamma/sqrt(ell), and T=O(ell/gamma). Frequency and time-tail errors are O(epsilon). Finite dyadic sigma and rational T are described in work-03.

6. A finite-range Lieb–Robinson iteration for Q^M has rate kappa<=C r0^4 at spatial weight 1/r0. Replace the time evolution of q_x' by that under the parent restricted to B_R(x). Omitted supports are at least R-3r0 from the initial support; use the safe R-4r0 margin. The summed Duhamel error is bounded using N omitted anchors and N initial anchors. These N^2 factors enter the logarithm determining R, rather than the gate count. Taking R<=C gamma^-1 ell^6 suffices. No restricted parent is assumed gapped.

The localized flow A_R=sum_x d_x has ||d_x||<=CT, support B_R(x), and ||d_x'||<=C(T+V_R T^2), V_R<=C gamma^-2 ell^12. Its oracle uses a uniform time register, the computable erfc flag and coherent simulations of the restricted parent. Finite grids cost logarithmically many bits; Hamiltonian coefficients are evaluated on the quantum time register. A first-derivative bound controls discretization without exponential derivative losses.

7. To avoid multiplying two global N costs, apply overlapping-region unitary decomposition to A_R, first along one coordinate and then the other. Its interaction rate is at most C R^4 T. An overlap W<=C gamma^-6 ell^31 makes the total decomposition error O(epsilon). There are at most C N patch propagators of at most V_W<=C gamma^-12 ell^62 sites. This uses the time-dependent locality decomposition, not a theorem that treats a growing patch as a free constant-dimensional site. If a patch exceeds the torus, use the whole system, whose size then already obeys this bound.

8. A selected symbolic Pauli is applied by scanning the V_W patch qubits and comparing coordinates with its polynomial-size support list. Coherent center labels do not require an additional scan over all possible centers. This costs C V_W B^a0, B=b+ceil(ell), for a fixed arithmetic/synthesis exponent a0. Inner real-time Taylor/Dyson simulation needs at most C V_R T B^2 calls. An outer localized-flow block encoding therefore costs C V_W V_R T B^(a0+2). Outer time-dependent Dyson simulation on the patch uses at most C V_W T B^2 calls. Consequently
 G_patch<=C V_W^2 V_R T^2 B^(a0+4),
 G_total<=C N gamma^-28 ell^138 B^(a0+4).
The same elementary circuit construction bounds classical preprocessing. Taylor/Dyson coefficient preparation avoids assuming an efficient black-box phase optimizer. Fresh or uncomputed work registers and squared internal top-left errors give clean-isometry accuracy, with only doubled precision bits.

9. Assign fixed fractions of epsilon to parent truncation, approximate transport, time cutoff, spatial restriction, patch decomposition, local simulation and finite arithmetic. Every required per-call accuracy is epsilon divided by a polynomial in N,gamma^-1,ell,b; its bit length is O(b+ell). Discard the second copy and work registers at the end. Contractivity of trace distance gives the actual Gibbs mixed state, not just a classical-output sampler. Finally gamma=c_gamma min(1,delta)^q0 and fixed powers of log(1/delta) are absorbed in one extra power of delta^-1. One may take A=a0+142 and q=28q0+1. Constants are deliberately unoptimized.

Literature and novelty boundary.
The algorithmic ingredients are established: spectral transport, overlapping-region simulation (Haah–Hastings–Kothari–Low, arXiv:1801.03922v4), and time-dependent Dyson simulation (Low–Wiebe, arXiv:1805.00675v2). The work here is the explicit access construction and quantitative substitution for this parent, not a new general simulation algorithm.
Rouze–Stilck Franca, arXiv:2610.03578v1, already gives weak-field Ising preparation at each fixed subcritical temperature. Its fixed-parameter theorem does not by itself supply the polynomial critical-distance window and costs targeted here; the earlier qualification and R87 dependency analysis remain necessary.
Sengoku–Cao–Watanabe, arXiv:2610.27939, uses related auxiliary parents and spectral flow. Section II.4 explicitly leaves elementary-gate efficiency unclaimed; its Ising preparation theorem is in a sufficiently high-temperature domain. Its low-temperature result excludes m=0. Thus neither auxiliary parents nor spectral flow should be advertised as our invention, and that theorem does not directly discharge this near-critical elementary-gate target.

Files work-01, work-02 and work-03 retain the detailed construction, estimates and finite-precision choices. This theorem's decisive unresolved dependency is the full R87 gap/stability candidate, particularly its quantitative stability constants and classical-to-quantum local-ground-space bounds. Until that dependency is rigorously accepted, this is a conditional central-interface result and G02 is not marked solved.
