R88 G02 working derivation 1 — efficient access to the connected parent expansion
Status: current-agent derivation, conditional on the R87 gap candidate. This file alone is not a preparation theorem.

The missing resource issue is real: an operator supported on a ball of radius polylog(N/epsilon) is not automatically efficiently implementable. Its dense matrix has exponential dimension. We instead keep the connected expansion as a coherent sum of short symbolic Pauli paths. The normalization, path selection, time integration, and local data access are paid below.

1. Fixed conventions and two-stage path.
Write t=beta/2, t_c=log(1+sqrt(2)), A0=-H(0), V=sum_j X_j,
 tau_s(O)=exp(s A0)O exp(-s A0),
 K_g=exp(t(A0+gV))exp(-tA0)=tau_t(U_I(t)),
 U_I'(s)=g tau_-s(V)U_I(s), U_I(0)=I.
For A_i=P_i or h_i(beta), let R_i(g)=K_g A_i K_g^-1 and Q_g=sum_i R_i(g)^dagger R_i(g), with the two annihilators kept separate.

First vary beta from 0 to the requested beta at g=0. All Q_0(beta) terms and their first two derivatives are explicit constant-size matrices. At beta=0, P_i+h_i=I-|Phi+><Phi+| on the doubled site, so the initial ground state is a product of Bell pairs, prepared from product qubits with O(N) gates. The R87 classical bound supplies a gap uniform along this first path. Second keep beta fixed and put g_s=sg, 0<=s<=1. The R87 finite-field gap candidate applies throughout the second path. A derivative jump at the join is handled by two separate evolutions. There is no need to prepare a classical coherent Gibbs vector by a free oracle.

2. Symbolic term expansion.
The interaction-picture formula gives
 R_i(g)=sum_{m>=0} g^m int_{0<=u_m<=...<=u_1<=t}
 tau_t(ad_{tau_-u1(V)} ... ad_{tau_-um(V)}(tau_-t(A_i))) du_1...du_m,
with the nesting convention chosen consistently with the time-ordered ODE. The m=0 term is A_i. All estimates below are invariant under reversing the displayed nesting/ordering convention together; the actual compiler must use the ODE's ordered convention.

A_i acts on at most five physical sites and one ancilla. Its at most 4^6 Pauli coefficients are elementary functions of beta: conditional probabilities and their square roots. They and their first two beta derivatives are bounded on [0,beta_c]. These coefficients can be computed to k bits with poly(k+b) classical bit operations. No system-size matrix diagonalization is required.

For a Pauli P and an Ising edge B=Z_u Z_v, conjugation by exp(aB) either leaves P fixed or gives
 exp(aB)P exp(-aB)=cosh(2a)P+sinh(2a)BP
when P anticommutes with B. BP is a Pauli with a known phase. Only edges touching the initial support matter, since all Ising edges commute; newly added Z factors do not require an uncontrolled recursive conjugation. There are at most 4|S| relevant edges for physical support S. Hence each conjugation is an explicitly computable binary Pauli branching process with O(|S|) elementary steps.

The initial tau_-t(A_i) has physical support at most 13. Each next star tau_-u(X_j) must intersect the current support to give a nonzero commutator. At step r there are at most b_r=5(13+4r) possible star centers, and a star adds at most four physical sites. A reversible list of the current support and its neighboring star centers can be generated, sorted and deduplicated in polynomial time in m and log N. Pad to b_r choices with zero contribution. We may branch into left/right multiplication for each commutator, attaching its sign. All subsequent calculations manipulate Pauli labels and coordinates, not their matrices.

3. A deliberately conservative LCU normalization.
To make the construction explicit even without optimal coefficient-state preparation, represent each binary Ising conjugation using a uniform two-choice register and a controlled scalar-amplitude flag. For |a|<=t/4 its two coefficients have modulus at most exp(t/2); normalization 2 exp(t/2) suffices. A commuting edge is represented by coefficient 1 on one branch and 0 on the other. Missing edges are padded in the same manner. The scalar coefficient, including its phase, is calculated reversibly; the flag has the required top-left amplitude, rather than being interpreted as a free successful postselection.

Let C_A be a fixed normalization for the constant-size expansion of A_i. Initial conjugation contributes at most C_A 2^20 exp(10t). A star contributes at most 2^4 exp(2t), each commutator an additional factor 2, and final conjugation on at most 13+4m physical sites contributes at most
 2^(52+16m) exp[2t(13+4m)].
Together with the center count and the time-simplex volume t^m/m!, the order-m LCU normalization is bounded by
 alpha_m <= C_A 2^72 exp(36t) q_LCU^m (13/4)_m/m!,
 q_LCU=40 t exp(10t) 2^20 |g|.                                        (1)
Choose a fixed compiler threshold
 g_LCU=[320 t_c exp(10t_c) 2^20]^-1,
so q_LCU<=1/8. Decreasing the constant c in the original allowed window to include this threshold is legitimate: the theorem seeks existence of a positive window, not an optimized physical value. This very conservative choice is not a claim of practical advantage at laboratory-sized fields.

The sum of alpha_m and each of its fixed polynomial-in-m weighted versions is bounded by a constant independent of N,delta,epsilon. The large factor C_A 2^72 exp(36t_c) is fixed. In particular the coefficient sum is not exponential in the chosen truncation order. Direct explicit enumeration of all paths would be expensive; coherent path preparation does not enumerate them.

For g_s=sg, derivatives in s introduce m g^m s^(m-1), or m(m-1)g^m s^(m-2). They are evaluated as polynomials, including at s=0. No division by g_s is used. Equation (1) with polynomial m factors bounds the normalized first and second derivative sums uniformly. The beta-varying first stage has g=0 and only constant-size terms, so no differentiation of a shrinking time simplex is needed there.

4. Time ordering without a cost proportional to the integration grid.
For order m, prepare m independent uniform grid times on [0,t], then sort them reversibly while retaining enough information to uncompute. For distinct continuous times each ordered simplex point has m! preimages, so the expectation of the time-ordered integrand is m!/t^m times its simplex integral. The factor t^m/m! is already included in alpha_m. On the finite grid, ties are allowed and ordered consistently. Sorted coordinates are Lipschitz in the original coordinates; the integrand at equal times is well defined. Bounding its derivatives before summing by the same majorant gives a Riemann error at most C (m+1)^2 alpha_m/Q for Q grid points per time. Summing m leaves a bound C/Q (or the looser C(M+1)^2/Q). Thus log Q=O(log(1/eta)+log(M+1)) bits suffice for operator error eta. No loop over Q or Q^m grid points is needed.

The coefficient sign and complex phase are computed from the symbolic Pauli products. A top-left scalar flag amplitude realizes any coefficient divided by its bound; padding branches encode zero. This is an ordinary approximate block encoding, with no outcome-dependent rejection probability inserted into the preparation complexity. A final Hermitian simulation will use the block encoding coherently. Finite gate synthesis of all arithmetic and rotations costs a polynomial in m,b,log N and log(1/eta); square-root sensitivity in a flag can be handled by computing its coefficient to O(eta^2) accuracy, doubling precision bits without a polynomial 1/eta gate cost.

5. Truncation and actual patch oracles.
Let R_i^M be the expansion through m=M, using the exact simplex integrals at this mathematical stage. The R83 norm majorant gives
 ||R_i-R_i^M|| + ||partial_s R_i-partial_s R_i^M||
 + ||partial_s^2 R_i-partial_s^2 R_i^M|| <= C (M+1)^2 2^-M,
after enlarging a constant and using q_LCU<=1/8. Each R_i^M is supported in radius r0=3+2M, including its one ancilla. Define the POSITIVE Hermitian parent
 Q^M(s)=sum_i (R_i^M)^dagger R_i^M,
again separately for P_i and h_i. It need not retain an exact zero-energy ground vector. Product block encodings with fresh work registers implement each Gram term; first and second derivatives follow by the product rule. Their LCU normalizations remain fixed constants.

The full norm error is <=C N(M+1)^2 2^-M. If the exact path has gap at least gamma, choose
 M=C_0 ceil(log(C_0 N/(epsilon gamma)))
with a sufficiently large fixed C_0. Then the whole truncated path has gap at least gamma/2 by Weyl, and its final ground projector differs from the exact one by O(epsilon) by the isolated-eigenvector perturbation bound. Initial g=0 terms are exact. This is a norm approximation with its N factor paid, not a claim that the truncated parent is frustration-free.

For a finite spatial patch B, define Q_B^M by including only anchored Gram terms whose declared radius-r0 support is wholly inside B. The declared support can overestimate the actual Pauli support; that only removes more boundary terms. A block encoding of Q_B^M has normalization <=C|B|. Its circuit may scan all anchors and qubits of B and select a Pauli path by ordinary controlled gates. Its cost is polynomial in |B|,M,b,log N,log(1/eta); there is no qRAM assumption. When |B| is polynomial in delta^-1 and log(N/epsilon), this cost has exactly the allowed form. A whole-system SELECT would cost O(N) per use and must not be nested blindly inside an O(N)-normalized outer simulation.

Established so far: efficient block encodings of the truncated parent, its first two path derivatives, and restrictions to explicitly specified patches. Still to be paid: an approximate spectral-flow filter; its global state error; spatial localization; simulation of the localized flow in near-linear total N cost; coherent time parameters; final finite precision and resource accounting.
