M05 qualification: on-site-uniform implementation of one interacting QBP filter
Author: /root/qualification_backlog, 22 October 2026.
Status: constructive scoped algorithm/interface, not full Gibbs preparation.

ORIGINAL AND PRIOR COMPARISON
M05 requests nearly linear system-size Gibbs preparation with single-exponential
temperature overhead, charging classical work, precision and all block oracles.
Read complete relevant R02/R03/R04, R12/R18/R23/R29/R30/R32/R36/R39 arguments.
Their missing interacting-block QBP coefficient interface is the present target.
R29's fixed two-site spectral response and R39's holomorphic parameter strip
do not implement the interacting filter. R03's collective Kraus encoding is
inherited. Pre-campaign semantic memory and H0 history were separately screened
in M27-prehistory-scope-v1.json; no matching filter mechanism was located.

The new bridge replaces an explicitly enumerated high-frequency time quadrature
by a binary quantile register, with its QBP-specific cumulative distribution
evaluated reversibly. The register may describe exponentially many nodes, but
none are tabulated. Interaction-picture simulation supplies the coherent node
operation from the actual local input matrices, including interactions.

SCOPED INPUT AND CLAIM
On a block of L fixed-dimensional sites let H=A+B, A=sum_i a_i on-site, and
B a finite-range sum with a known local-term block-encoding normalization
alpha_B=O(L J_int). Let alpha_A>=sum_i||a_i|| and alpha=alpha_A+alpha_B.
Let V be the boundary interaction, with a local-term encoding of V/J, J>0,
where J>=||V||. The boundary contains only a fixed number of terms in 1D.
All matrices and beta>0 are supplied as ordinary finite binary rational data
to specified precision; their input length is charged. This does not assume
a succinct power-tower representation of matrix entries or a free spectral oracle.

For 0<eps<1/4 define the source's half-temperature QBP filter
 F_H(V)=integral_R p_beta(t) exp(itH) V exp(-itH) dt,
 p_beta(t)=4/(pi beta) log coth(pi|t|/beta).
We construct a block encoding with normalization J and error at most J eps.
Its elementary-gate count and classical circuit-construction cost are bounded by
 poly(L,b,1+beta alpha_B,log(1/eps)),
where b includes input bits for beta, local entries and the specified bounds.
Equivalently, large alpha_A enters through log(1+alpha_A), already O(b+log L)
for ordinary binary local matrices, never as a multiplicative energy value.
The polynomial degree is fixed by local dimension/range and arithmetic choices.
The construction is coherent in an additional binary interpolation parameter
s when B=B0+sV: replace alpha_B by a uniform local-term bound and charge the
bits of s. A zero boundary V gives the zero encoding trivially.

1. A POSITIVE LAW WITH AN EFFICIENT CUMULATIVE DISTRIBUTION
For x>0, expand log coth x=2 sum_(k>=0) exp[-2(2k+1)x]/(2k+1).
The positive kernel is a mixture of two-sided Laplace distributions with
 rates lambda_k=2pi(2k+1)/beta,
 weights w_k=8/[pi^2(2k+1)^2], sum_k w_k=1.
The absolute-time survival function is consequently
 S_beta(x)=8/pi^2 sum_(k>=0) exp[-2pi(2k+1)x/beta]/(2k+1)^2
 =8/pi^2 [Li_2(q)-Li_2(q^2)/4], q=exp(-2pi x/beta).
Let F_beta=1-S_beta. The expression is used as a scalar arithmetic formula,
not by enumerating the mixture. In particular S_beta(x)<=exp(-2pi x/beta).

Li_2(y) for y in[0,1] is evaluable to P bits with poly(P) bit operations:
on[0,1/2], its power series sum y^j/j^2 has a geometric tail after O(P) terms;
on[1/2,1], use
 Li_2(y)=pi^2/6-log y log(1-y)-Li_2(1-y).
Endpoints are evaluated by their limits. Logarithms/exponentials use range
reduction and convergent series; cancellation is controlled by extra guard bits.
Inputs so close to endpoints that the limiting correction is below requested
absolute accuracy can use that limit. These operations need only polynomially
many input/precision bits. No spectral frequencies occur in this scalar law.

2. QUANTILE DISCRETIZATION WITHOUT AN ENUMERATED NODE TABLE
Choose integer K>=log(32/eps), K=O(log(1/eps)), and T=beta K. Replace |t|>T
by T with its sign retained. The changed probability mass is at most eps/32;
since the conjugated V/J has norm at most1, this changes the normalized filter
by at most eps/16.

Let q(u)=min{F_beta^{-1}(u),T}, 0<u<1. It is nondecreasing with range[0,T].
For Q=2^n and midpoint quantiles u_j=(j+1/2)/Q, coupling a uniform u in each
bin to u_j gives
 integral_0^1 |q(u)-q(u_j(u))| du <=T/Q.
Indeed the error in bin j is at most its width times the oscillation of q;
the oscillations telescope to at most T over all bins. The atom at T is included.
For A(t)=exp(itH)(V/J)exp(-itH),
 ||A(t)-A(t')||<=2alpha |t-t'|.
Taking Q>=64(1+alpha)T/eps makes this discretization error at most eps/32.
Its index length is only
 n=O(1+log(1+alpha)+|log beta|+log(1/eps)+log log(1/eps)).
Uniform amplitudes over j need n Hadamards; one further Hadamard gives the sign.

Each q(u_j) can be computed reversibly to time accuracy
 eta=eps/[64(1+alpha)] using bisection and the displayed CDF. On[0,T],
 F_beta'(x)=2p_beta(x)>=lambda_min=(16/(pi beta))exp(-2pi K)>0.
Use CDF absolute error at most lambda_min eta/16. At an ambiguous comparison,
the true quantile lies within O(eta) of the current midpoint; otherwise retain
the certified half interval. Use enough guard bits to obtain final error eta.
The branch above F_beta(T) returns T, with the same certified boundary rule.
Only O(log(T/eta)+1) bisection steps and
 O(K+|log beta|+log(1/eta)+1) precision bits are needed. Round eta/T upwards
in these expressions when eta>=T, where the constant time0 approximation
already suffices. The resulting time-rounding error is at most eps/32.
All arithmetic is uncomputed after the controlled evolution.

3. COHERENT REAL-TIME EVOLUTION FROM LOCAL MATRICES
External algorithmic input: Low--Wiebe, arXiv:1805.00675v2, Definition2,
Theorem3/Corollary4, Lemma6 and Eq25. Their truncated-Dyson construction uses
O((1+alpha_B T) polylog((2+alpha_B T)/xi)) calls to a binary-time encoding of
exp(iAt)B exp(-iAt)/alpha_B and on-site evolution. Time-grid and additional
gate-register sizes depend logarithmically on alpha_A+alpha_B. Its proof is
in projected-encoding norm, suitable for the product below. The theorem is an
imported algorithm, not a new Hamiltonian-simulation result.

Its oracle hypothesis is met here, not assigned unit cost. Fixed-dimensional
on-site matrices can be diagonalized to the required accuracy classically;
controlled exp(i a_i t) is implemented by binary arithmetic on eigenvalue
phases, fixed-size basis changes and rotation synthesis. Near-degenerate
eigenvalues may be grouped at accuracy O(xi/T), so no inverse eigenvalue-gap
cost is assumed. Reading integer bits and evaluating large phases modulo2pi
costs poly(b,log(1/xi),log(1+T)) operations per site. Thus exp(iAt) is a product
of L charged local circuits, including when t is supplied coherently.
The finite-range local-term sum gives an explicit encoding of B/alpha_B by
O(L) controlled constant-size encodings and coefficient arithmetic. Conjugate
it by those on-site circuits to realize the required time oracle.

To handle a coherent requested time t with |t|<=T, set x=t/T in[-1,1] and
simulate x(A+B) for the fixed duration T. Use common segment counts and uniform
norm bounds for all x. The oracle encodes
 x exp(ixAs)B exp(-ixAs)/alpha_B;
the scalar x is a bounded rotation and xs is reversible multiplication.
The derivative bound is uniformly at most2alpha_A alpha_B. The Dyson algorithm
therefore acts coherently on the x register with the same cost bound. Negative
times and the extra interpolation parameter s need no separate node tables.
The alpha_B=0 case is just on-site evolution. Approximate local oracles to
xi divided by the number of uses, so accumulated errors stay within xi.

Let U_t' be the top-left block of this circuit, satisfying
 ||U_t'-exp(itH)||<=xi uniformly and ||U_t'||<=1.
The product with separate ancillary registers encodes U_t'(V/J)U_t'^dagger.
Its difference from A(t) is at most2xi, by a two-term telescoping bound.
The success projection is part of a block encoding; no physical postselection
or discarded failure probability is silently charged as a prepared Gibbs state.

4. ASSEMBLY, NORMALIZATION AND PRECISION ACCOUNTING
Prepare the uniform index/sign registers, compute the approximate quantile,
apply the controlled product encoding above, uncompute the quantile, and
unprepare the uniform registers. Its projected block is exactly the arithmetic
average of the controlled encoded products. All weights are positive and sum
to1, so normalization remains J, with no Q or Hilbert-dimension factor.
Assign xi<=eps/64 and normalized V-encoding error<=eps/32. Tail, grid,
quantile, simulation and V errors sum to less than eps; use remaining slack
for arithmetic and final gate synthesis. The index precision is polynomial in
b, log L and log(1/eps), whereas the simulation cost is polynomial in
1+alpha_B beta log(1/eps). Combining these gives the claimed bound.

Errors in the supplied local matrices are also charged. If their block sum
differs by Delta_H in operator norm, real-time Duhamel gives an additional
filter error at most2 E|t| Delta_H <=C beta Delta_H. Thus absolute per-term
matrix precision eps/[C beta L] suffices for this interface, alongside reading
the integer bits; V/J requires its own normalized eps allocation. Beta input
error can be handled by writing t=beta y and bounding the same integral by
C alpha |Delta_beta|. This needs O(log alpha) additional mantissa bits, not
operations proportional to alpha. These are explicit input costs, not free data.

SOURCE SCOPE AND REMAINING ORIGINAL GAP
The source arXiv:2610.03578v1 AppendixB.1/B.2 supplies the kernel and uses an
explicit positive quadrature with interval resolution proportional to
1/(1+Lambda_R). The present coherent quantile construction removes enumerated
high-frequency node cost and implements the interacting filtered boundary
operator. Kernel identities and general block-encoding rules are inherited.
Primary full-text locators were checked on22 October2026. No global novelty
claim is made for quantile sampling or interaction-picture simulation.

This theorem concerns F_H(V), the coefficient of the QBP dressing ODE. It
does not by itself implement the complete nonunitary dressing, its inverse,
the recursive amplified purification, or the full coherent completion and
Lindblad evolution with the original total N,temperature,precision budget.
In particular a naive commutator encoding of the coherent correction still
contains an alpha_H normalization that this filtered-V theorem does not remove.
Those must be composed and re-audited rather than declared automatic. M05 is
still open. The route decision is to retain an on-site-uniform interacting
coefficient oracle and move the bottleneck to those specific remaining stages.

CHECKS
The companion checker validates the scalar law, inversion and register-size
accounting at high precision, including large on-site-frequency controls and
negative controls for a naively enumerated grid. These are implementation
diagnostics, not a substitute for the analytic uniform error/cost proof or a
compiled full Gibbs circuit. Freeze before adoption. Actual usage unknown.
