For every fixed T>0, the original learning problem has an affirmative answer. Given an upper bound m>=1, an n-qubit traceless Hamiltonian H with at most m nonidentity Pauli terms and ||H||<=1 can be learned to maximum coefficient error epsilon with probability at least 2/3. Total unknown evolution time and query count are tilde O_T(poly(m)/epsilon); known circuit and classical-processing costs are polynomial in n,m,1/epsilon. Every unknown call is ordinary forward evolution for a duration at least T. This construction completes the supplied coordinator proposal; it requires neither controlled unknown evolution nor an unknown inverse.

Assume 0<epsilon<1; otherwise output zero. Write d=2^n, |X>=(X tensor I)|Phi_d>, and use normalized Hilbert--Schmidt inner products. Pauli Choi vectors are an orthonormal basis. Fix an integer tau>=max(1,T). All constants below depend only on tau and are effectively computable.

1. Polynomial-cost initialization. Put f_P(t)=Tr(P exp(-iHt))/d. Its real-time derivatives have absolute value at most one. For L>=1, interpolate at t_j=tau+j/L, j=0,...,L, with derivative weights w_j=ell'_j(0). Writing b0=tau+1 and W=sum_j|w_j|, the cardinal-polynomial formula gives
W<=L b0^(L-1) L^L 2^L/L! <=(L/b0)(2e b0)^L.
Interpolation is exact on the degree-L Taylor polynomial, whose linear coefficient is -i a_P. Taylor's integral remainder therefore gives
|f'_P(0)-sum_j w_j f_P(t_j)|<=W b0^(L+1)/(L+1)! <=[2e^2 b0^2/L]^L.
Choose L at least 4e^2 b0^2 and log_2(4/z). The last bound is at most z/4. Consequently, if |a_P|>=z, some node has |f_P(t_j)|>=3z/(4W). A uniformly random node, ordinary Choi preparation, and Bell measurement samples this P with probability at least 9z^2/[16(L+1)W^2]. Thus O((L+1)W^2 z^-2 log(m/p)) samples discover all such labels except with probability p. This number is polynomial in 1/z for fixed tau; no label enumeration is used.

Estimate each discovered nonidentity label using two independently prepared Choi states x=|U(tau)> and y=|U(tau+delta)>. The known unitary O=(P on the first output)SWAP, swapping the complete Choi registers, has expectation
<x,y|O|x,y>=f_P(delta) conjugate(f_I(delta)).
A Hadamard test controls only this known O. Since Tr H=0, Taylor bounds give |f_P(delta)+i delta a_P|<=delta^2/2 and |f_I(delta)-1|<=delta^2/2. Also |f_P(delta)|<=1. Hence minus the imaginary expectation divided by delta estimates a_P with bias at most delta. Set delta=z/2. Hoeffding bounds give coefficient error at most z simultaneously with O(z^-4 log(Lc/p)) shots per candidate, where Lc is the candidate count.

Keep the m largest estimated magnitudes. Retained coefficients have error at most z. An omitted true label outside the list has magnitude below z. An omitted true label in the list forces at least one retained spurious label, whose estimated magnitude is at most z; thus its true magnitude is at most 2z. Therefore maximum coefficient error is at most 2z and operator error at most 4mz, since the union of the true and retained supports has size at most 2m. The initializer is polynomial in n,m,1/z and log(1/p).

2. A constructive residual filter. Suppose a known, traceless, m-term H0 satisfies ||H0||<=2 and ||Delta||<=eta, where Delta=H-H0. Set h=1/4, t1=tau, t2=tau+h, b=t2, and A=ad_H0. This self-adjoint operator on operator Hilbert space has spectrum in [-4,4]. Define
F_j(x)=integral_0^{t_j} exp(ixs) ds,
G(x)=(|F_1(x)|^2+|F_2(x)|^2)/2,
R_j(x)=conjugate(F_j(x))/G(x).
For |x|<=6, the inequality sin^2(u)+sin^2(u+xh/2)>=1-|cos(xh/2)| implies G(x)>=h^2/pi^2=:g, including x=0 by continuity. Thus |R_j|<=b/g=:M and (R_1F_1+R_2F_2)/2=1. Set kappa_j=R_j(0)=t_j/G(0)>0.

Here is an explicit dimension-independent implementation. Let chi=1 on [-4,4], zero outside [-6,6], and on the transition use 1-10u^3+15u^4-6u^5 with u=(|x|-4)/2. Extend chi R_j periodically with period 16. It is C^2. Its Fourier coefficients c_jk are real, because its value at -x is the conjugate of its value at x. Integration by parts twice gives
|c_jk|<=4||(chi R_j)''||_1/(pi^2 k^2), k!=0.
The derivative norm and the absolute coefficient sum are bounded by constants depending only on tau. Explicit bounds follow from |F_j^(ell)|<=b^(ell+1)/(ell+1), G>=g, differentiation of the quotient, and the displayed cutoff.

For accuracy rho<=1, truncate at D=O_tau(1/rho), compute real rational coefficients with total absolute error O(rho), and adjust the constant coefficient so Q_j(0)=kappa_j exactly. This gives sup_{[-4,4]}|Q_j-R_j|<=rho and a uniformly bounded coefficient absolute sum. Computation is polynomial in 1/rho: uniform quadrature with mesh O_tau(rho/D^2), using certified elementary-function evaluations, suffices by the integrands' first-derivative bounds. The adjustment is rational because t_j and kappa_j are rational.

Choose a fixed rational S>1 above these absolute sums. Add cancelling +I and -I coefficients to make the absolute sum S. Elementary LCU, preparing coefficient magnitudes, selecting signed known unitaries, and unpreparing, has success Kraus operator K_j=Q_j(A)/S. Its selected unitaries are
exp(i pi k H0/8) tensor exp(-i pi k H0^T/8).
In particular K_j|I>=(kappa_j/S)|I>=k_j|I> exactly, and ||K_j||<=1.

This construction has polynomial known-gate cost. H0 has Pauli absolute sum at most 2m. First-order product formulas approximate a known evolution of length v to error gamma using O(1+m^2 v^2/gamma) slices, each consisting of m Pauli rotations; this follows from the two-term Duhamel commutator bound and telescoping unitary products. Here |v|<=pi D/8. Compile paired factors as V tensor conjugate(V), which fixes |I> exactly. Coefficient-state preparation needs O(D) two-level rotations. Ideal known rotations give the exact Kraus operator above; finite gate-set rounding is included in the error budget below. No unknown oracle is controlled.

3. Residual amplification and measurement. Prepare the Choi state of W_j^r, where W_j=exp(iH0 t_j)exp(-iHt_j). Each factor uses one allowed unknown call. Concatenating the interaction-picture evolutions, the integrated generator norm is at most br eta. The second-order Duhamel remainder, with its final propagator kept unitary, yields
W_j^r=I-i r F_j(A)Delta+E_j, ||E_j||<=B(r eta)^2, B=b^2/2.
Let q=r eta. Define d_jP=(R_j(A)F_j(A)Delta)_P. Because R_jF_j=|F_j|^2/G is real and even, d_jP is real; moreover (d_1P+d_2P)/2=Delta_P and |d_jP|<=Mb eta.

The unnormalized successful filtered vector y_j has, for P!=I,
y_jI=k_j+e_jI, y_jP=-i(r/S)d_jP+e_jP.
The total error norm is at most E=Bq^2+rho bq/S. Its identity component satisfies |e_jI|<=k_j Bq^2: the linear term is traceless and Q_j(0)=kappa_j. Here we used normalized Hilbert--Schmidt norm bounded by operator norm.

Choose j uniformly, apply the filter and, on success, measure the Bell basis. Failure yields no label. Jensen and the triangle inequality show that every |Delta_P|>=zeta has unconditional sampling probability at least (r zeta/S-E)^2 when the parenthesis is positive.

For estimation, on success measure O_P=-i|I><P|+i|P><I|, with scores +1,-1,0; filter failure also scores zero. Its mean is 2 Im(conjugate(y_jI)y_jP). Report X=-S score/(2r k_j). The ideal mean is d_jP. Direct multiplication of the two displayed amplitudes bounds the bias by
(S/(r k_j))[k_j E+k_j Bq^2(rMb eta/S)+k_j Bq^2 E]
<=C0 eta(q+rho), for q,rho<=1.
Also |X|<=C1/r, where C1=S/(2 min_j k_j). Averaging uniform j estimates Delta_P. These observables require only polynomial known gates: inverse Bell transformation, comparison with two specified bit strings, and a two-level rotation implement them.

Choose constants K sufficiently large and c>0 sufficiently small, depending only on the preceding constants. Set a=1/(Km), assume eta<=a/2, take r=floor(a/eta), rho=c/m, and zeta=eta/(16m). Then a/2<=q<=a. Constants can be chosen so
E<=q/(64mS), C0(q+rho)<=1/(64m).
Indeed E/q<=B/(Km)+cb/(mS), and C0(q+rho)<=C0(1/K+c)/m. Discovery probability is consequently at least (3q/(64mS))^2>=c2/m^4. There are at most 2m true residual labels, so O_tau(m^4 log(4m/p)) experiments discover all those of magnitude at least zeta with failure at most p.

The estimation bias is at most zeta/4. Since r zeta>=1/(32Km^2), Hoeffding gives statistical error at most zeta/2 on any Lc specified labels using O_tau(Lc m^4 log(4Lc/p)) fresh experiments. This remains valid conditional on any discovery history.

Compilation does not change these scalings. Require per-experiment trace-distance error nu=c3/m^4, for a sufficiently small fixed c3. Label probabilities change by at most nu, retaining a constant multiple of m^-4. Score means change by at most 2nu, and scaled estimates by at most 2C1 nu/r, which can be made at most zeta/4. Compile each compensating evolution to O(nu/r), and the remaining preparation, LCU and measurement circuit jointly to O(nu). Telescoping proves the stipulated experiment error. Product formulas and finite-precision known rotations meet these accuracies with polynomial circuit and classical cost in n,m,r. Thus total estimation error is at most zeta. All filter failures have already been charged. Each experiment uses exactly r unknown calls.

4. Bootstrap and termination. Choose eta0=1/(2Km), and run Step 1 with z=eta0/(4m). This gives an m-term H0 with operator error at most eta0, hence ||H0||<=2, with polynomial initialization cost independent of final precision except failure logarithms. If epsilon>=eta0, return it. Otherwise iterate with deterministic bounds eta_s=eta0/2^s until eta_J<=epsilon.

At stage s apply Steps 2--3, and include every current H0 label in the discovered list. Add estimated residual coefficients to the current coefficients and retain the m largest magnitudes. Every listed provisional coefficient has error at most zeta, and every unlisted true coefficient has magnitude below zeta, since outside the list H0 is zero. The truncation argument of Step 1 gives maximum error at most 2zeta and operator error at most 4m zeta=eta_s/4, in particular at most eta_{s+1}. This establishes the induction and the final coefficient guarantee. On good histories all coefficients of H0 have magnitude at most 2. One may abort if this bound is violated; this never affects a good history and ensures bounded implementation cost on all histories.

There are two failure events for initialization and two for each stage. Assign each probability p=1/[12(J+1)], taking J=0 if no refinement is needed. A conditional union bound gives total failure at most 1/6, stronger than required.

Each stage has O_tau(m^4 log(4m/p)) candidates and therefore tilde O_tau(m^8) experiments. Each uses r_s<=a/eta_s allowed calls, of durations at most b. Since eta_{J-1}>epsilon, sum_{s<J} r_s<=2a/epsilon. Thus refinement evolution time and queries are tilde O_tau(m^8/epsilon), conservatively. Initialization is polynomial in m: L=O_tau(log m), W=poly_tau(m), and z^-1=O_tau(m^2). Absorb its polynomial cost into poly_tau(m)/epsilon. Logarithmic factors include the failure allocation and number of stages. All stored lists, controls, precision requirements and arithmetic have polynomial cost in n,m,1/epsilon. Every unknown duration is at least tau>=T, completing the original guarantee.