The $T$-count is a dominant cost of fault-tolerant Clifford$+T$ computation. We prove that every Hadamard-free Clifford$+T$ circuit with clean ancillas for a layer of $m$ disjoint CCZ gates, the diagonal core of a parallel Toffoli layer, uses at least $6m+1$ $T$ gates. Campbell and Howard gave a circuit with $6m+1$, and to our knowledge this is the first proof that it is optimal for general $m$. The proof combines Campbell and Howard's self-orthogonality of the columns of the parity matrix with a rank (flattening) bound, sharpened by one. More generally, the argument gives a lower bound on the $T$-count of every diagonal gate in the third level of the Clifford hierarchy, the isotropy floor, computable by Gaussian elimination. It is never below the stabilizer nullity $ν$ and reaches $2ν+1$ on non-Clifford gates whose phase is purely cubic. No bound that also holds for circuits with measurement and feedforward can do this, since such circuits implement CCZ ($ν=3$) with four $T$ gates and the layer with $4m$, where the floor gives seven and $6m+1$. The floor also recovers Campbell and Howard's count $4m+3$ for a fan-out of Toffolis. On the output of PyZX's TODD-based optimizer, relative to its Hadamard placement, it certifies $193$ of $311$ phase-polynomial blocks $T$-optimal once repeated parities are merged (at the best of five optimizer seeds), against $113$ for nullity, including $37$ of the $72$ blocks too large for exhaustive search.
In fault-tolerant Clifford+T circuits, a Boolean oracle costs T gates and extra qubits, and on four or more qubits every exact Toffoli gate needs an extra qubit. Gosset, Kothari and Zhang showed that sampling a fresh random circuit for each call makes the T-count of the Toffoli gate independent of its size, but a direct implementation of their sampler uses about one extra qubit per bit of precision. We show that computing the sampled parities in place, on the gate's own controls, removes every extra qubit once the gate is large enough: each random circuit then uses about 8 T gates per bit of precision and borrows only the gate's own idle inputs. This is one T gate more per circuit than the best measurement-free variant that keeps one clean ancilla. The same holds for exclusive-ors of indicators of affine subspaces, such as equality tests and checks of linear equations. For randomized circuits that do not measure, this is within a factor of eight of optimal up to an additive constant, even with extra qubits, and when each random circuit computes a Boolean function exactly, closing the factor asymptotically is equivalent to determining the leading constant in the T-count of exact multi-controlled Toffoli gates. Finally, the oracle of the indicator of a random set of fixed density needs exponentially many T gates at constant error, while that of an affine subspace of the same density needs a constant number, and extra qubits reduce the exponential cost only quadratically.
We present, to our knowledge, the first adaptation of Pauli Correlation Encoding (PCE) to quantum topological data analysis, reformulating Betti number estimation as a depth-efficient variational optimization over a compressed qubit register. From a Takens embedding and Vietoris-Rips filtration of S&P 500 returns, we extract combinatorial Laplacians and recast null-space counting as a continuous-PCE Rayleigh-quotient minimization with variational deflation, encoding $n_k$ simplex indices into $O(n_k^{1/κ})$ qubits with shallow, ancilla-free circuits. Because the resulting loss is rational rather than bilinear in the correlators, the barren-plateau bound of Sciorilli et al. does not transfer; empirically the gradient variance decays only polynomially, with no exponential barren plateau, over $n=4$-$12$ qubits. The classical stage matches ripser on all 190 sliding windows (2003-2010). On the real market Laplacians ($β_1=1$-$22$), PCE-VQE recovers no null vector from a random start, but warm-starting from a classical null-space surrogate recovers $β_1$ exactly on all seven windows tested ($n_k \le 259$), placing the obstacle in the optimization landscape rather than the encoding. Chronologically split classification gives in-regime ROC AUC 0.818, but out-of-distribution evaluation on the 2020 COVID shock and 2022 rate cycle (AUC 0.009, 0.515) shows the calibration does not generalize across crisis regimes.