Sum-of-squares compilation of matrix-model dynamics for NISQ and fault-tolerant hardware

The SU$(N)$ matrix-model potential $V=-\frac{g^2}{4}{\sum }_{I,J}Tr[X_I,X_J{]}^2$, the interaction of the BFSS/BMN family, whose thermal large-$N$ states are dual to black holes, is diagonal in the coordinate basis and is compiled in the literature as a phase polynomial with $O(d^2{N}^4{Q}^4)$ rotations per Trotter step. That cost is close to optimal for its gate class: we show that for a generic Trotter step the Walsh support $m_{\min }$ lower-bounds both the rotation count and the two-qubit count of every circuit over $\{CNOT,CZ,CPhase,R_{ZZ},SWAP,X,R_z\}$, ancillas included, and we build a parity network attaining $1.01$–$1.10m_{\min }$. Improvement therefore requires a gate outside that class, and the exact factorisation of $V$ into a sum of squared commutator entries supplies one in each era. For NISQ, Hadamard: each squared entry is added into a Fourier register and squared there, $\mathrm{\Theta }(d^2{N}^3{Q}^3)$ two-qubit gates. For fault tolerance, $T$: reversible arithmetic with one phase-gradient injection per squared entry, $\mathrm{\Theta }(d^2{N}^3{Q}^2)$ T gates and no arbitrary-angle rotation. Both are built and verified. With a native $R_{ZZ}$ the NISQ circuit uses fewer two-qubit gates than every parity network from $Q=6$ at SU(2) and $Q=2$ at SU(5); the oracle beats a precision-matched Hamming-weight-phasing incumbent at every point built except the smallest, by up to $11.9\times$ in T count and, at equal width, $25.5\times$ in surface-code spacetime. An ablation with the same arithmetic but no factorisation costs up to $21\times$ more and loses to that incumbent at $15$ of $17$ points, so the factorisation, not the arithmetic, produces the win. Both constructions survive the supersymmetric deformation to mini-BMN with fermions, and complete Trotter evolutions on a statevector simulator reproduce the exact product formula to $2\times {10}^{-12}$.