Sum-of-squares compilation of matrix-model dynamics for NISQ and fault-tolerant hardware
Abstract
The SU(N) matrix-model potential V=−g2/4∑I,JTr[XI,XJ]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(d2N4Q4) 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 mmin lower-bounds both the rotation count and the two-qubit count of every circuit over {CNOT,CZ,CPhase,RZZ,SWAP,X,Rz}, ancillas included, and we build a parity network attaining $1.01−−1.10\,m_{\min}$. 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, Θ(d2N3Q3) two-qubit gates. For fault tolerance, T: reversible arithmetic with one phase-gradient injection per squared entry, Θ(d2N3Q2) T gates and no arbitrary-angle rotation. Both are built and verified. With a native RZZ 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× in T count and, at equal width, 25.5× in surface-code spacetime. An ablation with the same arithmetic but no factorisation costs up to 21× 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 2×10−12.
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BibTeX
@misc{dutta2026sumofsquares,
title = {{Sum-of-squares compilation of matrix-model dynamics for NISQ and fault-tolerant hardware}},
author = {Ayanendu Dutta and Nabanita Sarkar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04731},
eprint = {2610.04731},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04731},
}
RIS
TY - UNPB
TI - Sum-of-squares compilation of matrix-model dynamics for NISQ and fault-tolerant hardware
AU - Ayanendu Dutta
AU - Nabanita Sarkar
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04731
UR - https://arxiv.org/abs/2610.04731
AB - The SU(N) matrix-model potential V=−g^2/4∑_I,JTr[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^2N^4Q^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 $1.01−−1.10\,m_{\min}$. 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, Θ(d^2N^3Q^3) two-qubit gates. For fault tolerance, T: reversible arithmetic with one phase-gradient injection per squared entry, Θ(d^2N^3Q^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× in T count and, at equal width, 25.5× in surface-code spacetime. An ablation with the same arithmetic but no factorisation costs up to 21× 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 2×10^−12.
ER -