aiqbench analysis
This research presents a novel compilation method for SU(N) matrix-model dynamics, significantly reducing the gate counts required for both NISQ and fault-tolerant quantum hardware by leveraging exact factorization of commutator potentials. The study demonstrates that this specific algebraic approach outperforms standard parity networks and Hamming-weight-phasing incumbents, offering up to a 25.5x reduction in surface-code spacetime overhead for native R_ZZ architectures. By proving optimality within its gate class and validating constructions on simulators, the work provides a concrete pathway to efficient quantum simulation of black hole duals and other high-energy physics models.
Sum-of-squares compilation of matrix-model dynamics for NISQ and fault-tolerant hardware
arXiv:2610.04731v1 Announce Type: new Abstract: The SU$(N)$ matrix-model potential $V=-\tfrac{g^2}{4}\sum_{I,J}Tr[X_I,X_J]^2$, 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 $O(d^2N^4Q^4)$ 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, $\Theta(d^2N^3Q^3)$ two-qubit gates. For fault tolerance, $T$: reversible arithmetic with one phase-gradient injection per squared entry, $\Theta(d^2N^3Q^2)$ 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 $11.9\times$ 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 $2\times10^{-12}$.
Abstract:The SU$(N)$ matrix-model potential $V=-\tfrac{g^2}{4}\sum_{I,J}Tr[X_I,X_J]^2$, 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 $O(d^2N^4Q^4)$ 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, $\Theta(d^2N^3Q^3)$ two-qubit gates. For fault tolerance, $T$: reversible arithmetic with one phase-gradient injection per squared entry, $\Theta(d^2N^3Q^2)$ 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 $11.9\times$ 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 $2\times10^{-12}$.
arXiv:2610.04731 [quant-ph]
(or arXiv:2610.04731v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2610.04731 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 358 words
