发表机构
Jadavpur University; Presidency University（贾达普大学; 总统大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对 SU(N) 矩阵模型 Trotter 演化，提出基于平方和分解的编译方法，在 NISQ 与容错硬件上分别以 Hadamard 和 T 门实现，显著降低门数并优于现有奇偶校验网络。
AI 中文摘要
SU$(N)$ 矩阵模型势能 $V=-\tfrac{g^2}{4}\sum_{I,J}Tr[X_I,X_J]^2$ 是 BFSS/BMN 家族的相互作用项，其热大-$N$ 态对偶于黑洞，该势能在坐标基下是对角的，并在文献中被编译为每个 Trotter 步包含 $O(d^2N^4Q^4)$ 个旋转的相位多项式。对于其门类而言，该成本接近最优：我们证明，对于一般的 Trotter 步，Walsh 支撑 $m_{\min}$ 下界限制了每个包含 $\{CNOT,CZ,CPhase,R_{ZZ},SWAP,X,R_z\}$ 门的电路（包括辅助比特）的旋转计数和双量子比特门计数，并且我们构建了一个达到 $1.01$--$1.10\\,m_{\min}$ 的奇偶校验网络。因此，改进需要该类之外的门，而将 $V$ 精确分解为平方对易子项之和在每一个时代都提供了这样的门。对于 NISQ，Hadamard：每个平方项被添加到傅里叶寄存器中并在其中平方，需要 $\Theta(d^2N^3Q^3)$ 个双量子比特门。对于容错，$T$：每个平方项使用一次相位梯度注入的可逆算术，需要 $\Theta(d^2N^3Q^2)$ 个 T 门且无任意角度旋转。两者均已构建并验证。使用原生 $R_{ZZ}$ 时，NISQ 电路在 SU(2) 的 $Q=6$ 和 SU(5) 的 $Q=2$ 起使用的双量子比特门少于所有奇偶校验网络；预言机在除最小点外的所有构建点上均优于精度匹配的汉明权重相位化现有方法，T 门数量最多减少 $11.9\times$，在相同宽度下表面码时空体积减少 $25.5\times$。使用相同算术但无分解的消融实验成本最多增加 $21\times$，并在 17 个点中的 15 个点上输给现有方法，因此是分解而非算术带来了优势。两种构造均能经受超对称形变到含费米子的 mini-BMN，并且在态矢量模拟器上完成的完整 Trotter 演化与精确乘积公式的差异达到 $2\times10^{-12}$。
英文摘要
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, $Θ(d^2N^3Q^3)$ 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 $11.9\times$ 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 $2\times10^{-12}$.
Comments39 pages, 15 figures, 22 tables
