Prateek P. Kulkarni

The Logical Clifford Synthesis (LCS) framework of Rengaswamy et al.\ (IEEE TQE, 2020) enumerates all physical Clifford circuits realizing a given logical Clifford operator for an $[[m,k]]$ stabilizer code, proving that exactly $2^{r(r+1)/2}$ symplectic solutions exist, where $r = m - k$. This result, however, does not extend to subsystem stabilizer codes, whose gauge group introduces degrees of freedom invisible to stabilizer-only theory. We formulate and solve the LCS problem for $[[m,k,g,d]]$ subsystem codes. Our main result proves that the group $\Hgauge$ of physical symplectic matrices that normalize the gauge group and act trivially on logical qubits splits as a direct product $\Hgauge \cong \Hstab \times \Sp(2g,\Ftwo)$, where $\Hstab$ is the stabilizer-sector group of order $2^{r(r+1)/2}$ with $r = m - k - g$. Consequently, every logical Clifford has exactly $2^{r(r+1)/2} \cdot |\Sp(2g,\Ftwo)|$ distinct physical realizations, exhibiting a product structure that permits independent optimization over stabilizer and gauge degrees of freedom. We supply a complete proof via a splitting of a short exact sequence, an efficient two-phase enumeration algorithm with $O(m^3)$ preprocessing and $O(m^2)$ per solution, and a verification on the $[[4,1,1,2]]$ Bacon–Shor code, recovering all $48$ distinct solutions versus the $8$ predicted by the naïve stabilizer count. We further derive closed-form orbit-size formulas, discuss conditions under which the splitting may fail for certain non-Abelian gauge groups, and outline implications for noise-adaptive compilation on near-term hardware.

We establish tight connections between entanglement entropy and the approximation error in Trotter-Suzuki product formulas for Hamiltonian simulation. For systems governed by local Hamiltonians with bounded entanglement entropy $\Smax$, we prove that the first-order Trotter error scales as $\BigO(t^2 \Smax \polylog(n)/r)$ rather than the worst-case $\BigO(t^2 n/r)$, where $n$ is the system size and $r$ is the number of Trotter steps. This yields an exponential improvement for area-law entangled systems where $\Smax = \BigO(\log n)$. We further establish a separation result showing that volume-law entangled systems fundamentally require $\Omega(\sqrt{n})$ more Trotter steps than area-law systems to achieve the same precision. Our analysis combines Lieb-Robinson bounds, tensor network methods, and novel commutator-entropy inequalities.