Logical Compilation for Multi-Qubit Iceberg Patches

Abstract: Recent advancements in quantum computing have enabled practical use of quantum error detecting and correcting codes. However, current architectures and future proposals of quantum computer design suffer from limited qubit counts, necessitating the use of high-rate codes. Such codes, with their code parameters denoted as $[[n, k, d]]$, have more than $1$ logical qubit per code (i.e., $k > 1$). This leads to reduced error tolerance of the code, since $\lceil (d-1)/2\rceil$ errors on any of the $n$ physical qubits can affect the logical state of all $k$ logical qubits. Therefore, it becomes critical to optimally map the input qubits of a quantum circuit to these codes, in such a way that the circuit fidelity is maximized. \par However, the problem of mapping program qubits to logical qubits for high-rate codes has not been studied in prior work. A brute force search to find the optimal mapping is super exponential (scaling as $O(n!)$, where $n$ is the number of input qubits), making exhaustive search infeasible past a small number of qubits. We propose a framework that addresses this problem on two fronts: (1) for any given mapping, it performs logical-to-physical compilation that translates input gates into efficiently encoded implementations utilizing Hadamard commutation and gate merging; and (2) it quickly searches the space of possible mappings through a merge-optimizing, noise-biased packing heuristic that identifies high-performing qubit assignments without exhaustive enumeration. To the best of our knowledge, our compiler is the first work to explore mapping and compilation for high-rate codes. Across 71 benchmark circuits, we reduce circuit depth by $34\%$, gate counts by up to $31\%$ and $17\%$ for one-qubit and two-qubit gates, and improve total variation distance by $1.75\times$, with logical selection rate improvements averaging $86\%$ relative to naive compilation.