Laura S. Herzog

Fault-tolerant quantum computation (FTQC) requires compiling logical quantum circuits encoded using a quantum error-correcting code into physical operations tailored to specific hardware architectures. Lattice surgery has emerged as a leading method to perform logical computation, initially motivated by superconducting qubit architectures with geometrically local connectivity. However, current lattice surgery techniques are limited due to certain paradigmatic assumptions that are widely regarded as standard. In our works we identify and address two of these limiting paradigms. First, prior work has predominantly focused on the surface code, even though other topological codes offer certain advantages. Second, compilation schemes usually follow a place-and-route paradigm where logical qubits remain fixed in space throughout the computation. We initiate a more flexible line of work that goes beyond both aforementioned paradigms. To address the first, we introduce the concept of a code substrate - a blueprint for realizing quantum error correction with topological quantum codes using lattice surgery. We formulate the problem using two layers of abstraction. The microscopic level specifies how lattice surgery operations are realized using distance-preserving ancilla regions, while the macroscopic level abstracts compilation as a “mapping” and “routing” problem on a coarse-grained routing graph. We exemplify this framework with detailed constructions for the color code and folded surface code. To challenge the second paradigm, we exploit movable logical qubits through teleportation during logical CNOT execution. Building on the color code substrate, we adapt the measurement-based CNOT scheme to incorporate logical qubit teleportations without additional time overhead. This enables data qubits to dynamically change positions during compilation – “mapping” and “routing” are thus not viewed as independent and subsequent steps, as previous methods have in an overly simplified manner. This flexibility has the potential to substantially reduce routed circuit depth. Thus, movable logical qubits can be exploited even when physical qubits remain static, making movement-based compilation applicable not only to trapped ion and neutral atom platforms - where physical qubits are dynamic by design - but also to superconducting architectures. In addition to the conceptual work, we provide a set of open-source tools for the compilation of logical circuits for the color code on GitHub https://github.com/munich-quantum-toolkit/qecc.