Minimizing the Number of Code Switching Operations in Fault-Tolerant Quantum Circuits

TL;DR

The paper tackles minimizing code-switching overhead in code-switching-based fault-tolerant quantum computing by reducing the minimal switching problem to a minimum $s$-$t$ cut on a graph derived from the circuit. It presents a flexible min-cut formulation that accommodates one-way CNOTs, idling during qubit idle periods, and biasing toward a preferred code, enabling simultaneous optimization of switching and other metrics. Empirical results show the approach scales to circuits with up to $1024$ qubits and millions of gates, yielding minimal switches efficiently and enabling depth reductions through idle-aware switching and code bias. The work provides an automated, logical-level compiler tool (part of the Munich Quantum Toolkit) and lays groundwork for evaluating when code switching is advantageous relative to magic-state distillation.

Abstract

Fault-tolerant quantum computers rely on Quantum Error-Correcting Codes (QECCs) to protect information from noise. However, no single error-correcting code supports a fully transversal and therefore fault-tolerant implementation of all gates required for universal quantum computation. Code switching addresses this limitation by moving quantum information between different codes that, together, support a universal gate set. Unfortunately, each switch is costly-adding time and space overhead and increasing the logical error rate. Minimizing the number of switching operations is, therefore, essential for quantum computations using code switching. In this work, we study the problem of minimizing the number of code switches required to run a given quantum circuit. We show that this problem can be solved efficiently in polynomial time by reducing it to a minimum-cut instance on a graph derived from the circuit. Our formulation is flexible and can incorporate additional considerations, such as reducing depth overhead by preferring switches during idle periods or biasing the compilation to favor one code over another. To the best of our knowledge, this is the first automated approach for compiling and optimizing code-switching-based quantum computations at the logical level.

Figures (6)

• Figure 1: Minimal code switching: (a) Transversal gates of the 2D and 3D color codes. CNOT gates are transversal in both codes. (b) Circuit implementation requiring $6$ switching operations. (c) Minimal solution requiring only $4$ switches.
• Figure 2: Network of the circuit in \ref{['fig:code_switching_example']}. Each qubit operation is represented as a graph node. The terminal nodes, depicted here as the green and red switches, represent the two different codes. The dashed line represents the minimum cut that separates the graph into two distinct subsets of nodes.
• Figure 3: Modeling extensions to the min-cut formulation and resulting impact on the min-cut. (a) One-way transversal CNOT: cutting only a single edge separates source from sink. (b) Prefer switching during qubit idling: a lower weight for idling edges pushes the min-cut algorithm to include these edges in the min-cut. (c) Code bias: adding additional bias edges and choosing capacities such that cutting bias edges of the 3D code results in lower overall costs, results in min-cuts that put more gate nodes on the 2D color code side of the cut.
• Figure 4: Average runtime of the proposed compilation methods across different circuit sizes and gate distributions.
• Figure 5: Average number of minimal switching operations across different circuit sizes and gate distributions.

Theorems & Definitions (8)

• Example 1
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