Magic states are rarely the best resource to optimize: An analytical tool for qubit resource estimation in concatenated codes

Abstract

Concatenated error-correction schemes are well-understood routes to fault-tolerant quantum computing, and research on such schemes continues, including recent claims that they may be competitive with surface codes, and show potential when combined with high-rate Quantum Low Density Parity Check codes. However, there are few tools to evaluate the qubit resources required by concatenated schemes. We propose such a tool here. Its equations are closed-form and remain simple for an arbitrary number of levels of concatenation, making it ideal for comparing and minimizing the resource costs of such schemes. We use this tool to evaluate the resources for gate operations that require the injection of so-called ``magic states'', needed to complete the set of logical operations. It was expected that the complexity of such ``magic operations" would make them dominate the resource costs of a calculation, with numerous works proposing optimizations of these cost. Our work reveals that this expectation is often inaccurate: Magic operations are rarely the dominant cost of concatenated schemes, mirroring similar conclusions from past work for surface codes. Optimizations affecting all operations naturally have more impact than those on magic operations alone, yet we unexpected find that the former can reduce qubit resources by a few orders of magnitude while the latter give only marginal reductions. We show this in detail for a 7-qubit concatenated scheme with Steane error-correction gadgets or flag-qubits gadgets, and argue that our findings are representative of most concatenated schemes.