Invariant Theory, Magic State Distillation, and Bounds on Classical Codes

TL;DR

This paper links the physical constraints of magic state distillation to classical error-correcting code properties, deriving new bounds and resolving longstanding open problems in coding theory.

Contribution

It introduces quantum consistency constraints on classical weight enumerators, leading to new bounds on code parameters and solving a major open problem in classical coding theory.

Findings

Derived new bounds on classical and quantum code parameters.

Proved the non-existence of certain extremal Hermitian self-dual codes over GF(4).

Performed exhaustive search of distillation protocols with n<20, finding no superior thresholds.

Abstract

We show that the physical consistency of magic state distillation imposes new constraints on the weight enumerators of classical error-correcting codes. We establish that for $|T\rangle$-state distillation protocols based on linear self-orthogonal $GF(4)$ codes, the distillation threshold and noise-suppression exponent are directly determined by the code's simple weight enumerator. By enforcing the physical consistency of the distillation process -- specifically, that the probability of successfully projecting onto the target state must be non-negative -- we derive a new set of constraints on classical weight enumerators. These ``quantum consistency'' constraints prove to be strictly stronger than those derived from classical invariant theory, yielding new upper bounds on the minimum distance of certain classical and quantum codes. Most notably, we show that these new constraints resolve a long-standing open problem in classical coding theory by proving the non-existence of extremal Hermitian self-dual codes over $GF(4)$ with parameters $[12m, 6m, 4m+2]$. Additionally, we use our formalism to perform an exhaustive search of distillation protocols based on linear $GF(4)$ codes with $n < 20$, finding no protocols with thresholds exceeding the 5-qubit code, and we derive analytic upper bounds on the noise-suppression exponents of such distillation routines as a function of $n$.