Invariant Theory, Magic State Distillation, and Bounds on Classical Codes
This work addresses a foundational problem in quantum computing and coding theory, providing stronger constraints that impact both classical and quantum code design.
The paper tackles the problem of deriving constraints on classical error-correcting codes from the physical consistency of magic state distillation, resulting in new upper bounds on code distances and resolving a long-standing open problem by proving the non-existence of extremal Hermitian self-dual codes with specific parameters.
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 $n$.