Invariant Theory and Magic State Distillation

TL;DR

This work links Bravyi–Kitaev |T⟩ state distillation to the simple weight enumerators of M3-codes (linear GF(4) self-orthogonal codes), enabling a rigorous framework that combines signed weight enumerators, invariant theory, and linear programming to bound the noise-suppression exponent ν and code distances. By deriving quantum consistency constraints (nonnegative projection probabilities and stabilization outside the stabilizer octahedron) in addition to classical positivity, the authors constrain the space of admissible weight enumerators and demonstrate nonexistence results for certain extremal classical codes. An exhaustive n<20 search shows no M3-code outperforms the 5-qubit code, while invariant-theoretic bounds and LP techniques yield broad restrictions on ν across many lengths and provide new insights into the distances of classical GF(4) codes. The findings clarify fundamental limits of qubit MSD, connect quantum coding constraints to classical code theory, and point to future directions for discovering higher-threshold MSD routines and extending the approach to qudits.

Abstract

We show that the performance of a linear self-orthogonal $GF(4)$ code for magic state distillation of Bravyi and Kitaev's $|T\rangle$-state is characterized by its simple weight enumerator. We compute weight enumerators of all such codes with fewer than 20 qubits and find none whose threshold exceeds that of the 5-qubit code. Using constraints on weight enumerators from invariant theory and linear programming, we establish bounds on the exponent characterizing noise suppression of a $|T\rangle$-state distillation protocol. We also obtain new non-negativity constraints on such weight enumerators by demanding consistency of the associated magic state distillation routine. These constraints yield new bounds on the distances of classical Hermitian self-dual and maximal self-orthogonal linear $GF(4)$ codes, notably proving the nonexistence of such codes with parameters $[12m, 6m, 4m+2]_{GF(4)}$.

Figures (14)

• Figure 1: Qubit magic states on the Bloch sphere. The red point denotes the $\mathinner{|{T}\rangle}$ magic state and the green point denotes the $\mathinner{|{H}\rangle}$ magic state.
• Figure 2: The distillation performance, $\epsilon_{\rm out}(\epsilon)$, of the 5-qubit code.
• Figure 3: $\epsilon_{\rm out}(\epsilon)$ computed for all $[[n,1]]$$M_3$-codes with $n=17$ (above) and $n=19$ (below) with nonzero success probability. While many $[[17,1]]$ codes distill the $\mathinner{|{T}\rangle}$ state, no code has threshold exceeding that of the 5-qubit code, $0.172673$.
• Figure 4: Linear programming bounds for $n=5$. The dashed blue line corresponds to the classical linear programming constraints in Equation \ref{['5-qubit-eqconstraint1']} and the yellow line corresponds to the bound the additional success probability constraint (Equation \ref{['5-qubit-eqconstraint2']}).The two black points are the two inequivalent $[[5,1]]$$M_3$-codes. the extremal weight enumerator, $d_0=-36$, for magic state distillation is far outside the allowed region.
• Figure 5: The distillation performance of a putative weight enumerator with $n=11$. While the weight enumerator satisfies positivity and integrality of $A_i$, $B_i$ and $C_i$, as well as consistency with the MacWilliams identity, the weight enumerator is clearly unphysical, as the threshold for distillation is deep within the stabilizer polytope.

Theorems & Definitions (17)

• Theorem 1: Rall's rule
• proof
• Theorem 2: Signed weight enumerators from simple weight enumerators
• Example 1
• Example 2
• Theorem 3
• Lemma 1
• proof
• Theorem 4
• Theorem 5