This article is a continuation of our recent work (Yin Chen and Runxuan Zhang, Shape enumerators of self-dual NRT codes over finite fields. SIAM J. Discrete Math. 38 (2024), no. 4, 2841-2854) in the setting of quantum error-correcting codes. We use algebraic invariant theory to study three weight enumerators of formally self-dual quantum codes over arbitrary finite fields. We derive a quantum analogue of Gleason's theorem, demonstrating that the weight enumerator of a formally self-dual quantum code can be expressed algebraically by two polynomials. We also show that the double weight enumerator of a formally self-dual quantum code can be expressed algebraically by five polynomials. We explicitly compute the complete weight enumerators of some special self-dual quantum codes. Our approach illustrates the potential of employing algebraic invariant theory to compute weight enumerators of self-dual quantum codes.

翻译：本文是我们近期工作（Yin Chen and Runxuan Zhang, Shape enumerators of self-dual NRT codes over finite fields. SIAM J. Discrete Math. 38 (2024), no. 4, 2841-2854）在量子纠错码背景下的延续。我们运用代数不变理论研究任意有限域上形式自对偶量子码的三种重量计数子。我们推导了Gleason定理的一个量子类比，证明了形式自对偶量子码的重量计数子可以由两个多项式代数表示。我们还证明了形式自对偶量子码的双重量计数子可以由五个多项式代数表示。我们显式计算了一些特殊自对偶量子码的完全重量计数子。我们的方法展示了运用代数不变理论计算自对偶量子码重量计数子的潜力。