Quantum codes do not fix isotropic errors

Abstract

In this work, we prove that quantum error correcting codes do not fix isotropic errors (Theorem 5), even assuming that their correction circuits do not introduce new errors. We say that a quantum code does not fix a quantum computing error if its application does not reduce the variance of the error. We also prove for isotropic errors that, if the correction circuit of a quantum code detects an error, the corrected logical m-qubit has uniform distribution (Theorem 3) and as a result, it already loses all the computing information.

Appendix

The values of the integrals that have been used throughout the article are included in this appendix:

$$\begin{aligned} \begin{array}{l} \displaystyle \int _{0}^{\pi }\sin ^{k}(\theta )\mathrm{d}\theta = \left\{ \begin{array}{l} \displaystyle 2\,\frac{(k-1)!!}{k!!}\quad \quad k=1,\,3,\,5,\,\,\dots \\ \\ \displaystyle \pi \,\frac{(k-1)!!}{k!!}\quad \quad k=2,\,4,\,6,\,\,\dots \end{array}\right. , \\ \\ \displaystyle \int _{0}^{\frac{\pi }{2}}\cos ^a(\theta )\,\sin ^b(\theta )\,\mathrm{d}\theta = \left\{ \begin{array}{l} \displaystyle \frac{\pi }{2}\frac{(a-1)!!\,(b-1)!!}{(a+b)!!}\quad \text{ if } \text{ a } \text{ and } \text{ b } \text{ are } \text{ even } \\ \\ \displaystyle \frac{(a-1)!!\,(b-1)!!}{(a+b)!!}\quad \text{ in } \text{ another } \text{ case } \end{array}\right. , \\ \\ \displaystyle \int _{0}^{\pi }\frac{\sin ^{2d-2}(\theta _0)}{(1+\sigma ^2-2\sigma \cos (\theta _0))^d}\mathrm{d}\theta _0= \frac{(2d-3)!!}{(2d-2)!!}\frac{\pi }{(1-\sigma ^2)}\quad \quad d=1,\,2,\,3,\,\,\dots \, , \\ \\ \displaystyle \int _{0}^{\pi }\frac{\cos (\theta _0)\sin ^{2d-2}(\theta _0)}{(1+\sigma ^2-2\sigma \cos (\theta _0))^d}\mathrm{d}\theta _0= \frac{(2d-3)!!}{(2d-2)!!}\frac{\sigma }{(1-\sigma ^2)}\pi \quad \quad d=1,\,2,\,3,\,\,\dots \, , \\ \\ \displaystyle \int _{0}^{\pi }\frac{\sin ^{2d}(\theta _0)}{(1+\sigma ^2-2\sigma \cos (\theta _0))^d}\mathrm{d}\theta _0= \frac{(2d-1)!!}{(2d)!!}\pi \quad \quad d=0,\,1,\,2,\,\,\dots \, , \\ \\ \displaystyle \int _{0}^{\pi }\frac{\cos ^2(\theta _0)\sin ^{2d-2}(\theta _0)}{(1+\sigma ^2-2\sigma \cos (\theta _0))^d}\mathrm{d}\theta _0 = \pi \,\frac{(2d-3)!!}{(2d)!!}\,\frac{1+(2d-1)\sigma ^2}{1-\sigma ^2} \ \ d=1,\,2,\,3,\,\,\dots \, . \\ \\ \end{array} \end{aligned}$$

Starting from the first integral, the surface of a unit sphere of arbitrary even (2d) or odd ($2d-1$) dimension can be calculated:

$$\begin{aligned}&\begin{array}{ccl} |\mathcal{S}_{2d}| &{} = &{} \displaystyle \int _0^{\pi }\cdots \int _0^{\pi }\int _0^{2\pi }\sin ^{2d-1}(\theta _0)\,\cdots \,\sin ^{1}(\theta _{2d-2})\ \mathrm{d}\theta _0\,\cdots \,\mathrm{d}\theta _{2d-2}\mathrm{d}\theta _{2d-1} \\ \\ &{} = &{} \displaystyle 2\frac{(2d-2)!!}{(2d-1)!!}\ \frac{(2d-3)!!}{(2d-2)!!}\pi \ 2\frac{(2d-4)!!}{(2d-3)!!}\ \cdots \ \frac{(2-1)!!}{2!!}\pi \ 2\frac{(1-1)!!}{1!!}\ 2\pi \\ \\ &{} = &{} \displaystyle \frac{2(2\pi )^d}{(2d-1)!!}, \\ \end{array} \\&\begin{array}{ccl} |\mathcal{S}_{2d-1}| &{} = &{} \displaystyle \int _0^{\pi }\cdots \int _0^{\pi }\int _0^{2\pi }\sin ^{2d-2}(\theta _0)\,\cdots \,\sin ^{1}(\theta _{2d-3})\ \mathrm{d}\theta _0\,\cdots \,\mathrm{d}\theta _{2d-3}\mathrm{d}\theta _{2d-2} \\ \\ &{} = &{} \displaystyle \frac{(2d-3)!!}{(2d-2)!!}\pi \ 2\frac{(2d-4)!!}{(2d-3)!!}\ \cdots \ \frac{(2-1)!!}{2!!}\pi \ 2\frac{(1-1)!!}{1!!}\ 2\pi \\ \\ &{} = &{} \displaystyle \frac{(2\pi )^d}{(2d-2)!!}. \end{array} \end{aligned}$$