Quantum codes do not increase fidelity against isotropic errors

TL;DR

This paper demonstrates that quantum error-correcting codes do not improve fidelity against isotropic errors, suggesting that avoiding coding may be more effective for such error types.

Contribution

The authors prove that quantum codes do not enhance fidelity for isotropic errors, challenging the assumed benefit of quantum error correction in this context.

Findings

Fidelity without coding is always greater or equal to that with coding.

Quantum codes do not increase fidelity against isotropic errors.

Optimal strategy is not to use quantum codes for isotropic errors.

Abstract

Given an $m-$qubit $\Phi_0$ and an $(n,m)-$quantum code $\mathcal{C}$, let $\Phi$ be the $n-$qubit that results from the $\mathcal{C}-$encoding of $\Phi_0$. Suppose that the state $\Phi$ is affected by an isotropic error (decoherence), becoming $\Psi$, and that the corrector circuit of $\mathcal{C}$ is applied to $\Psi$, obtaining the quantum state $\tilde\Phi$. Alternatively, we analyze the effect of the isotropic error without using the quantum code $\mathcal{C}$. In this case the error transforms $\Phi_0$ into $\Psi_0$. Assuming that the correction circuit does not introduce new errors and that it does not increase the execution time, we compare the fidelity of $\Psi$, $\tilde\Phi$ and $\Psi_0$ with the aim of analyzing the power of quantum codes to control isotropic errors. We prove that $F(\Psi_0) \geq F(\tilde\Phi) \geq F(\Psi)$. Therefore the best option to optimize fidelity against isotropic errors is not to use quantum codes.