11.4 Quantum errors

The most general qubit-environment interaction \begin{aligned} |0\rangle|e\rangle &\longmapsto |0\rangle|e_{00}\rangle + |1\rangle|e_{01}\rangle, \\|1\rangle|e\rangle &\longmapsto |1\rangle|e_{10}\rangle + |0\rangle|e_{11}\rangle, \end{aligned} where the states of the environment are neither normalised nor orthogonal, leads to decoherence \begin{aligned} \Big( \alpha|0\rangle + \beta|1\rangle \Big) |e\rangle \longmapsto & \Big( \alpha|0\rangle + \beta|1\rangle \Big) \frac{|e_{00}\rangle+|e_{11}\rangle}{2} \\+& \Big( \alpha|0\rangle - \beta|1\rangle \Big) \frac{|e_{00}\rangle-|e_{11}\rangle}{2} \\+& \Big( \alpha|1\rangle + \beta|0\rangle \Big) \frac{|e_{01}\rangle+|e_{10}\rangle}{2} \\+& \Big( \alpha|1\rangle - \beta|0\rangle \Big) \frac{|e_{01}\rangle-|e_{10}\rangle}{2}. \end{aligned} We can also write this as |\psi\rangle|e\rangle \longmapsto \mathbf{1}|\psi\rangle|e_{\mathbf{1}}\rangle + Z|\psi\rangle |e_Z\rangle +X|\psi\rangle |e_X\rangle + Y|\psi\rangle |e_Y\rangle. The intuition behind this expression is that four things can happen to the qubit:

  1. nothing (\mathbf{1}),
  2. phase-flip (Z),
  3. bit-flip (X), or
  4. both bit-flip and phase-flip (Y).

This is certainly the case when the states |e_{\mathbf{1}}\rangle, |e_X\rangle, |e_Y\rangle and |e_Z\rangle are mutually orthogonal, otherwise we cannot perfectly distinguish between the four alternatives.

What is important here is the discretisation of errors, and the fact that we can reduce quantum errors to two types: bit-flip errors X, and phase-flip errors Z.

In general, given n qubits in state |\psi\rangle and the environment in state |e\rangle the joint evolution can be expanded as |\psi\rangle|e\rangle \longmapsto \sum_i E_i|\psi\rangle|e_i\rangle, where the E_i are the n-fold tensor products of the Pauli operators and the |e_i\rangle are the corresponding states of the environment, which are not assumed to be normalised or mutually orthogonal. A typical operator E_i acting on five qubits may look like this, X\otimes Z \otimes \mathbf{1}\otimes \mathbf{1}\otimes Y \equiv XZ\mathbf{1}\mathbf{1}Y. We can say that E_i represents an error consisting of the bit (X) error on the first qubit, phase (Z) error on the second qubit and both bit and phase (Y) error on the fifth qubit. Again, this is not quite accurate if the corresponding states of the environment are not mutually orthogonal, but it gives the right kind of intuition nonetheless. Here the index i in E_i ranges from 1 to 4^5=1024, because there are 4^5 different Pauli operators acting on 5 qubits.