13.4 Digitising quantum errors

The most general qubit-environment interaction is of the form \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. This 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} which can be written 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)
  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, but if this is not so then we cannot perfectly distinguish between the four alternatives.262

We can reduce quantum errors in this general scenario to just two types: bit-flip errors X, and phase-flip errors Z.

In short, if we can correct Pauli errors then we can correct all errors.

In general, given n qubits in state |\psi\rangle, and an environment in state |e\rangle, the joint evolution can be expanded as263 |\psi\rangle|e\rangle \longmapsto \sum_{i=1}^{4^n} 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, again, are not assumed to be normalised or mutually orthogonal). For example, in the case n=5, a typical operator E_i may look like X\otimes Z \otimes \mathbf{1}\otimes \mathbf{1}\otimes Y \equiv XZ\mathbf{1}\mathbf{1}Y. We say that such an E_i represents an error consisting of the bit error (or X error) on the first qubit, phase error (or Z error) on the second qubit, and both bit and phase error (or Y error) on the fifth qubit.264

In terms of density operators, we have a quantum channel described by the Kraus operators E_i above \rho \longmapsto \sum_i E_i\rho E_i^\dagger that acts on any input state \rho, be it mixed or pure. In particular, this channel turns the pure state |\psi\rangle into a statistical mixture of states |\widetilde{\psi_i}\rangle=E_i|\psi\rangle. Note that the |\widetilde{\psi_i}\rangle are not normalised: p_i \coloneqq \langle\widetilde{\psi_i}|\widetilde{\psi_i}\rangle = \langle\psi|E_i^\dagger E_i|\psi\rangle is exactly the probability with which the normalised version of E_i|\psi\rangle appears in the mixture. This mixture may arise if one measures the environment in the |e_i\rangle basis and then forgets about the result.

  1. We will soon stop warning that this intuition is not entirely accurate, so keep it in mind!↩︎

  2. The sum is from i=1 to 4^n because there are 4^n different (tensor products of) Pauli operators acting on n qubits.↩︎

  3. One final time: this is not entirely accurate if the corresponding states of the environment are not mutually orthogonal, but it gives the right kind of intuition nonetheless.↩︎