13.3 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.

The important thing is the discretisation of errors, and the fact that we can reduce quantum errors in this scenario to two types: bit-flip errors X, and phase-flip errors Z.

In general, given n qubits in state |\psi\rangle, and an environment in state |e\rangle, the joint evolution can be expanded as257 |\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 (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 entirely accurate if the corresponding states of the environment are not mutually orthogonal, but it gives the right kind of intuition nonetheless.

Of course, we can always pick an orthonormal basis |u_j\rangle of the environment and express the system-environment evolution as \begin{aligned} |\psi\rangle|e\rangle \longmapsto &\sum_{i,j} E_i|\psi\rangle|u_j\rangle\langle u_j|e_i\rangle \\&= \sum_{j}\Big( \sum_i \langle u_j|e_i\rangle E_i\Big)|\psi\rangle|u_j\rangle \\&= \sum_j M_j|\psi\rangle|u_j\rangle. \end{aligned} The new “error” operators M_j satisfy \sum_j M_j^\dagger M_j =\mathbf{1} but are not, in general, unitary. Now, the evolution of the density operator |\psi\rangle\langle\psi| can be written as |\psi\rangle\langle\psi|\longmapsto \sum_j M_j|\psi\rangle\langle\psi| M_j^\dagger. Which particular errors you choose depends of your choice of the basis in the environment. If, instead of |u_j\rangle, you pick a different basis, say |v_k\rangle, then \begin{aligned} |\psi\rangle|e\rangle \longmapsto &\sum_j M_j|\psi\rangle|u_j\rangle \\&= \sum_j M_j |\psi\rangle\sum_k|v_k\rangle\langle v_k|u_j\rangle \\&= \sum_k \Big(\sum_j \langle v_k|u_j\rangle M_j \Big)|\psi\rangle|v_k\rangle \\&= \sum_k N_k|\psi\rangle|v_k\rangle, \end{aligned} and, consequently, |\psi\rangle\langle\psi|\longmapsto \sum_k N_k|\psi\rangle\langle\psi| N_k^\dagger. The new “errors” still satisfy \sum_k N_k^\dagger N_k = \mathbf{1}, and the error operators N_k and M_j are related by the unitary matrix U_{kj}=\langle v_k|u_j\rangle.

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