13.3 Discretisation of quantum errors

When a quantum computer interacts and becomes entangled with its environment, it impacts the environment in such a way that the environment maintains a physical record of how the computer arrived at the desired output. Here, in our simplistic diagram, we consider only two computational paths.

With decoherence present, quantum computation spills out the environment and results in not one, but two output states: O1Oe1=“computer shows output O, environment knows that path 1 was taken”O2Oe2=“computer shows output O, environment knows that path 2 was taken” \begin{aligned} |O_1\rangle &\coloneqq |O\rangle|e_1\rangle \\&= \text{``computer shows output $O$, environment knows that path 1 was taken''} \\|O_2\rangle &\coloneqq |O\rangle|e_2\rangle \\&= \text{``computer shows output $O$, environment knows that path 2 was taken''} \end{aligned}

The two final states O1O_1 and O2O_2 are identical if and only if e1e1=1|\langle e_1|e_1\rangle|=1. In this case, the environment does not know anything about what happened during the computation — there is quantum interference — and we add probability amplitudes corresponding to the two computational paths. In contrast, the two final states O1O_1 and O2O_2 are completely different if and only if e1e2=0|\langle e_1|e_2\rangle|=0. Then there is only one path to the output — there is no quantum interference — and there is nothing to add. Of course, there are also midway cases 0<e1e2<10<|\langle e_1|e_2\rangle|<1 corresponding to partial distinguishability of the final states.

If we wish to study the evolution of the qubit alone, then we can do so in terms of density operators: it evolves from the pure state ψψ|\psi\rangle\langle\psi| to a mixed state, which can be obtained by tracing over the environment. We know that the state vector ψ=α0+β1|\psi\rangle=\alpha|0\rangle+\beta|1\rangle evolves as (α0+β1)eα0e00+β1e11 \left( \alpha|0\rangle +\beta |1\rangle\right)|e\rangle \longmapsto \alpha |0\rangle|e_{00}\rangle +\beta |1\rangle |e_{11}\rangle and we can write this as the evolution of the projector ψψ|\psi\rangle\langle\psi|, and then trace over the environment to obtain ψψα200e00e00+αβ01e11e00+αβ10e00e11+β211e11e11. \begin{aligned} |\psi\rangle\langle\psi| \longmapsto & |\alpha|^2|0\rangle\langle 0| \langle e_{00}|e_{00}\rangle+ \alpha\beta^\star |0\rangle\langle 1|\langle e_{11}|e_{00}\rangle \\+ &\alpha^\star\beta |1\rangle\langle 0|\langle e_{00}|e_{11}\rangle + |\beta|^2|1\rangle\langle 1|\langle e_{11}|e_{11}\rangle. \end{aligned} Written in matrix form, this is [α2αβαββ2][α2αβe11e00αβe00e11β2]. \begin{bmatrix} |\alpha|^2 & \alpha\beta^\ast \\\alpha^\ast\beta & |\beta|^2 \end{bmatrix} \longmapsto \begin{bmatrix} |\alpha|^2 & \alpha\beta^\ast \langle e_{11}|e_{00}\rangle \\\alpha^\ast\beta \langle e_{00}|e_{11}\rangle & |\beta|^2 \end{bmatrix}. The off-diagonal elements (originally called coherences) vanish as e00e11\langle e_{00}|e_{11}\rangle approaches zero. This is why this particular interaction is called decoherence.

Notice that ψe1ψe1+ZψeZ, |\psi\rangle|e\rangle \longmapsto \mathbf{1}|\psi\rangle|e_{\mathbf{1}}\rangle+Z|\psi\rangle|e_Z\rangle, implies ψψ1ψψ1e1e1+ZψψZeZeZ, |\psi\rangle\langle\psi|\longmapsto \mathbf{1}|\psi\rangle\langle\psi| \mathbf{1}\langle e_{\mathbf{1}}|e_{\mathbf{1}}\rangle +Z|\psi\rangle\langle\psi| Z\langle e_Z|e_Z\rangle, only if e1eZ=0\langle e_{\mathbf{1}}|e_Z\rangle=0, since otherwise we would have additional cross terms 1ψψZ\mathbf{1}|\psi\rangle\langle\psi|Z and Zψψ1Z|\psi\rangle\langle\psi|\mathbf{1}. In this case (i.e. when e1eZ=0\langle e_{\mathbf{1}}|e_Z\rangle=0) we can indeed say that, with probability e1e1\langle e_{\mathbf{1}}|e_{\mathbf{1}}\rangle, nothing happens, and, with probability eZeZ\langle e_Z|e_Z\rangle, the qubit undergoes the phase-flip ZZ. We can also represent this with the Kraus operators E0=e1e11E1=eZeZZ \begin{aligned} E_0 &= \sqrt{|e_\mathbf{1}\rangle\langle e_\mathbf{1}|} \mathbf{1} \\E_1 &= \sqrt{|e_Z\rangle\langle e_Z|} Z \end{aligned} which can be shown to satisfy E0E0+E1E1=1E_0^\dagger E_0+E_1^\dagger E_1=\mathbf{1}.

The process of decoherence is continuous. It involves the environment gradually acquiring information about computational paths and the associated relative environmental states (e0|e_0\rangle and e1|e_1\rangle in our example above), which evolve over time to become increasingly orthogonal to one another. Despite this, we can perceive the influence of the environment on our system of interest — a collection of qubits in a quantum computer — in terms of discrete operations. In essence, we can digitise quantum errors.