# What is the clear definition of coherent versus incoherent errors?

The noise types in quantum computing are pivotal topics frequently discussed in academic papers.

Could you provide a clear definition of coherent versus incoherent errors? Additionally, I'm keen to understand the distinctions between unitary and non-unitary errors. Lastly, the concepts of local and non-local errors have been somewhat perplexing to me.

If these terminologies have well-established mathematical representations, is there a scalable experimental method to differentiate between them?

Should there be a thorough review or analysis concerning this subject, please highlight it. Your insights on this would be immensely beneficial to me.

Coherent vs incoherent: as a rule of thumb, coherent is unitary, incoherent is stochastic. The distinction is not entirely clear when your channel is a mixture of these. Thus, it makes sense to introduce a measure of coherence. These include unitarity and coherence angle, typically checking how quadratic the growth of "average loss of fidelity after applying the channel $$m$$ times" $$r_m$$ in the number of repetitions $$m$$. The Pauli Transfer Matrix representation of the channel is useful. For an $$n$$ qubit channel of dimension $$d=2^n$$, the PTM has two submatrices, the unital $$N_u \in \mathbb{R}^{(d^2-1) \times (d^2-1)}$$ and the non-unital $$N_n \in \mathbb{R}^{(d^2-1) \times 1}$$:

$$N=\begin{pmatrix} 1 & 0 \\ N_n & N_u \end{pmatrix} \in \mathbb{R}^{d^2 \times d^2}$$

For unital channels $$N_n=0$$. Incoherent channels (e.g. Pauli noise, amplitude damping) have a diagonal $$N_u$$, while coherent channels have off-diagonal elements in $$N_u$$. Thus, you could use the sum of the diagonal elements of $$N_u$$ as a measure of coherence as well.

Single qubit channel examples:

• Depolarizing channel (unital, incoherent):

$$\mathcal{E}_p(\rho)=(1-p) \rho + p Tr(\rho) \frac{I}{2}\\ N_{depol}=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1-p & 0 & 0 \\ 0 & 0 & 1-p & 0 \\ 0 & 0 & 0 & 1-p \\ \end{pmatrix}$$

• Amplitude damping channel (non-unital, incoherent) with Kraus operators $$K_0 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-\gamma}\end{pmatrix}, K_1 = \begin{pmatrix} 0 & \sqrt{\gamma} \\ 0 & 0\end{pmatrix}$$

$$N_{ampdamp, \gamma}=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & \sqrt{1-\gamma} & 0 & 0 \\ 0 & 0 & \sqrt{1-\gamma} & 0 \\ \gamma & 0 & 0 & 1-\gamma \\ \end{pmatrix}$$

• Rotation around X (coherent, unital), with "Kraus operator" $$U_X(\theta)=e^{-i\theta/2 X}$$:

$$N_{X, \theta}=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos(\theta) & \sin(\theta) \\ 0 & 0 & -\sin(\theta) & \cos(\theta) \\ \end{pmatrix}$$

The problem with experimental differentiation is that randomized benchmarking estimates the average loss of fidelity ($$r_1$$), but that doesn't tell you how coherent it is. There are proposals, e.g. Unitary RB (https://arxiv.org/abs/1808.00850), that aims to measure unitarity to tackle this issue.

Iverson and Preskill (2020) https://core.ac.uk/download/pdf/345074667.pdf - initial sections explain coherent noise really well, plus the Pauli Transfer Matrix.

unitary vs non-unitary: a $$U$$ operator with $$UU^\dagger=I$$ defines a unitary channel $$\mathcal{E}_U(\rho)=U\rho U^\dagger$$, while every other channel - e.g. Pauli noise (depolarizing, phase-flip, bitflip channels), amplitude damping channels are non-unitary. Unitary operators preserve the inner product (angles) between states. Related - basically synonymous - to coherent vs incoherent. See for a good answer on this: How can one check if a given quantum channel is unitary?

local vs non-local: $$k$$-locality - in the Kraus representation of the channel, if each Kraus operator is a product of operators acting on most $$k$$ subsystems (e.g. qubits), then it is $$k$$-local. This is different from the geometric locality, where the connectivity of subsystems comes into the picture as well. In that case, local operators should only act on neighboring subsystems. See also What is a local operator?.

• For this quote, "while coherent channels have off-diagonal elements. ", the PTM representation of damping channels has off-diagonal elements as well. Does that suggest that the definition is valid for only one-way? Oct 31, 2023 at 2:31
• I clarified in my response what I meant - the off-diagonal elements of the unital submatrix $N_u$. Even though the amplitude damping channel has "off-diagonals" in the larger matrix, that is in the non-unital part that does not count in this aspect. I hope this helps. Nov 20, 2023 at 19:13

The need for these terms might be helpful to understand the intuition.

Coherent errors are statistical characterizations while still respecting unitary evolution.

Incoherent errors are the traditional statistical descriptions (might not be necessarily real) which are not constrained by unitary evolution.

Thus, we are making a distinction between statistical characterization while still following rules of unitary evolution and statistical characterization without any constraints.