Why is a quantum erasure error & a Pauli error not the same?

Preliminary

This post assumes that the reader has some prior knowledge about quantum error correction and quantum computing in general, and thus if you are not well acquainted with it, it might be hard to grasp what is being explained. If you wish to get acquainted, I highly recommend the superb introductory guide to quantum error correction by (Roffe, 2019)¹.

The desire to write this post was spurred by the fruitful discussion I had yesterday with my colleagues on a paper that I’m working on.

The [[5,1,3]] quantum error correcting code

An arbitrary logical state in the [[5,1,3]] quantum error correcting code a.k.a the 5-qubit code, which is the smallest possible quantum error correcting code shown by (Laflamme et al., 1996)², is given by the linear superposition of the logical states and

where and . Note that this is a highly entangled state. Written out explicitly, the logical states and are

While it is not surprising that eqns () and () are orthogonal to each other, i.e., , it is interesting to note that the state () is essentially the linear superposition of all 5-qubit states with even (odd) parity. The stabiliser group of the 5-qubit code is

For the uninitiated, each element in a stabiliser group is called a stabiliser generator, and they have the property that when applied on a logical state, it leaves it invariant, i.e., . It is common in literature to say that these stabiliser generators result in the eigenstate with respect to the logical state .

Pauli errors

Its density matrix is given by

which is a pure state, i.e., . We can rewrite it by factoring the first qubit out

If we have an arbitrary Pauli error acting on the first qubit, then we simply write the resulting density matrix as the following

where is a Hermitian unitary matrix which represents the rotation of the qubit’s Bloch vector by some arbitrary amounts and around some arbitrary axis, i.e., an arbitrary single qubit Pauli error. Note that is still a pure state, i.e., since Pauli operators are trace preserving.

Erasure errors

Now imagine we have an erasure error on the first qubit (we effectively trace out the first qubit), and then we replace the lost qubit with a qubit in state , the resulting density matrix would become

From eqns () and (), we can see that there exist no combinations of single-qubit Paulis that can bring us from state to or vice versa. In fact, there exists no unitaries that can do that, i.e.,

where is the Lie group of unitary matrices with determinant 1. Furthermore, the state is no longer a pure state, but a mixed state, i.e., . This means that information about our original logical state, i.e., the and components, is irreversibly lost. This is the difference between an erasure error and a Pauli error.

Correcting erasure errors

However, there exists a theorem (Grassl et al., 1997)³, which states that it is possible to use any -error correcting code to correct for up to -erasure errors. The idea of using the same 5-qubit code that corrects for arbitrary single qubit Pauli errors to correct for erasure errors have been explored in my thesis (Wo, 2020)⁴ and it will also be explored in my upcoming paper that’s currently in the drafting stage, stay tuned! (Update: Paper out on arxiv). In the example by (Gingrich et al., 2003)⁵, they demonstrated a single qubit erasure error correction using the [[4,2,2]] code, which is actually not a quantum error correcting code, but a quantum error detection code, so it’s intriguing in its own right. This paper did a tremendous job of explaining quantum erasure error correction using an [[n,k,d]] code and I had much joy reading it. The same steps apply if you want to use the [[5,1,3]] code for erasure error correction, though there are some extra nuances present (might expound on this in another post).

An erasure error in the field of quantum error correction means that it is the loss of a physical qubit that was part of a group of physical qubits encoding a logical qubit, but we know the location of the loss.

Footnotes

1. Roffe, J., 2019. Quantum Error Correction: An Introductory Guide. Contemporary Physics 2019. Available at: https://arxiv.org/abs/1907.11157. ↩

2. Laflamme, R. et al., 1996. Perfect Quantum Error Correcting Code. Phys. Rev. Lett., 77(1), pp.198–201. Available at: https://link.aps.org/doi/10.1103/PhysRevLett.77.198. ↩

3. Grassl, M., Beth, Th. & Pellizzari, T., 1997. Codes for the quantum erasure channel. Physical Review A, 56(1), pp.33–38. Available at: https://doi.org/10.1103/physreva.56.33. ↩

4. Wo, K.J., 2020. Hybrid one-way quantum repeater concatenated with the [[5,1,3]] code. TUDelft Repository. Available at: https://repository.tudelft.nl/islandora/object/uuid:649f5fe9-0266-4be2-a3a5-4b5ffd13073e. ↩

5. Gingrich, R.M. et al., 2003. All Linear Optical Quantum Memory Based on Quantum Error Correction. Physical Review Letters, 91(21), p.217901. Available at: https://doi.org/10.1103/physrevlett.91.217901. ↩