Alright, this is the first official blog post type where I will break down a quantum computing article in terms that I can remember and understand. This may be a bit all over the place, but that sounds okay to me!
This is the article by J. Kelly, “State Preservation by Repetitive Error Detection in a Superconducting Quantum Circuit”, from the John Martinis group in USCB. The overall idea of this paper is to demonstrate the use of repeated quantum error correction through multiple cycles, rather than considering just a single cycle of error correction.
It’s obvious that quantum error correction is a fundamentally needed aspect for practical quantum computing, because of the inherent residences present in quantum systems. The number of errors that can happen at one time is related to the nth-order fault tolerance level, which allows for n errors to occur before the code breaks down. Furthermore, quantum error correction needs to be repeated over n cycles, not just in a single round. This kind of robust system requires quantum non-demolition (QND) parity measurements, where certain measurement qubits are able to detect errors in actual data qubits without disturbing the system. Observing the pattern of measurement qubits leads to determining the error and then correcting it.
This paper looks into a simple one-dimensional chain of qubits, a predecessor to a more complex two-dimensional surface code. For a 2D code, a checkerboard with (4n+1)^2 qubits is nth order fault tolerant, but a 1D chain of 9 qubits can go up to second-order fault tolerance. The Martinis group focuses on transmon superconducting qubits as well, which they use as a series of Xmon transmon qubits on a sapphire substrate. In their nine qubit code, they are using five data qubits and four measurement qubits in an alternating pattern, such that the measurement qubits are sandwiched in between data qubits. Instead of using the actual state of each measurement qubit as a way to determine errors, they use the relative flip of each measurement qubit over a cycle as a detection event. In other words, the researchers do not assume that there is no error in the measurement qubits in themselves.
In figure 2 of the paper, we have a very complex looking graph that shows many different cases where errors could occur. We see that the measurement qubit is the only qubit that is being measured upon, but regardless of the error, whether it’s an error on the measurement qubit or in the data qubit, the error is able to be caught over at most two cycles. By observing if the measurement qubit has flipped compared to its previous state, the original error is able to be deduced.
The paper makes several remarks on this note that I don’t fully understand. First, the paper refers frequently to minimum-weight perfect matching to decode the pattern of detection events. But it seems that the errors are fairly straightforwards to decode. Furthermore, there doesn’t actually appear to be an error correcting stage in this QEC path. For instance, in the standard five qubit QEC, there is a stage where the output from the measurement qubits are used to determine whether an X flip is needed. In a three qubit QEC, a CCNOT gate is used to correct arbitrary phase rotations. However, there is none of that in this system. The paper alludes to using post-processing as a way to correct the errors, but that seems to imply that the errors are corrected long after the system. How is the system able to identify the error and then process the entire system through with all the needed amplitudes afterwards? It seems that any arbitrary rotation will mess this up beyond belief!
Alright, getting back to the paper. I thought that the most interesting part of this paper was in the experimental studies that these researchers conducted. They begin with either a five-qubit or a nine-qubit code, and then iterate it over 8 cycles. For each number of cycles, they run the code 90,000 times. After each cycle, they compare the error corrected result to the actual state of the system, which is done by quantum state tomography. They show these results in Figure 4, where a five qubit code is able to be 2.7 times better than the non error-corrected fidelity, and the nine qubit code is 8.5 times better than the non corrected code. They see that the errors are able to be corrected in a non-exponential decay, which is explained by the increasing error rate with cycle number.
I think that the interesting aspect of this paper is their ability to do experiments and validate their QEC code under k cycles, up to 8 cycles. But I’m still unclear about how this QEC is actually done, and how the group is implementing their post-processing methodology.
Reference: Kelly, J. et al. State Preservation by Repetitive Error Detection in a Superconducting Quantum Circuit Nature 519 66-69 (2015)