arXiv subjects
Publications and source records attributed to Jesse Fern.
We study encodings that give the best known thresholds for the non-zero capacity of quantum channels, i.e., the upper bound for correctable noise, using an entropic approach to calculation of the threshold values. Our results show that Pauli noise is correctable up to the hashing bound. For a depolarizing channel, this approach allows one to achieve a non-zero capacity for a fidelity (probability of no error) of f=0.80870.
In this paper we calculate upper bounds on fault tolerance, without restrictions on the overhead involved. Optimally adaptive recovery operators are used, and the Shannon entropy is used to estimate the thresholds. By allowing for unrealistically high levels of overhead, we find a quantum fault tolerant threshold of 6.88% for the depolarizing noise used by Knill, which compares well to "above 3%" evidenced by Knill. We conjecture that the optimal threshold is 6.90%, based upon the hashing rate. We also perform threshold calculations for types of noise other than that discussed by Knill.
We examine the transformation of noise under a quantum error correcting code (QECC) concatenated repeatedly with itself, by analyzing the effects of a quantum channel after each level of concatenation using recovery operators that are optimally adapted to use error syndrome information from the previous levels of the code. We use the Shannon entropy of these channels to estimate the thresholds of correctable noise for QECCs and find considerable improvements under this adaptive concatenation. Similar methods could be used to increase quantum fault tolerant thresholds.
We find expectation values of functions of time integrated two-level telegraph noise. Expectation values of this noise are evaluated under simple control pulses. Both the Gaussian limit and $1/f$ noise are considered. We apply the results to a specific superconducting quantum computing example, which illustrates the use of this technique for calculating error probabilities.
We apply a dynamical systems approach to concatenation of quantum error correcting codes, extending and generalizing the results of Rahn et al. [1] to both diagonal and nondiagonal channels. Our point of view is global: instead of focusing on particular types of noise channels, we study the geometry of the coding map as a discrete-time dynamical system on the entire space of noise channels. In the case of diagonal channels, we show that any code with distance at least three corrects (in the infinite concatenation limit) an open set of errors. For Calderbank-Shor-Steane (CSS) codes, we give a more precise characterization of that set. We show how to incorporate noise in the gates, thus completing the framework. We derive some general bounds for noise channels, which allows us to analyze several codes in detail.
There are well known necessary and sufficient conditions for a quantum code to correct a set of errors. We study weaker conditions under which a quantum code may correct errors with probabilities that may be less than one. We work with stabilizer codes and as an application study how the nine qubit code, the seven qubit code, and the five qubit code perform when there are errors on more than one qubit. As a second application, we discuss the concept of syndrome quality and use it to suggest a way that quantum error correction can be practically improved.