Approach for the construction of non-Calderbank-Steane-Shor low-density-generator-matrix–based quantum codes

Abstract

The game took me to Saint Andrew's College and so we now turn to that marvelous place. I would not be who I am today had I not spent two years at SAC. True to its motto, the place molded me into a significantly more mature individual and taught me skills that have shined with brilliance during my time as a PhD student. To all my teachers, coaches, and friends at SAC, thank you. Special shoutouts to my friends in 1st Hockey and the so-called "Dawgz": Graham, Humza, David, YoungWoo, West, and Andy. Although we have not seen each other in a long time, I know our friendship still runs as deep as it did when we gamed away our days at the Manor. Following this theme of childhood friends, thank you to Isma, Guille, Luken, and Andrey. From the day we first met in kindergarten at Saint Patrick's, our passion for sports and later on fitness kept us united. I pray that our games of pádel never stop being so competitive.Barring chance or extremely good fortune, those I will thank next will likely never read this dissertation. Thank you to R. A Salvatore, Steven Erikson, Brandon Sanderson, Patrick Rothfuss, and all those other authors from whom I have learned so much. Thank you also to Satoshi Nakamoto. I do not doubt that your creation will change the world for the better. I must also thank the Counting Crows, Rise Against, Machine Gun Kelly, and the Chainsmokers for motivating me in all those instances when my discipline evaporated. In relation to this, I must also thank the people at Youtube and Nespresso (and Iñigo Gutierrez by association); procrastination via the classic video and black coffee combo will never get old. Finally, a special shoutout to all those online meetings, home workouts, and quarantine periods brought to me by the COVID-19 pandemic. Somehow, throughout this entire ordeal, my work has not been impeded at any point in time. I have been extremely fortunate.Having left the best for last, I must now turn to those closest to my heart. After spending days thinking about how to thank my parents appropriately, I found it best to use the following quote by the writer Chuck Palahniuk: "First your parents, they give you your life, but then they try to give you their life". Despite this, I am still left with a feeling of insufficiency. I guess my parents have done so much for me that it is not possible to put my gratitude into words. Gracias attatto y amatxo por todo. The same goes for my sister Idoia, to whom I owe more than can be expressed. Thank you for always being there, for always having time to talk, and most of all, for always being willing to listen. Through thick and thin, your presence has never failed to remind me that there is always light at the end of the tunnel. Lastly, thank you Moose for being such an energetic furball and for greeting me at the door everyday as if you had not seen me in years.Alas, it is now time for me to thank one final person. Maria, you, above anyone else, have lived through the highs and the lows of my PhD. You have seen the extent to which manuscript revisions can frustr...

Degeneracy and Its Impact on the Decoding of Sparse Quantum Codes

“…In this paper, as in most of the work conducted on quantum error correction, we will mainly consider the independent depolarizing channel model [19]- [21], [26], [28], [29], [53], [62]- [66]. This model is a specific instance of the Pauli channel in which the individual depolarizing probabilities are all equal, i.e., p x = p z = p y = p 3 , and the channel is characterized by the depolarizing probability p. When quantum states of N qubits are considered, the errors that take place belong to the effective N -fold Pauli group G N .…”

“…the commutativity constraint of stabilizer codes (recall that all the elements of the stabilizer must commute), which results in the QPCM of the code having an even number of row overlaps and in the appearance of length 4 cycles in the corresponding factor graph [18]. The negative effects of these cycles can be mitigated by using specific QLDPC construction strategies like the bicycle codes of [16], LDGM based CSS and non-CSS codes [19]- [21], or the quasi-cyclic constructions of [109]- [111]. Recently, some results have shown that there is some merit in preserving a number of these cycles [112], as they can help with spreading information throughout the factor graph during the decoding process.…”

On the Logical Error Rate of Sparse Quantum Codes

“…Unfortunately, although degeneracy may potentially improve performance, limited research exists on how to quantify the true impact that this phenomenon has on Quantum Low Density Parity Check (QLDPC) codes. This has resulted in the performance of QLDPC codes being assessed differently throughout the literature; while some research considers the effects of degeneracy by computing the metric known as the logical error rate [3], [15]- [20], other works employ the classical strategy of computing the physical error rate [21]- [26], a metric which provides an upper bound on the logical error rate of these codes since it ignores degeneracy. In the context of degenerate quantum codes, the discrepancy between results computed based on the physical error rate and the logical error rate can become significant.…”

“…Given the coset structure of sparse quantum codes, intuition would point towards approaching the issue of calculating the logical error rate by finding and comparing the stabilizer cosets of the estimated error sequences and the stabilizer cosets of the channel errors. Unfortunately, the task of computing stabilizer cosets has been shown to be computationally hard [4], [12], [24], which is the reason why the performance of some sparse quantum codes [21]- [26] has been assessed based on the physical error rate. This metric is computed by comparing the error sequence estimated by the decoder, Ê ∈ G 𝑁 , to the channel error, E ∈ G 𝑁 , where G 𝑁 denotes the 𝑁-fold effective Pauli group.…”

Time-varying quantum channel models for superconducting qubits

“…In consequence, the quantum information community has gone to extraordinary lengths to construct QECCs that are capable of efficiently reversing the deleterious effects that quantum information experiences due to the interaction between qubits and their surrounding environment. This has led to the design of several promising families of QECCs, such as quantum Reed-Muller codes 1 , quantum low-density parity-check (QLDPC) codes 2 , quantum low-density generator matrix codes [3][4][5] , quantum convolutional codes 6 , quantum turbo codes (QTCs) [7][8][9][10][11] , and quantum topological codes 12,13 . It is important to point out that the aforementioned QECCs are utilized in the context of conventional qubit-based quantum computation, in which the physical elements are realized by discrete two-level systems.…”