Quantum technologies are maturing by the day and making exciting advances across computing, communications, networking, sensing, and beyond. The critical path to scaling these technologies for a practical advantage over classical systems involves the implementation of fault tolerant procedures. The most established fault tolerance framework uses quantum error correcting codes and decoders. The theory of quantum error correction has recently produced codes with optimal parameters that could potentially reduce the resource overhead of fault tolerance. However, several challenges remain to be addressed before these theoretical advances lead to scalable, fault tolerant, practical quantum systems. Besides computing, error correction techniques are necessary for other applications as well.
A communication-efficient protocol is introduced over a many-to-one quantum network for Q-E-B-MDS-X-TPIR, i.e., quantum private information retrieval with MDS-X-secure storage and T-private queries. The protocol is resilient to any set of up to E unresponsive servers (erased servers or stragglers) and any set of up to B Byzantine servers. The underlying coding scheme incorporates an enhanced version of a Cross Subspace Alignment (CSA) code, namely a Modified CSA (MCSA) code, into the framework of CSS codes. The error-correcting capabilities of CSS codes are leveraged to encode the dimensions that carry desired computation results from the MCSA code into the error space of the CSS code, while the undesired interference terms are aligned into the stabilized code space. The challenge is to do this efficiently while also correcting quantum erasures and Byzantine errors. The protocol achieves superdense coding gain over comparable classical baselines for Q-E-B-MDS-X-TPIR, recovers as special cases the state of art results for various other quantum PIR settings previously studied in the literature, and paves the way for applications in quantum coded distributed computation, where CSA code structures are important for communication efficiency, while security and resilience to stragglers and Byzantine servers are critical.
Classical coding theory contains several techniques to obtain new codes from other codes, including puncturing and shortening. Both of these techniques have been generalized to quantum codes. Restricting to stabilizer codes, this paper introduces more freedom in the choice of the encoded states after puncturing. Furthermore, we also give an explicit description of the stabilizers for the punctured code. The additional freedom in the procedure also opens up for new ways to construct new codes from old, and we present several ways to utilize this in the search for codes with good or even optimal parameters. In particular, we use the construction to obtain codes whose parameters exceed the best previously known and which are better than what general puncturing guarantees. Lastly, the freedom in our puncture procedure allowed us to generalize the proof of the Griesmer bound from the classical setting to stabilizer codes for qudits of prime dimension since the proof relies heavily on the puncturing technique.
Locally recoverable codes (LRCs) with locality parameter r can recover any erased code symbol by accessing r other code symbols. This local recovery property is of great interest in large-scale distributed classical data storage systems as it leads to efficient repair of failed nodes. A well-known class of optimal (classical) LRCs are subcodes of Reed-Solomon codes constructed using a special type of polynomials called good polynomials. Recently, Golowich and Guruswami initiated the study of quantum LRCs (qLRCs), which could have applications in quantum data storage systems of the future. The authors presented a qLRC construction based on good polynomials arising out of subgroups of the multiplicative group of finite fields. In this paper, we present a qLRC construction method that can employ any good polynomial. We also propose a new approach for designing good polynomials using subgroups of affine general linear groups. Golowich and Guruswami also derived a lower bound on the minimum distance of their qLRC under the restriction that r+1 is prime. Using similar techniques in conjunction with the expander mixing lemma, we develop minimum distance lower bounds for our qLRCs without the r+1 prime restriction.