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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.

The non-local interactions in several quantum device architectures allow for the realization of more compact quantum encodings while retaining the same degree of protection against noise. Anticipating that short to medium-length codes will soon be realizable, it is important to construct stabilizer codes that, for a given code distance, admit fault-tolerant implementations of logical gates with the fewest number of physical qubits. To this aim, we construct three kinds of codes encoding a single logical qubit for distances up to 31. First, we construct the smallest known doubly even codes, all of which admit a transversal implementation of the Clifford group. Applying a doubling procedure arXiv:1509.03239 to such codes yields the smallest known weak triply even codes for the same distances and number of encoded qubits. This second family of codes admit a transversal implementation of the logical T-gate. Relaxing the triply even property, we obtain our third family of triorthogonal codes with an even lower overhead at the cost of requiring additional Clifford gates to achieve the same logical operation. To our knowledge, these are the smallest known triorthogonal codes for their respective distances. While not qLDPC, the stabilizer generator weights of the code families with transversal T-gates scale roughly as the square root of their lengths.

Kitaev’s toric code is one of the most prominent models for fault-tolerant quantum computation, currently regarded as the leading solution for connectivity constrained quantum technologies. Significant effort has been recently devoted to improving the error correction performance of the toric code under message-passing decoding, a class of low-complexity, iterative decoding algorithms that play a central role in both theory and practice of classical low-density parity-check codes. Here, we provide a theoretical analysis of the toric code under min-sum (MS) decoding, a message-passing decoding algorithm known to solve the maximum-likelihood decoding problem in a localized manner, for codes defined by acyclic graphs. Our analysis reveals an intrinsic limitation of the toric code, which confines the propagation of local information during the message-passing process. We show that if the unsatisfied checks of an error syndrome are at distance ≥5 from each other, then MS decoding is locally blind: the qubits in the direct neighborhood of an unsatisfied check are never aware of any other unsatisfied checks, except their direct neighbor. Moreover, we show that degeneracy is not the only cause of decoding failures for errors of weight at least 4, that is, the MS non-degenerate decoding radius is equal to 3, for any toric code of distance ≥9. Finally, complementing our theoretical analysis, we present a pre-processing method of practical relevance. The proposed method, referred to as stabiliser blowup, has linear complexity and allows correcting all (degenerate) errors of weight up to 3, providing quadratic improvement in the logical error rate performance, as compared to MS alone.

In this paper, we introduce the Union-Intersection Union-Find (UIUF) algorithm for decoding depolarizing errors in topological codes, combining the strengths of iterative and standard Union-Find (UF) decoding. While iterative UF improves performance at moderate error rates, it lacks an error correction guarantee. To address this, we develop UIUF, which maintains the enhanced performance of iterative UF while ensuring error correction up to half the code distance. Through simulations under code capacity, phenomenological, and biased noise models, we show that UIUF significantly outperforms UF, reducing the logical error rate by over an order of magnitude (at around 10-5). Moreover, UIUF achieves lower logical error rates than the Minimum Weight Perfect Matching (MWPM) decoder on rotated surface codes under both the code capacity and phenomenological noise models, while preserving efficient linear-time complexity.

We introduce a new erasure decoder that applies to arbitrary quantum LDPC codes. Dubbed the cluster decoder, it generalizes the decomposition idea of Vertical-Horizontal (VH) decoding introduced by Connolly et al. in 2022. Like the VH decoder, the idea is to first run the peeling decoder and then post-process the resulting stopping set. The cluster decoder breaks the stopping set into a tree of clusters, which can be solved sequentially via Gaussian Elimination. By allowing clusters of unconstrained size, this decoder achieves maximum-likelihood (ML) performance with reduced complexity compared with full Gaussian Elimination. When Gaussian Elimination is applied only to clusters whose sizes are less than a constant, the performance is degraded, but the complexity becomes linear in the block length. Our simulation results show that, for hypergraph product codes, the cluster decoder with constant cluster size achieves near-ML performance similar to VH decoding in the low-erasure-rate regime. For the general quantum LDPC codes we studied, the cluster decoder can be used to estimate the ML performance curve with reduced complexity over a wide range of erasure rates.

We propose a new systematic construction of CSS-T codes from any given CSS code using a map Ï•. When Ï• is the identity map I, we retrieve the construction of hu2021mitigating and use it to prove the existence of asymptotically good binary CSS-T codes, resolving a previously open problem in the literature, and of asymptotically good quantum LDPC CSS-T codes. We analyze the structure of the logical operators corresponding to certain non-Clifford gates supported by the quantum codes obtained from this construction (Ï•=I), concluding that they always result in the logical identity. An immediate application of these codes in dealing with coherent noise is discussed. We then develop a new doubling transformation for obtaining triorthogonal codes, which generalizes the doubling construction presented in jain2024. Our approach permits using self-orthogonal codes, instead of only doubly-even codes, as building blocks for triorthogonal codes. This broadens the range of codes available for magic state distillation.

CSS-T codes are a class of stabilizer codes introduced by Rengaswamy et al with desired properties for quantum fault-tolerance. In this work, we comprehensively study non-degenerate CSS-T codes built from Reed-Muller codes. These classical codes allow for constructing optimal CSS-T code families with nonvanishing asymptotic rates up to 12 and possibly diverging minimum distance when non-degenerate.

An m-uniform quantum state on n qubits is an entangled state in which every m-qubit subsystem is maximally mixed. Starting with an m-uniform state realized as the graph state associated with an m-regular graph, and a classical n,k,d≥m+1 binary linear code with certain additional properties, we show that pure [n,k,m+1]]2 quantum error-correcting codes (QECCs) can be constructed within the codeword stabilized (CWS) code framework. As illustrations, we construct pure [[22r-1,22r-2r-3,3]]2 and [[(24r-1)2,(24r-1)2-32r-7,5]]2 QECCs. We also give measurement-based protocols for encoding into code states and for recovery of logical qubits from code states.

The standard approach to universal fault-tolerant quantum computing is to develop a general purpose quantum error correction mechanism that can implement a universal set of logical gates fault-tolerantly. Given such a scheme, any quantum algorithm can be realized fault-tolerantly by composing the relevant logical gates from this set. However, we know that quantum computers provide a significant quantum advantage only for specific quantum algorithms. Hence, a universal quantum computer can likely gain from compiling such specific algorithms using tailored quantum error correction schemes. In this work, we take the first steps towards such algorithm-tailored quantum fault-tolerance. We consider Trotter circuits in quantum simulation, which is an important application of quantum computing. We develop a solve-and-stitch algorithm to systematically synthesize physical realizations of Clifford Trotter circuits on the well-known n,n-2,2error-detecting code family. Our analysis shows that this family implements Trotter circuits with optimal depth, thereby serving as an illuminating example of tailored quantum error correction. We achieve fault-tolerance for these circuits using flag gadgets, which add minimal overhead. The solve-and-stitch algorithm has the potential to scale beyond this specific example and hence provide a principled approach to tailored fault-tolerance in quantum computing.

In a recently introduced coset guessing game, Alice plays against Bob and Charlie, aiming to meet a joint winning condition. Bob and Charlie can only communicate before the game starts to devise a joint strategy. The game we consider begins with Alice preparing a 2m-qubit quantum state based on a random selection of three parameters. She sends the first m qubits to Bob and the rest to Charlie, and then reveals to them her choice for one of the parameters. Bob is supposed to guess one of the hidden parameters, Charlie the other, and they win if both guesses are correct. From previous work, we know that the probability of Bob’s and Charlie’s guesses being simultaneously correct goes to zero exponentially as m increases. We derive a tight upper bound on this probability and show how Bob and Charlie can achieve it. While developing an optimal strategy, we devised an encoding circuit using only and Hadamard gates, which builds CSS codes from EPR pairs using only local operations. We found that the role of quantum information that Alice communicates to Bob and Charlie is to make their responses correlated rather than improve their individual (marginal) correct guessing rates.

In this work, we present a generalization of the recently proposed quantum Tanner codes by Leverrier and Zémor, which contains a construction of asymptotically good quantum low-density parity-check codes. Quantum Tanner codes have so far been constructed equivalently from groups, Cayley graphs, or square complexes constructed from groups. We show how to enlarge this to graphs with labeled local views and a family of square complexes, which is the largest possible in a certain sense. We show that the proposed generalization contains a family of asymptotically good quantum codes that are based on non-Cayley Schreier graphs, i.e., a new family of (generalized) quantum Tanner codes is provided. Moreover, we evaluate the performance of the generalized codes and compare with those based on Cayley graphs both in terms of minimum distance and logical error rate on the depolarizing channel, demonstrating that the proposed generalized codes based on Schreier graphs outperform those based on Cayley graphs.