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Ma and Shi Tighten Bounds for Quantum Local-Recovery Codes
For a 20-qudit example, a paper by Shuowei Ma and Fei Shi derives successively tighter upper bounds on the code parameter k, from 6 to 2. Since the code-space dimension is K = q^k, those bounds correspond to K ≤ q⁶, q⁴, q³ and q², with the
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The 20-Qudit Example Falls from Six to Two
A 20-qudit example in a new paper shows how much a code’s structure can change the limit on its encoded information: successive bounds put k at no more than 6, 4, 3 and finally 2. Here k = logq K, where K is the code-space dimension, so the corresponding dimension bounds are K ≤ q6, q4, q3 and q2. The strongest bound applies to a particular class of codes with locality parameters r = δ = 3 and distance d = 6; it is not a universal bound for every quantum code with 20 qudits.
Shuowei Ma and Fei Shi’s paper, posted to arXiv on October 29, develops Singleton-like bounds for quantum locally recoverable codes, or qLRCs. These codes can recover erased qudits from a limited subset of other qudits. The new analysis treats recovery blocks that may overlap, and does not require the code to be pure or built as a stabilizer code.
The numerical comparison separates the gains. For the example, the prior general single-erasure bound gives 6 as the upper limit on k. The authors’ multiple-erasure bound reduces that limit to 4; integer optimization brings it to 3; and a further structural result for linear stabilizer codes at r = δ lowers it to 2. The paper says its construction attains k = 2 for every prime-power alphabet q ≥ 5.
Recovery from Several Erasures, Without Disjoint Blocks
In an (r, δ)-qLRC, each qudit belongs to a designated recovery block of size at most r + δ − 1. The block must support recovery after as many as δ − 1 erasures within it. When δ is greater than two, that capability contains more information than a guarantee for recovery from one erasure alone.
Ma and Shi use a selection lemma to turn that local capability into two disjoint sets of globally correctable erasures. Their argument chooses erasures across recovery blocks while accounting for overlap, then combines local recovery with the code’s global distance. The resulting bound applies to general quantum subspaces, including impure codes, rather than only to constructions with a particular algebraic form.
Ma and Shi, the paper’s authors, write:
An integer optimization yields an explicit remainder correction that tightens the bound by up to δ−2.
The correction comes from optimizing how many coordinates the proof assigns to each round of recovery; it is a refinement of the upper bound, not a measured improvement in a quantum-memory device.
Purity Adds a Stronger Constraint
The paper then derives a tighter expression for pure codes, using local decoupling and entropy arguments. Purity is a condition on how errors of weight below the code distance act on the code space; it is not an assumption the authors impose on their general result. Their pure-code bound extends an earlier parameter bound from particular stabilizer constructions to arbitrary pure quantum subspaces.
For disjoint recovery blocks, the authors also remove a partition assumption from several existing bounds and obtain another remainder correction through budget optimization. Those extensions matter for comparing code families on equal terms: a bound tied to equal-sized, disjoint blocks does not automatically cover codes whose recovery sets have different sizes or overlap.
Equality Reveals More Than a Dimension
The analysis also examines when codes meet the bounds. The authors identify three attaining families: pure codes with overlapping blocks and d = δ; impure codes with disjoint blocks that meet the general bound; and an existing pure family with overlapping blocks and d > δ that attains both the general and pure-code bounds.
At minimal locality, r = δ, the paper proves additional structure for linear stabilizer codes, including a decomposition into disjoint MDS inner codes and an outer stabilizer code after fixed pure-state coordinate factors are removed. The 20-qudit example illustrates the distinction between a broad bound and one that exploits that extra structure. The authors identify open work on when the bounds are attainable and which structural conclusions extend beyond linear stabilizer codes.