Good quantum locally testable codes
Construct stabilizer quantum locally testable codes with constant rate, linear distance, constant check weight and degree, and constant soundness . For checks on qubits, require $|\operatorname{syn}(E)|/m\ge s,\min_{F:\operatorname{syn}(F)=0}\ope
Problem statement
Construct stabilizer quantum locally testable codes with constant rate, linear distance, constant check weight and degree, and constant soundness . For checks on qubits, require for every Pauli error .
Lean 4
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Background
The syndrome counts violated checks. Distance in this soundness inequality is distance to the code space: errors are considered modulo zero-syndrome operators, including logical operators, not just modulo stabilizers. This distinguishes local testability from ordinary error-correction distance.
What is known: Panteleev–Kalachev constructed asymptotically good quantum LDPC codes, with constant rate and linear distance; later quantum Tanner constructions provide another route. Earlier fibre-bundle and balanced-product breakthroughs had different parameters and should not all be credited with the full good-code theorem.
Dinur–Lin–Vidick construct almost-good qLTCs with constant rate, bounded-size checks and inverse-polylogarithmic relative distance and soundness. Thus several parameters are close, but the simultaneous constant-parameter target above remains open. Good qLDPC codes and NLTS do not by themselves establish constant local-test soundness, and quantum PCP should not be identified with this particular stabilizer-code construction problem. Related: A6, A18.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
Recorded progress
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Discussion
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