Low-overhead universal logical gates on constant-rate qLDPC codes
In plain words
Storing quantum information in these codes is understood far better than computing with it. A way to apply every needed operation to the stored qubits without giving back the qubit savings is still missing in practice.
Precise statement
For a constant-rate qLDPC family with distance d, give a fault-tolerant implementation of a universal logical gate set (all logical Clifford gates plus one non-Clifford gate on any chosen logical qubits) that uses O(n) ancilla qubits, has space-time cost per logical gate O(d polylog d), and has a threshold under circuit-level noise. Candidate ingredients include code surgery, code automorphisms and codes with transversal CCZ.
What would settle it
An explicit gate scheme with stated overhead and a threshold proof or convincing circuit-level simulation at relevant sizes.
Status in the literature
Unverified note
Several 2024-2026 schemes cover parts of the gate set at moderate overhead.