Circuit-to-Hamiltonian approximate code[1] 

Approximate qubit block code that forms the ground-state space of a frustration-free Hamiltonian with non-commuting terms. Its distance and logical-qubit number are both of order \(\Omega(n/\log^5 n)\) [1; Thm. 3.1]. The code is an approximate non-stabilizer QLWC code since the Hamiltonian consists of non-commuting weight-ten non-Pauli projectors, with each qubit acted on by order \(O(\text{polylog}(n)\) projectors.

The code is constructed by converting the encoding circuit of a Brown-Fawzi random Clifford-circuit code into a Hamiltonian using the spacetime circuit-to-Hamiltonian construction [2,3] (a generalization of the Feynman-Kitaev clock construction [4]). The ground-state subspace of this Hamiltonian is the \(\epsilon\)-approximate code with infidelity of recovery \(\epsilon = O(1/\text{polylog}(n))\).

Using Markov-chain techniques, the gap of the Hamiltonian can be proven to be of order \(\Omega(D^{-2}n^{-3.09}\log^{-6} n)\) for an \(n\)-qubit input circuit of depth \(D\).