Balanced Product Quantum Codes

Abstract

This work provides the first explicit and nonrandom family of [[N, K, D]] LDPC quantum codes which encode K ∈ Θ(N 4 5 ) logical qubits with distance D ∈ Ω(N 3 5 ). The family is constructed by amalgamating classical codes and Ramanujan graphs via an operation called balanced product.Recently, Hastings-Haah-O'Donnell and Panteleev-Kalachev were the first to show that there exist families of LDPC quantum codes which break the polylog(N ) √ N distance barrier. However, their constructions are based on probabilistic arguments which only guarantee the code parameters with high probability whereas our bounds hold unconditionally.Further, balanced products allow for non-abelian twisting of the check matrices, leading to a construction of LDPC quantum codes that can be shown to have K ∈ Θ(N ) and that we conjecture to have linear distance D ∈ Θ(N ).

Degenerate Quantum LDPC Codes With Good Finite Length Performance

“…Consider a matrix A = (a ij ) m×n ∈ M m×n (R). We denote by C(A, b) the CSS code called generalized hypergraph product (GHP ) code with the following parity-check matrices 18 :…”

“…Therefore we see that the cardinality of the column space of the non-binary matrix ϕ i (A) over the field F i is equal to (2 di ) ri = 2 diri . Hence using isomorphism (18) we conclude that the number of different vectors in the column space of the matrix A is equal to…”

“…For a QLDPC code on n qubits with m stabilizers the matrix H is an m × 2n binary matrix 2. Since the first version of the current work was released, the codes from this family have been shown to have very large minimal distances[16,17,18] and found interesting applications in geometry[19]. In[17] they were further generalized and called lifted product codes.…”

“…Since the first version of this work appeared in arXiv, some lower bounds on the minimal distance have been obtained in[16,17,18] for special cases of QC GHP codes C(A, 1 + x).…”

Logical blocks for fault-tolerant topological quantum computation

“…In addition to its reduced overhead, the phase gate we present can be implemented in a static 2D planar (square) lattice of qubits using only the standard 4-qubit stabilizer measurements, without needing higher-weight stabilizer measurements [18,54] or modified code geometries [8,27] that are typically required for braiding twists. These logical blocks can be composed together to produce fault-tolerant circuits, which we illustrate by proposing an avenue for fault-tolerant quantum computation based on concatenating surface codes with more general quantum low-density parity check (LDPC) codes [55][56][57][58][59][60][61][62]. Such concatenated code schemes may offer the advantages of both the high thresholds of surface codes, with the reduced overheads of constant-rate LDPC codes-an attractive prospect for future generations of quantum computers.…”