Good approximate quantum LDPC codes from spacetime circuit Hamiltonians

Thomas C. Bohdanowicz, Elizabeth Crosson, Chinmay Nirkhe, Henry Yuen
2019 Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing - STOC 2019  
We study approximate quantum low-density parity-check (QLDPC) codes, which are approximate quantum error-correcting codes specified as the ground space of a frustration-free local Hamiltonian, whose terms do not necessarily commute. Such codes generalize stabilizer QLDPC codes, which are exact quantum error-correcting codes with sparse, low-weight stabilizer generators (i.e. each stabilizer generator acts on a few qubits, and each qubit participates in a few stabilizer generators). Our
more » ... tion is motivated by an important question in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes with constant rate, linear distance, and constant-weight stabilizers exist? We show that obtaining such optimal scaling of parameters (modulo polylogarithmic corrections) is possible if we go beyond stabilizer codes: we prove the existence of a family of [[N , k, d, ε]] approximate QLDPC codes that encode k = Ω(N ) logical qubits into N physical qubits with distance d = Ω(N ) and approximation infidelity ε = 1/polylog(N ). The code space is stabilized by a set of 10-local noncommuting projectors, with each physical qubit only participating in polylog N projectors. We prove the existence of an efficient encoding map and show that the spectral gap of the code Hamiltonian scales as Ω(N −3.09 ). We also show that arbitrary Pauli errors can be locally detected by circuits of polylogarithmic depth. Our family of approximate QLDPC codes is based on applying a recent connection between circuit Hamiltonians and approximate quantum codes (Nirkhe, et al., ICALP 2018) to a result showing that random Clifford circuits of polylogarithmic depth yield asymptotically good quantum codes (Brown and Fawzi, ISIT 2013). Then, in order to obtain a code with sparse checks and strong detection of local errors, we use a spacetime circuit-to-Hamiltonian construction in order to take advantage of the parallelism of the Brown-Fawzi circuits. Because of this, we call our codes spacetime codes.
doi:10.1145/3313276.3316384 dblp:conf/stoc/BohdanowiczCNY19 fatcat:cum2dlxhdvht5cgyonf4vf4pmq