Simulating LDPC code Hamiltonians on 2D lattices

TL;DR

This paper presents a method to simulate LDPC code Hamiltonians on 2D lattices with only nearest-neighbour interactions, enabling hardware-compatible quantum error correction with provable guarantees.

Contribution

It introduces a novel constructive tool for simulating long-range interactions with 1D chains, enabling efficient 2D lattice simulation of sparse Hamiltonians.

Findings

Achieves approximate ground state reproduction of LDPC codes in 2D.

Provides polynomial interaction strength bounds for simulation.

First epsilon-simulation of arbitrary sparse Hamiltonians on 2D lattices.

Abstract

While LDPC codes have been demonstrated with desirable error correcting properties, this has come at a cost of diverging from the geometrical constraints of many hardware platforms. Viewing codes as the groundspace of a Hamiltonian, we consider engineering a simulation Hamiltonian reproducing some relevant features of the code. Techniques from Hamiltonian simulation theory are used to build a simulation of LDPC codes using only 2D nearest-neighbour interactions at the cost of an energy penalty polynomial in the system size. We derive guarantees for the simulation that allows us to approximately reproduce the ground state of the code Hamiltonian, approximating a $[[N, \Omega(\sqrt{N}), \Omega(\sqrt{N})]]$ code in 2D. The key ingredient is a new constructive tool to simulate an $l$-long interaction between two qubits by a 1D chain of $l$ nearest-neighbour interacting qubits using $\mathrm{poly}( l)$ interaction strengths. This is an exponential advantage over the existing gadgets for this routine which facilitates the first $\epsilon$-simulation of \emph{arbitrary sparse} Hamiltonian on $n$ qubits with a Hamiltonian on a 2D lattice of $O(n^2)$ qubits with interaction strengths scaling as $O\left(\mathrm{poly}(n,1/\epsilon)\right)$.