Majorana loop stabilizer codes for error correction of fermionic quantum simulations

An apparatus comprising:

• one or more classical processors;
• and
• quantum hardware;
• wherein the apparatus is configured to perform quantum simulation of a fermionic system, wherein the fermionic system is characterized by a fermionic Hamiltonian comprising elements from a group of quadratic Majorana operators of even weights, wherein the quadratic Majorana operators satisfy a set of commutation relations and a dependence relation that implies that the quadratic Majorana operators are not independent, the quantum simulation comprising:
• representing the fermionic system as an interaction graph of vertices and edges, wherein each vertex represents a fermionic system fermionic mode and each edge represents an interaction between two respective fermionic modes;
• allocating a qubit to each edge in the interaction graph to form a qubit system;
• determining qubit operators that satisfy the set of commutation relations and the dependence relation, wherein the qubit operators are non-uniform with respect to the interaction graph vertices;
• determining stabilizer operators corresponding to products of quadratic Majorana operators on respective loops in the interaction graph, wherein a common eigenspace of the defined stabilizer operators defines a code subspace that encodes states of the fermionic system;
• and
• simulating the fermionic system by evolving the qubit system under a qubit Hamiltonian, wherein the qubit Hamiltonian comprises the determined qubit operators and the determined stabilizer operators.