Majorana Loop Stabilizer Codes for Local Fermionic Error Correction
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Solution Overview
Problem
Simulating complex quantum systems using classical techniques is untenable due to exponential resource scaling, and existing fermion-to-qubit mappings in quantum computing introduce non-local parity terms or geometric non-locality, which are not suitable for near-term quantum devices with short-ranged qubit interactions.
Innovation Solution
A Majorana loop stabilizer code that maps fermionic systems to qubit systems with geometric locality, allowing for error correction of single-qubit errors on interaction graphs with vertex degree 4, using non-uniform qubit operators and stabilizers defined by products of Majorana operators on loops, preserving locality and reducing simulation complexity.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If existing fermion-to-qubit mappings are used, then fermionic systems can be simulated on quantum computers, but non-local parity terms or geometric non-locality are introduced which are not suitable for near-term quantum devices with short-ranged qubit interactions
Solution Approach 1:
The fermionic system is segmented into Majorana modes that are distributed across the qubit system, with each Majorana mode represented by a pair of qubits. This segmentation allows local Majorana operators to be constructed from local qubit operators, eliminating the need for non-local parity terms while preserving the fermionic algebra structure.
Solution Approach 2:
The mapping implements local quality by ensuring that Majorana operators acting on local fermionic modes are represented by local qubit operators acting on adjacent qubits. This local representation is achieved through the stabilizer code construction where Majorana operators are expressed as products of Pauli operators with support only on neighboring qubits, matching the short-ranged interaction capability of near-term quantum devices.
2Reliability
If Majorana loop stabilizer code is used, then single-qubit errors can be corrected with geometric locality, but the system requires non-uniform qubit operators and stabilizers defined by products of Majorana operators on loops
Solution Approach 1:
The stabilizer code is constructed in advance with stabilizer operators defined as products of Majorana operators around loops in the interaction graph. This preliminary construction of the code structure with built-in error correction capability allows single-qubit errors to be detected and corrected through stabilizer measurements without requiring complex real-time operations during computation.
Solution Approach 2:
Majorana operators serve as intermediaries between the fermionic system and the qubit representation. By expressing stabilizer operators as products of Majorana operators on loops, the code achieves geometric locality in error correction while the non-uniform qubit operators naturally emerge from the loop structure of the interaction graph.
3Device complexity
If qubits are allocated to edges in the interaction graph, then geometric locality is preserved, but the qubit operators become non-uniform with respect to the interaction graph vertices
Solution Approach 1:
The mapping embraces asymmetry by allocating qubits to edges rather than vertices, which naturally leads to non-uniform qubit operators. This asymmetric allocation preserves geometric locality because edge-associated qubits naturally represent interactions between adjacent vertices, and the resulting non-uniform operators reflect the actual structure of the fermionic system's interaction graph.
Data Source
AI summary
Methods, systems, and apparatus for error correction of fermionic quantum simulation. In one aspect, a method includes representing a fermionic system as a graph of vertices and edges, where 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 graph to form a qubit system; determining qubit operators that satisfy a set of fermionic commutation and dependence relations, where the qubit operators are non-uniform with respect to the graph vertices; determining stabilizer operators corresponding to products of quadratic Majorana operators on respective loops in the graph, where a common eigenspace of the defined stabilizer operators defines a code subspace that encodes states of the fermionic system to be simulated; and simulating the fermionic system by evolving the qubit system under a qubit Hamiltonian that includes the determined qubit operators and stabilizer operators.