Majorana Loop Stabilizer Codes for Local Fermion 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 like Jordan-Wigner and Bravyi-Kitaev transformations introduce non-local parity terms and geometric non-locality, which are inefficient for near-term quantum devices with short-ranged qubit interactions.
Innovation Solution
A Majorana loop stabilizer code that preserves geometric locality by mapping fermionic systems to qubit systems with short-ranged interactions, correcting all single-qubit errors on 2D lattices without additional vertices or edges, using products of Majorana operators on closed paths as stabilizers.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If Jordan-Wigner or Bravyi-Kitaev transformation is used to map fermionic systems to qubit systems, then fermionic states can be encoded, but non-local parity terms and geometric non-locality are introduced which increase device complexity and reduce ease of operation on near-term quantum devices
Solution Approach 1:
The fermionic system is segmented into local modes that can be independently mapped to qubits, with each mode's stabilizers defined locally. This segmentation allows the overall fermionic system to be represented without requiring non-local parity terms, as each local mode maintains its own stabilizer structure that preserves geometric locality.
Solution Approach 2:
The patent applies local quality by ensuring that each qubit mode has stabilizers that act only on local qubits rather than requiring global parity operations. This local quality approach means that the stabilizer for each fermionic mode is constructed from qubit operators that act locally, eliminating the need for non-local parity terms while maintaining the fermionic algebra.
2Device complexity
If geometric locality-preserving mappings are used to reduce device complexity, then short-ranged qubit interactions are achieved, but error correction capability is reduced to only detecting single-qubit errors rather than correcting them
Solution Approach 1:
The patent resolves this contradiction by introducing a new dimension to the stabilizer construction - using products of Majorana operators on closed paths (loops) rather than simple local operators. This dimensional change in the stabilizer structure allows the code to correct single-qubit errors while maintaining geometric locality, as the loop stabilizers provide the necessary redundancy without requiring long-range interactions.
Solution Approach 2:
The error correction capability is achieved through a composite stabilizer structure that combines multiple local Majorana operators into loop stabilizers. This composite approach creates a stabilizer code that has both the local interaction property (from individual Majorana operators) and the error correction capability (from the composite loop structure), effectively combining the benefits of both approaches.
3Reliability
If higher code distance is achieved to improve error correction, then more qubits are required which increases device complexity, but the Majorana loop stabilizer code maps fermionic operators to lower-weight qubit operators
Solution Approach 1:
The patent changes the parameter of stabilizer construction from local operators to loop products of Majorana operators. This parameter change allows the code to achieve higher effective code distance for error correction while maintaining the same physical qubit footprint, because the loop stabilizers provide additional error detection capability without requiring additional qubits beyond what is needed for the fermionic mode representation.
Data Source
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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.