Rotated Surface Code for Fault-Tolerant Logical Hadamard Gates
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Solution Overview
Problem
Existing approaches to determining fault-tolerance of stabilizer channels in quantum computing are inadequate, lacking clear definitions, diagnostic tools for fault-tolerance issues, and guidance for improving designs.
Innovation Solution
A computing system and method for implementing a logical Hadamard gate using a rotated surface code, involving measurement of generators with expansion and contraction circuits, and applying a transverse Hadamard gate to data qubits, with tools to diagnose and modify stabilizer channels for increased fault-tolerance.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If a logical Hadamard gate is implemented using traditional surface code methods, then the gate functionality is achieved, but the fault-tolerance distance is insufficient
Solution Approach 1:
The implementation is divided into multiple stages: expansion stages that grow the code patch, contraction stages that shrink it, and a middle stage where the Hadamard gate is applied. Each stage measures specific stabilizer generators independently, allowing fault-tolerance to be maintained at each step while achieving the overall logical gate functionality with improved fault distance d.
Solution Approach 2:
The surface code patch is rotated by 45 degrees, transforming the boundary conditions and stabilizer structure. This rotation enables the logical X-string and Z-string operators to terminate at swapped boundaries, creating a new geometric configuration that supports the Hadamard gate with enhanced fault-tolerance properties.
2Adaptability or versatility
If the surface code patch is rotated to implement the Hadamard gate, then the logical operator boundaries are swapped, but the measurement circuit complexity increases
Solution Approach 1:
The measurement circuit is segmented into expansion circuits for growing the rotated code patch, contraction circuits for shrinking it, and a middle circuit for applying the Hadamard gate. Each segment measures specific stabilizer generators (first and second expansion/contraction stages) independently, making the complex rotated configuration manageable while maintaining adaptability.
3Reliability
If multiple expansion and contraction stages are used to rotate the surface code, then the fault-tolerance is improved, but the number of measurement operations increases
Solution Approach 1:
The expansion and contraction stages are designed to proceed continuously through the rotation process, with each stage measuring stabilizer generators that contribute to the final rotated configuration. This continuous progression through defined stages (first expansion, middle, first contraction, second contraction) ensures fault-tolerance is maintained throughout the transformation while completing the Hadamard gate implementation efficiently.
Data Source
AI summary
A method is presented for implementing a logical Hadamard gate with a fault distance of d. A patch of surface code is rotated such that boundaries where logical X-string operators terminate are swapped with boundaries where logical Z-string operators terminate. Rotating the patch of surface code comprises at least measuring generators of a first expansion stage with a first expansion circuit, and measuring generators of a second expansion stage with a first sub-circuit and a second sub-circuit of a second expansion circuit. Generators of a first contraction stage are measured with a first sub-circuit and a second sub-circuit of a first contraction circuit. Generators of a second contraction stage are measured with a second contraction circuit. A transverse Hadamard gate is applied to data qubits of the rotated patch of surface code. The patch of surface code is translated to a final position.