Fibonacci Anyon Braid Circuit Decomposition
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Solution Overview
Problem
Current methods for decomposing single-qubit quantum gates into braiding matrices require exponential time, making them impractical for achieving precision in long braids, which is necessary for topological quantum computing using Fibonacci anyons.
Innovation Solution
The development of methods to decompose single-qubit quantum gates into Fibonacci anyon braid matrices with a circuit depth of O(ln(1/ε)), where c=1 and ∈ is the desired precision, in expected polynomial time, using iterative procedures and exact synthesis algorithms to reduce circuit complexity.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Device complexity
If brute-force search methods are used to find optimal braid decompositions, then circuit depth can be reduced to O(ln(1/ε)) with c=1, but the computational time becomes exponential
Solution Approach 1:
The decomposition process is divided into two distinct phases: an iterative procedure that rapidly reduces circuit depth to near-optimal levels, followed by an exact synthesis algorithm that completes the decomposition with polynomial-time complexity. This segmentation allows the system to achieve shallow circuit depth without incurring exponential computational costs.
Solution Approach 2:
The iterative procedure performs preliminary decomposition work that reduces the circuit depth to O(ln(1/ε)) with c=1 before the exact synthesis step. By pre-processing the decomposition in this manner, the subsequent exact synthesis operates on a already-optimized structure, avoiding the need for exhaustive search of the entire decomposition space.
2Measurement precision
If conventional Solovay-Kitaev theorem implementation is used, then any single-qubit gate can be approximated to desired precision, but the circuit depth scales with c≈3.97 which is less efficient
Solution Approach 1:
The invention changes the fundamental parameter c in the circuit depth formula from approximately 3.97 (conventional Solovay-Kitaev) to exactly 1 (optimal Fibonacci anyon decomposition). This parameter improvement is achieved through the combination of iterative optimization and exact synthesis, which together enable shallower circuits while maintaining the same precision ε.
3Measurement precision
If brute-force search is used to achieve optimal circuit depth, then decomposition precision can be maximized, but the method becomes infeasible for long braids requiring high precision
Solution Approach 1:
The decomposition task is segmented into iterative reduction (achieving near-optimal depth) and exact synthesis (completing with polynomial time). This segmentation makes high-precision decomposition feasible for long braids by avoiding the need for exhaustive search across the entire precision range.
Solution Approach 2:
The iterative procedure performs preliminary optimization to reduce circuit depth before exact synthesis is applied. This pre-optimization ensures that the subsequent exact synthesis operates efficiently even for high-precision requirements, making the overall process feasible for long braids.
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AI summary
Methods for compiling single-qubit quantum gates into braid representations for non-Abelian quasiparticles described by the Fibonacci anyon model are based on a probabilistically polynomial algorithm that, given a single-qubit unitary gate and a desired target precision, outputs a braid pattern that approximates the unitary to desired precision and has a length that is asymptotically optimal (for a circuit with such property). Single-qubit unitaries that can be implemented exactly by a Fibonacci anyon braid pattern are classified, and associated braid patterns are obtained using an iterative procedure. Target unitary gates that are not exactly representable as braid patterns are first approximated to a desired precision by a unitary that is exactly representable, then a braid pattern associated with the latter is obtained.