Quantum Circuit Synthesis for Exact Qudit and Multi-Qubit Gates
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Solution Overview
Problem
Current methods for implementing operations on quantum systems, such as qubits and qudits, are expensive and lack efficient methods for selecting quantum gates to represent arbitrary operators, particularly for multi-qubit systems.
Innovation Solution
The development of methods for decomposing a unitary into a quantum circuit by gradually decreasing the complexity of the unitary or quantum state using the structure of torsion-free modules, represented by integer vectors of p-adic valuations, to find a circuit that prepares a given state on a quantum computer.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional methods are used to implement operations on quantum systems, then quantum computing capabilities can be achieved, but the implementation cost becomes expensive
Solution Approach 1:
The patent changes the parameter representation from continuous unitary matrices to discrete structures (torsion-free modules over rings, integer vectors of p-adic valuations). This discretization enables exact synthesis using finite gate sets, reducing implementation complexity and cost while maintaining quantum computing capabilities.
Solution Approach 2:
The patent replaces the traditional continuous mathematical approach (matrix decompositions) with an algebraic number theory approach (modules over rings, p-adic valuations). This substitution enables exact representation of quantum operations using discrete algebraic structures, leading to more efficient and less expensive circuit implementations.
2Adaptability or versatility
If arbitrary quantum operators are represented using available quantum gates, then complete quantum operations can be performed, but the gate selection process becomes complex
Solution Approach 1:
The patent transforms the gate selection problem from continuous matrix approximation to discrete algebraic synthesis. By representing quantum operators as elements of torsion-free modules and using p-adic valuations, the method provides a systematic algorithm for exact gate decomposition, greatly simplifying the selection process while maintaining complete operational capability.
Solution Approach 2:
The patent introduces torsion-free modules over rings and integer vectors of p-adic valuations as intermediary structures between arbitrary quantum operators and discrete quantum gates. These intermediaries provide a bridge that enables exact representation and systematic decomposition of quantum operations into basis gates.
3Adaptability or versatility
If quantum circuits are synthesized for qudit and multi-qubit systems, then enhanced quantum computing functionality is achieved, but the circuit complexity increases
Solution Approach 1:
The patent segments the synthesis problem by dimension, handling qudit (d-dimensional) and multi-qubit systems through systematic extension of the base algorithm. The method decomposes complex multi-dimensional unitaries into sequences of simpler operations using tensor product structures and modular arithmetic, reducing overall circuit complexity while maintaining enhanced functionality.
Solution Approach 2:
The patent creates a universal synthesis framework that works for qudits of any dimension d and multi-qubit systems of any size. The same core algorithm using torsion-free modules and p-adic valuations applies to all these cases, providing a multi-functional solution that reduces circuit complexity across different quantum system types.
Data Source
AI summary
Methods are provided for exact synthesis of unitaries for qudit and multi-qubit systems. In addition, state preparation methods are provided. The syntheses produce circuits that have lowest cost for a given cost function.