This research proposal develops a methodology for applying Quantum Index Algebra (QIA) to Grover’s search and Shor’s factoring algorithms. QIA represents quantum operators using Boolean code indexing rather than dense matrix arithmetic, enabling efficient computation of operator products and commutators. The proposal outlines reformulations of oracle construction and diffusion operators in Grover’s algorithm, optimization of the Quantum Fourier Transform and modular exponentiation in Shor’s algorithm, and a unified framework for analyzing classical simulability. Validation plans include circuit depth analysis, gate count optimization, and experimental benchmarks on simulated quantum hardware.
Key findings
QIA enables sparse representation of Grover's oracle operators using Boolean code indexing, allowing efficient classical evaluation of structured oracles.
The diffusion operator and Walsh-Hadamard transforms can be expressed as compact linear combinations of QIA basis elements D and T.
Algebraic index manipulations in QIA can optimize Quantum Fourier Transform components and modular exponentiation in Shor's algorithm.
The framework provides unified criteria for analyzing classical simulability of structured quantum circuits across both algorithms.
QIA-based formulations demonstrate potential for reducing gate counts and circuit depths compared to standard implementations.
Limitations & open questions
Validation remains theoretical and experimental; implementation on NISQ devices faces decoherence and noise challenges.
The framework assumes specific algebraic structures that may not generalize to all quantum algorithm variants or mixed state operations.
Classical simulability criteria require further theoretical development for noisy intermediate-scale quantum operations.