Space-Efficient and Noise-Robust Quantum Factoring
Name
ragavan-sragavan-sm-eecs-2024-thesis.pdf
Description
Thesis PDF
Size
830.09 KB
Format
Adobe PDF
Checksum (MD5)
6cb49be46366344c7a5d06e14661f0fe
Author(s)
Ragavan, Seyoon
Advisor(s)
Vaikuntanathan, Vinod
Date Issued
May 2024
Publisher
Massachusetts Institute of Technology
Abstract
We provide two improvements to Regev's quantum factoring algorithm (arXiv:2308.06572), addressing its space efficiency and its noise-tolerance.
Our first contribution is to improve the quantum space efficiency of Regev's algorithm while keeping the circuit size the same. Our main result constructs a quantum factoring circuit using O(n log n) qubits and O(n^(3/2) log n) gates. We achieve the best of Shor and Regev (upto a logarithmic factor in the space complexity): on the one hand, Regev's circuit requires O(n^(3/2)) qubits and O(n^(3/2) log n) gates, while Shor's circuit requires O(n^2 log n) gates but only O(n) qubits. As with Regev, to factor an n-bit integer N, we run our circuit independently ≈ sqrt{n} times and apply Regev's classical postprocessing procedure.
Our optimization is achieved by implementing efficient and reversible exponentiation with Fibonacci numbers in the exponent, rather than the usual powers of 2, adapting work by Kaliski (arXiv:1711.02491) from the classical reversible setting to the quantum setting. This technique also allows us to perform quantum modular exponentiation that is efficient in both space and size without requiring significant precomputation, a result that may be useful for other quantum algorithms. A key ingredient of our exponentiation implementation is an efficient circuit for a function resembling in-place quantum-quantum modular multiplication. This implementation works with only black-box access to any quantum circuit for out-of-place modular multiplication, which we believe is yet another result of potentially broader interest. Additionally, we show how to generalize our reversible exponentiation technique beyond the Fibonacci numbers to obtain constant-factor improvements in the number of qubits and/or gates.
Our second contribution is to show that Regev's classical postprocessing procedure can be modified to tolerate a constant fraction of the quantum circuit runs being corrupted by errors. In contrast, Regev's analysis of his classical postprocessing procedure requires all ≈ sqrt{n} runs to be successful. In a nutshell, we achieve this using lattice reduction techniques to detect and filter out corrupt samples.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
Copyright retained by author(s)
Persistent DSpace Link
