On Shor's Factoring Algorithm with More Registers and the Problem to Certify Quantum Computers

TL;DR

This paper questions the standard understanding of Shor's factoring algorithm by analyzing the implications of using multiple registers and highlights the need for physical verification of quantum computational claims.

Contribution

It introduces a novel perspective on the entanglement structure in Shor's algorithm with additional registers, challenging existing complexity assumptions.

Findings

Measured register states should be equal if original assumptions hold

Entanglement involving many registers may not be interpretable as EPR pairs

The claim of polynomial-time quantum factoring requires further physical validation

Abstract

Shor's factoring algorithm uses two quantum registers. By introducing more registers we show that the measured numbers in these registers which are of the same pre-measurement state, should be equal if the original Shor's complexity argument is sound. This contradicts the argument that the second register has $r$ possible measured values. There is an anonymous comment which argues that the states in these registers are entangled. If so, the entanglement involving many quantum registers can not be interpreted by the mechanism of EPR pairs and the like. In view of this peculiar entanglement has not yet been mentioned and investigated, we think the claim that the Shor's algorithm runs in polynomial time needs more physical verifications. We also discuss the problem to certify quantum computers.