Operator Imprecision and Scaling of Shor's Algorithm

TL;DR

This paper demonstrates that operator imprecision and input errors severely impair Shor's algorithm, challenging its polynomial scaling advantage and suggesting quantum computers may not outperform classical ones in factoring.

Contribution

It reveals that operator imprecision destroys Shor's algorithm's polynomial scaling and that quantum error correction cannot fully mitigate these errors.

Findings

Operator imprecision destroys polynomial scaling of Shor's algorithm.

Quantum error correcting codes are ineffective against operator imprecision.

Operator errors are equivalent to decoherence, limiting quantum advantage.

Abstract

Shor's algorithm (SA) is a quantum algorithm for factoring integers. Since SA has polynomial complexity while the best classical factoring algorithms are sub-exponential, SA is cited as evidence that quantum computers are more powerful than classical computers. SA is critically dependent on the Quantum Fourier Transform (QFT) and it is known that the QFT is sensitive to errors in the quantum state input to it. In this paper, we show that the polynomial scaling of SA is destroyed by input errors to the QFT part of the algorithm. We also show that Quantum Error Correcting Codes (QECC) are not capable of suppressing errors due to operator imprecision and that propagation of operator precision errors is sufficient to severely degrade the effectiveness of SA. Additionally we show that operator imprecision in the error correction circuit for the Calderbank-Shor-Steane QECC is mathematically equivalent to decoherence on every physical qubit in a register. We conclude that, because of the effect of operator precision errors, it is likely that physically realizable quantum computers will be capable of factoring integers no more efficiently than classical computers.