Effects of Quantum Noise on Shor’s Algorithm

Abstract

Shor's quantum factoring algorithm(Shor's algorithm) can solve factorization problem of large integers using a fully-operational quantum computer with the complexity of polynomial level, thereby cracking a series of encryption algorithms whose security is guaranteed by the factorization of large integers being a hard problem, such as Rivest-Shamir-Adleman encryption algorithm, Diffie-Hellman key exchange protocol, etc. We are currently in the noisy intermediate-scale quantum era, which means we can only operate on quantum computers with limited number of qubits and we have to take care of the effects of quantum noise. Quantum states are susceptible to quantum noise when running on a quantum computer due to low-fidelity gates or interaction between qubits and environment, which will lead to the incorrect results by measurements. We study the effects of quantum noise on Shor's algorithm from 3 typical quantum noise channels respectively: depolarizing channel, state preparation and measurement channel and thermal relaxation channel. We successfully simulate factoring 15, 21 and 35 into their corresponding prime factors based on the quantum circuit we have constructed on a classical computer. Our research then simulates running quantum circuit of Shor's algorithm in a noisy environment with different level of noise for one certain type of noise channel and yields numerical results. We can obtain precise distribution of measurement by calculating the statevector before measurement which contributes to higher effiency instead of simulating and measuring for a large amount of times. Each experiment is conducted for 1000 times to diminish discrepancy. Our research indicates that Shor's algorithm is easily affected by quantum noise. Success rate of Shor's algorithm decreases exponentially with increasing level of noise in depolarizing channel, where succuss rate is an indicator we propose in this research to quantify the effect of noise on Shor's algorithm, meanwhile noise in both state preparation and measurement channel and thermal relaxation channel can linearly affect the success rate of Shor's algorithm. There are $O(n^4)$ quantum gates in the circuit, each of which is disrupted by noise in depolarizing channel during running the circuit, meanwhile there are only $O(n)$ interruption caused by noise in state preparation and measurement channel since we only do measurements for $O(n)$ times in the circuit where $n$ is the number of bits of the integer about to be factored. Linear relationship in thermal relaxation channel is mainly due to the large gap between quantum gate time and relaxation time even if every gate in the circuit is disrupted by noise in thermal relaxation channel like depolarizing channel. Our paper can be used for subsequent error correction, improving Shor's algorithm and providing guidance to requirements for fidelity in engineering implementation of Shor's algorithm.