Experimental realisation of Shor's quantum factoring algorithm using qubit recycling

Abstract

Quantum algorithms are computational routines that exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms. Shor's quantum factoring algorithm [1] is a key example and the prime motivator in the international effort to realise a quantum computer. However, due to the large number of resources required, to date, there have been only four small scale demonstrations [2][3][4][5].Here we address this resource demand and demonstrate a scalable version of Shor's algorithm in which the n qubit control register is replaced by a single qubit that is recycled n times: the total number of qubits is one third of that required in the standard protocol [6]. Encoding the work register in higher-dimensional states, we implement a two-photon compiled algorithm to factor N = 21 [7]. Significantly, the algorithmic output exhibits structure that is distinguishable from noise, in contrast to previous demonstrations. 0 1 0 1 2 00 01 10 11 0 1 8 1 4 3 8 000 011 101 0 0.35 0000 0101 1011 0 0.35 0 0 0 0 0 0 1 Fig. 1. (a) Measurement of the control qubit after each controlled unitary gives the next most significant bit in the output and the outcome is fed forward to the iterated (semi-classical) Fourier transform, which applies either the identity operation I or the appropriate phase gate R, prior to the Hadamard H. (b) As the number of iterations increases the precision increases. (c) For two bits of precision the controlled unitary operations can be constructed with this arrangement of controlled-swap gates.The correct algorithmic output from the quantum order finding circuit for factoring N = 21 is confirmed by the experimental results with a fidelity of 99 ± 4% with the ideal probability distribution. The experimental output has a critical dependence on decoherence: phase instability drives the output toward a uniform probability distribution, in contrast with previous experimental demonstrations of Shor's algorithm, all of them for N = 15, in which a uniform probability distribution is the expected outcome. We confirmed this analysis experimentally [7].These results point to larger-scale implementations of Shor's algorithm by harnessing substantial but scalable resource reductions applicable to all physical architectures.

Quantum algorithms: an overview

“…Problem solved Shor's algorithm Bulk optics [68] Factorisation of 21 Grover's algorithm NMR [95] Unstructured search, N = 8 Quantum annealing D-Wave 2X [55] Ising model on a "Chimera" graph with 1097 vertices HHL algorithm Bulk optics [25,10], NMR [72] 2 × 2 system of linear equations Table 3: Some proof-of-concept experimental implementations of quantum algorithms. Table only includes some "largest" problem instances solved thus far.…”

Teaching Quantum Computing with the QuIDE Simulator

“…In the recent years the idea of quantum computation is really pushing forward as the significant progress has been made in the field of constructing a quantum computer [14,15]. The main concept is to use of quantum-mechanical phenomena to perform operations on data considerably faster than with classical computers.…”

Quantum Factoring Algorithm: Resource Estimation and Survey of Experiments

“…From the theoretical viewpoint, it has been evaluated how much resources are needed for the prime factorization of composite number of the currently used sizes (1024-bit, 2048-bit) (Häner 2017;Kunihiro 2005). However, from the experimental viewpoint, several experiments have been performed for the prime factorization of small composite numbers such as 15 and 21 (Lucero et al 2012;Martin-Lopez et al 2012;Monz et al 2016;Politi 2009;Vandersypen 2001). In addition, commercial services for small-scale quantum computers such as IBM Q ( 2020) are beginning to be launched, and it is expected that the Noisy Intermediate-Scale Quantum (NISQ) technology might be available in the near future (Preskill 2018).…”

“…This paper presents a detailed survey of actual quantum experiments for prime factorization based on Shor's algorithm (Lucero et al 2012;Martin-Lopez et al 2012;Monz et al 2016;Politi 2009;Vandersypen 2001). We give a detailed explanation of the circuits used in the experiments.…”