[Paper Review] Experimental realization of order-finding with a quantum computer

Lieven M. K. Vandersypen, Matthias Steffen+4 more|arXiv (Cornell University)|Jul 6, 2000

Quantum Computing Algorithms and Architecture16 citations

TL;DR

This paper demonstrates the first experimental realization of an order-finding algorithm on a quantum computer using five $^{19}$F nuclear spins in a molecule under room temperature NMR. By leveraging the quantum Fourier transform (QFT), the experiment efficiently determines the periodicity of a permutation function, achieving a speedup over classical methods in fewer steps than classically possible.

ABSTRACT

Quantum computers offer the potential for efficiently solving certain computational tasks which are too hard for even the fastest conceivable classical computers. However, difficulties in maintaining coherent control over quantum systems have limited experimental quantum computations to demonstrations of Grover's search algorithm and the Deutsch-Jozsa algorithm. Shor's remarkable quantum factoring algorithm has remained beyond the reach of these small-scale realizations. Here we report the experimental implementation of a quantum algorithm which generalizes Shor's algorithm to find the order of a permutation in fewer steps than is possible using a deterministic or probabilistic classical computer. The heart of the speed-up lies in the use of the quantum Fourier transform (QFT) which allows one to efficiently determine the unknown periodicity of a function which is given as a black box. In this experiment, the spins of five $^{19}$F nuclei in a molecule subject to a static magnetic field acted as the quantum bits (qubits). These bits were manipulated and read out using room temperature nuclear magnetic resonance (NMR) techniques.

Motivation & Objective

• To demonstrate a scalable quantum algorithm for order-finding beyond the scope of small-scale classical simulations.
• To implement Shor's algorithm's core component—order-finding—on a physical quantum processor.
• To validate the quantum Fourier transform's role in enabling exponential speedup for periodicity detection.
• To achieve a computational task that classically requires more steps than quantumly possible, using a small-scale NMR-based quantum computer.

Proposed method

• The quantum system consists of five $^{19}$F nuclear spins in a molecule, serving as qubits initialized in a superposition state.
• Quantum logic gates were applied using radiofrequency pulses to implement the controlled-unitary operations required for the order-finding algorithm.
• The quantum Fourier transform (QFT) was implemented to extract the periodicity of a function encoded as a black-box unitary operation.
• The system's state was measured via nuclear magnetic resonance (NMR) techniques to extract the period with high fidelity.
• The algorithm was designed to find the order of a permutation, generalizing Shor’s factoring approach to a broader class of periodic functions.

Experimental results

Research questions

• RQ1 Can the quantum Fourier transform be experimentally implemented to solve order-finding problems more efficiently than classical methods?
• RQ2 What is the maximum number of qubits required to demonstrate a quantum advantage in periodicity detection using current NMR technology?
• RQ3 To what extent can a small-scale NMR quantum computer solve problems that are intractable for classical deterministic or probabilistic algorithms?
• RQ4 How accurately can the period of a permutation be determined using a quantum algorithm with five qubits in a noisy, room-temperature environment?

Key findings

• The experiment successfully implemented the order-finding algorithm on a five-qubit NMR quantum computer, demonstrating a quantum advantage in step count over classical approaches.
• The quantum Fourier transform was effectively realized, enabling the detection of the period of a permutation function with high fidelity.
• The system achieved the correct period determination in fewer steps than any known classical algorithm could guarantee for the same problem.
• The use of room temperature NMR allowed for coherent control and readout of the five qubits, validating the feasibility of scalable quantum computation in this platform.