Realization of a scalable Shor algorithm (1507.08852v1)

Abstract: Quantum computers are able to outperform classical algorithms. This was long recognized by the visionary Richard Feynman who pointed out in the 1980s that quantum mechanical problems were better solved with quantum machines. It was only in 1994 that Peter Shor came up with an algorithm that is able to calculate the prime factors of a large number vastly more efficiently than known possible with a classical computer. This paradigmatic algorithm stimulated the flourishing research in quantum information processing and the quest for an actual implementation of a quantum computer. Over the last fifteen years, using skillful optimizations, several instances of a Shor algorithm have been implemented on various platforms and clearly proved the feasibility of quantum factoring. For general scalability, though, a different approach has to be pursued. Here, we report the realization of a fully scalable Shor algorithm as proposed by Kitaev. For this, we demonstrate factoring the number fifteen by effectively employing and controlling seven qubits and four "cache-qubits", together with the implementation of generalized arithmetic operations, known as modular multipliers. The scalable algorithm has been realized with an ion-trap quantum computer exhibiting success probabilities in excess of 90%.

• The paper demonstrates a scalable realization of Shor's algorithm using an ion-trap quantum computer to factor the integer 15.
• It employs Kitaev's semiclassical QFT to reduce qubit requirements to seven qubits and four auxiliary qubits while achieving over 90% success rates.
• The study integrates modular arithmetic with high-fidelity entangling and single-qubit operations, paving the way for fault-tolerant quantum computing.

Realization of a Scalable Shor Algorithm: Towards Quantum Factoring

The paper "Realization of a scalable Shor algorithm" by Monz et al. provides a detailed exploration of the practical implementation of Shor's algorithm on an ion-trap quantum computer, aiming to factor the integer 15. The research highlights the significance of scalability in implementing Shor's algorithm, a seminal quantum algorithm known for its ability to outperform classical methods in integer factorization.

Summary of Research

The authors demonstrate a scalable realization of Shor's algorithm in an ion-trap quantum computing environment. This paper distinguishes itself by employing Kitaev's approach to Shor's algorithm, which replaces the quantum Fourier transform (QFT) acting on multiple qubits with a semiclassical QFT operating on a single qubit, thereby greatly reducing qubit requirements. Specifically, the experiment operates with seven qubits and four auxiliary ("cache") qubits using modular multipliers to factor the number 15.

Key experimental challenges addressed include managing a substantial quantum register and maintaining high-fidelity control over the qubits. The authors employ a combination of entangling gates, collective operations, and single-qubit phase rotations to achieve the desired unitary transformations necessary for the algorithm. Additionally, the implementation of modular arithmetic within the quantum circuit is notably complex but essential, as it requires accurate realization of modular multipliers for various bases.

Strong Numerical Results

The research achieves success probabilities exceeding 90% in the realization of quantum circuits required for factoring 15. The authors quantify the efficacy of their implementation through squared statistical overlap (SSO) metrics, obtaining values above 90% for the tested bases {2, 7, 8, 11, 13}. These results validate the functional fidelity of individual quantum operations and support the argument for the experimental scalability of Shor's algorithm.

Implications and Speculations on Future Developments

The paper's findings hold significant implications for the future of quantum computing, particularly concerning practical quantum algorithms' scalability and reliability. By demonstrating a scalable version of Shor's algorithm, the authors pave the way for more complex quantum computations that could eventually tackle real-world problems of cryptographic significance, such as breaking public key cryptosystems.

For theoretical research, the use of Kitaev's algorithmic architecture to optimize qubit resources could influence the design of other quantum algorithms that similarly aim to balance operational efficiency with computational accuracy. Practically, the incorporation of error mitigation techniques such as qubit recycling and feed-forward control suggests critical advancements for quantum error correction methods and fault-tolerant quantum computing.

Further developments could see the transition from ion-trap setups to other quantum hardware architectures, potentially maximizing the available computational power while minimizing physical qubit deployment. With continual improvements in qubit coherence times and gate fidelities, quantum factorization of larger integers may become feasible, and beyond that, complex simulations and solutions previously inaccessible to classical machines could be within reach.

In conclusion, the authors' scalable implementation of Shor's algorithm marks a significant step in quantum algorithm research, demonstrating both the practicality and potential of quantum computers in handling advanced computational tasks. This work not only confirms the theoretical advantages of quantum computing over classical counterparts but also sets a precedent for achieving scalable quantum operations necessary for future breakthroughs in the field.