Factoring an integer with three oscillators and a qubit (2412.13164v1)

Abstract: A common starting point of traditional quantum algorithm design is the notion of a universal quantum computer with a scalable number of qubits. This convenient abstraction mirrors classical computations manipulating finite sets of symbols, and allows for a device-independent development of algorithmic primitives. Here we advocate an alternative approach centered on the physical setup and the associated set of natively available operations. We show that these can be leveraged to great benefit by sidestepping the standard approach of reasoning about computation in terms of individual qubits. As an example, we consider hybrid qubit-oscillator systems with linear optics operations augmented by certain qubit-controlled Gaussian unitaries. The continuous-variable (CV) Fourier transform has a native realization in such systems in the form of homodyne momentum measurements. We show that this fact can be put to algorithmic use. Specifically, we give a polynomial-time quantum algorithm in this setup which finds a factor of an $n$-bit integer $N$. Unlike Shor's algorithm, or CV implementations thereof based on qubit-to-oscillator encodings, our algorithm relies on the CV (rather than discrete) Fourier transform. The physical system used is independent of the number $N$ to be factored: It consists of a single qubit and three oscillators only.

• The paper presents a novel hybrid quantum algorithm that factors integers using a fixed setup of one qubit and three oscillators.
• It leverages continuous-variable systems and GKP state techniques to perform efficient pseudomodular power computations in polynomial time.
• This resource-efficient framework shifts quantum computation paradigms by reducing the need for scalable qubit networks and easing experimental implementation.

Hybrid Quantum Algorithm for Integer Factorization with Three Oscillators and a Qubit

The paper "Factoring an integer with three oscillators and a qubit" presents a novel approach to integer factorization, leveraging a hybrid quantum system that employs three harmonic oscillators (bosonic modes) and a single qubit. This system executes a polynomial-time quantum algorithm capable of factoring an $n$-bit integer $N$ with a fixed physical setup, comprising significantly fewer resources than traditional methods like Shor’s algorithm. Crucially, this work shifts the paradigm of quantum computation, stepping away from the notion that scalability requires an extensively increasing count of physical qubits.

Traditional quantum computation frameworks emphasize scaling qubits in line with the problem size, as evidenced by Shor’s algorithm. Shor’s algorithm scales both in terms of the number of qubits and operational gates as the integer to be factored grows. Contrarily, the proposed hybrid paradigm maintains a constant system size of one qubit and three oscillators, trading off circuit size for the invariant physical setup. This is achieved by capitalizing on continuous-variable (CV) quantum systems, particularly employing the native continuous-variable Fourier transform through linear optics operations and qubit-controlled Gaussian unitaries.

Technical Contributions

• Continuous-Variable Systems: The algorithm exploits the inherent properties of CV systems, particularly leveraging the native realization of continuous-variable Fourier transforms, sidestepping finite-dimensional modular arithmetic using intrinsic bosonic operations.
• GKP State Utilization: Gottesman-Kitaev-Preskill (GKP) states, known for their utility in error correction, are innovatively used to simulate a modular measurement, crucial for implementing pseudomodular power computations which underpin the factorization process.
• Algorithmic Framework: The algorithm derives efficiencies by embedding integer factorization tasks within a hybrid quantum framework, employing approximate Gaussian states and utilizing squeezing operations to substitute extensive qubit networks with fewer qubits and harmonic oscillators.
• Declarative Physical Operations: By enumerating a detailed set of operations such as linear optics gates and qubit-boson interactions, the paper delineates an achievable path towards implementing a sophisticated quantum algorithm within modern experimental quantum optics setups.

Implications and Future Research

The feasibility of using a fixed-size quantum system for factoring opens new avenues for quantum algorithm design, prioritizing native physical operations over qubit scalability. This approach may stimulate further investigation into non-qubit-centric quantum computations, potentially leading to more resource-efficient quantum information processing.

• Complexity-Theoretic Insights: The reduction of hardware requirements posits intriguing questions in quantum complexity theory, notably whether traditionally hard problems could be redefined under this new hybrid paradigm.
• Experimental Implementation: While theoretically sound, practical realization mandates addressing challenges such as preparing highly accurate GKP states and ensuring stability in bosonic operations, ongoing foci in experimental quantum physics.
• Cross-Platform Applications: The conceptual framework presented could foreseeably extend beyond quantum computing to encompass other fields such as quantum communication or metrology where hybrid systems may afford significant advantages.

In conclusion, the paper signifies a pivotal step in quantum algorithmic design, advocating for a nuanced perspective that marries computational efficiency with physical realism, and catalyzing future explorations into CV-quantum systems as viable contenders for quantum computational supremacy.