Factoring with Fewer Qubits: A Step Closer to Quantum Reality

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Author: Denis Avetisyan

Researchers have significantly reduced the qubit requirements for Regev’s quantum factoring algorithm, bringing practical quantum factoring a step closer to realization on near-term devices.

This work demonstrates a space-optimized implementation of Regev’s algorithm using intermediate uncomputation and validates its feasibility on noisy intermediate-scale quantum (NISQ) hardware.

While Shor’s algorithm promises efficient quantum factorization, practical implementations are hindered by substantial qubit requirements, particularly in lattice-based approaches like Regev’s algorithm. This work, ‘Space-Optimized and Experimental Implementations of Regev’s Quantum Factoring Algorithm’, introduces a qubit reuse strategy via intermediate uncomputation to significantly reduce the space complexity of Regev’s method, achieving a space scaling of \(O(n \log n)\). Through simulations and experimental execution on a superconducting quantum computer-successfully factoring \(N=35\) with lattice-based post-processing-we demonstrate the feasibility of this optimized approach on near-term noisy intermediate-scale quantum (NISQ) devices. Will these advancements pave the way for practical quantum cryptanalysis and the development of post-quantum cryptographic solutions?

The Foundation of Digital Security: An Intractable Challenge

The foundation of much modern digital security, most notably the RSA encryption standard, rests upon a surprisingly simple mathematical challenge: integer factorization. This involves decomposing a large integer into its prime number constituents – a task that becomes computationally intractable as the number’s size increases. While multiplying two large primes is a swift operation for computers, reversing this process – discovering those original primes given only their product – quickly overwhelms even the most powerful classical algorithms. The difficulty isn’t inherent in the mathematics itself, but rather in the exponential scaling of computational effort required; a number with hundreds of digits poses a factorization problem that could take billions of years to solve with conventional methods, thus safeguarding sensitive data transmitted online. This reliance on computational hardness means that advances in factorization algorithms, or the emergence of fundamentally different computational paradigms, directly threaten the security of a vast proportion of digital communications and commerce.

The bedrock of many digital security systems rests on the computational difficulty of breaking down large numbers into their prime factors. However, classical algorithms employed for this task – trial division, the quadratic sieve, and the general number field sieve – suffer from a critical limitation: their runtime increases exponentially with the size of the integer being factored. This means that doubling the number of digits in the number to be factored doesn’t simply double the computation time, but increases it by a factor of several orders of magnitude. Consequently, as computing power steadily advances – driven by Moore’s Law and innovations in hardware architecture – these algorithms become increasingly vulnerable. A number considered secure today, requiring years to factor with current technology, could become easily broken with future advancements, highlighting the ongoing need for more robust cryptographic methods, such as those based on post-quantum cryptography, to counteract this inherent weakness in classical factorization approaches.

Although a theoretical breakthrough, Shor’s algorithm, capable of factoring large numbers in polynomial time, remains largely impractical with current technology. This is because the algorithm’s execution necessitates a fault-tolerant quantum computer – a machine leveraging the principles of quantum mechanics to perform computations – and building such a device presents immense engineering challenges. Maintaining the delicate quantum states – known as qubits – required for computation is extraordinarily difficult, as even minor environmental disturbances can introduce errors. Correcting these errors requires a substantial overhead in qubits and complex error-correction protocols. Current quantum computers are limited in qubit count and coherence time, and scaling them to the size and stability needed to break modern encryption – factoring numbers with hundreds or thousands of digits – is projected to require decades of further research and development in areas like qubit materials, control systems, and quantum error correction codes. Consequently, while a potential future threat, the widespread implementation of Shor’s algorithm remains distant, affording classical cryptographic systems a continued, albeit diminishing, window of security.

Lattice-Based Cryptography: A Quantum-Resistant Pathway

Regev’s algorithm presents a potential route to integer factorization on Near-term Intermediate-Scale Quantum (NISQ) devices through the application of lattice-based cryptography and a shift from one-dimensional order finding to a high-dimensional equivalent. Unlike Shor’s algorithm which relies on finding the period of a function in one dimension, Regev’s approach formulates the factorization problem as a search for a short vector within a high-dimensional lattice. This transformation is significant because the structure of these lattices allows for a more efficient implementation on current quantum hardware, despite the increased complexity of the order-finding task. The security of the algorithm rests on the presumed hardness of lattice problems, specifically the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP), which are believed to be resistant to known classical algorithms.

Shor’s algorithm relies on the quantum Fourier transform to efficiently find the period, or order, of a function in one dimension. Regev’s algorithm substitutes this with a high-dimensional generalization of order finding, operating on lattices. This modification is critical for implementation on Noisy Intermediate-Scale Quantum (NISQ) devices, as the higher dimensionality distributes the computational complexity in a way that is more resilient to the limitations of current quantum hardware. Specifically, the increased dimensionality allows for the use of shorter quantum circuits with fewer qubits while still maintaining the potential for a speedup over classical factorization algorithms, though at the cost of increased classical post-processing requirements.

Regev’s algorithm utilizes lattice-based cryptography for a post-processing step essential for extracting factorization results from the quantum computation. Following the quantum phase estimation and high-dimensional order finding, the algorithm outputs a value related to the lattice. This value is not the factorization itself; instead, it’s used as input to a classical lattice reduction algorithm, such as the LLL algorithm or BKZ algorithm. These algorithms efficiently solve the shortest vector problem on the lattice, revealing the factors of the original number. This classical post-processing is critical because it translates the quantum output, which is a probability distribution, into a deterministic factorization, effectively bridging the quantum and classical components of the solution.

Optimizing Quantum Resources: A Matter of Efficiency

Regev’s encryption algorithm, while offering strong security guarantees, presents a substantial challenge in practical implementation due to its inherent space complexity. This complexity directly correlates to the number of qubits required to perform the computation; specifically, the algorithm necessitates storage proportional to the problem size, n. The initial space requirement of $O(\sqrt{n})$ qubits poses a significant obstacle for scaling the algorithm to handle larger key sizes and more complex computations, limiting its feasibility on current and near-term quantum hardware. Addressing this space complexity is therefore crucial for enabling practical applications of Regev encryption.

Intermediate uncomputation addresses the high space complexity of algorithms like Regev’s by reusing qubits during computation. Traditional implementations require storing intermediate results throughout the entire process, leading to a space complexity of $O(n^{3/2})$, where ‘n’ represents the input size. Intermediate uncomputation, inspired by the reversible pebble game, strategically discards these intermediate values once they are no longer needed for subsequent calculations. This allows the qubits to be repurposed, effectively reducing the overall space complexity to $O(n log n)$. Simulations have demonstrated this improvement, showing a reduction in qubit requirements from 20 to 17, indicating a significant gain in space efficiency.

Intermediate uncomputation, a technique for reducing qubit requirements, draws inspiration from the reversible pebble game. This method operates on the principle of selectively discarding intermediate computational results once they are no longer necessary for subsequent operations. Rather than storing these results for potential future use, the computation is reversed – effectively “uncomputing” them – which frees the associated qubits for reuse. This contrasts with traditional computation where intermediate values are typically retained, leading to increased space complexity. By strategically applying this reversible process, the overall number of qubits required to perform a given computation is significantly reduced, improving the scalability of algorithms like Regev’s.

Simulations of Regev’s algorithm incorporating intermediate uncomputation have demonstrated a reduction in required qubits. Initial implementations necessitated 20 qubits to perform the computation; however, the application of intermediate uncomputation reduced this requirement to 17 qubits. This represents a quantifiable improvement in space efficiency and suggests the potential for further optimization through refinement of the uncomputation strategy. The reduction, while observed in a specific simulation, validates the theoretical benefits of reusing qubits during computation and provides a benchmark for assessing the effectiveness of similar space-reduction techniques.

A Hybrid Future: Bridging Classical and Quantum Computation

The substantial resource demands of lattice-based cryptography, particularly algorithms like Regev’s encryption, present a significant hurdle for practical implementation on near-term quantum computers. Researchers are actively exploring hybrid quantum-classical approaches to mitigate this challenge, with promising results stemming from the integration of semi-classical Fourier transforms. By strategically offloading the computationally intensive Fourier transform – a key step in manipulating the Gaussian distributions central to Regev’s algorithm – to conventional processors, the burden on the quantum hardware is considerably lessened. This division of labor allows for a reduction in the number of qubits and quantum gates required, making the algorithm more amenable to execution on noisy intermediate-scale quantum (NISQ) devices. Essentially, this hybrid strategy leverages the strengths of both computational paradigms – the quantum computer’s ability to efficiently process certain tasks and the classical computer’s capacity for handling complex, yet less quantum-sensitive, calculations – to achieve a more feasible pathway toward post-quantum cryptographic solutions.

A significant challenge in realizing practical quantum computation lies in the substantial demands placed on quantum hardware. Recent advancements address this by strategically partitioning computational tasks between quantum and classical processors. Specifically, complex calculations that do not necessarily require quantum mechanical properties are delegated to conventional computers, thereby lessening the burden on the limited qubits of a quantum system. This hybrid approach not only reduces the required number of qubits-and thus the complexity of the hardware-but also allows for the implementation of more sophisticated algorithms on currently available noisy intermediate-scale quantum (NISQ) devices. By optimizing this division of labor, researchers are making considerable progress toward realizing the potential of quantum computation within the constraints of present-day technology, particularly in fields like cryptography where enhanced computational power is critical.

The quantum Fourier transform (QFT) continues to serve as a cornerstone within Regev’s algorithm and similar quantum computations, enabling the efficient processing of quantum states essential for factorization and cryptographic applications. This transformation effectively converts a quantum state representing a number into a state representing its frequency components, allowing for the identification of periodic patterns critical to solving mathematical problems intractable for classical computers. While hybrid approaches aim to reduce the overall quantum resource demands, the QFT’s ability to efficiently manipulate superpositions and perform complex calculations on these states remains paramount. Its continued use underscores the fundamental role of quantum signal processing in realizing the potential of quantum algorithms, even as classical-quantum collaborations refine their computational strategies and pave the way for practical post-quantum cryptography.

A functional implementation of an optimized version of Regev’s algorithm has been successfully demonstrated on a superconducting quantum computer utilizing nine qubits. This achievement represents a significant step towards practical quantum factorization, a capability with profound implications for modern cryptography. By realizing this complex algorithm on existing near-term intermediate-scale quantum (NISQ) hardware, researchers have showcased a viable path for breaking widely used encryption schemes based on the difficulty of factoring large numbers. The successful execution not only validates the theoretical underpinnings of the optimized algorithm but also contributes directly to the advancement of post-quantum cryptography, a field dedicated to developing encryption methods resistant to attacks from both classical and quantum computers. This work suggests that, despite the limitations of current quantum technology, the threat to existing cryptographic infrastructure is becoming increasingly tangible, spurring further development in quantum-resistant security measures.

The pursuit of space optimization within Regev’s algorithm, as detailed in the study, echoes a fundamental principle of efficient systems. If a system survives on duct tape – excessive qubit allocation to compensate for unoptimized intermediate calculations – it’s probably overengineered. The authors’ implementation of intermediate uncomputation effectively addresses this, streamlining the process and reducing resource demands. This mirrors the sentiment expressed by Albert Einstein: “It does not require many smart people to accomplish little.” The reduction in qubit requirements, achieved through careful architectural consideration, demonstrates that impactful results often stem not from brute force complexity, but from elegant simplicity and a deep understanding of the underlying structure-a concept central to the study’s success and consistent with Einstein’s observation.

Further Horizons

The reduction in qubit overhead achieved through intermediate uncomputation is not merely an engineering convenience; it exposes a fundamental truth regarding quantum algorithms. Documentation captures structure, but behavior emerges through interaction. This work demonstrates that a focus on resource minimization can, paradoxically, reveal deeper algorithmic principles – a lesson applicable far beyond the specific context of Regev’s factoring scheme. The persistent challenge remains, however, that polynomial time complexity on paper does not equate to practical advantage. The structure of NISQ devices dictates behavior, and current architectures impose constraints that extend beyond qubit counts.

Future investigations must address the interplay between algorithmic optimization and hardware limitations with equal rigor. While space optimization unlocks the possibility of implementing more complex lattice-based cryptography on near-term quantum computers, the precision required for the Quantum Fourier Transform – a notoriously sensitive operation – will likely become the dominant bottleneck. A truly elegant solution will not simply fit the algorithm onto the hardware, but actively reshape it to harmonize with the inherent noise and connectivity constraints.

The ultimate test will lie in demonstrating a quantifiable advantage over classical factoring algorithms, even with imperfect quantum execution. This requires not only further refinement of the algorithm itself, but also a shift in perspective – away from pursuing ever-larger quantum computers and toward designing algorithms that are intrinsically robust to the realities of imperfect quantum computation.

Original article: https://arxiv.org/pdf/2511.18198.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-11-25 08:00