Researchers Factor 551 Using Novel Feedback Quantum Control with 9 Qubits

Scientists are tackling the notoriously difficult problem of prime factorization with a novel quantum approach. Hari Krishnan KB, Vishal Varma, and T. S. Mahesh, all from the Indian Institute of Science Education and Research, Pune, demonstrate an experimental method utilising feedback quantum control to factor numbers , sidestepping the limitations of both circuit-based algorithms like Shor’s and traditional Hamiltonian optimisation techniques. Their research, detailed in a new paper, presents an all-measurement-based feedback loop that iteratively guides a quantum system towards the solution, crucially removing the need for extensive classical computation of drive parameters after the initial Hamiltonian is set. By successfully factoring the biprime 551 with a three-qubit NMR register and simulating larger factorisations, this work signifies a potentially scalable pathway towards more efficient quantum algorithms for number theory and cryptography.

This breakthrough relies on a Lyapunov-inspired iterative algorithm called FALQON, which leverages measurements of the system’s instantaneous state to determine optimal drive parameters for each subsequent iteration. The team experimentally verified this method by successfully factoring the biprime number 551 using a three-qubit NMR quantum register, alongside numerical analyses assessing the method’s resilience to control field errors.

The study establishes a significant advancement over existing quantum factorization methods, which often demand high-fidelity quantum gates or substantial classical post-processing. While Shor’s algorithm requires an impractical number of gates with extremely low error rates, Hamiltonian optimization schemes necessitate complex calculations to determine the control parameters needed to reach the ground state representing the factors. This new approach, however, sidesteps these challenges by employing a feedback loop that dynamically adjusts the system’s evolution based on real-time measurements. Experiments show that the FALQON algorithm effectively steers the quantum system towards the desired ground state, offering a pathway to scalable prime factorization without relying on computationally intensive classical optimization.

Furthermore, the researchers demonstrated the scalability of their method through numerical simulations, successfully implementing the FALQON factorization of larger biprimes, 9,167 and 2,106,287, using 5 and 9 qubits, respectively. The problem Hamiltonian is constructed to encode the biprime number, with its prime factors represented as degenerate ground states. By minimizing the energy of this Hamiltonian, the algorithm iteratively converges towards the solution, effectively revealing the prime factors. This approach utilizes digital-analog quantum computing (DAQC), employing single-qubit rotations interspersed with natural multi-qubit evolutions like Ising interactions, to efficiently realize the problem Hamiltonian.

The work opens exciting possibilities for quantum computation in the noisy intermediate-scale quantum (NISQ) era, where imperfect state preparations and imprecise gate realizations pose significant hurdles. FALQON’s inherent self-correction against control errors enhances its robustness, making it particularly well-suited for experimental implementation. By eliminating the need for pre-calculated control field profiles, the method simplifies the quantum control process and reduces the computational burden. Recent applications of FALQON include solving graph partitioning problems, estimating eigenvalues using superconducting qubits, and optimizing controlled-phase gates in neutral atoms, highlighting its versatility and potential for broader quantum algorithm development.

Quantum Factoring via Measurement-Based Feedback

Scientists pioneered a novel, all-quantum approach to prime factorization, circumventing the limitations of both circuit-based algorithms like Shor’s and traditional Hamiltonian optimization techniques. This work introduces a measurement-based feedback method that iteratively guides a quantum system towards the ground state representing the prime factors, crucially eliminating the need for extensive classical computation of drive parameters after the problem Hamiltonian is established. The team experimentally demonstrated this method by factoring the biprime 551 using a three-qubit Nuclear Magnetic Resonance (NMR) register, validating the core principles of their approach. To begin, researchers constructed a problem Hamiltonian where the degenerate ground states directly encode the desired prime factors, a key innovation for simplifying the factorization process.

This Hamiltonian was then utilized within the Feedback Algorithm for Lyapunov-inspired Quantum Optimization (FALQON), an iterative process that leverages measurements of the instantaneous quantum state to refine control parameters for each subsequent iteration. The study employed digital-analog quantum computing (DAQC) to efficiently implement this time-independent Hamiltonian, utilizing single-qubit rotations interspersed with natural multi-qubit Ising interactions. Experiments involved precise control and measurement of the three-qubit NMR register, carefully calibrating radiofrequency pulses to enact the necessary single-qubit rotations. The team meticulously designed the measurement protocol to accurately determine the system’s state at each iteration, providing the feedback signal for adjusting the control parameters.

Robustness against control field errors was then assessed through numerical simulations, revealing the algorithm’s resilience to experimental imperfections, a critical step towards practical implementation. Furthermore, the researchers extended their analysis through numerical simulations, successfully implementing FALQON factorization of the larger biprimes 9,167 and 2,106,287 using 5 and 9 qubits respectively, demonstrating the scalability of the method. This innovative methodology achieves iterative convergence to the target ground state, starting from almost any initial state, and relies solely on experimental measurements to determine subsequent drive parameters. The approach enables self-correction against control errors, offering a significant advantage over methods requiring extensive classical pre-computation and demonstrating a pathway towards robust quantum factorization in the noisy intermediate-scale quantum (NISQ) era.

NMR Factorisation of 551 via Quantum Feedback

Scientists achieved the prime factorization of the biprime 551 using a novel, measurement-based feedback approach with a three-qubit Nuclear Magnetic Resonance (NMR) register. This breakthrough eliminates the need for classical optimization of drive parameters, a significant hurdle in traditional Hamiltonian optimization methods for prime factorization. Experiments revealed that the iterative steering of the system towards the target ground state successfully identified the prime factors without pre-calculated drive parameters, demonstrating a fundamentally new pathway for quantum computation. The team measured the commutator observable, ⟨C⟩ρ, using a pulse sequence detailed in Figure 0.1(d) to refine the drive parameters during each iteration.

Results demonstrate a decreasing energy trend, as evidenced by the experimentally obtained drive parameter βj plotted against the iteration number j in Figure 0.1(f). The energy of the state, E(ρ), was calculated as Tr{Hpρ} = X m emρmm, confirming the system’s progression towards the ground state. Data shows that after an initial oscillation, the βj values approached zero, as anticipated by the FALQON scheme, with the scaling factor set to c = 0.25. Furthermore, the probability of the solution space, psolj = ⟨011 |ρj| 011⟩+ ⟨100 |ρj| 100⟩, increased from approximately 0.2 to 0.4, indicating a growing likelihood of finding the correct factors.

Tests prove that the probabilities of basis states, visualized in the heatmap of Figure 0.1(h), concentrated on states corresponding to the prime factors 19 (10011) and 29 (11101). This concentration, maintained over the last 16 iterations, convincingly demonstrates the successful factorization of 551. Researchers numerically implemented the FALQON factorization of larger biprimes, 9,167 using 5 qubits and 2,106,287 using 9 qubits, showcasing the scalability of the method. Measurements confirm that FALQON converges faster than the adiabatic method, particularly when subjected to control field errors (δθ, δφ), as illustrated in Figure 0.2.

The robustness of the method was further analyzed by examining the minimum energy reached over 100 iterations against varying RF amplitude (ν1) and RF inhomogeneity (∆ν1/ν1). Colormaps in Figure 0.3 demonstrate that the GRAPE realization of the problem Hamiltonian (Hp) offers a wider operational area and greater robustness against RF inhomogeneity compared to the DAQC realization. Finally, analysis of ten independent trajectories of the solution space probability, psol, revealed that a smaller step size (c = 0.25) enhances the method’s resilience to random noise, as shown in Figure 0.4. This work delivers a promising new approach to prime factorization with potential applications in cryptography and quantum algorithm development.

FALQON Factoring Achieved with Quantum Optimisation

Scientists have demonstrated a novel quantum prime factorization method called FALQON, an iterative Hamiltonian optimization technique that distinguishes itself from existing approaches like Shor’s algorithm and adiabatic quantum computation. This all-quantum algorithm iteratively steers a system towards the ground state representing the prime factors, crucially eliminating the need for classical computation to determine drive parameters after the initial problem Hamiltonian is established. The researchers experimentally verified this method by factoring the biprime 551 using a three-qubit nuclear magnetic resonance (NMR) register, and further validated its scalability through numerical simulations factoring larger biprimes, 9,167 and 2,106,287, with 5 and 9 qubits respectively. The significance of this work lies in its potential to overcome limitations inherent in current prime factorization methods; unlike adiabatic methods, FALQON exhibits less sensitivity to initial states, allowing operation directly from a thermal state.

Furthermore, numerical analyses revealed that FALQON converges faster than adiabatic methods and displays fewer oscillations compared to the Quantum Approximate Optimization Algorithm (QAOA) under various noise conditions. The authors acknowledge limitations regarding the systematic truncation of problem Hamiltonians, which remains an open research question, and the sensitivity of the problem Hamiltonian realization to imperfections in π pulses, though the GRAPE technique offers robust implementation. Future research directions include exploring the application of FALQON to other quantum information tasks and investigating potential hybridizations with alternative quantum control strategies.