⚛️ quantum physics

Experimental prime factorization via the feedback quantum control

This paper presents an all-quantum, measurement-based feedback approach for prime factorization that eliminates the need for classical parameter optimization, demonstrating its feasibility by experimentally factoring 551 on a three-qubit NMR processor and numerically scaling the method to larger biprimes.

Original authors: K. B. Hari Krishnan, Vishal Varma, T. S. Mahesh

Published 2026-01-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: K. B. Hari Krishnan, Vishal Varma, T. S. Mahesh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, locked safe containing a secret code. The only way to open it is to find two specific keys that, when multiplied together, create the number on the safe's dial. In the world of mathematics, this is called prime factorization. For very large numbers, this is incredibly hard for regular computers to do quickly.

This paper describes a new way to crack these codes using a quantum computer, but with a clever twist: instead of following a rigid, pre-written script, the computer "learns" its way to the answer through a process of trial, error, and feedback.

Here is a simple breakdown of how they did it and what they found:

The Problem: The "Perfect Path" vs. The "Feedback Loop"

Usually, quantum computers try to solve problems like this in two ways:

The Scripted Route (Shor's Algorithm): This is like trying to walk a tightrope. You need perfect balance and incredibly precise steps. If you wobble even a tiny bit (due to noise or errors), you fall. It requires extremely high-quality equipment that we don't fully have yet.
The Slow Crawl (Adiabatic/Annealing): This is like slowly melting a block of ice to find a hidden gem inside. It's more forgiving, but it requires a lot of heavy lifting by a regular computer to figure out the melting schedule beforehand.

The New Approach (FALQON):
The authors propose a "feedback loop" method called FALQON. Think of this like steering a car in the dark.

You don't need a perfect map of the road ahead.
Instead, you drive a little bit, check your position (measure the system), and then adjust the steering wheel based on what you just felt.
If you drift left, you steer right. If you drift right, you steer left.
By constantly checking and adjusting, the car naturally steers itself toward the destination (the correct factors) without needing a pre-calculated map.

The Experiment: Factoring 551

To prove this works, the team used a small quantum computer made of three tiny magnets (specifically, three fluorine atoms in a liquid molecule) inside a machine called an NMR spectrometer.

The Target: They wanted to find the two prime numbers that multiply to make 551. (The answer is 19 and 29).
The Process: They started with the atoms in a random, "warm" state (like a cup of coffee sitting on a table). They didn't need to cool them down to absolute zero or prepare them perfectly.
The Loop:
1. They applied a "push" (a control signal) to the atoms.
2. They measured where the atoms were.
3. Based on that measurement, they calculated the next push needed to get closer to the answer.
4. They repeated this over and over.

The Result:
After about 22 rounds of this "push-measure-adjust" cycle, the atoms settled into a state that clearly represented the numbers 19 and 29. The system naturally "found" the factors without needing a supercomputer to plan the steps in advance.

Why This Is Special: It's Tough and Flexible

The paper highlights two major advantages of this method:

It's Resilient (Like a Self-Correcting Compass):
Real-world quantum computers are "noisy." The control signals aren't perfect; they might be slightly too strong or slightly off-angle.
- Analogy: Imagine trying to walk a straight line while someone is gently pushing you from the side. A rigid method would make you stumble. But because FALQON checks its position after every step, it immediately corrects for the push. The paper shows that even with "messy" signals, the method still found the answer.
- They also found that using a specific technique called GRAPE (which designs very robust pulses) made the system even more resistant to these errors, similar to how a shock absorber on a car smooths out a bumpy road.
It Scales Up (The "Big Numbers" Test):
While they only physically tested the number 551, they used computer simulations to see if this would work for much larger numbers.
- They simulated factoring 9,167 (using 5 qubits) and 2,106,287 (using 9 qubits).
- The simulation showed that the method still worked. Interestingly, they discovered that for these larger numbers, they didn't even need the full, complex "map" of the problem. They could use a simplified, "truncated" version of the rules, and the feedback loop still found the correct factors.

The Bottom Line

The researchers successfully demonstrated that you can factor numbers using a quantum computer by letting the system "steer itself" toward the answer through constant measurement and adjustment.

No perfect preparation needed: You can start with a messy, random state.
No pre-calculated map: The computer figures out the next step on the fly.
Error-tolerant: It handles the "noise" of real-world experiments better than other methods.

This suggests a promising path for solving hard math problems on today's imperfect quantum machines, without waiting for the perfect, error-free machines of the future.

Technical Summary: Experimental Prime Factorization via Feedback Quantum Control

Problem Statement
Current quantum prime factorization faces significant hurdles in the Noisy Intermediate-Scale Quantum (NISQ) era. Circuit-based approaches, such as Shor's algorithm, require high-fidelity gates and deep circuits ( $O(l^3)$ gates with extremely low infidelities), limiting demonstrations to very small numbers. Conversely, Hamiltonian optimization methods (adiabatic, annealing, or variational) encode factors as degenerate ground states of a problem Hamiltonian ( $H_p$ ). However, these methods often suffer from slow evolution, strict initial state requirements, and the need for expensive classical post-processing to determine time-dependent control parameters.

Methodology: FALQON
The authors propose and implement FALQON (Feedback-based Algorithm for Quantum Optimization), a measurement-based, all-quantum iterative approach.

Core Mechanism: FALQON utilizes a Lyapunov-inspired iterative scheme. Instead of pre-calculating control fields, the algorithm measures the instantaneous state of the quantum system at each iteration to determine the drive parameters for the subsequent step.
Control Logic: The system evolves under a total Hamiltonian $H_{total} = H_p + \beta(t)H_d$ , where $H_p$ is the problem Hamiltonian and $H_d$ is a drive Hamiltonian. The scalar control parameter $\beta_{j+1}$ for iteration $j+1$ is determined by measuring the expectation value of the commutator observable $C = i[H_p, H_d]$ at iteration $j$ . The update rule is $\beta_{j+1} = c \langle C \rangle_{\rho_j}$ , where $c$ is a scaling factor. This ensures the energy $\langle H_p \rangle$ decreases monotonically (or converges asymptotically) toward the ground state.
Experimental Implementation: The authors implement this on a three-qubit Nuclear Magnetic Resonance (NMR) register using $^{19}$ $^{19}$ F nuclei in 1,1,2-trifluoro-2-iodoethene.
- Hamiltonian Realization: The problem Hamiltonian for the biprime 551 is realized using Digital-Analog Quantum Computing (DAQC). To mitigate sensitivity to pulse imperfections, the authors employ Gradient Ascent Pulse Engineering (GRAPE) to generate robust RF sequences.
- Measurement Efficiency: Rather than full-state tomography, the method utilizes "diagonal tomography" to efficiently measure the commutator observable $\langle C \rangle$ and the system energy, requiring only linear measurements.
- Initial State: Unlike adiabatic methods requiring specific ground states, FALQON starts from the NMR thermal state, converting it to transverse magnetization without elaborate pseudopure state preparation.

Key Contributions

Experimental Demonstration: The paper reports the first experimental factorization of a biprime number (551) using a feedback-based quantum algorithm on an NMR processor.
All-Quantum Control: The work demonstrates a method that eliminates the need for classical computation of drive parameters once $H_p$ is realized, as the feedback loop self-determines the control field.
Robustness Analysis: The authors provide a detailed analysis of the algorithm's resilience against control-field errors, RF inhomogeneities, and random noise, comparing FALQON favorably against adiabatic and QAOA methods.
Scalability Projections: Through numerical simulation, the authors demonstrate the method's scalability by factoring larger biprimes (9,167 and 2,106,287) using 5 and 9 qubits, respectively.

Results

Factorization of 551: The experiment successfully factored 551 into its prime components, 19 and 29. After 22 iterations, the system showed a clear concentration of probability in the basis states corresponding to these factors (after bit-reversal and padding). The energy decreased and saturated, while the solution-space probability rose from ~0.2 to ~0.4.
Robustness:
- Noise: FALQON demonstrated faster convergence than the adiabatic method and less oscillation than QAOA under random control errors.
- Pulse Imperfections: The study confirmed that while DAQC realization of $H_p$ is sensitive to $\pi$ -pulse imperfections, the GRAPE-optimized implementation is highly robust against RF inhomogeneity and limited amplitude.
- Step Size: The analysis revealed a trade-off: smaller step sizes ( $c=0.25$ ) offer robustness against noise, while larger step sizes ( $c=0.75$ ) increase sensitivity to noise, highlighting the importance of parameter selection.
Truncation: Numerical results for 9,167 and 2,106,287 showed that "truncated" problem Hamiltonians (containing fewer terms than the full expansion) retained the same degenerate ground states, allowing successful factorization with reduced complexity.

Significance and Claims
The paper positions FALQON as a viable alternative for NISQ-era factorization that avoids the gate-depth constraints of Shor's algorithm and the classical overhead of variational methods. The authors claim that the method is "all-quantum" in its dynamic control, as it relies solely on experimental measurements to steer the system. They emphasize that the approach is resilient to experimental imperfections and does not require specific initial state preparation, making it suitable for current NMR and potentially other architectures. The work concludes that while the experimental demonstration is limited to small numbers, the numerical scalability and robustness suggest a promising pathway for factoring larger integers, provided efficient Hamiltonian realization and systematic truncation methods are developed.

The Problem: The "Perfect Path" vs. The "Feedback Loop"

The Experiment: Factoring 551

Why This Is Special: It's Tough and Flexible

The Bottom Line

Technical Summary: Experimental Prime Factorization via Feedback Quantum Control

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