Experimental prime factorization via the feedback quantum control

TL;DR

The paper introduces FALQON, a measurement-based feedback quantum control method that directs a quantum system toward the ground state of a problem Hamiltonian $H_p$ encoding factorization, without requiring classical optimization of drive parameters. It demonstrates an all-quantum, DAQC-enabled factoring of the biprime $n=551$ on a three-qubit NMR register and provides numerical evidence of robustness to control-field errors and scalability to larger biprimes using 5 and 9 qubits with truncated Hamiltonians. Compared to adiabatic and QAOA approaches, FALQON offers faster convergence and inherent correction to control imperfections, enabling scalable, all-quantum factorization workflows. The results support potential applications to broader quantum information tasks and suggest practical truncation strategies for large-scale factoring.

Abstract

Prime factorization on quantum processors is typically implemented either via circuit-based approaches such as Shor's algorithm or through Hamiltonian optimization methods based on adiabatic, annealing, or variational techniques. While Shor's algorithm demands high-fidelity quantum gates, Hamiltonian optimization schemes, with prime factors encoded as degenerate ground states of a problem Hamiltonian, generally require substantial classical post-processing to determine control parameters. We propose an all-quantum, measurement-based feedback approach that iteratively steers a quantum system toward the target ground state, eliminating the need for classical computation of drive parameters once the problem Hamiltonian is determined and realized. As a proof of principle, we experimentally factor the biprime 551 using a three-qubit NMR quantum register and numerically analyze the robustness of the method against control field-errors. We further demonstrate scalability by numerically implementing the FALQON factorization of larger biprimes, 9,167 and 2,106,287, using 5 and 9 qubits, respectively.

Figures (5)

• Figure 1: (a) Molecular structure of 1,1,2-trifluoro-2-iodoethene and table of NMR parameters, namely resonance offsets $\nu_i$ (diagonal elements), and scalar couplings $J_{ij}$ (off-diagonal elements), and the relaxation times $T_1$ and $T_2^*$. (b) The FALQON experimental scheme. (c) DAQC implementation of $\exp(-i~H_p~0.2)$. Here, durations $t_1$ to $t_8$ are 1.72, 1.2, 0.82, 1.54, 0.8, 0.88, 0.62, 1.14 ms. (d) Measuring the commutator observable $C$ by a set of three diagonal state tomography experiments and thereby evaluating the drive parameter $\beta_j$. (e) Measuring the energy $E_j$ using the diagonal state tomography. (f) Experimentally obtained drive parameter $\beta_j$ (in degrees) plotted versus $j$. (g) Experimentally measured energy $E_j$ and the solution-space probability $p_\mathrm{sol}$. (h) Heatmap of probabilities of basis states (bit-reversed and padded with bit-1 in the least and most significant places). Note that the states corresponding to the factors 19 and 29 are clearly more populated than the other states.
• Figure 2: Energy $E_j$ plotted versus the iteration number $j$ and control errors $(\delta \theta, \delta\phi)$ for three optimization algorithms as indicated.
• Figure 3: Colormap of the minimum energy reached over 100 FALQON iterations versus RF amplitude $\nu_1$ and the RF inhomogeneity $\Delta \nu_1/\nu_1$ using (a) $H_p$ realized by DAQC, (b) $H_p$ realized by GRAPE, and (c) theoretical $H_p$.
• Figure 4: Ten independent trajectories of probability of solutions $p_\mathrm{sol}$ versus FALQON iteration number $j$ for different parameters $(c,\delta \theta)$ controlling the step size and control-field noise. The energy $E_j$ is indicated by the filling colors of the circles.
• Figure 5: (a-f) Numerical The progression of drive parameter $\beta_j$, energy $E_j$, solution-space probability $p_{\mathrm{sol}_j}$, and final probabilities for the FALQON factoring of 9,167 with full (a-c) and truncated (d-f) Hamiltonians. (g-l) Similar progressions and final probabilities for the FALQON factoring of 2,106,287 with full (g-i) and truncated (j-l) Hamiltonians.