Integer Programming Using A Single Atom

TL;DR

The paper addresses the difficulty of solving integer programming (IP), which is NP-hard, by proposing a direct quantum encoding that uses the superposition principle on a single multi-level atom, avoiding the traditional binary-encoding QUBO route. The method maps each integer variable to a manifold of internal states and encodes problem constraints through time-dependent internal and external couplings, with coefficient optimization performed via quantum optimal control and solution read out from population measurements. It demonstrates microsecond-scale solution times for prototypical ILP and NLIP problems (up to eight variables and four constraints) and benchmarks against classical branch-and-bound, highlighting potential advantages and a clear path to hybrid quantum–classical strategies for larger instances. The work suggests scalable extensions with more atoms or decomposition techniques (e.g., Benders) to tackle industrial-sized IPs and positions this non-QUBO approach as a complementary route to quantum optimization that may outperform classical solvers on certain non-linear IP instances.

Abstract

Integer programming (IP), as the name suggests is an integer-variable-based approach commonly used to formulate real-world optimization problems with constraints. Currently, quantum algorithms reformulate the IP into an unconstrained form through the use of binary variables, which is an indirect and resource-consuming way of solving it. We develop an algorithm that maps and solves an IP problem in its original form to any quantum system possessing a large number of accessible internal degrees of freedom that are controlled with sufficient accuracy. This work leverages the principle of superposition to solve the optimization problem. Using a single Rydberg atom as an example, we associate the integer values to electronic states belonging to different manifolds and implement a selective superposition of different states to solve the full IP problem. The optimal solution is found within a few microseconds for prototypical IP problems with up to eight variables and four constraints. This also includes non-linear IP problems, which are usually harder to solve with classical algorithms when compared to their linear counterparts. Our algorithm for solving IP is benchmarked by a well-known classical algorithm (branch and bound) in terms of the number of steps needed for convergence to the solution. This approach carries the potential to improve the solutions obtained for larger-size problems using hybrid quantum-classical algorithms.

Figures (7)

• Figure 1: The schematic plot represents a typical integer programming problem consisting of two variables $x_1$ and $x_2$ with three constraints ($C_1$, $C_2$, and $C_3$). The cost function $C_f$ is maximized to solve the problem while satisfying the constraints. The dotted region depicts the allowed solution space (feasible region)
• Figure 2: The Figure summarizes the algorithm for solving IP problems using a multi-level system. (a) As explained in the text, the three manifolds correspond to three individual integer variables. The external couplings $\tilde{\Omega}$ are between the manifolds $M=1,2,3$ while the internal couplings $\Omega$ are between the states ($\ket{\psi_{i0}},\ket{\psi_{i1}},\ket{\psi_{i2}}$) residing in a particular manifold $M=i$. The value that each variable can take is associated with the population of the dominant internal level belonging to a manifold. (b) Schematically shows the coupling Hamiltonians $\hat{H_1}(\tau_1)$ and $\hat{H_2}(\tau_2)$ used to evolve an initial state, used for describing constraints as shown in Equations (\ref{['H1eq']}-\ref{['H2eq']}). The height of the shaded rectangle indicates the strength of the coupling while the width is the evolution time, both of which are varied in the optimization procedure. During the optimization, the coupling Hamiltonians are applied repeatedly in layers indicated by $L$. (c) Shows the population transfer from one state to another during the first layer of the protocol in time $\tau_1+\tau_2$. $P_{11}$ and $P_{32}$ correspond to the population of the states $\psi_{11}$ and $\psi_{32}$ (shown as states with thick blue lines in panel a) respectively, assigning the values to the variables as $x_1=1$ and $x_3=2$.
• Figure 3: The table in (a) shows the chosen sample integer programming problems that are encoded and solved using multi-levels of a single Rydberg atom. The problems with varied complexity (as explained in Sub-section \ref{['Complexity']}) depend on the problem type and the number of variables. The corresponding benchmark metrics kronqvist2019review are shown in (b-d) panels, where the black dot indicates the complexity of the problems $P_1-P_4$ as defined in panel (a).
• Figure 4: Integer linear programming problem: maximize $C_f(\mathbf{x}) = 3x_1 + 2x_2 + x_3$ where constraints are $x_i \in \{0,1,2\}$, $C_1:2x_1+x_2 \leq 3$ and $C_2: x_2 + x_3 \leq 2$. $C_f=6$ is the true solution (global maxima in this case) to the problem calculated by considering all the possible values of the variables. (a) Depicts the value of the cost function of the problem with the shaded region (both in purple and green with slanted lines) corresponding to the time intervals when all the inequalities are satisfied simultaneously. Specifically, the shaded green-slanted-line region represents the time intervals for which the optimal value (maxima) of the cost function is reached while satisfying the constraints. Panels (b-c) show the implementation of the two constraints of the problem, both constraints are implemented using coupling Hamiltonians given by Equations (\ref{['H1eq']}-\ref{['H2eq']}). The shaded regions in (b) and (c) correspond to the time intervals when the constraint inequalities $C_1,C_2$ are individually satisfied respectively. The two vertical blue-dashed lines mark the time interval during which the population of the levels in (d-f) for different manifolds are shown. The state with the highest population assigns the value to the variables which are then used to calculate $C_f, C_1$, and $C_2$.
• Figure 5: Non-linear integer programming problem: maximize $C_f(\mathbf{x}) = 2x_2x_3$ where constraints are $x_i \in \{0,1,2\}$, $C_1:x_1x_2 + 2x_3 \leq 2$ and $C_2: x_2 + x_1x_3 \leq 4$. $C_f=4$ is the true solution (global maxima) to the problem found by considering all the possible values of the variables. In (a) the value of the non-linear cost function of the problem is shown with the shaded region (including the ones with slanted lines) corresponding to the time interval when both inequalities are satisfied simultaneously. Panels (b-c) show the implementation of the constraints of the problem. The shaded regions in (b-c) correspond to the time intervals when the constraint inequalities are satisfied. The population of the levels for different manifolds as shown in (d-f) is during the time interval marked by the two vertical dashed-blue lines.