Towards solving industrial integer linear programs with Decoded Quantum Interferometry

TL;DR

This work develops an end-to-end pipeline that transforms industrial 0-1 ILPs into max-XORSAT instances, then solves them via Decoded Quantum Interferometry (DQI) with a coherent binary belief-propagation decoder integrated into the quantum circuit. The authors provide a full gate-level BP1 circuit, embed it in the DQI flow, and apply it to an automotive option-pack pricing problem, comparing performance against Gurobi and random baselines. They offer detailed resource estimates and small-scale circuit simulations, revealing favorable quantum-resource scaling but current classical solvers outperforming DQI at tested sizes, while highlighting a potential crossover as problem scales grow. The study also documents a low-distance LDPC code construction (d=3) arising from the carry-based encodings and discusses paths to improved distance and decoder designs for future quantum advantage in industrial optimization.

Abstract

Optimization via decoded quantum interferometry (DQI) has recently gained a great deal of attention as a promising avenue for solving optimization problems using quantum computers. In this paper, we apply DQI to an industrial optimization problem in the automotive industry: the vehicle option-package pricing problem. Our main contributions are 1) formulating the industrial problem as an integer linear program (ILP), 2) converting the ILP into instances of max-XORSAT, and 3) developing a detailed quantum circuit implementation for belief propagation, a heuristic algorithm for decoding LDPC codes. Thus, we provide a full implementation of the DQI algorithm using Belief Propagation, which can be applied to any industrially relevant ILP by first transforming it into a max-XORSAT instance. We also evaluate the effectiveness of our implementation by benchmarking it against both Gurobi and a random sampling baseline.

Figures (16)

• Figure 1: Success rate of our implementation of the BP1 decoder for the zero code-word for $\ell$ randomly generated bit flips as a function of $\ell$ and the problem size, i.e, the size of $B$. We use $10^4$ sampled code words for each value of $\ell$ and $B$. We set $T=5$. The success rate of the algorithm rapidly decays as we require it to decode larger errors.
• Figure 3: First iteration of the quantum‐circuit implementation of the Binary Belief Propagation algorithm, BP1, for the $m=3, n=2$ example defined by the parity‐check matrix in Eq. \ref{['eq:matrix small example']}. Here, $C$ denotes Qiskit’s IntegerComparator subroutine and $H$ denotes the Hamming‐weight subroutine shown in Fig. \ref{['fig: hamming circuit']}. At the right of the barrier, we find the gates corresponding to the subroutine updating the syndrome.
• Figure 4: Second iteration of the quantum‐circuit implementation of the Binary Belief Propagation algorithm, BP1, for the $m=3, n=2$ example defined by the parity‐check matrix in Eq. \ref{['eq:matrix small example']}. Here, $C$ denotes Qiskit’s IntegerComparator subroutine and $H$ denotes the Hamming‐weight subroutine shown in Fig. \ref{['fig: hamming circuit']}.
• Figure 5: Final step of the quantum‐circuit implementation of the BP1 algorithm for the $m=3, n=2$ example defined by the parity‐check matrix in Eq. \ref{['eq:matrix small example']}. The label Inv indicates the inverse of the entire circuit—excluding the final CNOT gates controlled by the flip registers (fi) on the message (y) register.
• Figure 6: Schematic for building an encoding for integer addition (an "integer adder circuit") using CARRY and CARRY1 gadgets, using addition of two 3-bit integers $x=x_0x_1x_2$ and $y=y_0y_1y_2$ as an example.