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IEEE Transactions on Computers

RSQC: Recursive Sparse QUBO Construction for Quantum Annealing Machines

Jianwen Luo, Yuhao Shu, Yajun Ha

Published Apr 4, 2025• 2 citations
Publisher

Abstract

Quantum annealing algorithms have shown commercial potential in solving some instances of combinatorial optimization problems. However, existing mapping for general optimization problems into a compatible format for quantum annealing yields dense topology and complicated weighting, which limits the size of solvable problems on practical quantum annealing machines. To address this issue, we propose a novel mapping framework with three new techniques. First, to address the issue from general constraints, we introduce a recursive methodology to map constraints into interconnected Boolean gates and small algebraic cliques, which yields sparse topology and hardware-friendly biases/interactions. Second, to better address frequently-used constraints, we introduce a specialized penalty set based on this methodology with detailed optimizations. Third, to address the issue from the objective, we reformulate the complicated objective into a single multi-bit variable and apply binary search to its range, which turns each search step into a constraint-only problem. Compared with the state-of-the-art, experimental results and analysis over an exhaustive scan for operand bit-widths from 1 to 64 show that: (1) the growth order of the number of physical qubits with regard to operand bit-widths is reduced fromO(w2) toO(w), while the number is reduced by a factor of 10-1 in the best case; (2) the dynamic range of biases/interactions is reduced fromO(22w) to−2in the best case. For the same optimization problem, our framework reduces the requirement of the number of physical qubits and machine precision, and shortens the time from problem to machine.

DOI
10.1109/tc.2025.3557965

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To date, conventional computers have never been able to efficiently handle certain tasks, where the number of computation steps is likely to blow up as the problem size increases. As an emerging technology and new computing paradigm, quantum computing has a great potential to tackle those hard tasks efficiently. Among all the existing quantum computation models, quantum annealing has drawn significant attention in recent years due to the realization of the commercialized quantum annealer, sparking research interests in developing applications to solve problems that are intractable for classical computers.However, designing and implementing algorithms that manage to harness the enormous computation power from quantum annealer remains a challenging task. Generally, it requires mapping of the given optimization problem into quadratic unconstrained binary optimization(QUBO) problem and embedding the subsequent QUBO onto the physical architecture of quantum annealer. Additionally, practical quantum annealers are susceptible to errors leading to low probability of the correct solution.In this study, we focus on solving Boolean satisfiability (SAT) problem using quantum annealer while addressing practical limitations. We have proposed a mapping technique that maps SAT problem to QUBO, and we have further devised a tool flow that embeds the QUBO onto the architecture of a quantum annealing device. Additionally, We have optimized the proposed embedding flow to reduce run-time in addition to shortening the qubit chain length, leading to robust quantum annealing. To further improve the reliability of quantum annealing, we have also developed a post processing embedding technique that enlarges the energy gap between ground state and the first excited state. To demonstrate the effectiveness of proposed methods, we have conducted experiments on real quantum annealing devices manufactured by D-Wave Systems, showing compelling result of using quantum annealer to solve SAT problem.
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This paper discusses techniques for solving discrete optimization problems using quantum annealing. Practical issues likely to affect the computation include precision limitations, finite temperature, bounded energy range, sparse connectivity, and small numbers of qubits. To address these concerns we propose a way of finding energy representations with large classical gaps between ground and first excited states, efficient algorithms for mapping non-compatible Ising models into the hardware, and the use of decomposition methods for problems that are too large to fit in hardware. We validate the approach by describing experiments with D-Wave quantum hardware for low density parity check decoding with up to 1000 variables.
Factorization of large semiprimes remains one of the most challenging problems for classical computers. Shor’s algorithm offers a quantum approach that reduces computational complexity, but its practical application is currently limited by hardware constraints. Meanwhile, as a provisional approach, quantum annealing (QA) has been explored through formulations of the quadratic unconstrained binary optimization (QUBO) problem. Among existing methods, the blockwise partial-product approach effectively reduced the QUBO variable count but was limited to semiprimes up to 21 bits. To extend factorization to larger semiprimes, this paper addresses key engineering challenges in constructing efficient QUBO formulations for prime factorization. We propose five techniques to reduce variable counts and improve scalability with current QA hardware: (1) dividing the problem into subproblems with partial constraints; (2) applying analytical reductions near the LSB; (3) applying analytical reductions near the MSB; (4) exploiting special patterns in semiprimes, with odd bit widths and long MSB-side zero sequences; and (5) balancing variable usage across both sides of the subproblem. Integrated into a QUBO converter, these methods enable stable factorization of semiprimes up to 47-bits within 20 seconds and can extend to special 2049-bit instances with 1001 consecutive MSB-side zeros.
Quantum Annealing (QA) is a quantum computing paradigm for solving combinatorial optimization problems formulated as Quadratic Unconstrained Binary Optimization (QUBO) problems. An essential step in QA is minor embedding, which maps the problem graph onto the sparse topology of the quantum processor and then adjusts the problem weights. The process of mapping the problem variables to the hardware is computationally expensive and scales poorly with increasing problem size and hardware complexity. Existing heuristics are often developed for specific problem graphs or hardware topologies and are difficult to generalize. To address this limitation, we explore the use of machine learning methods, which would allow a much greater degree of flexibility, in particular of Reinforcement Learning (RL). RL offers a promising alternative by treating minor embedding as a sequential decision-making problem, where an agent learns to construct minor embeddings by iteratively mapping the problem variables to the hardware qubits. We propose a RL-based approach to minor embedding using a Proximal Policy Optimization agent, testing its ability to embed both fully connected and randomly generated problem graphs on two hardware topologies, Chimera and Zephyr. The results show that our agent consistently produces valid minor embeddings even when they span over more than a thousand qubits, in particular on the more modern Zephyr topology. Our proposed approach is also able to scale to moderate problem sizes and adapts well to different graph structures, highlighting RL’s potential as a flexible and general-purpose framework for minor embedding in QA but also pointing to limitations that will need to be addressed, for example in reducing the number of qubits required.
Abstract It has been demonstrated that Binary Integer Programming (BIP) formulations can be reformulated to a corresponding quadratic unconstrained binary optimization (QUBO) problem. The reformulation allows for BIPs to run as QUBOs on adiabatic quantum annealing hardware. Current BIP to QUBO reformulation techniques propagate dense QUBO structures that are not ideal for current hardware where quantum bits, and the connections between them, are scarce. We propose a methodology for BIP to QUBO reformulation that results in a sparse QUBO that, when used to embed set-partitioning problems onto a D-Wave Pegasus topology, requires 80% fewer qubits and is embedded is 41 times faster. The same methodology for set-covering problems requires 77% fewer qubits and is embedded 26 times faster. We discuss the limitations of the technique and avenues for future extensions.