QUBO Instances for Multiple Object Tracking

• Multiple Object Tracking QUBO Instances are formulations that encode tracking assignments using binary variables and quadratic cost functions to enforce constraints.
• They integrate quantum annealing and simulated bifurcation, enabling efficient resolution of combinatorial tracking challenges in real time.
• These frameworks support multi-objective optimization and robust constraint handling, paving the way for scalable, high-accuracy tracking in complex scenarios.

Multiple Object Tracking (MOT) QUBO instances refer to formulations of the multi-object tracking problem—particularly the combinatorial association and assignment subproblems—in the form of Quadratic Unconstrained Binary Optimization (QUBO). This approach is motivated by the need to solve tracking data association, hypothesis selection, and constraint-enforcing tasks that are fundamentally combinatorial, requiring optimization methods capable of handling large binary decision spaces. QUBO formulations are central to leveraging quantum computing and quantum-inspired hardware (such as Ising machines and simulated bifurcation devices) for real-time, large-scale, and robust MOT. A QUBO instance encodes the MOT decision space via binary variables and quadratic cost functions, automatically mapping the association constraints and objective metrics to a structure amenable to both classical and quantum optimization.

1. Structural Principles of QUBO in Multiple Object Tracking

MOT systems typically operate by estimating object trajectories over time, requiring the assignment of detections (or measurements) to tracks at each frame under constraints of exclusivity and temporal consistency. The core discrete assignment task can be encoded as a QUBO problem:

• Define binary variables $b_{t,d} \in \{0,1\}$ indicating if track $t$ is matched with detection $d$.
• The assignment cost is formulated as a quadratic function, combining match-likelihoods and constraint penalties:

$$H_\mathrm{cost} = H_\mathrm{object} + c \left( H_\mathrm{penalty1} + H_\mathrm{penalty2} \right)$$

where $H_\mathrm{object} = -\sum_{t,d} S_{t,d} b_{t,d}$ maximizes the overall similarity (e.g., via intersection-over-union), and $H_\mathrm{penalty1,2}$ enforce row/column sum constraints for one-to-one or many-to-one assignments.

• These cost functions are quadratic due to cross-variable penalty terms $(\sum_k b_{tk} - 1)^2$.

This QUBO encoding allows expressive representation of assignment constraints, association costs, and allows “softening” of constraints to support flexible matching (e.g., during occlusion) (Tatsumura et al., 18 Oct 2024, Ihara, 27 Mar 2024).

2. QUBO-to-Ising and Quantum Annealing Implementations

QUBO problems are naturally mapped to the Ising model for solution on quantum annealers and Ising machines:

• Binary variables are transformed via $s_i = 2b_i - 1$.
• Ising energy: $$H_{\text{Ising}} = -\frac{1}{2} \sum_{i,j} J_{ij}s_is_j + \sum_i h_i s_i$$.
• The quadratic and linear terms in the QUBO yield the $J_{ij}$ and $h_i$ couplings after expansion.

Quantum annealing proceeds via time-dependent Hamiltonian evolution $H(t) = (1-t/T)H_0 + (t/T)H_1$, where $H_1$ encodes the QUBO, and $H_0$ initializes a uniform superposition (Ihara, 27 Mar 2024). Reverse annealing starts from a classical candidate solution and locally explores lower-energy states, providing rapid refinement with very short annealing times (e.g., $3\,\mu$s per iteration) (Ihara, 27 Mar 2024).

Simulated bifurcation (SB) algorithms provide a quantum-inspired, hardware-embeddable method for rapidly minimizing the QUBO cost on FPGAs, simulating classical nonlinear oscillator networks whose final digitized states correspond to the QUBO ground state (Tatsumura et al., 18 Oct 2024).

3. Multi-Objective and Constraint-Rich QUBO Frameworks

QUBO structures can naturally accommodate multiple objectives and complex inequality constraints present in real-world MOT tasks:

• Weighted sum for multi-objective QUBOs: Multiple quadratic objectives $c_j(x)$ are aggregated as $E(x) = \sum_{j=1}^m \lambda_j c_j(x)$ with scalarization weights $\lambda_j$ (Ayodele et al., 2023). Adaptive schemes for $\lambda$ support diverse Pareto front exploration, allowing the solver to balance association cost, identity consistency, and other tracking metrics.
• MOQA (Multi-Objective Quantum Approximations): More general, MOQA encodes the maximum of several objectives as a polynomial sum $h_{(p)}(b) = \sum_{m=1}^M [h_m(b)]^p$, with sandwich bounds ensuring the minimum of the composite Hamiltonian approximates the “worst-case” cost (Egginger et al., 15 Oct 2025). This is crucial for robust assignment under adversarial occlusion or appearance ambiguity.

Inequality constraints, such as mutually exclusive assignments or temporal continuity, can be regularized and incorporated into the QUBO structure via penalty terms, further extending the expressiveness and applicability of QUBO-based formulations to complex tracking scenarios (Egginger et al., 15 Oct 2025).

4. Real-Time Implementation and Hardware Acceleration

QUBO-based MOT frameworks have demonstrated real-time performance on embedded hardware platforms:

• Ising machines and SB algorithms implemented on FPGAs can solve QUBO assignment instances in sub-millisecond time per frame, enabling throughputs of $23$ FPS for in-vehicle MOT (Tatsumura et al., 18 Oct 2024).
• Quantum annealers (e.g., D-Wave Advantage2) have been shown to yield high tracking accuracy (MOTA, IDF1, low ID switches) with annealing times as low as $3\,\mu$s (Ihara, 27 Mar 2024).

Hardware acceleration is particularly beneficial for handling NP-hard assignment problems with flexible (many-to-one) matching required in the presence of occlusions, and for integrating multiple sources of appearance and motion metrics in ensemble tracking (Ihara, 27 Mar 2024).

5. Integration with Classical and Deep Learning-Based MOT Methods

QUBO/MOQA formulations enable seamless integration with data association and feature extraction pipelines in modern MOT frameworks:

• Similarity matrices derived from learned feature embeddings (e.g., bi-softmax appearance metrics in QDTrack (Fischer et al., 2022), or multi-level feature objectives (Henschel et al., 2016)) can directly define QUBO costs.
• The match-selection or track-hypothesis selection steps in hypothesis-rich frameworks (e.g., GLMB-based (Papi et al., 2014) or TOMHT-style (Chen et al., 2017)) can be cast as binary optimization tasks, where the quadratic cost encodes hypothesis compatibility or sharing penalties.

In transformer-based and self-supervised trackers, while the network may produce initial association scores, downstream QUBO optimization can be used to enforce global, combinatorial constraints on assignments, supporting more robust and globally optimal data association (Chen et al., 2022, Korbar et al., 2022, Zhao et al., 2023).

6. Challenges, Scalability, and Theoretical Considerations

Despite their expressiveness, QUBO-based approaches face fundamental and practical challenges:

• The QUBO solution space grows exponentially with the number of association variables, leading to scaling limitations on both current quantum and quantum-inspired hardware (McCormick et al., 2021, Zaech et al., 2022).
• Embedding logical QUBO graphs onto restricted hardware topologies often requires chain embeddings or sparse couplers, which can complicate parameter tuning and degrade performance (e.g., chain strength calibration in quantum annealers (McCormick et al., 2021)).
• Overly high penalty coefficients enforcing constraints may narrow the spectral gap or impede convergence in adiabatic algorithms, requiring careful hyperparameter tuning (Zaech et al., 2022).
• The combinatorial expansion for multi-objective or multi-body polynomial Hamiltonians, as in MOQA, yields polynomial scaling in the number of nonzero terms $O(n^{2p})$ but requires exploiting sparsity to remain tractable in practice (Egginger et al., 15 Oct 2025).

Nonetheless, systematic approximations (such as p-norm for max objective composition), adaptive weighting for multi-objective QUBO, and robust hardware design have enabled compelling results for tracking benchmark problems and real-world datasets, demonstrating the feasibility and competitiveness of QUBO-based MOT (Egginger et al., 15 Oct 2025, Tatsumura et al., 18 Oct 2024).

7. Applications and Future Directions

Applications of QUBO-based MOT include in-vehicle autonomous tracking, urban traffic light control, industrial quality inspection, and intelligent video analysis, where real-time combinatorial optimization and robust association under uncertainty are paramount (Ihara, 27 Mar 2024, Tatsumura et al., 18 Oct 2024). Further anticipated developments involve:

• Advanced quantum hardware and Ising machines with higher connectivity, native multi-body interaction gates, or improved thermal isolation.
• Algorithms for exploiting sparsity and structure in multi-objective QUBOs and MOQA Hamiltonians to enable larger problem instances (Egginger et al., 15 Oct 2025, Ayodele et al., 2023).
• Integration with self-supervised and end-to-end deep tracking pipelines, leveraging QUBO optimization to enforce harder constraints on association and cross-modal consistency.

A plausible implication is that as quantum hardware matures and the theory of Hamiltonian encodings advances, QUBO-based multi-object tracking will become a default framework for large-scale, robust, and low-latency tracking in complex environments requiring global assignment optimization.