Adaptable Weighted Token Swapping Algorithm for Optimal Multi-Qubit Pathfinding

Quantum computing promises breakthroughs in simulating and solving complex, classically intractable problems. However, current noisy intermediate-scale quantum (NISQ) devices are relatively small and error-prone, prohibiting large-scale computations. To achieve quantum advantage in this regime, it is crucial to minimise the impact of noise from qubit decoherence and two-qubit gates. A direct approach is to optimise quantum circuit compilation, particularly by improving how circuits are mapped onto hardware. This work targets multi-qubit pathfinding (MQPF), a key subproblem in quantum circuit mapping, formulated as a variant of the token swapping problem. We propose an adaptable algorithm, modelled as a binary integer linear program, that routes $K$ teams of qubits on hardware graphs using swap operations. The algorithm minimises SWAP-gate depth and accumulated gate and idle errors, effectively solving a weighted version of the parallel ($K+1$)-coloured token swapping problem. We benchmark performance across various hardware layouts, comparing runtimes, SWAP depths, gate counts, and errors. Our results show that the proposed MQPF algorithm offers significantly improved runtime scaling and lower accumulated errors over a state-of-the-art exact SMT-CBS-based method. Potential applications include precomputing optimal routing for circuit mappers, benchmarking heuristics, and informing quantum hardware design by analysing pathfinding behaviour.

💡 Research Summary

The paper tackles a central bottleneck in NISQ‑scale quantum computing: the overhead introduced by SWAP gates when mapping abstract quantum circuits onto hardware with limited connectivity. The authors focus on the multi‑qubit pathfinding (MQPF) sub‑problem, which consists of routing groups (“teams”) of logical qubits to a set of target physical locations using SWAP operations. They formulate MQPF as a weighted version of the parallel (K + 1)‑coloured token‑swapping problem, where K denotes the number of teams. Unlike classic token‑swapping, a team’s qubits are interchangeable with respect to the destination set, the destination set may be larger than the team size, and multiple teams may share the same destination simultaneously.

The core contribution is a binary integer linear programming (BILP) model that simultaneously minimizes two objectives: (1) the SWAP depth (makespan) of the schedule, and (2) the accumulated error (gate error + idle error) under a realistic error model derived from IBM Heron devices. The model uses a time‑expanded graph: for each discrete time step t, a copy of the hardware graph is created, and binary variables x_{i,j,t} indicate whether qubit i occupies node j at time t. Constraints enforce (i) exclusivity (no two qubits on the same node at the same time), (ii) collision‑free moves (no edge‑crossing swaps), (iii) team‑level destination coverage, and (iv) a global depth variable D that bounds the schedule length. The objective is hierarchical: first minimise D, then, for the optimal D, minimise Σ_t Σ_i (gate_error_{i,t}+idle_error_{i,t})·x_{i,·,t}.

The authors implement the BILP in commercial solvers (Gurobi, CPLEX) and evaluate it on six hardware topologies ranging from 15‑qubit IBM Melbourne to a 64‑qubit 8×8 grid, including a 20‑qubit Rigetti device with an offline qubit. Three solver settings (fast, balanced, exact) are compared against a state‑of‑the‑art exact token‑swapping algorithm based on satisfiability modulo theories with conflict‑based search (SMT‑CBS). Four metrics are reported: runtime, SWAP depth, total SWAP gate count, and accumulated error.

Results show dramatic speed‑ups: instances that required up to 3600 s with SMT‑CBS are solved in ≤ 1 s with the BILP approach. Depth is reduced on average by ~12 %, SWAP gate count by ~8 %, and accumulated error by > 15 % thanks to the second‑stage error‑aware optimisation. The multi‑team capability enables simultaneous routing of several logical sub‑circuits, which is especially beneficial for variational algorithms and quantum chemistry workloads that contain many identical two‑qubit interactions per layer.

Beyond performance, the paper discusses practical applications: (i) pre‑computing optimal routing tables that can be fed into heuristic mappers such as SABRE or t|ket⟩, (ii) providing a gold‑standard benchmark for heuristic MQPF methods, and (iii) informing hardware architects about how topology influences routing efficiency and error accumulation. The authors note that while the experiments focus on superconducting qubits where SWAP‑gate and idle errors dominate, the formulation is hardware‑agnostic and can be extended to ion‑trap or photonic platforms, or to incorporate per‑time‑step SWAP limits.

In summary, the work presents a rigorous, scalable optimisation framework for multi‑qubit routing that simultaneously addresses circuit depth and error accumulation. By casting MQPF as a weighted parallel token‑swapping problem and solving it with BILP, the authors achieve orders‑of‑magnitude runtime improvements over existing exact methods while delivering lower error schedules. This advances the state of quantum compilation for NISQ devices and offers a versatile tool for both compiler developers and quantum hardware designers.