Hybrid Sequential Quantum Computing

• Hybrid Sequential Quantum Computing (HSQC) is a computational paradigm that interleaves classical and quantum methods to tackle complex HUBO problems.
• It sequentially employs simulated annealing, BF‑DCQO for quantum tunneling, and classical local refinement methods for optimal solution recovery.
• Empirical tests on 156-qubit HUBO instances demonstrate up to 700× speed enhancement, validating HSQC’s effectiveness and robustness.

Hybrid Sequential Quantum Computing (HSQC) is a computational paradigm for combinatorial optimization in which classical and quantum methods are systematically interleaved in a structured, stage-wise workflow. This architecture is designed to integrate the respective strengths of classical heuristics (superior global exploration) and quantum algorithms (capable of tunneling through local minima) to achieve optimal or near-optimal solutions significantly faster than either standalone approach. The practical relevance and empirical success of HSQC are demonstrated in challenging higher-order unconstrained binary optimization (HUBO) problems on advanced superconducting quantum processors, where HSQC achieves up to two orders of magnitude improvement in runtime quantum advantage relative to state-of-the-art classical solvers (Chandarana et al., 7 Oct 2025).

1. Concept and Motivation

HSQC is defined as a hybrid optimization procedure that involves an arbitrarily structured sequence of classical and quantum computational episodes, with the principal operational criterion being that the overall composite workflow outperforms each constituent standalone solver. Specifically, in HSQC, the workflow starts with classical optimizers (typically simulated annealing, SA) to broadly probe the solution landscape. Quantum optimization (notably, bias-field digitized counterdiabatic quantum optimization, BF-DCQO) is then used to refine and possibly tunnel through energy barriers that would otherwise confound classical methods. The circuit concludes with a classical refinement step—most commonly, a memetic tabu search (MTS) or a second round of SA—to exploit the now-improved configuration and recover exact or near-optimal states. This sequence capitalizes on the global search capabilities and low overhead of classical heuristics while harnessing the non-classical dynamics available to the quantum layer. The design aims to address and compensate for the respective limitations of each computational paradigm, such as the stagnation in classical local minima or quantum decoherence and noise (Chandarana et al., 7 Oct 2025).

2. Structured Stage-Wise Workflow

The HSQC protocol orchestrates optimization in three principal stages:

1. Initial Classical Optimization (Simulated Annealing):

• The problem is first encoded as a HUBO problem, which is mapped to the Ising Hamiltonian

$$F(z) = \sum_{r=1}^p \sum_{(a_1,\ldots,a_r) \in P_k} W_{a_1\ldots a_r} z_{a_1} \ldots z_{a_r}$$

where $z_a \in \{0,1\}$ and $p$ denotes the maximum interaction order.

• Simulated annealing is used to search for low-energy candidate bitstrings. SA leverages stochastic sweeps and energy-lowering moves, with rare uphill transitions for basin escape.

2. Quantum Refinement (BF‑DCQO):

• The best solution from SA seeds the quantum routine. BF‑DCQO performs a stepwise, counterdiabatic quantum evolution, employing a mixer Hamiltonian

$$H_m = \sum_{i=1}^N \left( h^x_i \sigma^x_i + h^b_i \sigma^z_i \right)$$

where transverse fields $h^x_i$ are fixed and longitudinal fields $h^b_i$ are set by the observed quantum measurement outcomes, $h^b_i = \pm \langle \sigma^z_i \rangle$.

• The quantum step utilizes phenomena such as tunneling to overcome energy barriers, potentially escaping local minima unreachable by classical methods.

3. Classical Local Refinement (MTS or Second SA):

• The quantum-processed solution is further refined by a high-performing classical local search (MTS or SA).
• MTS combines memetic (evolutionary crossover, adaptive mutation) and tabu (bit-flip with forbidden moves) mechanisms; the mutation rate is dynamically decayed per generation:

$$\mu_g = \mu_\text{end} + (\mu_\text{start} - \mu_\text{end}) \left[\frac{\ln(G_\text{max} + 1 - g)}{\ln(G_\text{max} + 1)}\right]$$

ensuring a controlled balance between exploration and exploitation.

This modular composition ensures that each stage compensates for the known limitations and noise sensitivities of the others, maximizing robustness and solution quality (Chandarana et al., 7 Oct 2025).

3. Component Algorithmic Roles

A detailed breakdown of HSQC components shows precise functional delineation:

Stage	Solver Type	Functional Role
Initial Exploration	Classical (SA)	Rapid landscape probing, thermal basin escape, warm start for quantum module
Quantum Enhancement	Quantum (BF‑DCQO)	Counterdiabatic evolution, quantum tunneling, refinement of classical candidate
Final Exploitation	Classical (MTS/SA)	Local search to correct quantum-stage imperfections, recover or approach the ground state

In practice, SA covers the global space efficiently but is susceptible to local traps, BF‑DCQO improves escape probability via physical quantum effects, and the MTS or a second SA removes quantum-induced noise or residual suboptimality.

4. Empirical Performance and Quantum Advantage

The paper presents empirical benchmarks on dense 156-qubit HUBO instances mapped to heavy‑hexagonal architectures. Notable findings include:

• Consistent ground-state solution recovery within approximately 1.5 seconds per run.
• Speedup factors: Up to $700\times$ faster than SA and $9\times$ relative to standalone MTS. In comparison, even a highly optimized exact solver (CPLEX) required up to 23 seconds for full proof of optimality.
• Optimality gap metrics:

$$\mathcal{E}_\text{gap} = 100 \times \frac{E_\text{best} - E_\text{GS}}{|E_\text{GS}|}$$

show that HSQC not only achieves optimal or near-optimal energy values but does so with markedly reduced variance over multiple random seeds and trials.

The observed quantum advantage arises primarily from the quantum routine’s tunneling capabilities, which shorten the time-to-solution drastically for landscapes that otherwise trap classical solvers in exponential regimes.

5. Quantum and Classical Integration Mechanisms

Integral to HSQC is the controlled transition of data—specifically, candidate solutions and measurement-derived observables—between classical and quantum modules. For BF‑DCQO, the protocol relies on:

• Initializing the quantum device on the experimentally observed classical bitstring.
• Iterative feedback: Quantum measurements set $h^b_i$, biasing further quantum evolution toward improved fidelity.
• Post-quantum classical stages treat the output bitstring as a starting point for further deterministic search.

These mechanisms make HSQC highly modular, allowing for parameter sweeps and batch processing suitable for high-throughput, scalable quantum-classical environments.

6. Demonstrated Limitations and Robustness

While the HSQC protocol achieves substantial improvements, certain limitations are acknowledged:

• Decoherence and device noise in BF‑DCQO stages may revert bitstrings or introduce errors, requiring classical post-processing for correction.
• The protocol remains probabilistic; statistical variability in both quantum and classical stages can cause rare suboptimal outcomes.
• Approximate digitization and finite measurement statistics in BF‑DCQO mean that even quantum-enhanced candidates may require additional classical refinement.

Nevertheless, the sequential redundancies in HSQC are effective in preventing outright failure—a critical property when scaling deployments to larger, industrial instances.

7. Extensibility and Future Impact

The HSQC paradigm is designed to be extensible:

• Each stage can be substituted with domain-specialized solvers or quantum routines (e.g., the quantum step could be replaced with quantum annealing or alternative counterdiabatic methods as hardware and algorithmic advances permit).
• The modular workflow is amenable to parallelization, parameter optimization, and orchestration within distributed HPC environments.
• The significant reductions in time-to-solution, already achieved on commercially available quantum hardware, position HSQC as a candidate protocol for near-term practical quantum advantage in combinatorial optimization.

By enabling a flexible, dynamically staged approach, HSQC represents a scalable template for hybrid quantum-classical algorithms—one that is especially effective in complex, noisy intermediate-scale quantum (NISQ) settings and anticipated to remain relevant as quantum hardware evolves (Chandarana et al., 7 Oct 2025).