Quantum-inspired dynamical models on quantum and classical annealers

1. Problem Statement

Simulating the real-time evolution of many-body quantum systems is a fundamental challenge in condensed matter physics, quantum chemistry, and quantum technologies. Classical simulations become infeasible beyond a few dozen qubits due to the exponential growth of the Hilbert space. While quantum hardware (specifically quantum annealers like those from D-Wave) promises to overcome this, rigorous benchmarking has been hindered by two main issues:

1. Mapping Difficulty: There has been no efficient, scalable method to map the time-dependent Schrödinger equation (a sequence of unitary operators) onto the static Quadratic Unconstrained Binary Optimization (QUBO) format required by current annealers.
2. Benchmarking Gap: Previous studies lacked standardized, physics-derived benchmark suites, making it difficult to distinguish genuine quantum speedups from artifacts caused by easy problem instances or favorable hardware embeddings.

The authors aim to close this gap by creating a solver-agnostic benchmarking framework to quantitatively compare modern quantum annealers against state-of-the-art classical heuristics.

2. Methodology

A. Parallel-in-Time QUBO Encoding

The core innovation is a method to convert continuous-time quantum dynamics into a combinatorial optimization problem:

• Discretization: The time interval $[t_0, t_f]$ is discretized into $N$ steps. The evolution is described by a "history state" $|\Psi\rangle = \sum |t_n\rangle \otimes |\psi(t_n)\rangle$.
• Clock Hamiltonian: A Hermitian "clock" operator $C$ is constructed such that its ground state corresponds to the correct time-evolved history.
• QUBO Formulation: By adding an initial state projector, the problem is transformed into minimizing a positive-definite quadratic form $f(x) = \frac{1}{2}\langle x|A|x\rangle - \langle x|\phi\rangle$.
• Binary Conversion: Real-valued variables are encoded into fixed-point binary variables, resulting in a QUBO instance with $O(LNR)$ spins (where $L$ is system dimension, $N$ is time steps, $R$ is precision bits).
• Generality: This framework handles both Hermitian and non-Hermitian (including PT-symmetric) Hamiltonians.

B. Benchmark Suite

The authors defined eight representative dynamical models to stress-test the framework:

1. Single-qubit systems: Y-rotation, oblique drive, and pair-flip (some native to Pegasus topology, others requiring embedding).
2. Multi-qubit entangling gates: Bell state generation, GHZ state generation, and cluster states.
3. Non-Hermitian systems: PT-symmetric qubits.

C. Solvers and Hardware

The study compares three distinct computational paradigms on identical QUBO instances:

1. Quantum Annealers:
   • D-Wave Advantage (Pegasus topology): Previous generation.
   • D-Wave Advantage2 (Zephyr topology): Newer generation with 33% higher connectivity (20 links/qubit vs. 15).
   • Strategies: Forward annealing and cyclic annealing (a refinement strategy involving back-and-forth schedules).
2. Classical Heuristics:
   • VeloxQ: A GPU-accelerated, physics-inspired solver based on dynamical systems theory. It does not require minor-embedding and scales to $2 \times 10^8$ variables.
   • Simulated Annealing (SA): Implemented on both CPU and GPU as a standard reference.

D. Metrics

• Success Probability: Fraction of runs finding the exact ground state.
• Time-to-Solution (TTS): Time required to find the ground state with 99% confidence ($TTS_{99}$).
• Scaling Exponent ($\beta$): Derived from fitting $TTS \propto \exp(N/\beta)$. A higher $\beta$ indicates better scaling performance.
• Time-to-Epsilon ($TT_\epsilon$): Used for large-scale instances ($N \sim 10^5$) where exact ground states are unreachable, measuring time to reach a solution within an optimality gap $\epsilon$.

3. Key Results

A. Hardware Progress (Advantage vs. Advantage2)

• Success Probability: The Advantage2 processor consistently outperformed its predecessor. For native problems, the median success probability improved by a factor of 4–5; for non-native (embedded) problems, the improvement was up to an order of magnitude (up to 26x for specific schedules).
• Scaling: Advantage2 showed improved scaling exponents. While Advantage had a uniform $\beta \approx 9$, Advantage2 exhibited two regimes: $\beta \approx 15$ for $N \lesssim 100$ and $\beta \approx 33$ for $N \gtrsim 100$ on native problems.
• Cyclic Annealing: While cyclic annealing improved the best-found energy quickly in the first few cycles, it did not significantly alter the overall exponential scaling of TTS compared to forward annealing, suggesting it is a useful local refinement tool but not a fundamental scaling fix.

B. Quantum vs. Classical Performance

• Current Dominance of Classical Solvers: Despite hardware improvements, VeloxQ (GPU-accelerated classical) retained the shortest absolute runtimes and the best scaling exponent ($\beta \approx 100$) across all problem sizes accessible to current quantum hardware.
• Connectivity Constraints: Quantum annealers struggled significantly with non-native problems (e.g., System H3 and H8) requiring minor embedding. The hardware failed to find the ground state for H8 entirely in some configurations, whereas VeloxQ solved all instances without connectivity constraints.
• Large-Scale Scaling: For massive instances ($N$ up to $10^5$), VeloxQ demonstrated power-law scaling and significantly outperformed GPU-based Simulated Annealing. Both classical solvers showed a transition to exponential complexity as the target error $\epsilon \to 0$, but VeloxQ remained superior.

4. Key Contributions

1. Unified Benchmarking Framework: Introduced a rigorous, physics-inspired protocol to map real-time quantum dynamics to QUBO, enabling "like-for-like" comparisons between fundamentally different hardware (quantum annealers vs. classical GPUs).
2. Comprehensive Dataset: Provided a suite of 8 diverse dynamical models, including non-Hermitian and entangling systems, spanning native and embedded topologies.
3. Quantitative Assessment: Established that while D-Wave Advantage2 represents significant hardware progress (better connectivity and control), current quantum annealers are still outperformed by optimized classical heuristics (VeloxQ) for the problem sizes tested.
4. Classical Baseline: Established a high-performance classical baseline ($N \sim 10^5$) to serve as a target for future quantum advantage claims.

5. Significance and Conclusion

The paper concludes that the path to quantum advantage in dynamical simulation is not yet realized for the tested problem classes. The results highlight a "balance of power" where algorithmic ingenuity (VeloxQ) and massive parallelism on classical GPUs currently outweigh the raw quantum speedups offered by current annealers.

However, the framework provides a transparent yardstick for the community. The authors note that as qubit lattices become denser and coupler graphs wider, medium-scale simulations (tens to hundreds of degrees of freedom) may soon become accessible within a single annealing run. The open-source benchmark suite released with the paper allows the community to track progress as both hardware and algorithms evolve.

In summary: The study validates the feasibility of mapping quantum dynamics to QUBO but demonstrates that, for now, state-of-the-art classical solvers on GPUs remain the superior choice for solving these specific dynamical problems, setting a high bar for future quantum hardware to surpass.