Quantum-inspired dynamical models on quantum and classical annealers

TL;DR

The paper addresses the challenge of benchmarking real-time quantum dynamics on quantum hardware versus classical methods by mapping the dynamics of an $n$-qubit (potentially non-Hermitian) system to a solver-agnostic QUBO via a parallel-in-time encoding, enabling direct comparison between quantum annealers and classical optimizers. The authors introduce a rigorous workflow that generates QUBO instances from discretized quantum evolution, and evaluate them on two generations of D-Wave annealers and the GPU-based VeloxQ, along with SA baselines, across eight dynamical models including PT-symmetric cases. Key contributions include an open-source benchmark suite, analysis of native versus embedded problem performance, and large-scale benchmarks up to $N \\sim 10^5$, revealing that Advantage2 improves ground-state success and scaling relative to its predecessor, while VeloxQ often dominates in absolute runtime on large instances. The work establishes a physically motivated baseline for tracking progress toward quantum-competitive dynamical simulation and clarifies the current hardware- and algorithmic-gap landscape, highlighting that continued advances in both quantum hardware connectivity/error mitigation and classical optimization heuristics will shape the path to quantum advantage.

Abstract

We propose a practical, physics-inspired benchmarking suite to challenge both quantum and classical computers by mapping real-time quantum dynamics to a common optimization format. Using a parallel-in-time encoding, we convert the real-time propagator of an $n$-qubit, possibly non-Hermitian, Hamiltonian into quadratic unconstrained binary optimization (QUBO) instances that are executable in a solver-agnostic manner on quantum annealers and classical optimizers alike. This enables direct, like-for-like performance comparisons across fundamentally different computational paradigms.To stress-test the framework, we consider eight representative dynamical models spanning single-qubit rotations, multi-qubit entangling gates (Bell, GHZ, cluster), and PT-symmetric and other non-Hermitian generators, and evaluate success probability and time-to-solution as standard benchmarking metrics. Applying this methodology to two generations of D-Wave quantum annealers and to state-of-the-art classical solvers (Simulated Annealing and the GPU-accelerated VeloxQ), we find that Advantage2 consistently outperforms its predecessor, while VeloxQ retains the shortest absolute runtimes, reflecting the maturity of classical heuristics.We further extend the benchmarks to large-scale instances ($N \simeq 10^{5}$), establishing a demanding classical baseline for future hardware. Together, these results position the parallel-in-time QUBO framework as a versatile and physically motivated testbed for quantitatively tracking progress toward quantum-competitive simulation of dynamical systems.

Figures (7)

• Figure 1: Overview of the paper workflow. (a) A quantum state $\ket{\psi_0}$ of dimension $N$ evolves under the (possibly non-Hermitian) generator $H(t)$ according to the Schrödinger equation $\partial_t\ket{\psi(t)} = H(t)\ket{\psi(t)}$. (b) The real-time interval $[t_0,t_f]$ is discretised into $N$ equal Trotter slices of width $\Delta t$; each slice is propagated by the short-time operator $U(t_{n+1},t_n)=\exp\bigl[H(t_n)\,\Delta t\bigr]$, so that the sequence $\{U(t_{n+1},t_n)\}_{n=0}^{N-2}$ fully specifies the history. (c) Discretised dynamics are encoded in the block-tridiagonal clock Hamiltonian $C$, Eq. \ref{['clock']}; appending the initial-state projector yields the positive-definite matrix $A$. Writing the quadratic form $\tfrac{1}{2}\langle x|A|x\rangle-\langle x\ket{\phi}$ in fixed-point binary variables ($R$ bits over the range $[-2^{D},2^{D}]$) produces a quadratic-unconstrained binary optimisation (QUBO) with $\mathcal{O}(LNR)$ spins, Eq. \ref{['qubo']}. (d) The QUBO instance is submitted to a D-Wave quantum annealer (Advantage and Advantage2) and to the GPU-accelerated classical heuristic VeloxQveloxq. Both solvers return low-energy spin configurations that decode unambiguously into the time-ordered amplitudes $\{|\psi(t_n)\rangle\}_{n=0}^{N-1}$; from these, physical observables---e.g. $\langle\sigma_z\rangle(t)$ shown on the right---are reconstructed without further fitting. Both solvers produce the same observable dynamics. This pipeline therefore maps continuous-time quantum dynamics to a solver-agnostic combinatorial problem, enabling a like-for-like performance comparison between quantum and classical optimisation hardware.
• Figure 2: (a) Time-dependent energy scales executed by the D-Wave quantum annealer. The coefficient $A(s)$ (red) multiplies the transverse-field term that enables quantum tunnelling and therefore falls rapidly as the run progresses. The coefficient $B(s)$ (blue) multiplies the problem Hamiltonian and rises smoothly, so that the device moves from a tunnelling-dominated regime at $s=0$ to a problem-dominated regime at $s=1$. (b) Native connectivity map for the Pegasus architecture (D-Wave Advantage). Circles mark individual superconducting qubits; blue lines are programmable couplers that lie within the main module, and orange lines link neighbouring modules. (c) Connectivity map for the newer Zephyr architecture (D-Wave Advantage2), which increases the number of couplers per qubit and introduces longer-range links.
• Figure 3: Exemplary plateau behavior observed in cyclic annealing for instance $H_3$ with three time points. The blue line indicates the mean over the minimum energy found over each iteration, while the red dashed line shows the mean of the energies found by standard forward annealing. Shadowed box indicates the lower and upper quartile for all samples for forward annealing, and for lowest cycle energy for cyclic annealing. For this plot, we performed three cyclic annealing runs over 30 cycles and 30 forward annealing runs on Advantage2_system 1.10. The percentage in the box corresponds to the percentage with which lowest energy samples were found by each method.
• Figure 4: Left column: coloured chains illustrate how the logical qubits of two representative model Hamiltonians are mapped onto the physical Pegasus P16 topology of the D-Wave hardware. Panel (a) shows the native single-qubit problem $H_{1}$, Eq. \ref{['sys1']}, which embeds without chains. Panel (d) shows the non-native single-qubit problem $H_{3}$, Eq. \ref{['sys3']}, where minor embedding produces chains that span several unit cells. Middle column: ground-state success probability obtained on the two quantum annealers-Advantage2 (solid lines) and the earlier Advantage system (dashed lines)-as a function of the annealing time. Circles correspond to a temporal discretisation with two time points, triangles to three. Each marker averages $5000$ annealing runs; if no ground-state sample is observed the marker is omitted. Right column: expectation value $\langle\sigma_{z}\rangle$ reconstructed from the annealer output of the three lowest energy states of the QUBO model of the target Hamiltonian. Blue triangles denote the ground state, green squares and red circles the first and second excited states, respectively. The black dashed curve is the exact unitary evolution computed with QuTiP qutip1qutip2qutip3qutip4qutip5 and serves as a baseline.
• Figure 5: Scaling properties of $\text{TTS}_{99}$ for Pegasus-native systems $(H_1,H_2,H_5,H_6,H_7)$, \ref{['sys1', 'sys2', 'sys5', 'sys6', 'eq:ptsymmetric']}, on the Advantage and Advantage2 QPU, as well as the classical solvers: Simulated Annealing (SA), both CPU and GPU based, and the GPU based VeloxQ. $N$ denotes the number of problem variables. Data points indicate values $\text{TTS}_{99}$, averaged over system size for all considered systems. Missing data points indicate failure to find the ground state. 5 (20) runs with 1000 samples each were performed on quantum (classical) solvers, for each system at time points $t\in\{2,3,\ldots, 14\}$.