Circuit compression for 2D quantum dynamics

TL;DR

The paper tackles the challenge of simulating large 2D quantum dynamics with near-term quantum hardware by deploying a variational circuit compression strategy. It replaces deep, noisy dynamics with a shallower variational circuit whose parameters are optimized to closely approximate the target unitary via a scalable local risk derived from Pauli propagation. The approach yields order-of-magnitude improvements in accuracy over standard Trotterization at the same circuit depth, demonstrated numerically for 2D lattices up to 30×30 and experimentally on the Quantinuum H1 chip with hard-core boson diffusion. This compression framework enables longer-time simulations with reduced quantum resources, advancing the prospect of practical quantum advantage in dynamics. The method relies on a Pauli transfer matrix formulation, a meet-in-the-middle evaluation of local observables, and truncation schemes to keep computations tractable, while benefiting from translation invariance to scale to large systems.

Abstract

The study of out-of-equilibrium quantum many-body dynamics remains one of the most exciting research frontiers of physics, standing at the crossroads of our understanding of complex quantum phenomena and the realization of quantum advantage. Quantum algorithms for the dynamics of quantum systems typically require deep quantum circuits whose accuracy is compromised by noise and imperfections in near-term hardware. Thus, reducing the depth of such quantum circuits to shallower ones while retaining high accuracy is critical for quantum simulation. Variational quantum compilation methods offer a promising path forward, yet a core difficulty persists: ensuring that a variational ansatz $V$ faithfully approximates a target unitary $U$. Here we leverage Pauli propagation techniques to develop a strategy for compressing circuits that implement the dynamics of large two-dimensional (2D) quantum systems and beyond. As a concrete demonstration, we compress the dynamics of systems up to $30 \times 30$ qubits and achieve accuracies that surpass standard Trotterization methods by orders of magnitude at identical circuit depths. To experimentally validate our approach, we execute the compiled ansatz on Quantinuum's H1 quantum processor and observe that it tracks the system's dynamics with significantly higher fidelity than Trotterized circuits without optimization. Our circuit compression scheme brings us one step closer to a practical quantum advantage by allowing longer simulation times at reduced quantum resources and unlocks the exploration of large families of hardware-friendly ansätze.

Figures (7)

• Figure 1: Overview of the VQC approach leveraged in this work. a) Starting in the Schrödinger picture, we approximate the Hilbert-Schmidt product between $U$ and $V$ by a global risk function obtained from sampling $U^\dagger V$'s action on product states. b) Shifting to the Heisenberg picture, we analytically show that the global risk function \ref{['eq:R_prod']} can be reduced to an evaluation of expectation with local observables, $P_j$, as shown in \ref{['eq:locHaarpref']}. In practice we compute $\left\langle\left\langle P_j \right.\right|\bm{V(\vec{\theta})}$ and $\bm{U^\dagger}\left|\left. P_j \right\rangle\right\rangle$ separately, and then take their inner product. c) The compressed circuit can then be applied $k$ times, accurately following the system's dynamics for long times $T$, and requiring much less quantum resources compared to a deep accurate Trotterization (bottom right).
• Figure 2: Compression for large 2D systems and comparison with Trotterization. At time $t$, the target $U$ cosists of $L_U\cdot t/\Delta t$ layers of Trotterization, whereas $V$ consists of $L_V$ layers at all times. a) Illustration of $\mathcal{T}_{n_x\times n_y}, \mathcal{T}_{\overline{n_x}\times \overline{n_y}}, \mathcal{T}_{n,\mathrm{h.-h.}}$. In b)-d) we compare costs for the Trotter and compressed circuits. b) compression of the TFIM Hamiltonian, for $\mathcal{T}_{127,\mathrm{h.-h.}}$ ($L_V=4, \Delta t=0.1, L_U=10, h_i=0,h_f=2$), $\mathcal{T}_{13\times 13}$ ($L_V=2, \Delta t=0.06, L_U=8, h_i=0, h_f=2$) and $\mathcal{T}_{\overline{30}\times \overline{30}}$ ($L_V=2, \Delta t=0.06, L_U=6, h_i=0, h_f=1.6$). c) compression of the Floquet Hamiltonian for two sizes of square periodic lattices, $\mathcal{T}_{\overline{8}\times \overline{8}}, \mathcal{T}_{\overline{10}\times \overline{10}}$. In both cases $L_V=2, \Delta t=\tau/7, L_U=8$. We observe that at odd periods the Trotterization has a very similar cost to the compressed ansatz, but that in general the Trotterization is very inaccurate, whereas the compressed ansatz outperforms it by many orders of magnitude. d) Improvement over Trotterization for different ramp speeds for the TFIM with identical choices of $h_i=0, h_f=2$ on the $\mathcal{T}_{11\times11}$ topology. Steeper ramps lead to shorter overall evolution time, and hence to larger improvement over Trotterization. e) Improvement over Trotterization for the data presented in panel b).
• Figure 3: Dynamics of hard-core bosons with the compressed circuit on the H1 chip. a) Illustration of $\mathcal{T}_{5\times \overline 4}$. The two red dots correspond to the positions of the bosons at $T=0$, and we compress the dynamics for fixed $t=0.4$. b) H1 results for $r_J:=J_y/J_x=0.2$ c) H1 results for $r_J=J_y/J_x=1$. For both b-c): top row: expectation value of the occupation number at increasing times (from left to right) obtained from our exact statevector simulation. Bottom row: occupation number experimentally obtained by running the compressed circuit. d) Mean error in the occupation number for states prepared at different times using the compressed circuit and the Trotter circuit in real experiments. C stands for "compressed", T for "Trotter".
• Figure 4: a) Growth of Pauli paths for $\bm {V}\left|\left. P \right\rangle\right\rangle$ and $\bm U\left|\left. P \right\rangle\right\rangle$ for different simulation times $t$. Here we illustrate weight truncation. Longer simulation times produce much bigger propagation trees. b) Comparison, at $\mathcal{T}_{\overline 4\times \overline 3}$, of $R^\mathrm{loc}_{\mathcal{Q}}$ and $C_\text{HST}$ for compressing the TFIM Hamiltonian. We observe that $R^\mathrm{loc}_{\mathcal{Q}}(\theta^\star_{W_\text{ref}})$ and $C_\text{HST}(\theta^\star_{W_\text{ref}})/n$ differ by a constant. Inset: relative error $e_{W} = \vert (R^\mathrm{loc}_{\mathcal{Q}}(\theta^\star_{W}) - R^\mathrm{loc}_{\mathcal{Q}}(\theta^\star_{W_\text{ref}}))/R^\mathrm{loc}_{\mathcal{Q}}(\theta^\star_{W_\text{ref}}) \vert$ for different weight truncations, with $W_\text{ref} = 12$. We cut off data for the $e_{W}$ inset at $10^{-15}$.
• Figure 5: As in \ref{['fig:truncations']} we consider the TFIM Hamiltonian with the topology $\mathcal{T}_{\overline4\times\overline3}$. Left: convergence of $e_W$ as a function of $W$ at different times. Right: convergence in $W$ of $e_\theta$ at different times.