Classical surrogate simulation of quantum systems with LOWESA

TL;DR

LOWESA offers a classical surrogate approach to quantum simulation by constructing a full expectation landscape over circuit parameters, enabling rapid re-evaluation for families of Hamiltonians and observables after an upfront cost. The method uses Pauli-transfer-matrix formalism and a Fourier-like path expansion over Clifford plus RZ circuits, with truncation techniques to keep computations tractable. In a 127-qubit heavy-hex transverse-field Ising model, the surrogate reproduces key dynamics and enables high-resolution parameter surfaces with fast evaluations, indicating competitive performance against tensor-network and related classical methods. The work positions LOWESA as a complementary tool for verification, meta-learning, and rapid scanning of quantum-system families, with potential for further improvements in truncation strategies and symmetry exploitation.

Abstract

We introduce LOWESA as a classical algorithm for faithfully simulating quantum systems via a classically constructed surrogate expectation landscape. After an initial overhead to build the surrogate landscape, one can rapidly study entire families of Hamiltonians, initial states and target observables. As a case study, we simulate the 127-qubit transverse-field Ising quantum system on a heavy-hexagon lattice with up to 20 Trotter steps which was recently presented in Nature 618, 500-505 (2023). Specifically, we approximately reconstruct (in minutes to hours on a laptop) the entire expectation landscape spanned by the heavy-hex Ising model. The expectation of a given observable can then be evaluated at different parameter values, i.e. with different onsite magnetic fields and coupling strengths, in fractions of a second on a laptop. This highlights that LOWESA can attain state-of-the-art performance in quantum simulation tasks, with the potential to become the algorithm of choice for scanning a wide range of systems quickly.

Figures (6)

• Figure 1: Framework schematic of LOWESA for classically constructing surrogate expectation landscapes. The task is to simulate the expectation value of an observable $O$ for an initial state $\psi_0$ evolved under a unitary quantum circuit $U$ with parameters $\bm\theta$. Typically, such parameters are only introduced in the context of Variational Quantum Algorithms. However, depending on the particular task, they can also be viewed as parametrizing a change of the observable or the initial state. Another possibility is that they can indirectly be defined by the coefficients of a Hamiltonian for the purposes of studying time evolution dynamics. While simulation methods conventionally assign numerical values to the parameters $\bm\theta$ before simulation, LOWESA instead aims to build a classical surrogate of the entire expectation landscape $f(\bm\theta)$. The initial overhead of this approach may be high, but re-evaluation of the surrogate landscape for different parameter values can be exceptionally fast.
• Figure 2: Simulating 127-qubit TFI dynamics. The observables measured correspond to Figs. 3a, 3b, 3c and 4a from Ref. kim2023evidence, which also provides the exact curves and the mitigated quantum computer (QC) results. The observables are the magnetization $M_z = \frac{1}{127}\sum_i\langle Z_i\rangle$, $\langle X_{13,29,31}Y_{9,30}Z_{8,12,17,28,32}\rangle$, $\langle X_{37,41,52,56,57,58,62,79}Y_{75}Z_{38,40,42,63,72,80,90,91}\rangle$, and $\langle X_{37,41,52,56,57,58,62,79}Y_{38,40,42,63,72,80,90,91}Z_{75}\rangle$, respectively. After reconstructing the expectation landscape of these observables with $L=5$ (a-c) or $L=6$ (d) trotter steps, we are able to replicate close to exact expectation curves for the correlated angle case $\theta_h^{(i)}=\theta_h$. Note that the exact curves are mostly hidden because of the high-quality approximation. In panel a) we use $\ell=40, W=8$, and in panel b) $\ell=32, p=0.05$. For this restricted 1D "slice" of the expectation landscape, the trivial paths corresponding to the trigonometric monomials $\sin^{25}$ and $\sin^{34}$ were sufficient to provide near-exact dynamics in panel c) and d), respectively. The 31-qubit lightcone simulations were generated with the code accompanying Ref. kechedzhi2023effective.
• Figure 3: Simulating 127-qubit TFI dynamics with $L=20$ Trotter steps. Panel a) showcases the example of Fig. 4b from Ref. kim2023evidence, for which we are able to generate very plausible results that are broadly within error bars of the quantum hardware results and close to the approximate results demonstrated in Ref. kechedzhi2023effective. Since we approximately reconstructed the entire expectation landscape spanned by the Hamiltonian coefficients $h^{(i)}$ in Eq. \ref{['eq:TFI']}, we are able to generate additional $158\times 158$ pixel expectation surfaces measuring $\langle Z_{62}\rangle$ for alternating magnetic field values on the qubits (b) and for random noise on the magnetic fields $h^{(i)}$, drawn from a normal distribution with standard distribution $\sigma/\Delta t$ on each qubit (c). Both panels b) and c) contain the curve in panel a) as special case, which is along the diagonal of panel b), and the x-axis of panel c) with $\sigma=0$.
• Figure 4: Simulating 127-qubit TFI dynamics with individually controllable angles $\theta_J^{(i,j)}, \theta_h^{(i)}$. Extending the case of Fig. \ref{['fig:reproduce']}a, we now employ the circuit in Eq. \ref{['eq:trotter-circuit']} with fully flexible angles, i.e., non-Clifford entangling gates. Here we show how each single-site magnetization evolves in time given the initial condition labeled by $\Delta t = 0$, and either $J^{(i, j)}=1$ between all qubits (a), or a diagonal linear ramp (b) from $J^{(0, 1)} = 0$ at the top left qubit to $J^{(125, 126)}= -3$ at the lower right (indicated by the thickness of the connecting edges). Having reconstructed the expectation landscapes for all 127 single-qubit $Z$ operators, recalculation of a snapshot in time can be done in seconds.
• Figure 5: Improving approximation with higher truncation weight $W$. For the system considered in Fig. \ref{['fig:4b_and_more']}a, we show the convergence of the expectation curve, and thus the surrogate landscape, with operator weight truncation $W$, given fixed values for the cutoff frequency $\ell=36$ and a truncation probability of $p=0.05$.