Inverse Quantum Simulation for Quantum Material Design

TL;DR

The paper reframes quantum simulation as an inverse design task by encoding target material properties into a cost functional and minimizing it on quantum hardware to prepare a desired many-body state. It then uses Hamiltonian learning to reconstruct a physically interpretable $\hat H_{\rm opt}$ whose ground state best approximates the optimized state, enabling material design guidance. The authors demonstrate the framework across three axes: (i) enhancing $d$-wave pairing in fermionic Hubbard ladders, (ii) continuous-phase Hamiltonian learning to stabilize and extend topological phases such as in the CIM, and (iii) spectral Hamiltonian learning to design and infer dynamical properties from frequency-resolved data. Together, these results show that quantum simulators can be used not only to explore known models but to actively design and discover new quantum materials with tailored static and dynamical properties.

Abstract

Quantum simulation provides a powerful route for exploring many-body phenomena beyond the capabilities of classical computation. Existing approaches typically proceed in the forward direction: a model Hamiltonian is specified, implemented on a programmable quantum platform, and its phase diagram and properties are explored. Here we present a quantum algorithmic framework for inverse quantum simulation, enabling quantum material design with desired properties. Target material characteristics are encoded as a cost function, which is minimized on quantum hardware to prepare a many-body state with the desired properties in quantum memory. Hamiltonian learning is then used to reconstruct a low-energy Hamiltonian for which this state is an approximate ground state, yielding a physically interpretable model that can guide experimental synthesis. As illustrative applications, we outline how the method can be used to search for high-temperature superconductors within the fermionic Hubbard model, enhancing $d$-wave correlations over a broad range of dopings and temperatures, design quantum phases by stabilizing a topological order through continuous Hamiltonian modifications, and optimize dynamical properties relevant for photochemistry and frequency- and momentum-resolved condensed-matter data. These results extend the scope of quantum simulators from exploring quantum many-body systems to designing and discovering new quantum materials.

Figures (13)

• Figure 2: Designing a fermionic Hamiltonian with enhanced $d$-wave correlations. (a) Spatial profile of the $d$-wave correlation function $\langle \hat{\Delta}^\dagger_{1}\hat{\Delta}_{1 + r}\rangle$ in a four-site Hubbard ladder for circuits optimized at various depths $d$, where $\hat{\Delta}_i$ denotes the singlet annihilation operator on the rung at site $i$. Deeper circuits, targeting $-\sum_{r} \langle \hat{\Delta}^\dagger_{1}\hat{\Delta}_{1 + r}\rangle$, naturally yield stronger $d$-wave correlations. (b) Learning maps. Shown are the variance (left) of the learned Hamiltonian $\hat{H}_{\rm opt}$ and the fidelity (right) with respect to the ground state of $\hat{H}_{\rm opt}$, across $(\lambda,d)$. The cost function is $C[\lambda] = \langle \hat{H}_0\rangle - \lambda \sum_{r} \langle \hat{\Delta}^\dagger_{1}\hat{\Delta}_{1 + r}\rangle$, where $\hat{H}_0$ is the reference Hubbard model with $t_x=t_y=-1$ (defining the energy unit) and $U=4$; see panel (c). Learning degrades at large $\lambda$ or $d$ as the Hamiltonian ansatz lacks sufficient non-locality; extending it with additional non-local terms can restore performance. (c) Learned translationally invariant Hamiltonians for $d=5$ and $d=10$. Off-diagonal hoppings and off-site density–density interactions are identified as the dominant terms enhancing $d$-wave correlations; here, $\lambda=2$ and the interactions are restricted to be repulsive. (d) $d$-wave correlations remain enhanced across the full doping range of a $2\times 24$-site ladder (relative to the ground state of $\hat{H}_0$), despite using the learned Hamiltonian $\hat{H}_{\rm opt}$ (pentagon symbols in (b)) obtained on a four-site ladder at quarter filling. Enforcing all off-site density–density interactions to be repulsive (orange) still yields appreciable enhancement; allowing them to be attractive (green) can lead to a stronger effect.
• Figure 3: Quantum phase design. (a) Schematic of the CPHL algorithm. Starting from the phase diagram of a many-body system, the goal is to extend a target (possibly fragile) quantum phase. A region in parameter space is selected and discretized into a grid of points. At each point, a variational circuit is optimized with a tailored cost function designed to enhance the target phase. From the optimized circuits, Hamiltonian coefficients are learned for each grid point and subsequently smoothed into a continuous manifold. The phase diagram is updated with the resulting continuous Hamiltonian family, and the procedure is iterated until convergence. (b1) Application to the CIM in Eq. \ref{['eqn::main_CIM']}, a minimal 1D setting exhibiting a topological phase transition as the tuning parameter $g$ is varied from the topological phase ($g = -1$) to the ferromagnetic phase ($g = 1$). The CPHL algorithm successfully extends the topological phase at the expense of competing ferromagnetic correlations. (b2) The key stabilizing terms are identified as the AFM next-nearest-neighbor coupling $ZIZ$ and quartic interaction $ZZZZ$, both of which suppress ferromagnetism. (c) Variational staircase circuit with tunable two-qubit unitary $U(\boldsymbol{\theta})$ captures both ends of the CIM phase diagram. The circuit can be naturally implemented using periodic teleportation and qubit reset. Dashed lines in panel (b1) show results from circuit simulations optimized to maximize fidelities with the exact ground states, indicating that the topological phase extension could, in principle, be achieved through the staircase circuit alone.
• Figure 4: Design of linear dynamical properties. (a) Tangent space of the variational manifold ${\cal M}$ capturing weak time-dependent perturbations around $\ket{\psi(\boldsymbol{\theta})}$, used to evaluate frequency-resolved dynamical cost functions. On quantum hardware, this reduces to preparing $\ket{\psi(\boldsymbol{\theta})}$ and measuring a fixed set of Hermitian operators (see Methods). (b)-(c) Benchmark with three spins-$1/2$, where the task is to learn a single-parameter Hamiltonian containing the ring-exchange interaction $J_{\rm ring}$ from a target response function (red spectrum in (c)). Starting from an initial guess with a single (incorrect) peak (green spectrum in (c)), the algorithm rapidly converges (b) to the correct three-peak structure and recovers the target $J_{\rm ring}$ (dashed line). (d)-(e) Applications to frequency- and momentum-resolved condensed-matter data. (d) The dynamical spin structure factor of a ferromagnetic chain, ${\cal S}_{\rm FM}(\omega, k)$, exhibits sharp peaks at the magnon (left inset) dispersion. Given only this spectrum, the algorithm recovers the original two-parameter ferromagnetic Hamiltonian (right inset). (e) For the AFM Heisenberg chain, the dynamical structure factor, ${\cal S}_{\rm AFM}(\omega, k)$, exhibits a continuum of excitations (shaded region) bounded by lower and upper branches (dashed orange lines). For a finite chain, ${\cal S}_{\rm AFM}$ is discrete but still fills the region between the branches. Learning the correct Hamiltonian (right inset) from such spectra provides a stringent benchmark of the framework.
• Figure :
• Figure S1: Energy of the variational state $\exp(-i\theta \hat{A})\ket{\psi_0}$ as a function of $\theta$ for the Hubbard model with $t_x = t_y = -1$ (these set the unit of energy) and $U=8$. The reference state $\ket{\psi_0}$ is the ground state of the $U=4$ Hubbard Hamiltonian with the same hoppings. For the three natural Hubbard operators in the pool—hopping along $x$, hopping along $y$, and on-site interaction—we find that the gradient \ref{['eqn::grad_ADAPTVQE']} vanishes, while the second derivative is positive ($\partial^2_\theta E>0$), which makes the state $\ket{\psi_0}$ appear optimized. However, Hessian analysis [Eq. \ref{['eqn::Hessian']}] reveals a saddle point with an unstable direction ($\partial^2_\theta E<0$). Quadratic expansions (green crosses) are overlaid with exact numerics (blue diamonds), thereby providing a benchmark for the Hessian calculations.

Theorems & Definitions (1)

• proof