Studying energy-resolved transport with wavepacket dynamics on quantum computers

TL;DR

This work introduces wavepacket dynamics as a pathway to energy-resolved transport studies on quantum hardware, leveraging Gaussian wavepackets to access tunable energy with reduced variance. It demonstrates a finite-size mobility edge in the 2D Anderson model by comparing time evolution of low- and high-energy wavepackets on Quantinuum's H2-2, and mitigates noise with an IID-MLE scheme that outperforms post-selection. It also develops a quantum algorithm for preparing quasiparticle wavepackets in a 1D XXZ model, enabling exploration of interacting transport with modest quantum resources. Overall, the results show that wavepacket-based probes, coupled with efficient error mitigation, can reveal energy-dependent transport phenomena on near-term quantum devices and lay groundwork for future many-body transport studies.

Abstract

Probing energy-dependent transport in quantum simulators requires preparing states with tunable energy and small energy variance. Existing approaches often study quench dynamics of simple initial states, such as computational basis states, which are far from energy eigenstates and therefore limit the achievable energy resolution. In this work, we propose using wavepackets to probe transport properties with improved energy resolution. To demonstrate the utility of this approach, we prepare and evolve wavepackets on Quantinuum's H2-2 quantum computer and identify an energy-dependent localization transition in the Anderson model on an 8x7 lattice--a finite-size mobility edge. We observe that a wavepacket initialized at low energy remains spatially localized under time evolution, while a high-energy wavepacket delocalizes, consistent with the presence of a mobility edge. Crucial to our experiments is an error mitigation strategy that infers the noiseless output bit string distribution using maximum-likelihood estimation. Compared to post-selection, this method removes systematic errors and reduces statistical uncertainty by up to a factor of 5. We extend our methods to the many-particle regime by developing a quantum algorithm for preparing quasiparticle wavepackets in a one-dimensional model of interacting fermions. This technique has modest quantum resource requirements, making wavepacket-based studies of transport in many-body systems a promising application for near-term quantum computers.

Figures (11)

• Figure 1: Energy-dependent localization revealed through wavepacket dynamics. Left panel: a) The IPR of the energy eigenstates $|\psi_E\rangle$ for a variety of disorder strengths $W$. The IPR is defined in Eq. \ref{['eq:IPR']}, and a larger value corresponds to a more localized wavefunction. b) The overlap of wavepackets with a variety of momentum $\vec{k}_0$ onto energy eigenstates with $W=3$. The wavepacket is defined in Eq. \ref{['eq:2dWP']}, and a momentum spread of $\vec{\sigma}_p=(0.1,0.1)$ has been used. All results in this panel were computed for a $50\times 50$ lattice and averaged over 2000 disorder realizations. Energies have been rescaled to lie in the interval $E\in[0,1]$. Right panels: The time-evolution of the probability density $p_{x,y}$ of a low-energy wavepacket (top) with ${\vec{k}}_0 = (0,0)$ and a high-energy wavepacket (bottom) with ${\vec{k}}_0 = (0.75\pi,0.75\pi)$ at times $t=0$ and $t=250$ for a single disorder instance with $W=3$. The time dependence of the IPR (right plots) reveals that the low-energy wavepacket settles to a finite value, i.e. it remains localized, whereas the high-energy wavepacket quickly decays to a small value, i.e. it becomes delocalized.
• Figure 2: Overview of quantum simulations. a) The 2D torus is mapped to 56 qubits on an $8\times7$ lattice. The circuit begins with initialization layers that prepare the single-particle wavepacket of Eq. \ref{['eq:2dWP']} (see Appendix \ref{['a:qcircs']} for their explicit construction). Then, $n_T$ steps of Trotterized time evolution under the 2D Anderson Hamiltonian in Eq. \ref{['eq:H_tightbind_spin']} are applied. Each Trotter step has four layers of nearest-neighbor hopping terms followed by a layer of $R_Z$ gates implementing the disorder. The hopping terms are represented by gold square dumbbells that implement $\exp(i \delta t (XX+YY)/2)$. The decomposition into gates is given by the $XXZ(-\delta t,0)$ circuit block in Fig. \ref{['fig:singletWP']}b). b) The single-particle probability density $p_{x,y}$ is determined from z-basis measurements. Low- (high-) energy wavepackets exhibit more (less) localized probability densities due to the finite-size mobility edge.
• Figure 3: The IPR obtained from H2-2. The IPR evaluated at a selection of simulation times for both the low-energy (blue) and high-energy (red) wavepackets. The IPR is defined in Eq. \ref{['eq:IPR']}, and a larger value corresponds to a more localized wavefunction. The dashed lines are the results of noiseless classical simulations. The points with error bars are the results obtained from Quantinuum's H2-2 quantum computer, and at $t=0,1$ have been offset for clarity. The circles and triangles correspond to error mitigation using post-selection (PS) and max-likelihood estimation (MLE), respectively. The error bars are statistical and are determined by bootstrap resampling over the measured bit strings. These results are tabulated in Table \ref{['tab:quantum_result_IPR']}.
• Figure 4: a) The single-particle energy spectrum in the 1D $XXZ$ model with no interactions. The ground state at half-filling has all states below the Fermi surface (dashed line) occupied. There are four degenerate lowest energy excitations corresponding to the particle hole excitations represented by the gray arrows. b) The many-body spectrum of the lowest energy excitations for each momentum (the $|\psi_k\rangle$). The energies are determined from exact diagonalization for $N=26$ and are shown for a selection of interaction strengths $\Delta$. The ground state energy has been subtracted.
• Figure 5: a) The structure of a quantum circuit that prepares a quasiparticle wavepacket in the 1D $XXZ$ model. The wavefunction at intermediate points are marked with blue dashed lines and defined in the main text. b) The definition of the circuit blocks, with $XXZ(\theta_i, \theta_j) = e^{-\frac{i}{2}\left [ \theta_i(XX+YY)+\theta_j ZZ\right ]}$. For $\theta_i=-\delta t,\theta_j=0$ the gold circuit simplifies to two CNOTs and implements the evolution of the nearest neighbor hopping term represented by the gold square dumbbell in Fig. \ref{['fig:Quantum_results']}. Circuits that prepare $|W(k_0)\rangle$ are given in Appendix \ref{['a:qcircs']}.