Simulating Quantum Many-Body Systems on Noisy Intermediate-Scale Quantum Computers

During the last twenty years, ultracold atoms in optical lattices have emerged as very versatile and powerful Quantum Simulators to study the many-body physics of interacting particles in periodic potentials. Not only can they faithfully reproduce many prototypical effects from condensed matter physics, they also enable radically new systems with fascinating physics and hold promise for wider quantum information applications. After a brief review of fundamental properties and key experiments that already reach far beyond what can be computed classically, this talk will present an outlook into current and coming developments and discuss future prospects of this platform in and beyond the NISQ era.

Our quantum experiment consists of a chain of 171Yb+ ions with individual Raman beam addressing and individual readout [1]. This fully connected system can be configured to run any sequence of single- and two-qubit gates, making it in effect an arbitrarily programmable digital quantum computer. The high degree of control can be used for digital, but also for analog and hybrid quantum simulations. We also add a classical optimization layer to our quantum stack to realize variational optimization methods. Using such an approach we produce so-called thermofield doubles [2], which allow the efficient creation of finite temperature quantum states. We recently employed these states to perform the first measurement at finite temperature of out-of-time-ordered correlators, which are are powerful tools to probe quantum information scrambling in quantum many-body systems [3]. We also present results from the digital simulation of the real-time dynamics of a lattice gauge theory in 1+1 dimensions, i.e., the lattice Schwinger model, and demonstrate effects such as pair creation for times much longer than previously accessible [4]. Finally, we show the first realization of para-particle dynamics using an analog simulation involving both spin and motional degrees of freedom of a trapped ion. These exotic particles are unlikely to be found in nature, but their dynamics have been used to study dark matter and excitations in solids, and our technique is a promising tool for other quantum many-body models. Finally, work towards scaling up the architecture will be discussed. [1] S. Debnath et al., Nature 563:63 (2016) [2] D. Zhu et al., Proc. Natl. Acad. Sci. 117:41 (2020) [3] A. M. Green et al., arXiv:2112.02068 (2021) [4] N. H. Nguyen et al., arXiv:2112.14262 (2021) [5] C. Huerta Alderete et al., arXiv:2108.05471 (2021)

We present recent theoretical and experimental developments on a hardware-efficient quantum computing architecture requiring only fixed couplings and single-qubit gates [1, 2, 3]. The control strategy saves the cost of calibrating pulsed two-qubit gates by replacing the entangling operation with the free evolution of the native Hamiltonian, inspired by dynamical decoupling techniques developed in the Nuclear Magnetic Resonance (NMR). We introduce algorithms for designing pulse sequences that can leverage static Ising Hamiltonians to perform quantum simulation, with favourable total time duration and pulse count scaling [2, 3]. We then show how the pulse sequence generated by this method can be applied to hardware-efficient Noisy Intermediate-Scale Quantum (NISQ) algorithms and quantum error mitigation. We then discuss the practicality of such quantum computing strategy on superconducting circuits by showing a proof-of-principle experiment on a two transmon qubit system, using the always-on residual ZZ coupling to perform a Variational Quantum Eigensolver algorithm. We will address practical challenges in controlling and measuring a device with always-on couplings and outline future circuit design improvements to circumvent these issues. [1] A. Parra-Rodriguez, P. Lougovski, L. Lamata, E. Solano, and M. Sanz, Phys, Rev. A 101, 022305 (2020) [2] G. Bhole, T. Tsunoda, P. J. Leek, and J. A. Jones, Phys. Rev. Applied 13, 034002 (2020). [3] T. Tsunoda, G. Bhole, S. A. Jones, J. A. Jones, and P. J. Leek, Phys, Rev. A 102, 032405 (2020).

Entanglement is the crucial ingredient of quantum many-body physics, and characterizing and quantifying entanglement in closed system dynamics of quantum simulators is an outstanding challenge in today's era of intermediate scale quantum devices. Here we discuss an efficient tomographic protocol for reconstructing reduced density matrices and entanglement spectra for spin systems. The key step is a parametrization of the reduced density matrix in terms of an entanglement Hamiltonian involving only quasi local few-body terms. We show the validity and efficiency of the protocol and demonstrate experimentally the measurement of the evolution of the entanglement spectrum in quench dynamics in trapped ion quantum simulators. [1] C. Kokail, R. van Bijnen, A. Elben, B. Vermersch and P. Zoller, Nature Physics 17, 936 (2021)

In this talk, I will discuss a hybrid quantum-classical approach to learn the entanglement Hamiltonian for subsystems of quantum states prepared in quantum simulation experiments. The approach is based on time-evolving subsystems of the quantum state under a deformed Hamiltonian which is physically realized on the programmable quantum device. Subsequent many-body spectroscopy allows for a direct measurement of the entanglement spectrum, which I will discuss for a Hubbard model in a topological phase.

The accurate implementation of quantum gates is essential for the realisation of quantum algorithms and digital quantum simulations. We demonstrate a method by which this accuracy may be increased on noisy hardware through variational optimisation techniques, using a hierarchy of fidelity approximations to overcome the difficulty of estimating the process fidelity of a quantum gate. We demonstrate the utility of this protocol through the successful optimisation of a quantum gate on a commercial quantum processor.

Heating to high-lying states strongly limits the experimental observation of driving induced non-equilibrium phenomena, particularly when the drive has a broad spectrum. Here we show that, for entire families of structured random drives known as random multipolar drives, particle excitation to higher bands can be well controlled even away from a high-frequency driving regime. This opens a window for observing drive-induced phenomena in a long-lived prethermal regime in the lowest band.

The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. It will be shown how to transform the ABA into a quantum circuit that can be implemented on a quantum computer. We illustrate our method in the spin 1/2 XX and XXZ models running numerical simulations, for systems of up to 24 qubits and 12 magnons. Furthermore, we run small-scale error-mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models in 4 sites.

Randomized measurements provide a novel toolbox to probe today’s noisy intermediate scale quantum devices. In this talk, I will mostly focus on the study of many-body quantum chaos in quantum simulation experiments. For quantum spin models, I will present a protocol to access the spectral form factor revealing properties of the energy eigenvalue statistics. Furthermore, I will define partial spectral form factors which refer to subsystems of the many-body system. I will show that partial spectral form factors give unique insights into energy eigenstate statistics determining thermalization in closed quantum systems. The presented measurement protocol allows to access both, SFF and pSFFs, from a single experimental dataset. It provides a unified testbed to probe many-body quantum chaotic behavior and thermalization, determined by properties of energy eigenvalues and eigenstates, in quantum simulators.

The realisation of a large-scale error corrected quantum computer relies on our ability to reproducibly manufacture qubits that are fast, highly coherent, controllable and stable. The promise of achieving this in a highly manufacturable platform such as silicon requires a deep understanding of the materials issues that impact device operation. In this talk I will demonstrate our progress to engineer every aspect of device behaviour in atomic qubits in silicon. This will cover the use of atomic precision lithography to achieve fast, controllable exchange coupling, qubit initialisation and read-out; high quality epitaxial growth to create all epitaxial gate structures allowing for highly stable qubits; and unique imaging and modelling techniques that provide a deep understanding of the impact of the solid state environment on qubit designs and operation.

I would like to present a workflow to use current quantum computing hardware for solving quantum many-body problems, using the example of the fermionic Hubbard model. Concretely, we study a four-site Hubbard ring that exhibits a transition from a product state to an intrinsically interacting ground state as hopping amplitudes are changed. We locate this transition and solve for the ground state energy with high quantitative accuracy using a variational quantum algorithm executed on an IBM quantum computer. Our results are enabled by a variational ansatz that takes full advantage of the maximal set of commuting $\mathbb{Z}_2$ symmetries of the problem and a Lanczos-inspired error mitigation algorithm. They are a benchmark on the way to exploiting near term quantum simulators for quantum many-body problems.

Novel dynamical phases that violate ergodicity have been a subject of extensive research in recent years. A periodically driven system is naively expected to lose all memory of its initial state due to thermalization, yet this can be avoided in the presence of many-body localization. A discrete time crystal represents a driven system whose local observables spontaneously break time translation symmetry and retain memory of the initial state indefinitely. Here, we report the observation of a discrete time crystal on a chain consisting of 57 superconducting qubits on a state-of-the-art quantum computer. We probe random initial states and compare the cases of vanishing and finite disorder to distinguish many-body localization from prethermal dynamics. We further report results on the dynamical phase transition between the discrete time crystal and a thermal regime, which is observed via critical fluctuations in the system’s subharmonic frequency response and a significant speedup of spin depolarization.

The recent progress in quantum hardware has sparked great interest in harnessing the power of machine learning in quantum computation. An efficient procedure for loading the data to the quantum devices becomes a crucial first step towards this goal. In this work, we propose a set of parameterized quantum circuit ansatzes, inspired by the matrix-product-state structure, for both data loading and data classification. The ansatzes are particularly suited for near-term realization on NISQ devices. Compression on both the input data and the classifier can be achieved by varying the depth of the quantum circuits. We demonstrate their effectiveness by classical simulation on the full Fashion-MNIST set and discuss their implications. If time permits, in the second part of the talk, we will discuss some recent progress of an ongoing work on quantum phase recongnition by training quantum circuits, where the input data are ground states of quantum many-body systems.

The sequential generation of tensor network states provides a way to deterministically prepare entangled states on both matter-based and photon- based quantum devices. In this talk, first, we discuss two implementations to sequentially generate photonic matrix product states, one based on a Rydberg atomic array [1], and another based on a microwave cavity dispersively coupled to a transmon [2]. We show both implementations can generate a large number of entangled photons. Then, we introduce plaquette projected entangled-pair states (p-PEPS)[3], a class of states in a lattice that can be generated by applying sequential unitaries acting on plaquettes of overlapping regions. They satisfy area- law entanglement, possess long-range correlations, and naturally generalize other relevant classes of tensor network states. We identify a subclass that can be more efficiently prepared in a radial fashion and that contains the family of isometric tensor network states. We also show how such subclass can be efficiently prepared using an array of photon sources, and devise a physical realization by extending the above cavity-transmon setup [2]. [1] Zhi-Yuan Wei et al., Physical Review Research, 3(2), 023021. [2] Zhi-Yuan Wei et al., arXiv:2109.06781 [3] Zhi-Yuan Wei et al., PRL 128 010607 (2022)

Variational quantum algorithms are promising algorithms for achieving quantum advantage on near-term devices. The quantum hardware is used to implement a variational wave function and measure observables, whereas the classical computer is used to store and update the variational parameters. The optimization landscape of expressive variational ansätze is however dominated by large regions in parameter space, known as barren plateaus, with vanishing gradients which prevents efficient optimization. In this work we propose a general algorithm to avoid barren plateaus in the initialization and throughout the optimization. To this end we define a notion of weak barren plateaus (WBP) based on the entropies of local reduced density matrices. The presence of WBPs can be efficiently quantified using recently introduced shadow tomography of the quantum state with a classical computer. We demonstrate that avoidance of WBPs suffices to ensure sizable gradients in the initialization. In addition, we demonstrate that decreasing the gradient step size, guided by the entropies allows to avoid WBPs during the optimization process. This paves the way for efficient barren plateau free optimization on near-term devices.

The discovery of topological order has revised the understanding of quantum matter and provided the theoretical foundation for many quantum error–correcting codes. Realizing topologically ordered states has proven to be challenging in both condensed matter and synthetic quantum systems. We prepared the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measured a topological entanglement entropy near the expected value of –ln2 and simulated anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigated key aspects of the surface code, including logical state injection and the decay of the nonlocal order parameter. Our results demonstrate the potential for quantum processors to provide insights into topological quantum matter and quantum error correction.

Confinement describes the phenomenon when the attraction between two particles grows with their distance, most prominently found in quantum chromodynamics (QCD) between quarks. In condensed matter physics, confinement can appear in quantum spin chains, for example, in the one dimensional transverse field Ising model (TFIM) with an additional longitudinal field. Here, I will discuss the capabilities of state-of-the-art quantum computers to simulate confinement physics in spin chains. We report quantitative confinement signatures of the TFIM on an IBM quantum computer observed via two distinct velocities for information propagation from domain walls and their mesonic bound states. We also find the confinement induced slow down of entanglement spreading by implementing randomized measurement protocols for the second order Rényi entanglement entropy. I will discuss the importance of error mitigation protocols for using digital quantum computers as reliable quantum simulators.