The dynamical structure factor is one of the experimental quantities crucial in scrutinizing the validity of the microscopic description of strongly correlated systems. However, despite its long-standing importance, it is exceedingly difficult in generic cases to numerically calculate it, ensuring that the necessary approximations involved yield a correct result. Acknowledging this practical difficulty, we discuss in what way results on the hardness of classically tracking time evolution under local Hamiltonians are precisely inherited by dynamical structure factors and, hence, offer in the same way the potential computational capabilities that dynamical quantum simulators do: We argue that practically accessible variants of the dynamical structure factors are bounded-error quantum polynomial time ([Formula: see text])-hard for general local Hamiltonians. Complementing these conceptual insights, we improve upon a novel, readily available measurement setup allowing for the determination of the dynamical structure factor in different architectures, including arrays of ultra-cold atoms, trapped ions, Rydberg atoms, and superconducting qubits. Our results suggest that quantum simulations employing near-term noisy intermediate-scale quantum devices should allow for the observation of features of dynamical structure factors of correlated quantum matter in the presence of experimental imperfections, for larger system sizes than what is achievable by classical simulation.
The dynamical structure factor is one of the experimental quantities crucial in scrutinizing the validity of the microscopic description of strongly correlated systems. However, despite its long-standing importance, it is exceedingly difficult in generic cases to numerically calculate it, ensuring that the necessary approximations involved yield a correct result. Acknowledging this practical difficulty, we discuss in what way results on the hardness of classically tracking time evolution under local Hamiltonians are precisely inherited by dynamical structure factors and, hence, offer in the same way the potential computational capabilities that dynamical quantum simulators do: We argue that practically accessible variants of the dynamical structure factors are bounded-error quantum polynomial time ([Formula: see text])-hard for general local Hamiltonians. Complementing these conceptual insights, we improve upon a novel, readily available measurement setup allowing for the determination of the dynamical structure factor in different architectures, including arrays of ultra-cold atoms, trapped ions, Rydberg atoms, and superconducting qubits. Our results suggest that quantum simulations employing near-term noisy intermediate-scale quantum devices should allow for the observation of features of dynamical structure factors of correlated quantum matter in the presence of experimental imperfections, for larger system sizes than what is achievable by classical simulation.
The dynamical structure factor is a central observable in condensed-matter physics, providing direct insight into the excitation spectrum and dynamical response of quantum materials. While it is traditionally accessed in real materials through inelastic neutron scattering, recent works have shown that analogous information can be extracted in quantum simulators using suitable dynamical protocols. So far, these approaches have been demonstrated mainly in paradigmatic integrable one-dimensional models. In this work, we use numerical emulations of a neutral-atom quantum processing unit to assess the feasibility of measuring the dynamical structure factor in a broader class of Ising-like spin systems. Beyond the standard one-dimensional transverse-field Ising chain, we benchmark the protocol in chains with dimerized interactions and in the two-dimensional transverse-field Ising model, where spectral properties are difficult to access with classical numerical methods at large scales. We further analyze the robustness of the protocol under realistic experimental conditions, including finite simulation times, pulse modulation, positional disorder, and laser noise. Our results show that neutral-atom quantum simulators can provide a practical route to probing dynamical response functions in regimes where classical simulations become increasingly demanding, paving the way toward experimental implementation.
Quantum science and engineering increasingly target the exploitation and control of large-scale dynamical systems. As system size grows, novel phenomena and quantum advantages emerge, but the resulting many-body dynamics become correspondingly harder to monitor and characterize. In particular, the underlying Hamiltonian often lacks symmetries, and the induced many-body dynamics quickly become hard to track. Focusing on one of the most important quantum features---entanglement---we demonstrate that, for a broad class of common Hamiltonian models, one can design effective witnesses that detect both the presence and structure of entanglement over relatively long evolution times. Inspired by stabilizer-based simulation, we track the evolution of generalized stabilizers of the underlying state and use them to construct linear entanglement witnesses. Through numerical experiments on Heisenberg-type Hamiltonians with up to twelve qubits, we show that the required number of measurement settings can be kept at a modest scale as the system size increases, and entanglement detection remains effective over evolution times that significantly exceed those accessible to naive approaches. The corresponding calculations can be carried out largely symbolically. Furthermore, unlike accurate state simulation, the stabilizer-based design of entanglement witnesses can be performed in a much more coarse-grained, digitized manner. We hope these results will stimulate further exploration of the quantum control of large systems.
Learning the Hamiltonian of a many-body system from its dynamics is a central task in quantum science, yet the algorithms with the strongest provable guarantees assume some level of quantum control--fast, arbitrary single-qubit gates interleaved with time evolution, and measurements in arbitrary bases--that is beyond the capabilities of near-term analog quantum simulators. Motivated by analog atom- and ion-based platforms, we study Hamiltonian learning under minimal access models.
Uniform state preparation and measurements: We first consider the setting where in every experiment, one can rotate each qubit to the same state, perform short-time evolution, and measure every qubit in the same basis. Surprisingly, we show that for generic 2-local Hamiltonians on any interaction graph, all of the parameters can be reconstructed from such experiments.
Computational basis state preparation and measurements: We then consider a similarly constrained setting, but where state preparation and measurement are restricted to the computational basis. For nearest-neighbor Hamiltonians with only Pauli X/Z interactions, a class which captures contemporary Rydberg atom platforms, we show that over 1D and 2D rectangular lattices, all of the parameters can be reconstructed from such experiments up to unavoidable gauges.
Our protocols introduce new techniques for solving structured polynomial systems over an extensive number of parameters. Taken together, our results suggest that one can learn a great deal from the dynamics of quantum many-body systems even under the most stringent experimental constraints.
In the past decades, rapid advances in the experimental control of quantum systems have opened up unparalleled capabilities of engineering exotic quantum states. Nowadays, we may consider ourselves witnesses of a second quantum revolution as strongly correlated quantum matter is regularly created, on a daily basis, within the most diverse platforms, e.g. arrays of Rydberg atoms, ultra-cold atoms in optical lattices, superconducting qubits, trapped ions, and quantum dots. In this era, dubbed Noisy-Intermediate Scale Quantum (NISQ) era, the effort in pursuing research for realizing quantum technologies with practical purposes, such as quantum computing, simulations, communication and metrology, has greatly accelerated, and we are just starting to experience the immense progress that could be achieved. The remarkable breakthrough we are facing builds upon seminal theoretical and experimental advances in quantum physics. The key achievements in atomic, molecular, and optical (AMO) physics have opened up the possibility of controlling, trapping, and measuring single atoms, one by one, with high accuracy and reliability. In parallel, from a theoretical point of view, the study of quantum entanglement and correlations has bridged AMO physics and quantum information with crucial proposals for the realization of universal quantum computers. In this context, entanglement has emerged as one of the key tools to characterize and to exploit quantum many-body systems for quantum information purposes. We will study quantum entanglement in several scenarios to investigate and probe complex quantum many-body systems. We will consider examples ranging from generic mixed states in equilibrium to out-of-equilibrium dynamics, with and without dissipation, and topologically non-trivial systems. The leitmotif of this work will be how entanglement and correlations can be exploited to characterize the many-body quantum state describing a physical system and how entanglement can be detected in an experimentally efficient manner.
The accurate and efficient simulation of many body systems has been a long-standing challenge for quantum chemists and physicists. At the heart of this challenge lies the correlation between quantum particles, which is crucial in understanding various physical phenomena, but is numerically hard to calculate. Over the decades, there has been remarkable progress in simplifying this problem aided by mathematical and chemical approximations and the availability of powerful computers. However, an exact solution to the problem still remains intractable as it requires exponential re- sources with respect to the system size. Recent advances in the field of quantum computing have shown promise in Feynman’s idea of simulating quantum mechanics on quantum computers. It has been proposed that an ideal fault-tolerant quantum computer can reduce the scaling of these sim- ulations from exponential to polynomial. However, initial quantum algorithms have shown that there is a large gap between the capabilities of current hardware and the resources required for simulating quantum systems of interest. The work of this thesis focuses on reducing this gap by introducing two novel algorithms for quantum xiv chemistry simulations, which are suitable for near-term hardware. First, we intro- duce a hybrid algorithm to decrease the total number of operations required for the quantum simulations. We use a classical computer to generate an effective Hamilto- nian containing O(n2) terms compared to the O(n4) terms of the full Hamiltonian, where n is the number of orbitals. This sparse Hamiltonian can then be used with a high-level method to recover the ground state energies. We demonstrate that this sparsification of the Hamiltonian reduces the number of quantum operations required for the simulation by an order of magnitude, thus making it accessible for the near- term Hardware. Our second algorithm aims to make the calculation of dynamic correlation functions (Green’s functions) more feasible on these quantum machines. This algorithm is designed with the motive of avoiding the use of time-evolution and reduced use of two-qubit gates. We demonstrate that we can reproduce the Green’s function within reasonable error limits using this approach. We have also discussed the use of re-sampling techniques for proper error-propagation of the stochastic data obtained from the quantum machines. The general structure of this thesis is as follows. We begin by motivating this work in Chapter 1. Chapters 2 and 3 provide a general introduction of quantum computing and Green’s function based methods, respectively. In chapter 4, the development of effective Hamiltonian is discussed in detail. Chapter 5 demonstrates the use of these effective Hamiltonians on the quantum machine for time evolution. In chapter 6, we have discussed the algorithm to calculate Green’s functions on a quantum machine. This is followed by conclusions and future directions in Chapters 7 and 8, respectively.
Recent progress in the development of quantum technologies has enabled the direct investigation of the dynamics of increasingly complex quantum many-body systems.This motivates the study of the complexity of classical algorithms for this problem in order to benchmark quantum simulators and to delineate the regime of quantum advantage.Here, we present classical algorithms for approximating the dynamics of local observables and nonlocal quantities such as the Loschmidt echo, where the evolution is governed by a local Hamiltonian.For short times, their computational cost scales polynomially with the system size and the inverse of the approximation error.In the case of local observables, the proposed algorithm has a better dependence on the approximation error than algorithms based on the Lieb-Robinson bound.Our results use cluster-expansion techniques adapted to the dynamical setting, for which we give a novel proof of their convergence.This has important physical consequences besides our efficient algorithms.In particular, we establish a novel quantum speed limit, a bound on dynamical phase transitions, and a concentration bound for product states evolved for short times.