Energy spectroscopy is a powerful tool with diverse applications across various disciplines. Variational quantum-classical algorithms based on programmable digital quantum simulators have emerged as promising approaches for conducting spectroscopy on various models using a single device, despite facing significant quantum and classical resource overheads. Here, we experimentally demonstrate multilevel variational spectroscopy for fundamental many-body Hamiltonians using a superconducting digital quantum simulator. By exploiting symmetries and the subspace search method, we achieve full spectroscopy for a four-qubit Heisenberg spin chain, yielding an average energy deviation as small as 0.13 from the theoretical values, assuming unity coupling strength. Our method, when extended to eight-qubit Heisenberg and transverse-field Ising Hamiltonians, successfully determines the three lowest-energy levels. In achieving the above, we introduce a circuit-agnostic compilation method that enhances the robustness of our simulator against signal crosstalk. Our study highlights the combination of the subspace search method and symmetry-assisted resource efficiency in variational quantum algorithms and lays the foundation for practical spectroscopy on near-term quantum simulators, with potential applications in quantum chemistry and condensed matter physics. Published by the American Physical Society 2024
Energy spectroscopy is a powerful tool with diverse applications across various disciplines. Variational quantum-classical algorithms based on programmable digital quantum simulators have emerged as promising approaches for conducting spectroscopy on various models using a single device, despite facing significant quantum and classical resource overheads. Here, we experimentally demonstrate multilevel variational spectroscopy for fundamental many-body Hamiltonians using a superconducting digital quantum simulator. By exploiting symmetries and the subspace search method, we achieve full spectroscopy for a four-qubit Heisenberg spin chain, yielding an average energy deviation as small as 0.13 from the theoretical values, assuming unity coupling strength. Our method, when extended to eight-qubit Heisenberg and transverse-field Ising Hamiltonians, successfully determines the three lowest-energy levels. In achieving the above, we introduce a circuit-agnostic compilation method that enhances the robustness of our simulator against signal crosstalk. Our study highlights the combination of the subspace search method and symmetry-assisted resource efficiency in variational quantum algorithms and lays the foundation for practical spectroscopy on near-term quantum simulators, with potential applications in quantum chemistry and condensed matter physics. Published by the American Physical Society 2024
Utilizing quantum computer to investigate quantum chemistry is an important research field nowadays. In addition to the ground-state problems that have been widely studied, the determination of excited-states plays a crucial role in the prediction and modeling of chemical reactions and other physical processes. Here, we propose a non-variational full circuit-based quantum algorithm for obtaining the excited-state spectrum of a quantum chemistry Hamiltonian. Compared with previous classical-quantum hybrid variational algorithms, our method eliminates the classical optimization process, reduces the resource cost caused by the interaction between different systems, and achieves faster convergence rate and stronger robustness against noise without barren plateau. The parameter updating for determining the next energy-level is naturally dependent on the energy measurement outputs of the previous energy-level and can be realized by only modifying the state preparation process of ancillary system, introducing little additional resource overhead. Numerical simulations of the algorithm with hydrogen, LiH, H2O and NH3 molecules are presented. Furthermore, we offer an experimental demonstration of the algorithm on a superconducting quantum computing platform, and the results show a good agreement with theoretical expectations. The algorithm can be widely applied to various Hamiltonian spectrum determination problems on the fault-tolerant quantum computers.
One of the key applications for the emerging quantum simulators is to emulate the ground state of many-body systems, as it is of great interest in various fields from condensed matter physics to material science. Traditionally, in an analog sense, adiabatic evolution has been proposed to slowly evolve a simple Hamiltonian, initialized in its ground state, to the Hamiltonian of interest such that the final state becomes the desired ground state. Recently, variational methods have also been proposed and realized in quantum simulators for emulating the ground state of many-body systems. Here, we first provide a quantitative comparison between the adiabatic and variational methods with respect to required quantum resources on digital quantum simulators, namely the depth of the circuit and the number of two-qubit quantum gates. Our results show that the variational methods are less demanding with respect to these resources. However, they need to be hybridized with a classical optimization which can converge slowly. Therefore, as the second result of the paper, we provide two different approaches for speeding the convergence of the classical optimizer by taking a good initial guess for the parameters of the variational circuit. We show that these approaches are applicable to a wide range of Hamiltonian and provide significant improvement in the optimization procedure.
Estimating the eigenvalue or energy gap of a Hamiltonian H is vital for studying quantum many-body systems. Particularly, many of the problems in quantum chemistry, condensed matter physics, and nuclear physics investigate the energy gap between two eigenstates. Hence, how to efficiently solve the energy gap becomes an important motive for researching new quantum algorithms. In this work, we propose a hybrid non-variational quantum algorithm that uses the Monte Carlo method and real-time Hamiltonian simulation to evaluate the energy gap of a general quantum many-body system. Compared to conventional approaches, our algorithm does not require controlled real-time evolution, thus making its implementation much more experimental-friendly. Since our algorithm is non-variational, it is also free from the "barren plateaus" problem. To verify the efficiency of our algorithm, we conduct numerical simulations for the Heisenberg model and molecule systems on a classical emulator.
Quantum computing promises to revolutionize several scientific and technological domains through fundamentally new ways of processing information. Among its most compelling applications is digital quantum simulation, where quantum computers are used to replicate the behavior of other quantum systems. This could enable the study of problems that are otherwise intractable on classical computers, transforming fields such as quantum chemistry, condensed matter physics, and materials science. Despite this potential, realizations of practical quantum advantage for relevant problems are hindered by imperfections of current devices. This also affects quantum hardware based on superconducting circuits which is among the most advanced and scalable platforms. The envisaged long-term solution of fault-tolerant quantum computers that correct their own errors remains out of reach mainly due to the associated qubit number overhead. As a result, the field has developed strategies that combine quantum and classical resources, exploit hardware-native operations, and employ error mitigation techniques to extract meaningful results from noisy data. This doctoral thesis contributes to this broader effort by exploring methods for advancing quantum simulation across the full computational stack, including hardware-level innovations, refined techniques for noise modeling and error mitigation, and algorithmic improvements enabled by efficient measurement processing.
Determining quantum excited states is crucial across physics and chemistry but presents significant challenges for variational methods, primarily due to the need to enforce orthogonality to lower-energy states, often requiring state-specific optimization, penalty terms, or specialized ansatz constructions. We introduce a novel variational principle that overcomes these limitations, enabling the \textit{simultaneous} determination of multiple low-energy excited states. The principle is based on minimizing the trace of the inverse overlap matrix multiplied by the Hamiltonian matrix, Tr(S−1H), constructed from a set of \textit{non-orthogonal} variational states {∣ψi⟩}. Here, Hij=⟨ψi∣H∣ψj⟩ and Sij=⟨ψi∣ψj⟩ are the elements of the Hamiltonian and overlap matrices, respectively. This approach variationally optimizes the entire low-energy subspace spanned by {∣ψi⟩} without explicit orthogonality constraints or penalty functions. We demonstrate the power and generality of this method across diverse physical systems and variational ansatzes: calculating the low-energy spectrum of 1D Heisenberg spin chains using matrix product states, finding vibrational spectrum of Morse potential using quantics tensor trains for real-space wavefunctions, and determining excited states for 2D fermionic Hubbard model with variational quantum circuits. In all applications, the method accurately and simultaneously obtains multiple lowest-lying energy levels and their corresponding states, showcasing its potential as a unified and flexible framework for calculating excited states on both classical and quantum computational platforms.
The real-time simulation of large many-body quantum systems is a formidable task, that may only be achievable with a genuine quantum computational platform. Currently, quantum hardware with a number of qubits sufficient to make classical emulation challenging is available. This condition is necessary for the pursuit of a so-called quantum advantage, but it also makes verifying the results very difficult. In this paper, we flip the perspective and utilize known theoretical results about many-body quantum critical dynamics to qualitatively benchmark digital quantum hardware and various error mitigation techniques. In particular, we benchmark against known universal scaling laws in the Hamiltonian simulation of a time-dependent transverse-field Ising Hamiltonian of up to 133 qubits. Incorporating only basic error mitigation and suppression methods, our study shows reliable control up to a two-qubit gate depth of 28, featuring a maximum of 1396 two-qubit gates, before noise becomes prevalent. These results are transferable to applications such as Hamiltonian simulation, variational algorithms, optimization, or quantum machine learning. We demonstrate this on the example of digitized quantum annealing for optimization and identify an optimal working point in terms of both circuit depth and time step on a 133-site optimization problem. Published by the American Physical Society 2024