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.
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.
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 nonvariational 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 for near-term quantum devices and early fault-tolerant quantum computers. To verify the efficiency of our algorithm, we conduct numerical simulations for the Heisenberg model and molecule systems on a classical emulator.
Determining the energy gap in a quantum many-body system is critical to understanding its behavior and is important in quantum chemistry and condensed matter physics. The challenge of determining the energy gap requires identifying both the excited and ground states of a system. In this work, we consider preparing the prior distribution and circuits for the eigenspectrum of structured, time-independent Hamiltonians (e.g., local, sparse, or symmetry-constrained), which can benefit both classical and quantum algorithms for solving eigenvalue problems. The proposed algorithm unfolds in three strategic steps: Hamiltonian transformation, parameter representation, and classical clustering. These steps are underpinned by two key insights: the use of quantum circuits to approximate the ground state of transformed Hamiltonians and the analysis of parameter representation to distinguish between eigenvectors. The algorithm is showcased through applications to the 1D Heisenberg system and the LiH molecular system, highlighting its potential for both near-term quantum devices and fault-tolerant quantum devices, restricted to structured Hamiltonians where approximate state preparation is feasible. The paper also explores the scalability of the method and its performance across various settings, setting the stage for more resource-efficient quantum computations that are both accurate and fast. The findings presented here mark a new insight into hybrid algorithms, offering a pathway to overcoming current computational challenges.
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.
The variational quantum eigensolver (VQE) typically optimizes variational parameters in a quantum circuit to prepare eigenstates for a quantum system. Its application to many problems may involve a group of Hamiltonians, e.g., a Hamiltonian of a molecule is a function of the nuclear configurations. In this paper we incorporate derivatives of the Hamiltonian into the VQE and develop some hybrid quantum-classical algorithms, which explore both Hamiltonian and wave-function spaces for solving quantum chemistry problems more efficiently. For geometry optimization, we propose a mutual gradient-descent algorithm that updates the parameters of the Hamiltonian and wave function alternately, which can give a rapid convergence towards equilibrium structures of molecules as demonstrated in numerical simulations. Moreover, to speed up the calculation of the energy potential surface, we establish differential equations that govern how the optimized variational parameters of the wave function change with the intrinsic parameters of the Hamiltonian. Our study suggests a direction for a hybrid quantum-classical algorithm for solving quantum systems more efficiently by considering spaces of both the Hamiltonian and the wave function.
Estimating the energy spectra of quantum many-body systems is a fundamental task in quantum physics, with applications ranging from chemistry to condensed matter. Algorithmic shadow spectroscopy is a recent method that leverages randomized measurements on time-evolved quantum states to extract spectral information. However, implementing accurate time evolution with low-depth circuits remains a key challenge for near-term quantum hardware. In this work, we propose a hybrid quantum-classical protocol that integrates Time Evolution via Probabilistic Angle Interpolation (TE-PAI) into the shadow spectroscopy framework. TE-PAI enables the simulation of time evolution using shallow stochastic circuits while preserving unbiased estimates through quasiprobability sampling. We construct the combined estimator and derive its theoretical properties. Through numerical simulations, we demonstrate that our method accurately resolves energy gaps and exhibits enhanced robustness to gate noise compared to first-order Trotter-based shadow spectroscopy. We further validate the protocol experimentally on up to 20 qubits using IBM quantum hardware. This makes TE-PAI shadow spectroscopy a promising tool for spectral analysis on noisy intermediate-scale quantum (NISQ) devices.
Computational methods are central to modern physics, yet many problems, especially those involving Quantum Many-Body Systems (QMBSs), remain challenging at scale. The core difficulty of simulating QMBSs is the exponential growth of the solution space with system size. This renders exact treatments intractable and pushes approximate methods to their limits when tackling two- and higher-dimensional systems and long-time dynamics. Quantum computers offer a quantum-native platform for representing and manipulating such systems. Recent algorithmic and technological advancements have enabled experiments on quantum processors in the quantum utility regime, a scale at which exact classical reproduction is no longer feasible. This represents the first step towards realizing quantum advantage, where quantum computers solve specific, practically relevant problems more efficiently than any classical approach. While advancing towards the goal of quantum advantage, useful computations on quantum processors are expected to be realised through hybrid quantum-classical algorithms, which combine quantum experiments and classical computational resources. A prominent example are variational quantum algorithms, where a Parametrized Quantum Circuit (PQC) prepares a trial quantum state whose parameters are iteratively updated by a classical optimisation routine to minimize a target cost function (e.g., the energy of a target Hamiltonian). However, the optimisation of such PQCs is affected by trainability issues, such as barren plateaus in the parameter landscape, which can render the optimisation increasingly difficult for larger system sizes. This motivates the development of novel hybrid approaches overcoming the limits of PQCs. In this thesis, we introduce a new class of approaches based on hybrid Tensor Networks (TNs). A hybrid TN is obtained by integrating quantum states prepared on a quantum processor within TNs -- one of the state-of-the-art approaches for simulating quantum systems. We first show how hybrid TNs can be used for ground-state searches within a variational setting. Then, we further extend the framework to non-variational hybrid TN ansätze for which the quantum tensors are prepared via quantum annealing and subsequently post-processed by classical TNs. In the context of quantum machine learning, which also relies on PQCs, we further investigate heuristic best practices that mitigate the same exponential concentration effects in kernel-based models for anomaly detection. Another class of hybrid algorithms are quantum error-mitigation workflows, where classical post-processing is used to reconstruct expectation values from noise-affected circuit executions. Such methods generally feature a trade-off between the bias and the variance: the price to pay for mitigating noise in a bias-free fashion is that the number of measurements usually grows exponentially with the depth of the circuit in order to achieve a fixed accuracy. In this thesis, we develop a low-resource error-mitigation strategy that reduces the measurement overhead in practical digital Hamiltonian-simulation workflows. We apply this method to Trotterized dynamics in a lattice-gauge-theory model, enabling the extraction of expectation values with improved accuracy at time-scales that allow the observation of post-collision scattering dynamics.