Quantum simulation of quantum chemistry is one of the most compelling applications of quantum computing. It is of particular importance in areas ranging from materials science, biochemistry, and condensed matter physics. Here, we propose a full quantum eigensolver (FQE) algorithm to calculate the molecular ground energies and electronic structures using quantum gradient descent. Compared to existing classical-quantum hybrid methods such as variational quantum eigensolver (VQE), our method removes the classical optimizer and performs all the calculations on a quantum computer with faster convergence. The gradient descent iteration depth has a favorable complexity that is logarithmically dependent on the system size and inverse of the precision. Moreover, the FQE can be further simplified by exploiting a perturbation theory for the calculations of intermediate matrix elements and obtaining results with a precision that satisfies the requirement of chemistry application. The full quantum eigensolver can be implemented on a near-term quantum computer. With the rapid development of quantum computing hardware, the FQE provides an efficient and powerful tool to solve quantum chemistry problems.

$\mathcal{PT}$-symmetric systems are one of the most interesting research fields in modern quantum physics, where various theoretical and experimental progress has been made. In our work, we experimentally demonstrate how to realize a general $\mathcal{PT}$-symmetric two-level operation in a quantum computation frame with a universal circuit model. It is based on enlarging the system with ancillary qubits and encoding the subsystem with the non-Hermitian Hamiltonian with postselection. Furthermore, we use the general formula to demonstrate one particular interesting issue, entanglement restoration induced by local $\mathcal{PT}$-symmetric operation in our four-qubit liquid nuclear magnetic resonance platform, and a theoretical explanation for the physical characteristics has been proposed. We also extend the protocol to the realization of an arbitrary two-level Hamiltonian evolution without a Hermitian restriction by an appropriate modification of the original quantum circuit. Our work lays the foundation for quantum simulation of the general $\mathcal{PT}$-symmetric problem in realistic quantum simulators, and the scheme can be extended to other quantum computation platforms.

Imaginary-time evolution is a powerful tool in the study of quantum physics. However, existing classical algorithms for simulating imaginary-time evolution suffer from high computational complexity as the dimension of the quantum system increases. In this study we propose a quantum algorithm for implementing imaginary-time evolution using nonunitary quantum circuits with one ancillary qubit. The success probability of our algorithm is a polynomial function of the output error and can be enhanced by reorganizing the terms of the Hamiltonian. To illustrate the practicality of our algorithm on current quantum devices, we conduct a demonstration on superconducting and trapped-ion quantum processors to calculate the ground-state energy and determine the most stable molecular structure of ${\mathrm{H}}_2$. Additionally, we validate the feasibility of our algorithm by numerically simulating the ground-state energies of LiH molecules and the quantum Ising chain. In contrast to existing algorithms, our method provides a systematic approach to construct the required nonunitary circuits using universal quantum gates, making it suitable for experimental implementation. Our algorithm opens up possibilities for exploring other physical phenomena such as finite-temperature properties and non-Hermitian systems.

The Yang-Baxter equation (YBE) has created many applications in the integrable models of the statistical mechanics. It is recently shown that the YBE can be further applied in quantum information processing. Conversely, quantum simulation also provides novel approaches for effectively checking the YBE in quantum devices. Here, we adopted the method inspired by the previous work to experimentally perform the simulation of the YBE on a four-qubit nuclear magnetic resonance quantum processor. Whether the YBE is satisfied or not can be preliminarily determined by measuring the ancillary qubit without the assistance of quantum state tomography, and two hand sides of the YBE are concurrently simulated, not in separate forms involved in the existing protocols. The experimental result shows that the discrimination accuracy is over 98.1% when the condition of YBE is satisfied. The global and local entanglements in the YBE system are also studied by the numerical simulation and experiments. It is found that the YBE is not only a tool to yield the global entanglement but also the symptom when the local entanglement is equal to each other.

Gradient-based algorithms provide the fundamental tools for the optimizations and are the essential components of many modern technologies. Theoretically, the extreme points of a cost function can be found by iterative descent(ascent) of the variable along the direction of the gradient. However, the time complexity to calculate a gradient is at a level of $O(poly\left(d\right))$ for the $d$-dimension problems. For processing the high-dimension case, especially modern machine-learning engineering with the number of optimized parameters being in billions, quantum techniques are introduced. Via the dressed amplitude encoding and nonunitary module, a gradient-based quantum optimization algorithm for general polynomials is proposed, aiming at solving the fast-convergence problems with both time and memory complexity in $O(poly(logd))$. Furthermore, numerical simulations are carried out to inspect the performance of the newly born protocol by considering the noises or perturbations during the initialization, operations, and truncations. For the potential values in high-dimension optimizations, this gradient-based quantum optimization algorithm is supposed to facilitate the polynomial optimizations, being a subroutine for future practical quantum computers.