The variational method is a versatile tool for classical simulation of a variety of quantum systems. Great efforts have recently been devoted to its extension to quantum computing for efficiently solving static many-body problems and simulating real and imaginary time dynamics. In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel variational principle, the McLachlan's variational principle, and the time-dependent variational principle, for simulating real time dynamics. We focus on the simulation of dynamics and discuss the connections of the three variational principles. Previous works mainly focus on the unitary evolution of pure states. In this work, we introduce variational quantum simulation of mixed states under general stochastic evolution. We show how the results can be reduced to the pure state case with a correction term that takes accounts of global phase alignment. For variational simulation of imaginary time evolution, we also extend it to the mixed state scenario and discuss variational Gibbs state preparation. We further elaborate on the design of ansatz that is compatible with post-selection measurement and the implementation of the generalised variational algorithms with quantum circuits. Our work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.
The variational method is a versatile tool for classical simulation of a variety of quantum systems. Great efforts have recently been devoted to its extension to quantum computing for efficiently solving static many-body problems and simulating real and imaginary time dynamics. In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel variational principle, the McLachlan's variational principle, and the time-dependent variational principle, for simulating real time dynamics. We focus on the simulation of dynamics and discuss the connections of the three variational principles. Previous works mainly focus on the unitary evolution of pure states. In this work, we introduce variational quantum simulation of mixed states under general stochastic evolution. We show how the results can be reduced to the pure state case with a correction term that takes accounts of global phase alignment. For variational simulation of imaginary time evolution, we also extend it to the mixed state scenario and discuss variational Gibbs state preparation. We further elaborate on the design of ansatz that is compatible with post-selection measurement and the implementation of the generalised variational algorithms with quantum circuits. Our work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.
Variational quantum algorithms have been proposed to solve static and dynamic problems of closed many-body quantum systems. Here we investigate variational quantum simulation of three general types of tasks-generalized time evolution with a non-Hermitian Hamiltonian, linear algebra problems, and open quantum system dynamics. The algorithm for generalized time evolution provides a unified framework for variational quantum simulation. In particular, we show its application in solving linear systems of equations and matrix-vector multiplications by converting these algebraic problems into generalized time evolution. Meanwhile, assuming a tensor product structure of the matrices, we also propose another variational approach for these two tasks by combining variational real and imaginary time evolution. Finally, we introduce variational quantum simulation for open system dynamics. We variationally implement the stochastic Schrödinger equation, which consists of dissipative evolution and stochastic jump processes. We numerically test the algorithm with a 6-qubit 2D transverse field Ising model under dissipation.
Variational quantum time evolution is a fundamental issue in physics and engineering. It provides a strong framework for modeling the dynamics of quantum states, which is essential for efficiently running quantum networks. It presents an excellent toolkit that can improve the efficacy, reliability, resilience and performance of quantum networks, opening the door to more sophisticated distributed computer systems, quantum communication, and other quantum technologies. The imaginary time evolution of quantum systems is one useful tool that enables the generation of ground or thermal states. In this work, we address standard variational principles, such as the Dirac and Frenkel variational principles, the McLachlan variational principle and the time-dependent variational principle to model real-time dynamics, as well as the Rayleigh–Ritz approach to address static challenges. We also present the variational quantum time evolution VarQITE and VarQRTE formalism in the density matrix method, which is not influenced by the global phase of the quantum state. We calculate the ground-state energy of our quantum Ising spin chain, using VarQITE with different interaction energies J , where three cases are considered. In the non-interacting ( NI ) case ( J = 0 ), we show that the energy of the ground state accomplished with VarQITE converges to an energy of − 2 . 00 Hartree at t = 1 . 1 seconds. In the Ferromagnetic ( FM ) interaction case ( J > 0 ), four scenarios are used to obtain the energy of the Hamiltonian. We show that the convergence and the energy 〈 H 〉 are sensitive to the energy of interaction J . Furthermore, the influence of the negative values of J < 0 , which correspond to the antiferromagnetic ( AFM ) case, on the energy of the ground state is also studied. We find that VarQITE and the exact classical solution are matched. We also investigated the expectation values for magnetization 〈 m z 〉 , employing VarQRTE and the exact classical solution. The NI, FM, and AFM cases are accomplished here. We find that when the spins are not interacting, 〈 m z 〉 = 0 , while it oscillates over time t when J > 0 and J < 0 . We believe that our outcomes will improve our knowledge of the variational quantum time evolution and lead to a better understanding of real and imaginary time evolution of quantum systems (which form a quantum network) and therefore, to more information on the time in networks.
Quantum algorithms for quantum dynamics simulations are traditionally based on implementing a Trotter approximation of the time-evolution operator. This approach typically relies on deep circuits and is therefore hampered by the substantial limitations of available noisy and near-term quantum hardware. On the other hand, variational quantum algorithms (VQAs) have become an indispensable alternative, enabling small-scale simulations on present-day hardware. However, despite the recent development of VQAs for quantum dynamics, a detailed assessment of their efficiency and scalability is yet to be presented. To fill this gap, we applied a VQA based on McLachlan's principle to simulate the dynamics of a spin-boson model subject to varying levels of realistic hardware noise as well as in different physical regimes, and discuss the algorithm's accuracy and scaling behavior as a function of system size. We observe a good performance of the variational approach used in combination with a general, physically motivated wave function ansatz, and compare it to the conventional first-order Trotter evolution. Finally, based on this, we make scaling predictions for the simulation of a classically intractable system. We show that, despite providing a clear reduction of quantum gate cost, the variational method in its current implementation is unlikely to lead to a quantum advantage for the solution of time-dependent problems.
Abstract Imaginary time evolution is a powerful tool for studying quantum systems. While it is possible to simulate with a classical computer, the time and memory requirements generally scale exponentially with the system size. Conversely, quantum computers can efficiently simulate quantum systems, but not non-unitary imaginary time evolution. We propose a variational algorithm for simulating imaginary time evolution on a hybrid quantum computer. We use this algorithm to find the ground-state energy of many-particle systems; specifically molecular hydrogen and lithium hydride, finding the ground state with high probability. Our method can also be applied to general optimisation problems and quantum machine learning. As our algorithm is hybrid, suitable for error mitigation and can exploit shallow quantum circuits, it can be implemented with current quantum computers.
We propose a general-purpose, self-adaptive approach to construct a variational wave-function ansatz for highly accurate quantum dynamics simulations based on McLachlan’s variational principle. The key idea is to dynamically expand the variational ansatz along the time-evolution path such that the “McLachlan distance”, which is a measure of the simulation accuracy, remains below a set threshold. We apply this adaptive variational quantum dynamics simulation (AVQDS) approach to the integrable Lieb-Schultz-Mattis spin chain and the nonintegrable mixed-field Ising model, where it captures both finite-rate and sudden post-quench dynamics with high fidelity. The AVQDS quantum circuits that prepare the time-evolved state are much shallower than those obtained from first-order Trotterization and contain up to 2 orders of magnitude fewer cnot gate operations. We envision that a wide range of dynamical simulations of quantum many-body systems on near-term quantum-computing devices will be made possible through the AVQDS framework.
We propose a general-purpose, self-adaptive approach to construct a variational wave-function ansatz for highly accurate quantum dynamics simulations based on McLachlan's variational principle.The key idea is to dynamically expand the variational ansatz along the time-evolution path such that the "McLachlan distance", which is a measure of the simulation accuracy, remains below a set threshold.We apply this adaptive variational quantum dynamics simulation (AVQDS) approach to the integrable Lieb-Schultz-Mattis spin chain and the nonintegrable mixed-field Ising model, where it captures both finite-rate and sudden post-quench dynamics with high fidelity.The AVQDS quantum circuits that prepare the timeevolved state are much shallower than those obtained from first-order Trotterization and contain up to 2 orders of magnitude fewer CNOT gate operations.We envision that a wide range of dynamical simulations of quantum many-body systems on near-term quantum-computing devices will be made possible through the AVQDS framework.