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We propose a general-purpose, self-adaptive approach to construct variational wavefunction ansatze for highly accurate quantum dynamics simulations based on McLachlans 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 two 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.
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By design, the variational quantum eigensolver (VQE) strives to recover the lowest-energy eigenvalue of a given Hamiltonian by preparing quantum states guided by the variational principle. In practice, the prepared quantum state is indirectly assessed by the value of the associated energy. Novel adaptive derivative-assembled pseudo-trotter (ADAPT) ansatz approaches and recent formal advances now establish a clear connection between the theory of quantum chemistry and the quantum state ansatz used to solve the electronic structure problem. Here we benchmark the accuracy of VQE and ADAPT-VQE to calculate the electronic ground states and potential energy curves for a few selected diatomic molecules, namely H$_2$, NaH, and KH. Using numerical simulation, we find both methods provide good estimates of the energy and ground state, but only ADAPT-VQE proves to be robust to particularities in optimization methods. Another relevant finding is that gradient-based optimization is overall more economical and delivers superior performance than analogous simulations carried out with gradient-free optimizers. The results also identify small errors in the prepared state fidelity which show an increasing trend with molecular size.
An adaptive variational quantum imaginary time evolution (AVQITE) approach is introduced that yields efficient representations of ground states for interacting Hamiltonians on near-term quantum computers. It is based on McLachlans variational principle applied to imaginary time evolution of variational wave functions. The variational parameters evolve deterministically according to equations of motions that minimize the difference to the exact imaginary time evolution, which is quantified by the McLachlan distance. Rather than working with a fixed variational ansatz, where the McLachlan distance is constrained by the quality of the ansatz, the AVQITE method iteratively expands the ansatz along the dynamical path to keep the McLachlan distance below a chosen threshold. This ensures the state is able to follow the quantum imaginary time evolution path in the system Hilbert space rather than in a restricted variational manifold set by a predefined fixed ansatz. AVQITE is used to prepare ground states of H$_4$, H$_2$O and BeH$_2$ molecules, where it yields compact variational ansatze and ground state energies within chemical accuracy. Polynomial scaling of circuit depth with system size is demonstrated through a set of AVQITE calculations of quantum spin models. Finally, it is shown that quantum Lanczos calculations can also be naturally performed alongside AVQITE without additional quantum resource costs.
Quantum state tomography is a key process in most quantum experiments. In this work, we employ quantum machine learning for state tomography. Given an unknown quantum state, it can be learned by maximizing the fidelity between the output of a variational quantum circuit and this state. The number of parameters of the variational quantum circuit grows linearly with the number of qubits and the circuit depth, so that only polynomial measurements are required, even for highly-entangled states. After that, a subsequent classical circuit simulator is used to transform the information of the target quantum state from the variational quantum circuit into a familiar format. We demonstrate our method by performing numerical simulations for the tomography of the ground state of a one-dimensional quantum spin chain, using a variational quantum circuit simulator. Our method is suitable for near-term quantum computing platforms, and could be used for relatively large-scale quantum state tomography for experimentally relevant quantum states.
We introduce a novel approach for a fully quantum description of coupled electron-ion systems from first principles. It combines the variational quantum Monte Carlo (QMC) solution of the electronic part with the path integral (PI) formalism for the quantum nuclear dynamics. On the one hand, the PI molecular dynamics includes nuclear quantum effects by adding a set of fictitious classical particles (beads) aimed at reproducing nuclear quantum fluctuations via a harmonic kinetic term. On the other hand, variational QMC can provide Born-Oppenheimer (BO) potential energy surfaces with a precision comparable to the most advanced post Hartree-Fock approaches, and with a favorable scaling with the system size. To deal with the intrinsic QMC noise, we generalize the PI molecular dynamics using a Langevin thermostat correlated according to the covariance matrix of QMC nuclear forces. The variational parameters of the QMC wave function are evolved during the nuclear dynamics, such that the BO potential energy surface is unbiased. Statistical errors on the wave function parameters are reduced by resorting to bead grouping average, which we show to be accurate and well controlled. Our general algorithm relies on a Trotter breakup between the dynamics driven by ionic forces and the one set by the harmonic interbead couplings. The latter is exactly integrated even in presence of the Langevin thermostat, thanks to the mapping onto an Ornstein-Uhlenbeck process. This framework turns out to be very efficient also in the case of deterministic ionic forces. The new implementation is validated on the Zundel ion by direct comparison with standard PI Langevin dynamics calculations made with a coupled cluster potential energy surface. Nuclear quantum effects are confirmed to be dominant over thermal effects well beyond room temperature giving the excess proton an increased mobility by quantum tunneling.
Quantum computers hold promise to greatly improve the efficiency of quantum simulations of materials and to enable the investigation of systems and properties more complex than tractable on classical architectures. Here, we discuss computational frameworks to carry out electronic structure calculations of solids on noisy intermediate scale quantum computers using embedding theories and effective many-body Hamiltonians. We focus on a specific class of materials, i.e., spin defects in solids, which are promising systems to build future quantum technologies, e.g., computers, sensors and devices for quantum communications. Although quantum simulations on quantum architectures are in their infancy, promising results for realistic systems appear within reach.
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