The evolution operator of a quantum system in a time-dependent potential is factorized in unitary exponential operators at order 6. This expression is derived with the time-ordering method. It is compared with lower-order factorizations on several simple one-dimensional examples. Better accuracies are reached at sixth order for a given time step than at lower orders. Due to a significant increase of computation duration per time step, the sixth-order approximation is mainly useful when high accuracies are required.
The evolution operator of a quantum system in a time-dependent potential is factorized in unitary exponential operators at order 6. This expression is derived with the time-ordering method. It is compared with lower-order factorizations on several simple one-dimensional examples. Better accuracies are reached at sixth order for a given time step than at lower orders. Due to a significant increase of computation duration per time step, the sixth-order approximation is mainly useful when high accuracies are required.
The time-dependent Schrödinger Equation (TDSE) is a parabolic partial differential equation (PDE) comparable to a diffusion equation but with imaginary time. Due to its first order time derivative, exponential integrators or propagators are natural methods to describe evolution in time of the TDSE, both for time-independent and time-dependent potentials. Two splitting methods based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracies in the time step △t. The unitary factorization of exponentials to high order accuracy depends on commutators of kinetic energy operators with potentials. Fourth-order accuracy propagators can involve negative or complex time steps, or real time steps only but with gradients of potentials, i.e. forces. Extending the propagators of TDSE's to imaginary time allows to also apply these methods to classical many-body dynamics, and quantum statistical mechanics of molecular systems.
We propose an approach to factorize the time-evolution operator of a quantum system through a (finite) sequence of elementary operations that are time-ordered. Our proposal borrows from previous approaches based on Lie algebra techniques and other factorization procedures, and requires a set of optimization operations that provide the final result. Concretely, the algorithm produces at each step three optimal quantities, namely the optimal duration of the desired unitary operation, the optimal functional dependence of the driving function on the optimal time, and the optimal elementary Hermitian operation that induces the additional unitary operation to be implemented. The resulting sequence of unitary operations that is obtained this way is sequential with time. We compare our proposal with existing approaches, and highlight which key assumptions can be relaxed for practical implementations.
The time-evolution operator for an explicitly time-dependent Hamiltonian is expressed as the product of a sequence of unitary operators. These are obtained by successive time-dependent unitary transformations of the Hilbert space followed by the adiabatic approximation at each step. The resulting adiabatic product expansion yields a generalization of the quantum adiabatic approximation. Furthermore, it leads to an infinite class of exactly solvable models. PACS number: 03.65.Bz Consider the dynamics of a quantum mechanical system whose Hamiltonian H = H(τ) is explicitly time-dependent. The evolution of a state vector |ψ(τ) 〉 is governed by the Schrödinger equation: i¯h | ˙ ψ(τ) 〉 = H(τ) |ψ(τ) 〉 , |ψ(0) 〉 = |ψ0 〉 , (1) where a dot means a time-derivative. Alternatively, one has |ψ(τ) 〉 = U(τ)|ψ0〉, where i τ − H(t) dt U(τ) = T e ¯h 0 (2) is the time-evolution operator. Here T denotes the time-ordering operator. The purpose of this article is to express U(τ) as the product of a sequence of unitary operators each of
The solution of many physical evolution equations can be expressed as an exponential of two or more operators. Approximate solutions can be systematically derived by decomposing the exponential in a product form. For time-reversible equations, such as the Hamilton or the Schr\\"odinger equation, it is immaterial whether the decomposition coefficients are positive or negative. For time-irreversible systems, such as the Fokker-Planck equation or the quantum statistical propagator, only positive-coefficient decompositions which respect the time-irreversibility of the diffusion kernel can yield practical algorithms. These positive time step, or forward decompositions, are a highly restrictive class of factorization algorithms. This work proves a fundamental theorem: in order for a 2nth order forward algorithm to become (2n+2)th order, one must include a new, higher order commutator in the decomposition process. Since these higher order commuatators are highly complex, it seems difficult to produce practical forward algorithms beyond fourth order. This proof generalize the Sheng-Suzuki theorem for the case of n=1. In particular, this work shows that it is not possible to have a sixth order forward algorithm by including just the [V,[T,V]] commutator.
The exact ground state of a strongly interacting quantum many-body system can be obtained by evolving a trial state with finite overlap with the ground state to infinite imaginary time. In many cases, since the convergence is exponential, the system converges essentially to the exact ground state in a relatively short time. Thus a short-time evolved wave function can be an excellent approximation to the exact ground state. Such a short-time-evolved wave function can be obtained by factorizing, or splitting, the evolution operator to high order. However, for the imaginary time Schr\"odinger equation, which contains an irreversible diffusion kernel, all coefficients, or time steps, must be positive. (Negative time steps would require evolving the diffusion process backward in time, which is impossible.) Heretofore, only second-order factorization schemes can have all positive coefficients, but without further iterations, these cannot be used to evolve the system long enough to be close to the exact ground state. In this work, we use a newly discovered fourth-order positive factorization scheme which requires knowing both the potential and its gradient. We show that the resulting fourth-order wave function alone, without further iterations, gives an excellent description of strongly interacting quantum systems such as liquid 4He, comparable to the best variational results in the literature. This suggests that such a fourth-order wave function can be used to study the ground state of diverse quantum many-body systems, including Bose-Einstein condensates and Fermi systems.
Simulating quantum many-body systems (QMBS) is one of the long-standing, highly non-trivial challenges in condensed matter physics and quantum information due to the exponentially growing size of the system's Hilbert space. To date, tensor networks have been an essential tool for studying such quantum systems, owing to their ability to efficiently capture the entanglement properties of the systems they represent. One of the well-known tensor network architectures, namely matrix product states (MPS), is the standard method for simulating one-dimensional QMBS. Here, we propose a simple, yet efficient, method to augment the already available MPS algorithms to simulate the dynamics of time-dependent Hamiltonians with better accuracy and a faster convergence rate, giving a second-order convergence compared to the first-order convergence of the standard method. We apply our proposed method to simulate the dynamics of a chain of single spins associated with nitrogen-vacancy color centers in diamonds, which has potential applications for practical and scalable quantum technologies, and find that our method improves the average error for a system of few NV centers by a factor of about 1000 for moderate step sizes. Our work paves the way for efficient simulation of QMBS under the influence of time-dependent Hamiltonians.