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Register a new userPath integral in a magnetic field using the Trotter product formula
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The derivation of the Feynman path integral based on the Trotter product formula is extended to the case where the system is in a magnetic field.
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We treat the problem of normally ordering expressions involving the standard boson operators a, a* where [a,a*]=1. We show that a simple product formula for formal power series - essentially an extension of the Taylor expansion - leads to a double exponential formula which enables a powerful graphical description of the generating functions of the combinatorial sequences associated with such functions - in essence, a combinatorial field theory. We apply these techniques to some examples related to specific physical Hamiltonians.
Let $A$ and $B$ be positive semidefinite matrices. The limit of the expression $Z_p:=(A^{p/2}B^pA^{p/2})^{1/p}$ as $p$ tends to $0$ is given by the well known Lie-Trotter-Kato formula. A similar formula holds for the limit of $G_p:=(A^p,#,B^p)^{2/p}$ as $p$ tends to $0$, where $X,#,Y$ is the geometric mean of $X$ and $Y$. In this paper we study the complementary limit of $Z_p$ and $G_p$ as $p$ tends to $infty$, with the ultimate goal of finding an explicit formula, which we call the anti Lie-Trotter formula. We show that the limit of $Z_p$ exists and find an explicit formula in a special case. The limit of $G_p$ is shown for $2times2$ matrices only.
We argue that the natural way to generalise a tensor network variational class to a continuous quantum system is to use the Feynman path integral to implement a continuous tensor contraction. This approach is illustrated for the case of a recently introduced class of quantum field states known as continuous matrix-product states (cMPS). As an example of the utility of the path-integral representation we argue that the state of a dynamically evolving quantum field admits a natural representation as a cMPS. An argument that all states in Fock space admit a cMPS representation when the number of variational parameters tends to infinity is also provided.
Various Hamiltonian simulation algorithms have been proposed to efficiently study the dynamics of quantum systems using a universal quantum computer. However, existing algorithms generally approximate the entire time evolution operators, which may need a deep quantum circuit that are beyond the capability of near-term noisy quantum devices. Here, focusing on the time evolution of a fixed input quantum state, we propose an adaptive approach to construct a low-depth time evolution circuit. By introducing a measurable quantifier that describes the simulation error, we use an adaptive strategy to learn the shallow quantum circuit that minimizes the simulation error. We numerically test the adaptive method with the electronic Hamiltonians of $mathrm{H_2O}$ and $mathrm{H_4}$ molecules, and the transverse field ising model with random coefficients. Compared to the first-order Suzuki-Trotter product formula, our method can significantly reduce the circuit depth (specifically the number of two-qubit gates) by around two orders while maintaining the simulation accuracy. We show applications of the method in simulating many-body dynamics and solving energy spectra with the quantum Krylov algorithm. Our work sheds light on practical Hamiltonian simulation with noisy-intermediate-scale-quantum devices.
We consider the dynamics $tmapstotau_t$ of an infinite quantum lattice system that is generated by a local interaction. If the interaction decomposes into a finite number of terms that are themselves local interactions, we show that $tau_t$ can be efficiently approximated by a product of $n$ automorphisms, each of them being an alternating product generated by the individual terms. For any integer $m$, we construct a product formula (in the spirit of Trotter) such that the approximation error scales as $n^{-m}$. We do so in the strong topology of the operator algebra, namely by approximating $tau_t(O)$ for sufficiently localized observables $O$.
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