On the Lorentz-Invariance of the Dyson Series in Theories with Derivative Couplings

We study the Dyson series for the S-matrix, when the interaction depends on derivatives of the fields. We concentrate on two particular examples: the scalar electrodynamics and the renormalized ф4 theory. By using Wick’s theorem, we eventually give evidence that the Lorentz invariance is satisfied, and the usual Feynman rules can be applied to the interaction Lagrangian.

Numerical Contour Integration

We present a straightforward implementation of contour integration by setting options for Integrate and NIntegrate, taking advantage of powerful results in complex analysis. As such, this article can be viewed as documentation to perform numerical contour integration with the existing built-in tools. We provide examples of how this method can be used when integrating analytically and numerically some commonly used distributions, such as Wightman functions in quantum field theory. We also provide an approximating technique when time-ordering is involved, a commonly encountered scenario in quantum field theory for computing second-order terms in Dyson series expansion and Feynman propagators. We believe our implementation will be useful for more general calculations involving advanced or retarded Green’s functions, propagators, kernels and so on.

Digital quantum simulation of hadronization in Yang–Mills theory

A quantum algorithm of SU([Formula: see text]) Yang–Mills theory is formulated in terms of quantum circuits. It can nonperturbatively calculate the Dyson series and scattering amplitudes with polynomial complexity. The gauge fields in the interaction picture are discretized on the same footing with the lattice fermions in momentum space to avoid the fermion doubling and the gauge symmetry breaking problems. Applying the algorithm to the quantum simulation of quantum chromodynamics, the quark and gluon’s wave functions evolved from the initial states by the interactions can be observed and the information from wave functions can be extracted at any discrete time. This may help us understand the natures of the hadronization which has been an outstanding question of significant implication on high energy phenomenological studies.

Time-dependent Hamiltonian simulation with L1-norm scaling

The difficulty of simulating quantum dynamics depends on the norm of the Hamiltonian. When the Hamiltonian varies with time, the simulation complexity should only depend on this quantity instantaneously. We develop quantum simulation algorithms that exploit this intuition. For sparse Hamiltonian simulation, the gate complexity scales with the L1 norm ∫0tdτ‖H(τ)‖max, whereas the best previous results scale with tmaxτ∈[0,t]‖H(τ)‖max. We also show analogous results for Hamiltonians that are linear combinations of unitaries. Our approaches thus provide an improvement over previous simulation algorithms that can be substantial when the Hamiltonian varies significantly. We introduce two new techniques: a classical sampler of time-dependent Hamiltonians and a rescaling principle for the Schrödinger equation. The rescaled Dyson-series algorithm is nearly optimal with respect to all parameters of interest, whereas the sampling-based approach is easier to realize for near-term simulation. These algorithms could potentially be applied to semi-classical simulations of scattering processes in quantum chemistry.

Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

Fundamentally, it is believed that interactions between physical objects are two-body. Perturbative gadgets are one way to break up an effective many-body coupling into pairwise interactions: a Hamiltonian with high interaction strength introduces a low-energy space in which the effective theory appears k-body and approximates a target Hamiltonian to within precision $$\epsilon $$ϵ. One caveat of existing constructions is that the interaction strength generally scales exponentially in the locality of the terms to be approximated. In this work we propose a many-body Hamiltonian construction which introduces only a single separate energy scale of order $$\Theta (1/N^{2+\delta })$$Θ(1/N2+δ), for a small parameter $$\delta >0$$δ>0, and for N terms in the target Hamiltonian $$\mathbf H_\mathrm {t}=\sum _{i=1}^N \mathbf h_i$$Ht=∑i=1Nhi to be simulated: in its low-energy subspace, our constructed system can approximate any such target Hamiltonian $$\mathbf H_t$$Ht with norm ratios $$r=\max _{i,j\in \{1,\ldots ,N\}}\Vert \mathbf h_i\Vert / \Vert \mathbf h_j \Vert ={{\,\mathrm{O}\,}}(\exp (\exp ({{\,\mathrm{poly}\,}}N)))$$r=maxi,j∈{1,…,N}‖hi‖/‖hj‖=O(exp(exp(polyN))) to within relative precision $${{\,\mathrm{O}\,}}(N^{-\delta })$$O(N-δ). This comes at the expense of increasing the locality by at most one, and adding an at most poly-sized ancillary system for each coupling; interactions on the ancillary system are geometrically local, and can be translationally invariant. In order to prove this claim, we borrow a technique from high energy physics—where matter fields obtain effective properties (such as mass) from interactions with an exchange particle—and employ a tiling Hamiltonian to discard all cross-terms at higher expansion orders of a Feynman–Dyson series expansion. As an application, we discuss implications for QMA-hardness of the Local Hamiltonian problem, and argue that “almost” translational invariance—defined as arbitrarily small relative variations of the strength of the local terms—is as good as non-translational invariance in many of the constructions used throughout Hamiltonian complexity theory. We furthermore show that the choice of geared limit of many-body systems, where e.g. width and height of a lattice are taken to infinity in a specific relation, can have different complexity-theoretic implications: even for translationally invariant models, changing the geared limit can vary the hardness of finding the ground state energy with respect to a given promise gap from computationally trivial, to QMAEXP-, or even BQEXPSPACE-complete.

Operator Ordering and Solution of Pseudo-Evolutionary Equations

The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain time- and operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions for fractional differential equations. We apply the method to several examples, in which fractional calculus and a certain umbral image calculus play a role of central importance.

Edgeworth expansions for slow–fast systems with finite time-scale separation

We derive Edgeworth expansions that describe corrections to the Gaussian limiting behaviour of slow–fast systems. The Edgeworth expansion is achieved using a semi-group formalism for the transfer operator, where a Duhamel–Dyson series is used to asymptotically determine the corrections at any desired order of the time-scale parameter ε . The corrections involve integrals over higher-order auto-correlation functions. We develop a diagrammatic representation of the series to control the combinatorial wealth of the asymptotic expansion in ε and provide explicit expressions for the first two orders. At a formal level, the expressions derived are valid in the case when the fast dynamics is stochastic as well as when the fast dynamics is entirely deterministic. We corroborate our analytical results with numerical simulations and show that our method provides an improvement on the classical homogenization limit which is restricted to the limit of infinite time-scale separation.