Real-time Estimators for Scattering Observables: A full account of finite volume errors for quantum simulation

Abstract

The real-time correlators of quantum field theories can be directly probed through new approaches to simulation, such as quantum computing and tensor networks. This provides a new framework for computing scattering observables in lattice formulations of strongly interacting theories, such as lattice quantum chromodynamics. In this paper, we prove that the proposal of real-time estimators of scattering observables is universally applicable to all scattering observables of gapped quantum field theories. All finite-volume errors are exponentially suppressed, and the rate of this suppression is controlled by the regulator considered, namely, a displacement of the spectrum of the theory into the complex plane. A partial restoration of Lorentz symmetry by averaging over different boosts gives an additional suppression of finite volume errors. Our results also apply to the simulation of wavepacket scattering, where a similar averaging is performed to construct the wavepackets that regulate the finite volume effects. This result represents a necessary key step towards determining a broad class of scattering observables via quantum computing that are currently inaccessible via classical computing. Such observables are relevant for various applications, including hadron spectroscopy, hadron structure, and precision tests of the Standard Model. We also comment on potential applications of our results to traditional computational schemes.

Figures (4)

• Figure 1: Graphical description of the wavepacket (a) and RESOs (b) approaches for two-particle elastic scattering. In these, $a$ and $\eta$ are lattice spacings in the spatial and temporal directions. (a) Initially, two non-overlapping one-particle states are created. Then, through time evolution, the states overlap and the scattering process occurs. Finally, through further evolution, two non-overlapping wavepackets are recovered. (b) Initially, a definite-momentum one-particle state $\ket{P_i}$ is prepared. Then, two separated currents are applied to the state. Two-particle observables are recovered through the overlap of the resulting state with another definite-momentum one-particle state.
• Figure 2: (a) For this bubble diagram we can write $k^1 = \ell^1$ and $k^2 = -\ell^1 + P^1 + P^2$. (b) For this scarab diagram we can write $k^1 = \ell^1$, $k^4 = \ell^2$, $k^6 = \ell^3$, $k^2 = -\ell^1 - \ell^3 + P^1$, $k^3 = \ell^1 - \ell^2 + P^2$ and $k^5 = -\ell^2 - \ell^3 + P^1 + P^2$.
• Figure 3: Dependence on the volume of $I_1$ for the bubble diagram \ref{['fig:diagrams']}(a). Computed with $D=1+1$, $s = (P^1 + P^2)^2 = (2.5m)^2$ and $\epsilon = 0.1m^2$. The solid line corresponds to \ref{['eq:final_formula']} without averaging and in the center of momentum frame. The blue points correspond to the result with averaging by considering all possible kinematics with external momenta lower than $(2\pi/a)/8$ with $ma = 0.1$. Variations of $s$ are allowed with a relative tolerance of $1\%$.
• Figure 4: Convergence of \ref{['eq:PoissonSummation']} for the center of momentum frame of the bubble diagram of \ref{['fig:diagrams']}(a). In here $D = 1 + 3$, $s=(P^1 + P^2)^2 = (2.4m)^2$, $mL = 10$ and $\epsilon = 0$. The blue circles show the partial sums, which we order as a sum over shells of increasing radii. For the orange squares, we first turn this sum into a power series, with the coefficient of degree $n$ corresponding to the $n$-th shell. The orange boxes then correspond to the diagonal terms in the associated Padé table. The black line shows the result obtained through traditional methods (see, for example, Ref. Kim:2005gf, where it was labeled $F_s$).