The Cumulants Expansion Approach: The Good, The Bad and The Ugly

TL;DR

This paper analyzes the cumulants expansion (CEM) as a means to tame the exponential growth of quantum dynamics by truncating higher-order correlations. It benchmarks CEM against full quantum dynamics in two problems: a dipole-dipole chain of two-level emitters and a bi-prime factorization task implemented via adiabatic quantum computing, using a defined error measure to compare orders of expansion. The findings show a dual character: in the dipole-chain problem, higher-order cumulants can converge toward the exact dynamics (the Good), while in the bi-prime problem higher-order terms often introduce artifacts or instability (the Bad) or even chaotic behavior (the Ugly). The work highlights that CEM is not universally reliable and stresses the need for rigorous applicability criteria and problem-dependent diagnostics to guide its use in quantum many-body dynamics.

Abstract

The configuration space, i.e. the Hilbert space, of compound quantum systems grows exponentially with the number of its subsystems: its dimensionality is given by the product of the dimensions of its constituents. Therefore a full quantum treatment is rarely possible analytically and can be carried out numerically for fairly small systems only. Fortunately, in order to obtain interesting physics, approximations often very well suffice. One of these approximations is given by the cumulants expansion, where expectation values of products of operators are approximated by products of expectation values of said operators, neglecting higher-order correlations. The lowest order of this approximation is widely known as the mean field approximation and used routinely throughout quantum physics. Despite its ubiquitous presence, a general criterion for applicability and convergence properties of higher order cumulants expansions remains to be found. In this paper, we discuss two problems in quantum electrodynamics and quantum information, namely the collective radiative dissipation of a dipole-dipole interacting chain of atoms and the factorization of a bi-prime by annealing in an adiabatic quantum simulator. In the first case we find smooth, convergence behavior, where the approximation performs increasingly better with higher orders, while in the latter going beyond mean field turns out useless and, even for small system sizes, we are puzzled by numerically challenging and partly non-physical solutions.

Figures (10)

• Figure 1: Model Systems and Order of Expansion. Our toy models consist of $N$ interacting identical two-level systems. The dimension of a full quantum dynamics scales exponentially with $N$ as $2^N \times 2^N$ considering the Quantum Langevin Equation (QLE). The first order of the cumulants expansion, dubbed mean field approximation, scales only linearly with system size, u.e. $\mathcal{O}(N)$, but does not account for quantum correlations. Including all pairwise quantum correlations in second order gives $\mathcal{O}(N^2)$. Higher order cumulants bring back multi-particle quantum correlation at the cost of a polynomial increase in equation numbers as a function of the order of the expansion.
• Figure 2: Chain of dipole-dipole interacting two-level QEs equidistantly separated by $d$ and constantly driven via $\eta$. The dipole moments $\boldsymbol{\mu}_i$ of all QEs are aligned in the same direction (red arrows). We choose $\boldsymbol{\mu}_i = \mu_i\boldsymbol{\varepsilon}_{\perp,z}$ to be perpendicular to the distance vector $\boldsymbol{r}_{ij} = \boldsymbol{r}_i - \boldsymbol{r}_j = d(i-j)\boldsymbol{\varepsilon}_{z}$ between the $i$-th and $j$-th QE (green arrow).
• Figure 3: Adiabatic quantum algorithm for the bi-prime factorization problem. The red arrows represent the Bloch vector alignments of the corresponding qubits. The visualization of each vector is done by utilizing the Bloch representation $\{\ket{+}_m = 1/\sqrt{2}(\ket{0}_m + \ket{1}_m),i\ket{+}_m=1/\sqrt{2}(\ket{0}_m + i\ket{1}_m),\ket{0}_m\}\in\mathcal{H}_m\cong\mathbb{C}^2$ where the full quantum state of the system is a vector within the collective Hilbert space $\ket{\psi(s)} \in\mathcal{H}$. The lower $n = k+l$ qubit arrangement represents the ground state of $\hat{H}_0$ at $s = 0$. Turning on the adiabatic process, u.e. $s = 0\to1$, changes the system's internal Bloch vector alignments which leaves (ideally) an array of 'up'- or 'down'-pointing vectors (upper vector arrangement). Measuring the projectors $\hat{a}_i,\hat{b}_j$ results in a sequence of the classical bit-digits, fulfilling the bi-prime problem $\omega=ab.$
• Figure 4: The mean populations $\langle\hat{\sigma}^{22}\rangle$ for $N = 5$, $\eta/\Gamma = 2$, $\Gamma t\in[0,10]$ and $d/\lambda\in\{0.1,0.15,0.2,\cdots,1.0\}$ in different CEM expansion orders $o\in\{1,2,3,4,5\}$ are visualized. With each plot ((a)-(e)), $o$ is increased, hence, plot (a) corresponds to $o=1$, plot (c) to $o = 3$ and plot (e) to $o=5$. In plot (f), the FQD is shown.
• Figure 5: Comparison of the CEM in different orders $o\in\{1,2,3,4,5\}$ with the FQD of the expectation values $\langle\hat{\sigma}^{\tilde{k}}\rangle$ can be observed in the left plot column in sub-wavelength separation $d/\lambda = 0.15$. The right column depicts the corresponding SD $\Delta_{o,\tilde{k}}$.