Effective light cone and digital quantum simulation of interacting bosons

Abstract

The speed limit of information propagation is one of the most fundamental features in non-equilibrium physics. The region of information propagation by finite-time dynamics is approximately restricted inside the effective light cone that is formulated by the Lieb-Robinson bound. To date, extensive studies have been conducted to identify the shape of effective light cones in most experimentally relevant many-body systems. However, the Lieb-Robinson bound in the interacting boson systems, one of the most ubiquitous quantum systems in nature, has remained a critical open problem for a long time. This study reveals a tight effective light cone to limit the information propagation in interacting bosons, where the shape of the effective light cone depends on the spatial dimension. To achieve it, we prove that the speed for bosons to clump together is finite, which in turn leads to the error guarantee of the boson number truncation at each site. Furthermore, we applied the method to provide a provably efficient algorithm for simulating the interacting boson systems. The results of this study settle the notoriously challenging problem and provide the foundation for elucidating the complexity of many-body boson systems.

Figures (31)

• Figure 1: Schematic picture of the definition of $(\partial X)_s$ for positive and negative $s$.
• Figure 2: Proof outline.
• Figure 3: In the upper bound \ref{['assump_start_point0_re1']} on the moment function, the operator $\hat{n}_X(\tau)$ is bounded from above by $\left( \hat{n}_{X} + c_{\tau,1} \hat{\mathcal{D}}_{X} + c_{\tau,2} \right)$. Here, the operator $\hat{\mathcal{D}}_{X}$ is roughly approximated on the surface region $\partial X$. Hence, we expect that $\hat{n}_{X} + c_{\tau,1} \hat{\mathcal{D}}_{X} + c_{\tau,2} \approx \hat{n}_X \left( 1 + |\partial X|/|X| \right)$. To make the expectation rigorous, we need to consider the possibility that most of the bosons concentrate on $\partial X$, which spoils the above discussions. However, by carefully choosing the subsets $X[r]$ in \ref{['refined_uppe_rough_boson_00']}, we can obtain a refined upper bound as in \ref{['refined_uppe_rough_boson']}.
• Figure 4: We decompose the subset $X[\ell] \setminus X$ to $\tilde{p}+1$ pieces. In the picture, a one-dimensional case is considered. We define the extended subset of $X$ from $X_1$ to $X_p$ as $X_{0:p} := X \cup X_1 \cup \cdots \cup X_p$. Then, we upper-bound the moment function $M^{(1)}_X (\tau)$ by using $M^{(1)}_{X_{0:p}}(\tau)$ with an appropriate choice of $p$. In the choice, we need to suppress the contribution from the surface region (i.e., the term $\hat{\mathcal{D}}_{X_{0:p}}$).
• Figure 5: Schematic picture of our setup for Theorem \ref{['main_theorem0_boson_concentration']}. We consider the boson number operator $\hat{n}_X$ in a region $X$. After a time evolution, $\hat{n}_X(t)$ is upper-bounded by the boson number $\hat{n}_{X[R]}$ on an extended subset $X[R]$, where the error is given by $e^{-\Omega(R/t)}$. The obtained inequality is given in the form of the operator inequality [see \ref{['main_ineq:main_theorem0_boson_concentration']}], and hence it can be applied to an arbitrary initial state.

Theorems & Definitions (59)

• Lemma 1
• Definition 1: Short-range interactions
• Definition 2: Finite coupling constants
• Definition 3: Low boson density
• Definition 4
• Lemma 2
• Lemma 3
• Corollary 4
• Corollary 5
• Proposition 6