Estimating truncation effects of quantum bosonic systems using sampling algorithms

TL;DR

The paper addresses the challenge of truncating infinite-dimensional bosonic Hilbert spaces for quantum simulations. It develops a classical Markov Chain Monte Carlo framework in the coordinate-basis truncation to quantify digitization errors across a broad class of bosonic systems, including a 2D lattice scalar field theory. The method yields non-negative weights enabling efficient sampling, reproduces exact results for simple cases, and accurately captures digitization effects in a small lattice QFT, with truncation errors decaying roughly exponentially with the digitization parameters. This approach provides a practical path to estimate quantum-resource requirements and to validate quantum simulations, while offering avenues to extend to other digitization schemes, gauge theories, and variational Monte Carlo techniques for cross-checks and benchmarking.

Abstract

To simulate bosons on a qubit- or qudit-based quantum computer, one has to regularize the theory by truncating infinite-dimensional local Hilbert spaces to finite dimensions. In the search for practical quantum applications, it is important to know how big the truncation errors can be. In general, it is not easy to estimate errors unless we have a good quantum computer. In this paper, we show that traditional sampling methods on classical devices, specifically Markov Chain Monte Carlo, can address this issue for a rather generic class of bosonic systems with a reasonable amount of computational resources available today. As a demonstration, we apply this idea to the scalar field theory on a two-dimensional lattice, with a size that goes beyond what is achievable using exact diagonalization methods. This method can be used to estimate the resources needed for realistic quantum simulations of bosonic theories, and also, to check the validity of the results of the corresponding quantum simulations.

Figures (4)

• Figure 1: Simulation history of $T=0.1$, $a_{\rm dig}=0.5$, $\Delta=0.001$, $B_{\rm max}=\frac{K}{2}=5000$. The exact value obtained by exact diagonalization is 0.2539 in this case.
• Figure 2: Simulation history of $T=0.1$, $a_{\rm dig}=0.5$, $\Delta=0.001$, $B_{\rm max}=1$. Autocorrelation is much longer than the simulation with $B_{\rm max}=\frac{K}{2}=5000$ shown in Fig. \ref{['fig:V-for-quartic-potential_history']}.
• Figure 3: By using the values in Table \ref{['table:a-dependence-2d-lattice-4*4']}, $\frac{{\rm Exact}-{\rm MC}}{{\rm Exact}}$ is plotted, where '${\rm Exact}$' is the exact value \ref{['eq:width-mom-rep']} which should be obtained in the limit of $\Delta\to 0$, $R\to\infty$ and $a_{\rm dig}\to 0$ and '${\rm MC}$' is the numerical results in Table \ref{['table:a-dependence-2d-lattice-4*4']}. The horizontal axis is $\frac{1}{a_{\rm dig}}$. We performed a fit of the data in Table \ref{['table:a-dependence-2d-lattice-4*4']} using the function $\left\langle\hat{\tilde{\phi}}_{\vec{q}}\hat{\tilde{\phi}}_{-\vec{q}}\right\rangle=Ae^{-B/a_{\rm dig}}+C$. In this plot we fix $C$ to the exact value $1.081977$ for $\vec{q}=(0,0)$ and $0.184131$ for $\vec{q}=(\pi,\pi)$. This is different from Fig. \ref{['fig:2d-scalar-log-2']}, in which $C$ is treated as a fitting parameter. We obtained $A=-0.079(11)$, $B=0.983(66)$ for $\vec{q}=(0,0)$ from $a_{\rm dig}=0.25, 0.30, 0.40, 0.50$ and $A= -0.0430(69)$, $B= 0.794(58)$ for $\vec{q}=(\pi,\pi)$ from $a_{\rm dig}=0.20, 0.25, 0.30, 0.40$.
• Figure 4: By using the values in Table \ref{['table:a-dependence-2d-lattice-4*4']}, we performed a fit using the function $\left\langle\hat{\tilde{\phi}}_{\vec{q}}\hat{\tilde{\phi}}_{-\vec{q}}\right\rangle=Ae^{-B/a_{\rm dig}}+C$. Unlike Fig. \ref{['fig:2d-scalar-log']}, here we treated $C$ as a fitting parameter. For $\vec{q}=(0,0)$, we obtained $A=-0.096(47)$, $B=1.10(28)$ and $C=1.0814(11)$ from $a_{\rm dig}=0.25, 0.30, 0.40, 0.50$. For $\vec{q}=(\pi,\pi)$, we obtained $A=-0.0780(85)$, $B=1.094(50)$ and $C=0.18324(10)$ from $a_{\rm dig}=0.20, 0.25, 0.30, 0.40$. We plot $\frac{{\rm Fit}-{\rm MC}}{{\rm Fit}}$, where '${\rm Fit}$' is the fit value $C$. This is again in contrast to Fig. \ref{['fig:2d-scalar-log']} where we used the '${\rm Exact}$' value for $C$. The horizontal axis is $\frac{1}{a_{\rm dig}}$.