Sampling quantum states with inequality constraints

TL;DR

This paper addresses the difficulty of sampling quantum states under properties stated as inequalities in high-dimensional state spaces. It introduces Sequentially Constrained Monte Carlo (SCMC), which builds a bridge from an easily sampled reference distribution to a challenging target via intermediate distributions and soft-to-hard constraint enforcement, enabling efficient sampling even under complex positivity constraints. The authors demonstrate three key applications: large-scale generation of bound entangled states (including several thousand two-qutrit states in minutes), sampling from targeted distributions in multi-qubit systems with favorable scaling, and an SCMC-enabled benchmark of the Oh–Teo–Jeong (OTJ) method showing SCMC's superior efficiency and the OTJ bias. The results suggest SCMC is a versatile, scalable tool for quantum-state sampling with broad potential across quantum information tasks and experimental data analysis.

Abstract

Random samples of quantum states with specific properties are useful for various applications, such as Monte Carlo integration over the state space. In the high-dimensional situations that one encounters already for a few qubits, the quantum state space has a very complicated boundary, and it is challenging to incorporate the specific properties into the sampling algorithm. In this paper, we present the Sequentially Constrained Monte Carlo (SCMC) algorithm as a powerful and versatile method for sampling quantum states in accordance with any desired properties that can be stated as inequalities. We apply the SCMC algorithm to the generation of samples of bound entangled states; for example, we obtain nearly ten thousand bound entangled two-qutrit states in a few minutes -- a colossal speed-up over independence sampling, which yields less than ten such states per day. In the second application, we draw samples of high-dimensional quantum states from a narrowly peaked target distribution and observe that SCMC sampling remains efficient as the dimension grows. In yet another application, the SCMC algorithm produces uniformly distributed quantum states in regions bounded by values of the problem-specific target distribution; such samples are needed when estimating parameters from the probabilistic data acquired in quantum experiments.

Figures (7)

• Figure S1: The generation of bound entangled states using SCMC. The initial reference sample has $10^4$ states (red crosses) uniformly distributed with respect to the Hilbert--Schmidt distance shown here on an $R$ vs. $\min\{\mu^{\ }_{\mathrm{PT}}\}$ plot. The states after SCMC are indicated by the blue dots. Out of these 10,000 post-SCMC states, 8,530, 7,011, 2,211, and 4,013 states are bound entangled for the respective systems of dimensions $3\times3$, $3\times4$, $3\times5$, and $4\times4$. The insets show samples filtered through further MCMC iterations with different propagation kernels (by choosing different directions of the random walks) represented by dots of different colors.
• Figure S2: Search for bound entangled states with SCMC for the $2\times4$ system. The initial reference sample has $10^4$ states (red crosses) uniformly distributed with respect to the Hilbert--Schmidt distance shown here on an $R$ vs. $\min\{\mu^{\ }_{\mathrm{PT}}\}$ plot. The states after SCMC are indicated by the blue dots. There are no blue dots in the first quadrant (gray) where the criteria (\ref{['eq:BEcriteria']}) are obeyed. Enforcing only the PPT criterion results in the orange sample; enforcing only the CCNR criterion results in the green sample. The inset shows the orange sample and, in different colors, further samples obtained by SCMC steps toward enforcing the CCNR criterion.
• Figure S4: The content $c_\lambda$ evaluated for samples generated for (a) a three-qubit target distribution and (b) a four-qubit target distribution. The target distributions are given by ${A=3000}$ randomly generated detection events for product tetrahedron measurements 34qubitdata. The samples are obtained using SCMC with $N_s$ initial reference points drawn from the Wishart distribution, the uniform distribution, the Dirichlet distribution, or the Dirichlet distribution centered at the peak of $f(\rho)$. The SCMC algorithm is run for different numbers of intermediate distributions $N_\tau$. The inset in (a) is a blow-up of the marked rectangular area.
• Figure S5: OTJ algorithm illustrated. Top: Uniform samples on the $\lambda$-regions for single-qubit states on the equatorial disk of the Bloch ball. The target distribution refers to the experimental data of a distorted trine measurement Len2018 with ${(\alpha_1,\alpha_2,\alpha_3)=(1802,315,303)}$ in Equation (\ref{['eq:Dirichlet']}). The scattered points of different colors mark $10^4$ states each in the $\lambda$-regions with with ${\log_{10}(\lambda) = -\infty}$, $-400$, $-200$, $-100$, $-50$, $-10$, and $-1$, respectively. The initial distribution (${\lambda=0}$, black points) is uniform on the disk. Bottom: Content $c_{\lambda}^{\ }$ evaluated for the three-qubit target distribution of Figure \ref{['fig:34qbbkref']}(a). The black curve is computed directly from a target sample with ${N_s=10^6}$ points generated from a Wishart sample by SCMC. The other colored curves show results from several runs of the OTJ algorithm. The inset is a blow-up of the marked rectangular area.
• Figure A1: A simple illustration of SCMC in 1D. A single-peak reference sample is used to sample a target distribution with two peaks. The intermediate distributions are given by Equation (\ref{['eq:hgen']}) with ${N_\tau=10}$. Both the exact and the numerically estimated distribution for $10^4$ sample points are shown.