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 already encounters when working with 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 practical and versatile method for sampling quantum states in accordance with 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, compared with less than ten such states per day from independence sampling in our implementation. In the second application, we draw samples of high-dimensional quantum states from a narrowly peaked target distribution and observe, for the system sizes investigated, that SCMC sampling remains computationally manageable as the dimensions grow. 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.
Quasi-Monte Carlo (QMC) methods are known to achieve faster convergence rates than Monte Carlo (MC), but their effectiveness in high dimensions often relies on additional structure, such as low effective dimension or carefully chosen coordinate weights. Moreover, in many applications one has access only to random samples rather than deterministic QMC constructions. In this work, we extend the recent method of N. Bansal and H. Jiang and construct high-dimensional QMC point sets from random samples via combinatorial discrepancy. We establish error bounds for these constructions in weighted function spaces, including settings with low effective dimension in both the superposition and truncation senses. We also implement the resulting Subgaussian Transference algorithm and present numerical experiments to assess empirically the performance of these constructions.
Highly accurate results for the homogeneous electron gas (HEG) have only been achieved to date within a diffusion Monte Carlo (DMC) framework. Here, we introduce a recently developed stochastic technique, full configuration interaction quantum Monte Carlo (FCIQMC), which samples the exact wave function expanded in plane-wave Slater determinants. Despite the introduction of a basis-set incompleteness error, we obtain a finite-basis energy, which is significantly and variationally lower than any previously published work for the 54-electron HEG at
rs = 0.5 a.u., in a Hilbert space of
10108 Slater determinants. At this value of
rs, as well as of 1.0 a.u., we remove the remaining basis-set incompleteness error by extrapolation, yielding results comparable to state-of-the-art DMC backflow energies. In doing so, we demonstrate that it is possible to yield highly accurate results with the FCIQMC method in periodic systems.