Cross-State Monte Carlo Sampling

• The paper's main contribution is the development of methods that partition the state space into discrete macrostates to achieve statistically optimal variance reduction.
• It demonstrates the use of stratified sampling in Markov chains, transition-matrix Monte Carlo for rare events, and state/superstate fragmentation in quantum impurity solvers.
• The approach yields significant computational efficiency gains, with improved variance decay rates and accelerated sampling of high-dimensional or low-probability events.

Cross-state Monte Carlo sampling refers to a class of Monte Carlo methodologies designed for efficient exploration and sampling across macroscopic or high-level “states” of a system. These can include stratified ensemble approaches in Markov processes, transition-matrix and rare-event sampling in statistical physics, as well as state- or superstate-based fragmentation in quantum impurity solvers. These methods explicitly address inefficiencies of standard Monte Carlo by structuring sampling so as to ensure uniform or targeted coverage over different regions of state space, often with statistically optimal variance reduction and substantially accelerated sampling of rare or low-probability events.

1. Core Principles of Cross-State Sampling

The foundational concept underlying cross-state Monte Carlo methods is the partitioning of the sample space into discrete macrostates or strata, accompanied by explicit bookkeeping of transitions or weights associated to each. For discrete Markov chains, stratified sampling operates by decomposing the state space into a multidimensional grid and applying deterministic or quasi-random sampling schemes to ensure each region is represented. In transition matrix Monte Carlo (TMMC), binning is performed with respect to a macroscopic observable (e.g., energy), and statistics on transitions between these bins are accumulated. For quantum many-body approaches such as continuous-time hybridization expansion Monte Carlo (CT–HYB), cross-state approaches fragment the configuration space into sums over superstates or states, allowing stochastic sampling of these fragments (Fakhereddine et al., 2016, Yevick, 2015, Kowalski et al., 2018, Shang et al., 2014).

2. Stratified Sampling for Markov Chains

In the stratified or "cross-state" Monte Carlo simulation of Markov chains, the state space $E \subset \mathbb{R}^s$ and transitions are described by

$X_{p+1} = \varphi_{p+1}(X_p, U_{p+1}),$

where $U_{p+1}$ is uniform on the $d$-dimensional unit hypercube $I^d$. Cross-state sampling simulates $N=n^{s+d}$ copies of the chain in parallel, sorting these samples after each step into an $s$-dimensional grid (according to lexicographical order of state coordinates), and then advancing the ensemble by applying a stratified sample in $I^{s+d}$ to the transitions. The result is that each s-dimensional hypercube is equally represented, and stepwise transitions are driven by innovations with carefully distributed randomness.

Variance reduction is quantifiable: while classical Monte Carlo exhibits $O(N^{-1})$ variance decay, stratified and Sudoku sampling achieve

$\operatorname{Var}(Y) = O(N^{-(1+1/s)}),$

which, for the Markov chain context, compounds over $P$ steps such that the overall variance for a function of the chain endpoint is $O(N^{-(1+1/(s+d))})$, assuming suitable regularity in the test function and transition kernels. This facilitates variance reductions of orders of magnitude for finite computational budgets, as detailed in option pricing experiments and corresponding empirical slopes (Fakhereddine et al., 2016).

Sampling Scheme	Variance Decay Rate	Stratification Dimension
Classical MC	$O(N^{-1})$	—
Stratified (SMC/SS)	$O(N^{-(1+1/(s+d))})$	$s+d$

3. Transition-Matrix Monte Carlo for Rare Event Sampling

Transition-matrix Monte Carlo (TMMC) focuses on the empirical reconstruction of density of states functions across a macroscopic observable $E$ (“cross-states”). The allowed $E$ range is partitioned into $N+1$ bins; for each attempted simulation move from state $i$ to $j$ (accepted or rejected), a transition count $C_{ij}$ is recorded. The empirical row-normalized transition matrix is

$T_{ij} = \frac{C_{ij}}{\sum_{k=0}^N C_{ik}},$

and the stationary distribution (density of states) $\mathbf{p}$ is the unit eigenvector of $T$:

$\mathbf{T}\mathbf{p} = \mathbf{p}.$

Alternatively, in one-dimensional cases, the ratio recursion via detailed balance enables closed-form recovery.

To accelerate sampling of rare macrostates, a directional bias is employed: transitions are systematically forced upward to the high-energy endpoint, then downward, alternating. During sweeps, only forward moves (in the given direction) are accepted, but all attempted transitions (including rejected ones) are accumulated in $C_{ij}$, ensuring unbiased statistics. This mechanism ensures uniform sampling over the entire $E$ domain and rapid probing of rare event statistics (Yevick, 2015).

4. State and Superstate Sampling in Quantum Impurity Methods

For CT–HYB quantum impurity solvers, cross-state sampling is realized via decomposition of configuration weights into sums over superstates (blocks defined by conserved quantum numbers) or even into individual states within each block. Conventional evaluation of the local trace $\mathrm{Tr}_{\mathrm{loc}}$ is replaced by stochastic summation over these fragments:

• Superstate sampling: Each Monte Carlo configuration is a $(C, \mathcal{S})$ pair, weight $$w_{\mathrm{loc},\mathcal{S}}(C) w_{\mathrm{bath}}(C)$$; the outer sum over superstates is handled by Monte Carlo.
• State sampling: Sampling is refined further, with MC configurations as $(C, s)$ within a superstate, weight $w_{\mathrm{loc},s}(C) w_{\mathrm{bath}}(C)$.

Moves ensuring ergodicity involve both standard local updates and global $\tau$-shifts, allowing changes in the outer superstate or state index. These approaches maintain (or negligibly affect) the average sign in fermionic systems, deliver negligible autocorrelation penalty, and reduce the local trace bottleneck from $O(D^2)$ to $O(\dim \mathcal{S})$ or $O(1)$ per update in large multi-orbital problems. Speed-ups of $10^2$–$10^3$ are reported in five-orbital impurity benchmarks, without introducing additional truncation or approximation (Kowalski et al., 2018).

Sampling Variant	MC Configuration	Weight Calculation	Computational Cost
Conventional	$C$	$w_{\mathrm{loc}}(C)$	$O(kD^2)$
Superstate sampling	$(C, \mathcal{S})$	$w_{\mathrm{loc},\mathcal{S}}(C)$	$O(kd_{\max}^2)$, $d_{\max} \ll D$
State sampling	$(C, s)$	$w_{\mathrm{loc},s}(C)$	$O(k\dim \mathcal{S})$

5. Cross-State Sampling in Quantum State Space Monte Carlo

Sampling over the quantum state space subject to positivity and trace constraints constitutes another relevant context for cross-state Monte Carlo. Standard MCMC (Metropolis–Hastings on the $x$-sphere representation), rejection sampling, and importance sampling are systematically adapted to this constrained domain. Proposals are drawn in a reparameterized space $x_k$, ensuring $\sum_k x_k^2 = 1$ and $x_k \geq 0$, with acceptance/rejection based on quantum physicality checks.

In the rejection and importance sampling schemes, independent draws from a reference measure $g(p)$ are weighted or filtered according to the desired target $\pi(p)$, with unphysical samples assigned zero weight. Efficiency comparisons reveal that MCMC with tuned proposal width ($\sigma$) attains effective sample sizes orders of magnitude larger than rejection sampling for high-dimensional domains, with autocorrelation minimized when the acceptance rate is around 23–25% (Shang et al., 2014).

6. Variance Reduction, Efficiency, and Practical Limitations

Across representative domains—Markov chains, rare-event statistics, quantum impurity solvers—the principal advantage of cross-state Monte Carlo is radical improvement in variance for fixed sample count. The stratified approaches underpin variance decay rates beyond $O(N^{-1})$, with explicit dependence on the sum $s+d$ of the state and noise dimensions.

Computational complexity involves $O(N\log N)$ sorting at each iteration in multi-dimensional stratification, and memory requirements scale as $N = n^{s+d}$. Thus, for very large $s+d$, practical usage is limited by the curse of dimensionality, but substantial computational gains are achievable for moderate dimensionalities. In TMMC or state/superstate sampling, efficiency is determined by both the macroscopic binning strategy and the capacity to accelerate exploration with directional or stochastic fragmentation of the configuration space (Fakhereddine et al., 2016, Yevick, 2015, Kowalski et al., 2018).

7. Extensions and Generalizations

The cross-state stratification paradigm extends naturally to systems with complex or block-diagonal structure, especially where quantum numbers or symmetries define macroscopic sectors of the configuration space. In multi-orbital or full-Coulomb quantum impurity models, fragmentation is particularly beneficial: increasing system complexity raises the number of blocks (superstates) but reduces their average size, magnifying efficiency gains as the dimensionality increases. In classical MC or option pricing, stratified algorithms generalize to higher dimensions and a variety of test statistics, provided reasonable regularity of domain boundaries; irregular payoff functions or indicator sets may reduce theoretical gains but stratification remains advantageous up to practical limits (Fakhereddine et al., 2016, Kowalski et al., 2018).

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References:

(Shang et al., 2014): Monte Carlo sampling from the quantum state space. I (Yevick, 2015): Accelerated rare event sampling (Fakhereddine et al., 2016): Stratified Monte Carlo simulation of Markov chains (Kowalski et al., 2018): State- and superstate-sampling in hybridization-expansion continuous-time quantum Monte Carlo