Finite size scaling of bitstring probability distributions for Rydberg arrays

Technical Summary: Finite Size Scaling of Bitstring Probability Distributions for Rydberg Arrays

Problem Statement
In quantum simulations of particle, nuclear, and condensed matter physics, estimating observables with statistical accuracy requires a large number of measurements (shots). As the simulated system size ($N_q$, the number of qubits) increases, the number of possible bitstring states grows exponentially, causing individual state probabilities ($p_{\{n\}}$) to decrease and become denser in the low-probability regime. This raises a critical question: can the smallness of individual probabilities be compensated by their sheer number, and what is the minimal number of shots required to achieve a desired statistical accuracy for larger systems? Previous work suggests that characterizing systems via large-scale distributions can reveal structural behaviors, but explicit scaling laws for bitstring probabilities in Rydberg atom arrays have not been fully quantified.

Methodology
The authors investigate Rydberg atom arrays arranged in a ladder configuration (two rows with a vertical-to-horizontal spacing ratio of $\rho=2$). The system is described by a Hamiltonian involving Rabi frequency ($\Omega$), detuning ($\Delta$), and van der Waals interactions ($V_{ij}$).

To analyze the probability distributions, the authors utilize Matrix Product States (MPS) calculated via Density Matrix Renormalization Group (DMRG) methods. They generate dictionaries of bitstring probabilities ($p_{\{n\}}$) for systems ranging from 4 to 22 rungs (up to 44 atoms). The core of their analysis involves the cumulative probability distribution, defined as:
$$\Sigma(p_\Lambda, N_q) = \sum_{\{n\}: p_{\{n\}} \leq p_\Lambda} p_{\{n\}}$$
This function represents the total probability of observing any state with a probability less than or equal to a threshold $p_\Lambda$. The authors analyze the scaling behavior of $\Sigma$ across different system sizes ($N_q$) and shot totals ($N_{sh} = 10^9$).

Key Contributions and Results

1. Finite Size Scaling and Curve Collapse:
   The authors demonstrate that cumulative probability distributions for different system sizes exhibit a "repeated behavior," particularly in the range $0.2 < \Sigma < 0.8$. By identifying a characteristic probability $p^*$ (where the distribution reaches a fixed $\Sigma$ value) and rescaling the independent variable as $(p_\Lambda/p^*)^B$ (where $B$ is the slope of the linear region), the distributions for various $N_q$ collapse onto a single universal curve.

2. Fermi-Dirac Analogy (Region II):
   In the central region of the distribution (Region II), the collapsed curve closely resembles a Fermi-Dirac statistical distribution. The authors show that $\Sigma(p_\Lambda)$ can be approximated by:
   $$\Sigma \simeq \frac{1}{1 + e^{\beta(\epsilon - \mu)}}$$
   where $\epsilon = -\ln(p_\Lambda)$ and $\mu = -\ln(p^*)$. This analogy allows for the characterization of the "inflection point" (where $\Sigma = 1/2$, denoted as $p^*_{1/2}$). The scaling of $p^*_{1/2}$ with system size is found to be exponential, indicating that as $N_q$ increases, the probability mass becomes more uniformly distributed across a vast number of states.

3. Scaling of Maximal Probabilities (Region I):
   In Region I (high probability states), the authors track the maximal single-state probability ($p_{\max}$). They confirm that $p_{\max}$ scales exponentially with system size ($p_{\max} \sim 2^{-N_q}$). This establishes a fundamental limit on the system size accessible for a given shot total, as the minimal resolvable probability is bounded by $1/N_{sh}$.

4. Low Probability and Single-Occurrence States (Regions III & IV):

   • Region III: At lower probabilities, the distributions follow a power-law form $\Sigma(p_\Lambda) \simeq A p_\Lambda^B$ when plotted on a log-log scale. The authors note that the behavior here is heavily influenced by the finite shot total ($10^9$), which creates a hard cutoff at $p \approx 10^{-9}$.
   • Region IV: This region represents the "tail" of single-occurrence states. The authors quantify the number of states observed exactly once, showing that while this number grows with system size, it grows slower than the total Hilbert space dimension ($2^{N_q}$).

5. Shot Requirement Estimation:
   By analyzing the scaling of $p^*_{1/2}$ and $p_{\max}$, the authors estimate the shot requirements for future simulations.

   • Based on the $p_{\max}$ scaling, a system size of approximately 120 qubits could be simulated with $10^9$ shots.
   • Based on the $p^*_{1/2}$ scaling (where the cumulative probability reaches 0.5), the limit for a specific accuracy is estimated around 60 qubits for the same shot total.

Significance and Claims
The paper claims to provide a method for predicting necessary shot requirements for large quantum systems by analyzing the cumulative probability distributions of smaller, computationally accessible systems. The primary significance lies in the identification of an exponential cost associated with increasing system size, which is quantified through the scaling of $p^*_{1/2}$ and $p_{\max}$.

However, the authors are modest in their conclusions, noting that while the cost is exponential, it increases at a fraction of the rate of the system size increase itself. Consequently, this does not "hard cap" device usage but rather motivates the "intelligent usage of resources." The work establishes that the behavior of large systems can be described by the distributions of smaller systems via the proposed collapse and Fermi-Dirac analogy, offering a tool to estimate statistical errors and the feasibility of simulating increasingly larger Rydberg arrays.