Approximate Classical Sampling Algorithms

Updated 20 January 2026

Approximate classical sampling algorithms are computational methods that approximate quantum circuit outputs using techniques like Taylor expansions, tensor contractions, and Monte Carlo integration.
They are essential for benchmarking quantum advantage, dequantizing quantum machine learning, and simulating noisy quantum protocols.
Key strategies include polynomial truncation, ℓ²-norm sampling, and rejection sampling, each optimized for specific regimes of circuit complexity and noise.

Approximate classical sampling algorithms are a broad class of computational procedures designed to efficiently approximate the output distributions, expectation values, or specific observables of quantum circuits and complex probabilistic models using classical resources. These algorithms are critical for benchmarking quantum advantage, dequantizing quantum algorithms, and simulating noisy or lossy quantum protocols where exact classical computation may be infeasible but approximate methods enable scalable, rigorous classical treatment.

1. Mathematical Foundations and Problem Formulation

Approximate classical sampling problems typically seek to generate samples $x \in \mathcal{X}$ or expectation values $\mathbb{E}[O]$ such that the generated distribution $q(x)$ is close—in total variation, cross-entropy, or other statistical metrics—to a target distribution $p(x)$ arising from a quantum process, high-dimensional integration, or probabilistic inference. These tasks appear prominently in linear optical circuits (boson sampling, GBS) (Umanskii et al., 2024, Lim et al., 18 Feb 2025, Oh et al., 2023), random circuit sampling (Kalachev et al., 2021), matrix multiplication (Wu, 2018), low-rank approximation (Tang, 2018, Chia et al., 2019), vibronic spectra estimation (Oh et al., 2022), and broader high-dimensional integration (Nakatsukasa, 2018).

The typical workflow involves:

Defining the probability (or moment) to be computed, often as a trace, expectation, or output amplitude from a quantum circuit, e.g. $P(\mathbf{n}) = \mathrm{Tr}\{\rho_{\text{out}} |\mathbf{n}\rangle\langle \mathbf{n}| \}$ for photon-output probabilities.
Approximating this target via Taylor, polynomial, or tensor-network expansion schemes, enabling truncation or low-degree methods for scalable evaluation.
Using Monte Carlo, rejection sampling, sketching, or data-structure-driven sampling for practical implementation.

2. Core Algorithmic Strategies

Taylor/Polynomial Expansion & Truncation

In lossy Gaussian boson sampling, probabilities are expressed as expectations over Gaussian variables via the Hubbard–Stratonovich transform and subsequently expanded in a Taylor series. Truncation at order $K$ yields a polynomially efficient approximation $P_K(\mathbf{n})$ with error scaling as $\varepsilon^{K+1}/(1-\varepsilon)$ , where $\varepsilon$ is a "perturbation parameter" set by squeezing and loss (Umanskii et al., 2024).

Similarly, Kalai–Kindler-style low-degree truncations in boson sampling exploit rapid noise-induced suppression of high-degree polynomial terms, yielding sparse polynomial approximants $\bar{q}(z)$ that can be sampled efficiently for high noise rates (Oh et al., 2023).

Tensor-Network and Matrix-Product State Samplers

For high-loss or shallow quantum circuits, tensor-network formulations (e.g., MPS) facilitate decomposition of quantum states $\rho$ into factorizable components that admit approximation with tunable bond dimension $\chi$ , controlling both trace-norm error and runtime (Oh et al., 2023, Bravyi et al., 2023). Sampling from the approximate state is achieved via contraction and recursion, with complexity governed by entanglement entropy and circuit locality.

Monte Carlo and Function Approximation

Variance-reduced Monte Carlo integration approximates the integrand by polynomial, sparse-grid, or low-rank surrogates $p(x)$ , integrates $p$ exactly, and samples the residual. This methodology accelerates convergence compared to pure MC, with variance determined by $\|f - p^*\|_{L^2}$ (Nakatsukasa, 2018).

ℓ²-Norm Sampling: Dequantizing Quantum Algorithms

Quantum-inspired classical algorithms leverage ℓ²-norm sampling and query-access data structures to implement linear-algebraic routines central to quantum machine learning (e.g., recommendation systems, SVT) in time polylogarithmic in input size (Tang, 2018, Chia et al., 2019). Matrix-vector and matrix-matrix products are approximated via random sampling in Frobenius norm, supporting scalable low-rank approximations and polytime classical sampling from projected distributions.

Rejection Sampling and Composite Potential Methods

For sampling from distributions proportional to $\exp(-f(x))$ , inexact rejection, centered at stationary points and capped by bounding ratios, enables efficient sampling from restricted Gaussian oracles (RGO) even for semi-smooth or composite potentials. The acceptance criterion is crafted to control total variation error, and concentration inequalities for semi-smooth functions over Gaussians underpin error bounds and step size selection (Fan et al., 2023).

3. Complexity Analysis and Regimes of Applicability

The runtime and accuracy of approximate classical sampling algorithms depend critically on structural parameters of the underlying quantum model or probabilistic function:

Polynomial Scaling: For fixed truncation order, loss/squeezing in boson sampling, or sparsity in output probabilities, algorithms can sample or estimate probabilities in $\mathrm{poly}(N,1/\epsilon)$ time, where $N$ is system size and $\epsilon$ the error (Umanskii et al., 2024, Lim et al., 18 Feb 2025, Oh et al., 2023).
Quasi-Polynomial/Polylogarithmic: In random circuit sampling or peaked/shallow circuits, entanglement and circuit geometry dictate complexity, with quasipolynomial scaling $n^{O(\log n)}$ in generic cases, but polynomial time for low-rank, sparsity, or geometric locality (Kalachev et al., 2021, Bravyi et al., 2023).
Variance Reduction: Adaptive enrichment in MC-based integration can break the $1/\sqrt{N}$ barrier as functional approximation improves (Nakatsukasa, 2018).
Oversampling Factors: In quantum-inspired matrix algorithms, cancellation factors and oversampling constants affect the number of samples required for Frobenius or total-variation approximation within desired error (Chia et al., 2019).

4. Implementation Protocols and Pseudocode

Algorithmic procedures are characterized by:

Precomputation: Covariances, moments, tensor contractions, and data structures (e.g. balanced binary trees for ℓ²-sampling) are typically computed once for each instance, enabling sublinear per-query costs and amortization over multiple samples.
Sampling Loop: Monte Carlo samples are drawn from parameterized distributions; function expansions (e.g. Taylor/Kalai–Kindler/truncated polynomial) are evaluated; output patterns or observables are accumulated via recursive contraction, rejection, or marginal sampling.
Performance Tuning: Truncation order, Monte Carlo sample size, bond dimension in MPS, and data structure depth are optimized based on physical parameters (loss, squeezing, sparsity, aspect ratio, circuit geometry).

A representative pseudocode for a Taylor-truncated GBS sampler (Umanskii et al., 2024):

Precompute moments, contractions, factorial tables
for t in range(1, M+1):
    xi0 = draw_normal(0, sigma^2, N)
    S = U.T @ xi0
    for pattern n in patterns:
        p_t[n] = sum_{k=0}^K T_k(S, n)
Estimate P_K(n) = avg(p_t[n] over t)

5. Benchmarking, Empirical Results and Limits

Comprehensive benchmarks demonstrate that, in parameter regimes relevant to contemporary quantum experiments (e.g., moderate squeezing and loss in GBS, moderate output sparsity), polynomial-time or polylogarithmic-time classical samplers attain fidelity, cross-entropy, and correlation accuracy rivaling or exceeding quantum experiments (Umanskii et al., 2024, Oh et al., 2023, Kalachev et al., 2021). For instance, simulating a 40-mode GBS instance on a laptop in seconds is feasible when $\varepsilon \lesssim 0.2$ and $K=3$ –5 (Umanskii et al., 2024).

Tensor-network simulations for large-scale GBS ( $M \sim 288$ ) achieve total-variation errors below experimental noise with manageable computational resources (Oh et al., 2023). In random quantum circuits, partial slicing and tensor contraction combined with frugal rejection sampling allow for bounded-fidelity sampling on classical clusters for circuits previously deemed intractable (Kalachev et al., 2021).

However, limitations arise for partial distinguishability, nonuniform loss, high entanglement entropy, or fully general (Fock + squeezing + displacement) vibronic spectra—these domains may retain regimes where quantum advantage persists due to hardness of permanent/hafnian computation or entropic scaling (Oh et al., 2023, Oh et al., 2022).

6. Applications and Broader Impact

Approximate classical sampling algorithms are now foundational in:

Benchmarking Quantum Advantage: Their development is critical for interpreting claims of quantum supremacy in GBS, RQC, and linear optics (Oh et al., 2023, Kalachev et al., 2021).
Dequantizing Quantum Machine Learning: Techniques in ℓ²-norm sampling and matrix sketching refute exponential speedup claims in recommendation systems, PCA, SVT, supervised clustering, and regression (Tang, 2018, Chia et al., 2019, Shah et al., 2024).
Molecular Spectroscopy and Quantum Chemistry: Sampling algorithms now provide efficient classical computation of vibronic spectra that were previously believed to require quantum resources (Oh et al., 2022, Lim et al., 18 Feb 2025).
High-Dimensional Numerical Integration: Function approximation and adaptive MC methods offer scalable alternatives to brute-force quadrature/cubature in multidimensional domains (Nakatsukasa, 2018).
Simulating Shallow and Local Quantum Circuits: Classical protocols reach quasi-polynomial or better scaling in shallow, peaked, or geometrically local circuits, guiding resource allocation in quantum hardware development and algorithm selection (Bravyi et al., 2023).

7. Limitations, Open Problems, and Future Directions

Despite extensive progress, precise classical simulation of circuits with severe entanglement or certain realistic noise models (partial distinguishability, correlated losses, general loop-hafnians) remains computationally hard in worst-case scenarios. There is an ongoing need to refine marginal sampling techniques, expand polynomial approximant bases for new classes of quantum circuits, and extend tensor-network contraction beyond local or peaked regimes (Oh et al., 2023, Oh et al., 2022, Bravyi et al., 2023).

A plausible implication is that further hybridization of classical and quantum-inspired sampling (e.g. leveraging problem structure, sparsity, adaptive approximants) may continue to expand the boundaries separating classical and quantum computational regimes. Ultimately, the regime of strict classical intractability is shifting dynamically as novel approximate sampling algorithms evolve.

Key Citations:

(Umanskii et al., 2024) Classical modelling of a lossy Gaussian bosonic sampler
(Lim et al., 18 Feb 2025) Efficient classical algorithms for linear optical circuits
(Oh et al., 2023) Classical algorithm for simulating experimental Gaussian boson sampling
(Kalachev et al., 2021) Classical Sampling of Random Quantum Circuits with Bounded Fidelity
(Tang, 2018) A quantum-inspired classical algorithm for recommendation systems
(Chia et al., 2019) Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning
(Nakatsukasa, 2018) Approximate and integrate: Variance reduction in Monte Carlo integration via function approximation
(Wu, 2018) A Note on Random Sampling for Matrix Multiplication
(Bravyi et al., 2023) Classical simulation of peaked shallow quantum circuits
(Oh et al., 2023) On classical simulation algorithms for noisy Boson Sampling
(Oh et al., 2022) Quantum-inspired classical algorithm for molecular vibronic spectra
(Fan et al., 2023) Improved dimension dependence of a proximal algorithm for sampling
(Shah et al., 2024) Quantum (Inspired) $D^2$ -sampling with Applications