Optimal Quantum Speedups for Repeatedly Nested Expectation Estimation

We study the estimation of repeatedly nested expectations (RNEs) with a constant horizon (number of nestings) using quantum computing. We propose a quantum algorithm that achieves $\varepsilon$-error with cost $\tilde O(\varepsilon^{-1})$, up to logarithmic factors. Standard lower bounds show this scaling is essentially optimal, yielding an almost quadratic speedup over the best classical algorithm. Our results extend prior quantum speedups for single nested expectations to repeated nesting, and therefore cover a broader range of applications, including optimal stopping. This extension requires a new derandomized variant of the classical randomized Multilevel Monte Carlo (rMLMC) algorithm. Careful de-randomization is key to overcoming a variable-time issue that typically increases quantized versions of classical randomized algorithms.

💡 Research Summary

The paper addresses the problem of estimating repeatedly nested expectations (RNEs) when the nesting depth (horizon) D is fixed, a setting that captures many important tasks such as optimal stopping, credit risk valuation, and inference in probabilistic programs. Classical approaches, notably the randomized Multilevel Monte Carlo (rMLMC) method introduced in prior work, achieve an ε‑mean‑squared error with a sample complexity of O(ε⁻²) under Lipschitz smoothness assumptions. However, a direct quantum translation of rMLMC fails to retain the quadratic speedup because rMLMC is a variable‑time algorithm: the random truncation level N leads to a random runtime, which interferes with amplitude‑amplification techniques and can erase the expected quantum advantage.

The authors’ central contribution is a two‑step construction that overcomes this obstacle. First, they derandomize rMLMC. They replace the geometric distribution governing the truncation level with a deterministic schedule, truncating the geometric tail at a level B_d = Θ(log ε⁻¹). By carefully choosing the geometric rate r_d (satisfying a specific algebraic relation that balances bias and variance) they ensure that the bias introduced by truncation is at most O(ε·2^{d−D}) while keeping the per‑sample cost constant. Moreover, they allocate a predetermined number of samples M_d·P_d(n) to each level n, where M_d = Θ(ε⁻²·(1+δ/2^{d−1})) and P_d(n) ∝ (1−r_d)^n for n ≤ B_d. This deterministic level scheduling eliminates the variable‑time behavior and yields a classical algorithm (Theorem 1.4) with the same O(ε⁻²) complexity as the original rMLMC but with a fully controlled runtime.

Second, they quantize the derandomized algorithm using Quantum‑Accelerated Monte Carlo (QAMC), which is essentially Grover‑based quantum amplitude estimation (QAE). QAE can estimate the mean of a bounded random variable to ε accuracy with O(ε⁻¹·polylog(1/ε)) quantum queries. By applying QAE independently at each deterministic level, the authors obtain a quantum algorithm whose total cost is the sum over levels of O(ε⁻¹·log (1/ε)) multiplied by the number of levels (which is O(D), treated as a constant). The resulting overall complexity is ˜O(ε⁻¹)·log³(D), as stated in Theorem 1.6. This matches the known lower bound Ω̃(ε⁻¹) for quantum mean estimation, proving optimality up to polylogarithmic factors.

The technical analysis hinges on several probabilistic inequalities. The von Bahr‑Esseen inequality bounds the p‑th moment (with p∈