Quantum Query Complexity of Entropy Estimation

Abstract

Estimation of Shannon and Rényi entropies of unknown discrete distributions is a fundamental problem in statistical property testing and an active research topic in both theoretical computer science and information theory. Tight bounds on the number of samples to estimate these entropies have been established in the classical setting, while little is known about their quantum counterparts. In this paper, we give the first quantum algorithms for estimating α-Rényi entropies (Shannon entropy being 1-Renyi entropy). In particular, we demonstrate a quadratic quantum speedup for Shannon entropy estimation and a generic quantum speedup for α-Rényi entropy estimation for all α ≥ 0, including a tight bound for the collision-entropy (2-Rényi entropy). We also provide quantum upper bounds for extreme cases such as the Hartley entropy (i.e., the logarithm of the support size of a distribution, corresponding to α = 0) and the min-entropy case (i.e., α = +∞), as well as the Kullback-Leibler divergence between two distributions. Moreover, we complement our results with quantum lower bounds on α-Rényi entropy estimation for all α ≥ 0.Our approach is inspired by the pioneering work of Bravyi, Harrow, and Hassidim (BHH) [13] on quantum algorithms for distributional property testing, however, with many new technical ingredients. For Shannon entropy and 0-Rényi entropy estimation, we improve the performance of the BHH framework, especially its error dependence, using Montanaro's approach to estimating the expected output value of a quantum subroutine with bounded variance [41] and giving a fine-tuned error analysis. For general α-Rényi entropy estimation, we further develop a procedure that recursively approximates α-Rényi entropy for a sequence of αs, which is in spirit similar to a cooling schedule in simulated annealing. For special cases such as integer α ≥ 2 and α = +∞ (i.e., the min-entropy), we reduce the entropy estimation problem to the α-distinctness and the log n -distinctness problems, respectively. We exploit various techniques to obtain our lower bounds for different ranges of α, including reductions to (variants of) existing lower bounds in quantum query complexity as well as the polynomial method inspired by the celebrated quantum lower bound for the collision problem.

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“…1 Later, Belovs introduced a beautiful new framework for designing quantum algorithms [17] and used it to improve the upper bound for k-distinctness to O(n 3/4−1/(2 k+2 −4) ) [16]. Several subsequent works have used Belovs' k-distinctness algorithm as a black-box subroutine for solving more complicated problems (e. g., [58,61]).…”

“…Best prior upper bound Our lower bound Best prior lower bound k-Distinctness O(n 3/4−1/(2 k+2 −4) ) [16]Ω(n 3/4−1/(2k) )Ω(n 2/3 ) [3] Image size testing O( √ n log n) [9]Ω( √ n)Ω(n 1/3 ) [9] k-Junta testing O( √ k log k) [9]Ω( √ k)Ω(k 1/3 ) [9] SDU O( √ n) [23]Ω( √ n)Ω(n 1/3 ) [3,23] Shannon entropyÕ( √ n) [23,58]Ω( √ n)Ω(n 1/3 ) [58] such a result is not known for any other quantum query lower bound technique. More generally, using approximate degree as a lower bound technique for quantum query complexity has other advantages, such as the ability to show lower bounds for zero-error and small-error quantum algorithms [24], unbounded-error quantum algorithms [13], and time-space tradeoffs [48].…”

Optimizing quantum optimization algorithms via faster quantum gradient computation

“…By Corollary 42 we get that for all |ψ ∈ H O|ψ |0 ⊗(n+a) − O p |ψ |0 ⊗(n+a) ≤ 10ε , therefore we can choose ε := ε/10 to conclude the proof. Now we show a corollary of the above result, which can be relevant for quantum distribution testing [LW17].…”

“…One possible application which is relevant for quantum distribution testing [LW17] is the following. Suppose we are given access to some probability distribution via a quantum oracle…”

Measurement-induced entanglement transitions in many-body localized systems

“…This may therefore be an opportunity for NISQ-era [77] quantum simulators to probe new physics out of the reach of numerics. Direct measurement of the entanglement entropy associated with a single quantum trajectory may be difficult, owing to the need to perform the exponentially many experimental repetitions associated with the postselection of measurement outcomes [11] and the complexity of measuring an entropy [78,79]. However, there have been proposals for more experimentally feasible probes of the entanglement transition based on the Fisher information [11] and coupled ancilla qubits [15].…”