[Paper Review] Quantum algorithm for estimating Renyi entropies of quantum states

Sathyawageeswar Subramanian, Min-Hsiu Hsieh|arXiv (Cornell University)|Aug 14, 2019

Quantum Computing Algorithms and Architecture41 references7 citations

TL;DR

This paper presents a quantum algorithm to estimate the $α$-Renyi entropy of an unknown quantum state using the one-clean qubit ($\mathsf{DQC1}$) model and quantum singular value transformation. It achieves additive precision $\epsilon$ with expected query complexity $\mathcal{O}(1/(x\epsilon)^2)$, where $x = \frac{1}{d}\mathrm{Tr}(\rho^\alpha)$, offering a significant improvement over the $\Theta(d^2/\epsilon^2)$ sample complexity of state tomography.

ABSTRACT

We describe a quantum algorithm to estimate the $α$-Renyi entropy of an unknown density matrix $ρ\in\mathcal{C}^{d imes d}$ for $α eq 1$ by combining the recent technique of quantum singular value transformations with the method of estimating normalised traces in the one clean qubit model. We consider an oracular input model where the input state is prepared via a quantum oracle that outputs a purified version of the state, assumed to be non-singular. Our method outputs an estimate of the $α$-Renyi entropy to additive precision $ε$, using an expected total number $O\left(\frac{1}{(xε)^2} ight)$ of independent applications of a quantum circuit which coherently queries the input unitary $O\left(\frac{1}δ\log \frac{d}ε ight)$ times, in each case measuring a single output qubit. Here $δ$ is a lower cutoff on the smallest eigenvalue of $ρ$ and $x=\frac{1}{d}\!\mathop{Tr}{ρ^α}$. The expected number of measurements made in this method can be compared to results in the sample complexity model that generally require $Θ(d^2/ε^2)$ samples. Furthermore, we also show that multiplicative approximations can be obtained by iteratively using additive approximations, with an overhead logarithmic in the dimension $d$.

Motivation & Objective

• To develop an efficient quantum algorithm for estimating $\alpha$-Renyi entropies of unknown quantum states, particularly for $\alpha \neq 1$.
• To reduce the sample complexity compared to full quantum state tomography, which requires $\Theta(d^2/\epsilon^2)$ copies of the state.
• To leverage oracular access to a purified version of the density matrix via a unitary oracle, enabling estimation without full state reconstruction.
• To enable multiplicative approximations through iterative use of additive estimators with logarithmic overhead in dimension $d$.
• To provide a practical alternative to amplitude estimation-based methods by using shallow circuits and many measurements, suitable for near-term quantum devices.

Proposed method

• The algorithm uses block encoding techniques to apply quantum singular value transformations to the density matrix $\rho$, enabling efficient computation of $\mathrm{Tr}(\rho^\alpha)$.
• It combines the one-clean qubit ($\mathsf{DQC1}$) model with trace estimation to estimate $\mathrm{Tr}(\rho^\alpha)$, which is then used to compute the $\alpha$-Renyi entropy via $S_\alpha(\rho) = \frac{1}{1-\alpha}\log\left[\mathrm{Tr}(\rho^\alpha)\right]$.
• The method assumes access to a unitary oracle that prepares a purified version of $\rho$, with a lower bound $\delta$ on its smallest eigenvalue to ensure non-singularity.
• Additive precision $\epsilon$ is achieved using $\mathcal{O}(1/(x\epsilon)^2)$ independent applications of a quantum circuit that coherently queries the input unitary $\mathcal{O}(1/\delta \log(d/\epsilon))$ times per run.
• For multiplicative approximation, the algorithm iteratively applies the additive estimator, with at most $\mathcal{O}(\log(d/\delta))$ iterations, ensuring runtime depends logarithmically on the target entropy.
• The approach avoids deep quantum circuits typical of amplitude estimation, making it suitable for NISQ-era devices with shallow circuits and high measurement counts.

Experimental results

Research questions

• RQ1 Can $\alpha$-Renyi entropy be estimated with sub-tomographic sample complexity using oracular access to a purified quantum state?
• RQ2 How does the query complexity of this algorithm compare to that of full quantum state tomography or amplitude estimation methods?
• RQ3 Can multiplicative approximations of the entropy be obtained efficiently using iterative additive estimators in the $\mathsf{DQC1}$ framework?
• RQ4 What is the trade-off between circuit depth and number of measurements in this approach, and how does it compare to variational or amplitude estimation methods?
• RQ5 Is entropy estimation within the $\mathsf{DQC1}$ complexity class, or is it strictly harder due to the inverse scaling with $\epsilon$?

Key findings

• The algorithm estimates the $\alpha$-Renyi entropy to additive precision $\epsilon$ using an expected $\mathcal{O}(1/(x\epsilon)^2)$ applications of a quantum circuit, where $x = \frac{1}{d}\mathrm{Tr}(\rho^\alpha)$.
• The query complexity per circuit is $\mathcal{O}(1/\delta \log(d/\epsilon))$, with $\delta$ a lower bound on the smallest eigenvalue of $\rho$, ensuring numerical stability.
• The method achieves a significant improvement over state tomography, which requires $\Theta(d^2/\epsilon^2)$ samples, especially in high-dimensional settings.
• Multiplicative approximations are obtained via iterative use of the additive estimator, with at most $\mathcal{O}(\log(d/\delta))$ iterations, ensuring runtime depends quadratically on the inverse of the target entropy.
• The algorithm is designed for near-term quantum devices, using shallow circuits and many measurements, offering a practical alternative to deep amplitude estimation circuits.
• The expected number of measurements scales as $\mathcal{O}(1/(x\epsilon)^2)$, which is independent of $d$ in the $x$-dependent term, making it efficient when $x$ is not too small.