We propose a series of quantum algorithms for computing a wide range of quantum entropies and distances, including the von Neumann entropy, quantum Rényi entropy, trace distance, and fidelity. The proposed algorithms significantly outperform the prior best (and even quantum) ones in the low-rank case, some of which achieve exponential speedups. In particular, forN-dimensional quantum states of rankr, our proposed quantum algorithms for computing the von Neumann entropy, trace distance and fidelity within additive error ε have time complexity of Õ(r/ε2), Õ(r5/ε6) and Õ(r6.5/ε7.5), respectively. By contrast, prior quantum algorithms for the von Neumann entropy and trace distance usually have time complexity Ω(N), and the prior best one for fidelity has time complexity Õ(r12.5/ε13.5). The key idea of our quantum algorithms is to extend block-encoding from unitary operators in previous work to quantum states (i.e., density operators). It is realized by developing several convenient techniques to manipulate quantum states and extract information from them. The advantage of our techniques over the existing methods is that no restrictions on density operators are required; in sharp contrast, the previous methods usually require a lower bound on the minimal non-zero eigenvalue of density operators.
We propose a series of quantum algorithms for computing a wide range of quantum entropies and distances, including the von Neumann entropy, quantum Rényi entropy, trace distance, and fidelity. The proposed algorithms significantly outperform the prior best (and even quantum) ones in the low-rank case, some of which achieve exponential speedups. In particular, forN-dimensional quantum states of rankr, our proposed quantum algorithms for computing the von Neumann entropy, trace distance and fidelity within additive error ε have time complexity of Õ(r/ε2), Õ(r5/ε6) and Õ(r6.5/ε7.5), respectively. By contrast, prior quantum algorithms for the von Neumann entropy and trace distance usually have time complexity Ω(N), and the prior best one for fidelity has time complexity Õ(r12.5/ε13.5). The key idea of our quantum algorithms is to extend block-encoding from unitary operators in previous work to quantum states (i.e., density operators). It is realized by developing several convenient techniques to manipulate quantum states and extract information from them. The advantage of our techniques over the existing methods is that no restrictions on density operators are required; in sharp contrast, the previous methods usually require a lower bound on the minimal non-zero eigenvalue of density operators.
In quantum information, trace distance is a basic metric of distinguishability between quantum states. However, there is no known efficient approach to estimate the value of trace distance in general. In this paper, we propose efficient quantum algorithms for estimating the trace distance within additive error ε between mixed quantum states of rankr. Specifically, we first provide a quantum algorithm usingr· Õ(1/ε2) queries to the quantum circuits that prepare the purifications of quantum states. Then, we modify this quantum algorithm to obtain another algorithm using Õ (r2/ε5) samples of quantum states, which can be applied to quantum state certification. These algorithms have query/sample complexities that are independent of the dimensionNof quantum states, and their time complexities only incur an extraO(log(N)) factor. In addition, we show that the decision version of low-rank trace distance estimation is BQP-complete.
Entropy is a measure of the randomness of a system. Estimating the entropy of a quantum state is a basic problem in quantum information. In this paper, we introduce a time-efficient quantum approach to estimating the von Neumann entropyS(ρ) and Rényi entropySα(ρ) of anN-dimensional quantum state ρ, given access to independent samples of ρ. Specifically, we provide the following quantum estimators. • A quantum estimator forS(ρ) with time complexity Õ (N2),1improving the prior best time complexity Õ(N6) by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). • A quantum estimator forSα(ρ) with time complexity Õ (N4/α−2) for 0N4−2/α) for α > 1, improving the prior best time complexity Õ(N6/α) for 0N6) for α > 1 by Acharya, Issa, Shende, and Wagner (2020), though at a cost of a slightly larger sample complexity. Moreover, these estimators are naturally extensible to the low-rank case. We also provide a sample lower bound Ω(max{N/ε,N1/α−1/ε1/α}) for estimatingSα(ρ). Technically, our method is quite different from the previous ones that are based on weak Schur sampling and Young diagrams. At the heart of our construction, is a novel tool calledsamplizer, which can “samplize” a quantum query algorithm to a quantum algorithm with similar behavior using only samples of quantum states; this suggests a new framework for estimating quantum entropies. Specifically, when a quantum oracleUblockencodes a mixed quantum state ρ, any quantum query algorithm usingQqueries toUcan be samplized to a δ-close (in the diamond norm) quantum algorithm using Θ(Q2/δ) samples of ρ. Moreover, this samplization is proven to be optimal, up to a polylogarithmic factor.
We investigate the computational hardness of estimating the quantum α-Rényi entropy SαR(ρ)=1−αlnTr(ρα) and the quantum q-Tsallis entropy SqT(ρ)=q−11−Tr(ρq), both of which converge to the von Neumann entropy as the order approaches 1. The promise problems Quantum α-Rényi Entropy Approximation (RényiQEAα) and Quantum q-Tsallis Entropy Approximation (TsallisQEAq) ask whether $ {\rm S}^ {\tt R}_α(ρ)$ or SqT(ρ), is at least τ1 or at most τ2, where τ1−τ2 is typically a positive constant. Previous hardness results cover only the von Neumann entropy (order 1) and some cases of the quantum q-Tsallis entropy, while existing approaches do not readily extend to other orders.
We establish that for all positive real α and q, and also for α=∞, the rank-2 variants Rank2RényiQEAα and Rank2TsallisQEAq are BQP-hard. Combined with prior (rank-dependent) quantum query algorithms in Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Liu and Wang (SODA 2025), as well as the one derived from O'Donnell and Wright (STOC 2016), our results imply:
- For all real orders α>0 or α=∞, and for all real orders 0<q≤1, LowRankRényiQEAα and LowRankTsallisQEAq are BQP-complete, where both are restricted versions of RényiQEAα and TsallisQEAq with ρ of polynomial rank.
- For all real order q>1, TsallisQEAq is BQP-complete.
Our hardness results stem from reductions based on new inequalities relating the α-Rényi or q-Tsallis binary entropies of different orders. These reductions differ substantially from previous approaches, and the inequalities are of independent interest.
Entropy is a measure of the randomness of a system. Estimating the entropy of a quantum state is a basic problem in quantum information. In this paper, we introduce a time-efficient quantum approach to estimating the von Neumann entropy S(ρ) and Rényi entropy Sα(ρ) of an N-dimensional quantum state ρ, given access to independent samples of ρ. Specifically, we provide the following: 1. A quantum estimator for S(ρ) with time complexity O~(N2), improving the prior best time complexity O~(N6) by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). 2. A quantum estimator for Sα(ρ) with time complexity O~(N4/α−2) for 0<α<1 and O~(N4−2/α) for α>1, improving the prior best time complexity O~(N6/α) for 0<α<1 and O~(N6) for α>1 by Acharya, Issa, Shende, and Wagner (2020), though at a cost of a slightly larger sample complexity. Moreover, these estimators are naturally extensible to the low-rank case. We also provide a sample lower bound for estimating Sα(ρ). Technically, our method is quite different from the previous ones that are based on weak Schur sampling and Young diagrams. At the heart of our construction, is a novel tool called samplizer, which can "samplize" a quantum query algorithm to a quantum algorithm with similar behavior using only samples of quantum states; this suggests a unified framework for estimating quantum entropies. Specifically, when a quantum oracle U block-encodes a mixed quantum state ρ, any quantum query algorithm using Q queries to U can be samplized to a δ-close (in the diamond norm) quantum algorithm using Θ~(Q2/δ) samples of ρ. Moreover, this samplization is proven to be optimal, up to a polylogarithmic factor.
Quantum Algorithms for Computing Maximal Quantum f-divergence and Kubo-Ando means
The development of quantum computation has resulted in many quantum algorithms for a wide array of tasks. Recently, there is a growing interest in using quantum computing techniques to estimate or compute quantum information-theoretic quantities such as Renyi entropy, Von Neumann entropy, matrix means, etc. Motivated by these results, we present quantum algorithms for computing the maximal quantum f-divergences and the operator-theoretic matrix Kubo--Ando means. Both of them involve Renyi entropies, matrix means as special cases, thus implying the universality of our framework.
We describe a quantum algorithm to estimate the α-Renyi entropy of an unknown density matrix ρ∈Cd×d for α≠1 by combining the recent technique of quantum singular value transformations with the method of estimating normalized 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 nonsingular. Our method outputs an estimate of the α-Renyi entropy to additive precision ϵ, using an expected total number O(1(xϵ)2) of independent applications of a quantum circuit which coherently queries the input unitary O(1δlogdϵ) times, in each case measuring a single output qubit. Here δ is a lower cutoff on the smallest eigenvalue of ρ and x=1dTr(ρα). The expected number of measurements made in this method can be compared to results in the sample complexity model that generally require Θ(d2ϵ2) samples. Furthermore, we also show that multiplicative approximations can be obtained by iteratively using additive approximations, with an overhead logarithmic in the dimension d.