Quantum algorithm for estimating 
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-Renyi entropies of quantum states

Subramanian, Sathyawageeswar; Hsieh, Min-Hsiu

doi:10.1103/physreva.104.022428

Cited by 24 publications

(27 citation statements)

References 27 publications

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“…(2.9) and (2.10)). [SH21] consider the estimation of Renyi entropies in the same purified query access model, to both additive and multiplicative precision. Their focus however is on how this task may be solved on restricted models of quantum computation (namely, DQC1), and they do not obtain optimal query complexities.…”

Section: Input Modelmentioning

confidence: 99%

Sublinear quantum algorithms for estimating von Neumann entropy

Gur¹,

Hsieh²,

Subramanian³

2021

Preprint

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Section: Input Modelmentioning

confidence: 99%

Sublinear quantum algorithms for estimating von Neumann entropy

Gur¹,

Hsieh²,

Subramanian³

2021

Preprint

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“…• For the quantum algorithms in [SH21] and [WZW22] that attempt to reduce the dependence on N , they introduce an extra dependence on κ, where κ is the reciprocal of the minimal non-zero eigenvalue of quantum states. Our quantum algorithms can be easily adapted to their settings by taking r = O(κ), but the converse seems not applicable 3 .…”

Section: • Von Neumann Entropymentioning

confidence: 99%

“…Recently, [GHS21] proposed a quantum algorithm for computing the von Neumann entropy within a multiplicative factor, which can reproduce the result of [GL20] within additive error. [SH21] found a method of computing the quantum α-Renyi entropy using O κ (xε) 2 log N ε queries to the oracle, where κ > 0 is given such that I/κ ≤ ρ ≤ I and x = tr(ρ α )/N .…”

Section: Introductionmentioning

confidence: 99%

New Quantum Algorithms for Computing Quantum Entropies and Distances

Wang¹,

Guan²,

Liu³

et al. 2022

Preprint

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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 best known (and even quantum) ones in the low-rank case, some of which achieve exponential speedups. In particular, for Ndimensional quantum states of rank r, our proposed quantum algorithms for computing the von Neumann entropy, trace distance and fidelity within additive error ε have time complexity of Õ r 2 /ε 2 , Õ r 5 /ε 6 and Õ r 6.5 /ε 7.5 1 , respectively. In contrast, the known algorithms for the von Neumann entropy and trace distance require quantum time complexity of Ω(N ) [AISW19,GL20,GHS21], and the best known one for fidelity requires Õ r 21.5 /ε 23.5 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. In particular, we introduce a novel technique for eigenvalue transformation of density operators and their (non-integer) positive powers, based on the powerful quantum singular value transformation (QSVT) [GSLW19]. 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 of the minimal non-zero eigenvalue of density operators. In addition, we provide some techniques of independent interest for trace estimation, linear combinations, and eigenvalue threshold projectors of (subnormalized) density operators, which will be, we believe, useful in other quantum algorithms.

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“…Regarding quantum computing methods, many proposals based on different models have been proposed [11,20,[22][23][24][25][26]. Specifically speaking, [20] studies the cost of estimating the von Neumann and Rényi entropies in a model where one can get independent copies of the state.…”

Section: Introductionmentioning

confidence: 99%

“…By allowing arbitrary measurements and classical post-processing, it shows that the cost of entropy estimation scales exponentially in the state size. Later, [24] and [25] study the von Neumann entropy and quantum Rényi entropy estimation in a quantum query model, respectively. The query model here is a quantum circuit that can prepare the purification of the input state, i.e., U ρ |0 A |0 B = |ψ ρ AB and tr A (|ψ ρ ψ ρ | AB ) = ρ.…”

Section: Introductionmentioning

confidence: 99%