Sublinear quantum algorithms for estimating von Neumann entropy

Gur, Tom; Hsieh, Min-Hsiu; Subramanian, Sathyawageeswar

doi:10.48550/arxiv.2111.11139

Cited by 9 publications

(21 citation statements)

References 13 publications

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“…[GL20] developed quantum algorithms for computing the von Neumann entropy and trace distance with query complexity O(N/ε 1.5 ) and O(N/ε), respectively, both of which have complexity exponential in the number of qubits. 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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Section: Introductionmentioning

confidence: 99%

New Quantum Algorithms for Computing Quantum Entropies and Distances

Wang¹,

Guan²,

Liu³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In these works, the times of using U ρ for estimating S(ρ) could be linear in the dimension (O(d) [24], where d is the dimension of system), while the results for R α (ρ) is comparable to the tomography (O(d 2 )) [25]. Recently, the work [27] has brought the number of using U ρ for estimating S(ρ) to be sub-linear in the system's dimension. Another work [28] has considered access to the purification of a state and used short-depth circuits, which generalize the swap test, to estimate tr(ρ k ).…”

Section: Introductionmentioning

confidence: 99%

“…Compared with algorithms of [11,[24][25][26][27], our circuits no longer depend on the quantum query model but use copies of the input state. Generally, constructing such a model is almost as hard as state tomography.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast, our algorithms need less information (about the minimal non-zero eigenvalue) and thus are more implementable in practice. Moreover, implementing algorithms of [11,[24][25][26][27] needs to devise corresponding circuits for different quantum states. In comparison, our methods can use fixed circuits to estimate all selected states.…”

Section: Introductionmentioning

confidence: 99%

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Quantum algorithms for estimating quantum entropies

Wang¹,

Zhao²,

Wang³

2022

Preprint

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“…Connection to Property Testing.-Fast methods for solving relaxations of decision problems by only locally accessing a small fraction of the input fall under the broad purview of property-testing algorithms [29]. More specifically, the testing problem in this work is formulated as a gapped promise problem in property testing, wherein algorithms are required to be able to decide with high probability whether the input scores above a threshold α or below a threshold β on some function of interest -for example, the algorithm may want to decide if a given probability distribution on n items has Shannon entropy larger than α log 2 n, or smaller than β log 2 n, by only looking at few samples out of the n items [30,31]. Promise gap here refers to the situation that we are free to output a random decision when the input falls into the region of no interest in between the two thresholds.…”

mentioning

confidence: 99%

Constant-time one-shot testing of large-scale graph states

Yamasaki¹,

Subramanian²

2022

Preprint

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Fault-tolerant measurement-based quantum computation (MBQC) with recent progress on quantum technologies leads to a promising scalable platform for realizing quantum computation, conducted by preparing a large-scale graph state over many qubits and performing single-qubit measurements on the state. With fault-tolerant MBQC, even if the graph-state preparation suffers from errors occurring at an unknown physical error rate, we can suppress the effect of the errors. Verifying graph states is vital to test whether we can conduct MBQC as desired even with such errors. However, problematically, existing state-of-the-art protocols for graph-state verification by fidelity estimation have required measurements on many copies of the entire graph state and hence have been prohibitively costly in terms of the number of qubits and the runtime. We here construct an efficient alternative framework for testing graph states for fault-tolerant MBQC based on the theory of property testing. Our test protocol accepts with high probability when the physical error rate is small enough to make fault-tolerant MBQC feasible and rejects when the rate is above the threshold of fault-tolerant MBQC. The novelty of our protocol is that we use only a single copy of the N -qubit graph state and single-qubit Pauli measurements only on a constant-sized subset of the qubits; thus, the protocol has a constant runtime independently of N . Furthermore, we can immediately use the rest of the graph state for fault-tolerant MBQC if the protocol accepts. These results achieve a significant advantage over prior art for graph-state verification in the number of qubits and the total runtime. Consequently, our work offers a new route to a fast and practical framework for benchmarking large-scale quantum state preparation.

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Sublinear quantum algorithms for estimating von Neumann entropy

Cited by 9 publications

References 13 publications

New Quantum Algorithms for Computing Quantum Entropies and Distances

New Quantum Algorithms for Computing Quantum Entropies and Distances

Quantum algorithms for estimating quantum entropies

Constant-time one-shot testing of large-scale graph states

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