Quantum algorithms for estimating quantum entropies

Wang, Youle; Zhao, Benchi; Wang, Xin

doi:10.48550/arxiv.2203.02386

Cited by 3 publications

(4 citation statements)

References 46 publications

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“…Proof Previous work [60] states that it is able to obtain an estimation of S α (ρ) up to precision ε, by an estimation of tr(ρ α ) within error ε ′ = |1−α| tr(ρ α ) 2 ε. Then we turn to demonstrate how to obtain tr(ρ α ) with error bounded by ε ′ .…”

Section: Supplementary Materialsmentioning

confidence: 99%

Quantum Phase Processing: Transform and Extract Eigen-Information of Quantum Systems

Wang¹,

Wang²,

Yu³

et al. 2022

Preprint

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Quantum computing can provide speedups in solving many problems as the evolution of a quantum system is described by a unitary operator in an exponentially large Hilbert space. Such unitary operators change the phase of their eigenstates and make quantum algorithms fundamentally different from their classical counterparts. Based on this unique principle of quantum computing, we develop a new algorithmic framework "Quantum phase processing" that can directly apply arbitrary trigonometric transformations to eigenphases of a unitary operator. The quantum phase processing circuit is constructed simply, consisting of single-qubit rotations and controlled-unitaries, typically using only one ancilla qubit. Besides the capability of phase transformation, quantum phase processing in particular can extract the eigen-information of quantum systems by simply measuring the ancilla qubit, making it naturally compatible with indirect measurement. Quantum phase processing complements another powerful framework known as quantum singular value transformation and leads to more intuitive and efficient quantum algorithms for solving problems that are particularly phase-related. As a notable application, we propose a new quantum phase estimation algorithm without quantum Fourier transform, which requires the least ancilla qubits and matches the best performance so far. We further exploit the power of our QPP framework by investigating a plethora of applications in Hamiltonian simulation, entanglement spectroscopy, and quantum entropies estimation, demonstrating improvements or optimality for almost all cases.

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Section: Supplementary Materialsmentioning

confidence: 99%

Quantum Phase Processing: Transform and Extract Eigen-Information of Quantum Systems

Wang¹,

Wang²,

Yu³

et al. 2022

Preprint

Self Cite

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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%

“…[AISW19] introduced a method of computing the von Neumann and quantum Rényi entropies of an N -dimensional quantum state using O(N 2 /ε 2 ) and O(N 2/α /ε 2/α ) copies, respectively. Recently, a new method of computing the von Neumann and quantum Rényi entropies was proposed in [WZW22] using Õ(κ 2 /ε 5 ) copies, where κ > 0 is given such that Π/κ ≤ ρ ≤ I for some projector Π. A distributed quantum algorithm for computing tr(ρσ), i.e., the fidelity of pure quantum states, was proposed in [ALL21] using O(max{ √ N /ε, 1/ε 2 }) copies.…”

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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“…However, the von Neumann entropy S(ρ) and its gradients are difficult to measure on real quantum hardware [18,22,23], with the cost scaling exponentially with system size, particularly in the case of Gibbs states, as eigenvalues of the target state ρ G are exponentially suppressed [24,25]. This makes the implementation of variational Gibbs state preparation too demanding for nearterm quantum processors, especially at low temperatures.…”

Section: Introductionmentioning

confidence: 99%

Adaptive variational algorithms for quantum Gibbs state preparation

Warren¹,

Zhu²,

Mayhall³

et al. 2022

Preprint

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The preparation of Gibbs thermal states is an important task in quantum computation with applications in quantum simulation, quantum optimization, and quantum machine learning. However, many algorithms for preparing Gibbs states rely on quantum subroutines which are difficult to implement on near-term hardware. Here, we address this by (i) introducing an objective function that, unlike the free energy, is easily measured, and (ii) using dynamically generated, problem-tailored ansätze. This allows for arbitrarily accurate Gibbs state preparation using low-depth circuits. To verify the effectiveness of our approach, we numerically demonstrate that our algorithm can prepare high-fidelity Gibbs states across a broad range of temperatures and for a variety of Hamiltonians.[1] M. Kieferová and N. Wiebe, Tomography and generative training with quantum Boltzmann machines,

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Quantum algorithms for estimating quantum entropies

Cited by 3 publications

References 46 publications

Quantum Phase Processing: Transform and Extract Eigen-Information of Quantum Systems

Quantum Phase Processing: Transform and Extract Eigen-Information of Quantum Systems

New Quantum Algorithms for Computing Quantum Entropies and Distances

Adaptive variational algorithms for quantum Gibbs state preparation

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