The quantum low-rank approximation problem

Ezzell, Nic; Holmes, Zoë; Coles, Patrick J.

doi:10.48550/arxiv.2203.00811

Cited by 4 publications

(11 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The history here is quite interesting. Lloyd et al's quantum PCA algorithm was proposed in 2014 and highlighted a potential exponential speedup over classical algorithms [3], and this also led to some near-term proposals for quantum PCA [4][5][6][7]. However, a spooky paper posted on Halloween of 2018 by Tang presented a "dequantized" classical algorithm that achieved the same asymptotic scaling as quantum PCA [25], suggesting that quantum speedup for quantum PCA was a false promise.…”

Section: Discussionmentioning

confidence: 99%

“…Natural future work would be to actually implement our method on real quantum hardware. Combining our method for preparing the covariance matrix with other near-term methods for extracting the spectrum [4][5][6][7] would lead to a near-term approach for quantum PCA. Indeed the nice feature of our method is how simple and easy-to-implement it is on near-term quantum hardware.…”

Section: Discussionmentioning

confidence: 99%

“…More recently, near-term approaches to quantum PCA have been developed [4][5][6][7] in the framework of variational quantum algorithms [8]. For example, Variational Quantum State Diagonalization [4] unitarily rotates ρ towards a diagonal form, while estimating the distance to being diagonal using two copies of ρ.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Covariance matrix preparation for quantum principal component analysis

Gordon¹,

Cerezo²,

Cincio³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Covariance matrix preparation for quantum principal component analysis

Gordon¹,

Cerezo²,

Cincio³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

“…If trained well, the learned state σ(α * , R) should be close to the solution to the quantum low-rank approximation problem (QLRAP) [18]. As discussed in [18], the unique optimal state that minimizes the HS distance, subject to a rank constraint, i.e. the state σ(α opt , R) where…”

Section: Quantum Mixed State Compiling (Qmsc) Algorithmmentioning

confidence: 99%

“…The compression of quantum data has a long history tracing back to the early days of quantum information theory [1,16,17]. More recently, the question of how well any given state may be approximated using a lower rank state was addressed analytically in [18] for the Hilbert-Schmidt (HS) and trace distances. Interestingly, the optimal lower rank approximation essentially corresponds to performing principal component analysis (PCA) (jump to equation (3) to look ahead) for the HS distance (but not for trace distance).…”

Section: Introductionmentioning

confidence: 99%

Quantum mixed state compiling

Ezzell

Ball

Siddiqui

et al. 2023

Quantum Sci. Technol.

View full text Add to dashboard Cite

The task of learning a quantum circuit to prepare a given mixed state is a fundamental quantum subroutine. We present a variational quantum algorithm (VQA) to learn mixed states which is suitable for near-term hardware. Our algorithm represents a generalization of previous VQAs that aimed at learning preparation circuits for pure states. We consider two different ansätze for compiling the target state; the first is based on learning a purification of the state and the second on representing it as a convex combination of pure states. In both cases, the resources required to store and manipulate the compiled state grow with the rank of the approximation. Thus, by learning a lower rank approximation of the target state, our algorithm provides a means of compressing a state for more efficient processing. As a byproduct of our algorithm, one effectively learns the principal components of the target state, and hence our algorithm further provides a new method for principal component analysis. We investigate the efficacy of our algorithm through extensive numerical implementations, showing that typical random states and thermal states of many body systems may be learnt this way. Additionally, we demonstrate on quantum hardware how our algorithm can be used to study hardware noise-induced states.

show abstract

Resource-efficient quantum principal component analysis

Wang,

Luo

2024

Quantum Sci. Technol.

View full text Add to dashboard Cite

Principal component analysis (PCA) is an important dimensionality reduction method in machine learning and data analysis. Recently, the quantum version of PCA has been established to diagonalize quantum states. Although these quantum algorithms promise quantum advantages, they require substantial resources beyond the reach of state-of-the-art quantum technologies. This work aims to reduce resource requirements and improve the efficiency of quantum principal component analysis. Assuming that the quantum state is accessed through a purified quantum query model and a sampling model, we propose quantum algorithms that use minimal resource requirements for ancillary qubits to reveal properties of eigenvectors and eigenvalues of a state. In particular, we show that estimating eigenvalue $\lambda$ with error $\epsilon$ and success probability larger than $\lambda(1-\eta)$ requests a query complexity {$\mathcal{\widetilde{O}}(\epsilon^{-1})$} and a sample complexity {$\mathcal{\widetilde{O}}(\epsilon^{-2}\eta^{-1})$}, respectively. To our knowledge, our result is the first quantum speedup that achieves asymptotic linear scaling in $1/\epsilon$ for quantum PCA. As applications, we discuss estimating the minimum relative entropy of entanglement of bipartite pure-states and performing quantum state discrimination tasks. We show that quantum speedups are maintained when the pure state has a low Schmidt number and states of discrimination have a low rank. This study opens up a new quantum principal component analysis method for high-dimensional quantum data analysis and discusses its application in quantum information processing tasks.

show abstract

The quantum low-rank approximation problem

Cited by 4 publications

References 15 publications

Covariance matrix preparation for quantum principal component analysis

Covariance matrix preparation for quantum principal component analysis

Quantum mixed state compiling

Resource-efficient quantum principal component analysis

Contact Info

Product

Resources

About