Emergent Statistical Mechanics from Properties of Disordered Random Matrix Product States

Haferkamp, Jonas; Bertoni, Christian; Roth, Ingo; Eisert, Jens

doi:10.1103/prxquantum.2.040308

Cited by 14 publications

(7 citation statements)

References 38 publications

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“…We define a unitarily embedded MPS [35,45,46] of local dimension d, system size n and virtual bond dimension D as:…”

Section: A Mps Architecturementioning

confidence: 99%

Barren plateaus from learning scramblers with local cost functions

Garcia¹,

Chen²,

Bu³

et al. 2022

Preprint

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“…We define a unitarily embedded MPS [35,45,46] of local dimension d, system size n and virtual bond dimension D as:…”

Section: A Mps Architecturementioning

confidence: 99%

Barren plateaus from learning scramblers with local cost functions

Garcia¹,

Chen²,

Bu³

et al. 2022

Preprint

View full text Add to dashboard Cite

“…For a large MPS, this is intrinsically related to the fact that, under random initialization, the state constitutes an approximate 2-design for which the expected gradients vanish [91]. It may be possible to overcome this problem by choosing a different initialization scheme [90], or by preconditioning the network with a different algorithm, such as DMRG. Formulating a local loss function should also alleviate the issue [91,110].…”

Section: Discussion and Summarymentioning

confidence: 99%

“…It is reasonable to expect the above picture to hold, to some extent, for a randomly initialized MPS. Indeed, a random MPS, viewed as a unitary embedding [89][90][91], forms an approximate 2-design [92]: as a random quantum circuit, the first and second moments approximate those of a Haar distribution. Moreover, Haar-distributed random unitary circuits possess WD ESS corresponding to the Gaussian unitary ensemble (GUE), β = 2 [82], although a precise relationship between ESS and k-designs remains an open question [84].…”

Section: B Entanglement Measuresmentioning

confidence: 99%

Entangling Solid Solutions: Machine Learning of Tensor Networks for Materials Property Prediction

Sommer¹,

Dunham²

2022

Preprint

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Progress in the application of machine learning techniques to the prediction of solid-state and molecular materials properties has been greatly facilitated by the development state-of-the-art feature representations and novel deep learning architectures. A large class of atomic structure representations based on expansions of smoothed atomic densities have been shown to correspond to specific choices of basis sets in an abstract many-body Hilbert space. Concurrently, tensor network structures, conventionally the purview of quantum many-body physics and quantum information, have been successfully applied in supervised and unsupervised learning tasks in computer vision and natural language processing. In this work, we argue that architectures based on tensor networks are well-suited to machine learning on Hilbert-space representations of atomic structures. This is demonstrated on supervised learning tasks involving widely available datasets of density functional theory calculations of metal and semiconductor alloys. In particular, we show that certain standard tensor network topologies exhibit strong generalizability even on small training datasets while being parametrically efficient. We further relate this generalizability to the presence of complex entanglement in the trained tensor networks. We also discuss connections to learning with generalized structural kernels and related strategies for compressing large input feature spaces.

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“…Finally, we comment on the effect of non-local or correlated errors, whose analysis remains challenging even for theory models based on RUCs. Using a novel analysis based on random matrix product states [35,56], we find that the validity of F d ≈ F undergoes a crossover as a function of the spatial extent of an error: when the size of an error exceeds what we call an entanglement correlation length, F d may not accurately approximate F . This leads us to design a slightly modified benchmark F e , which alleviates some of the aforementioned limitations -systems with a relatively small D β , the presence of nonlocal or correlated errors, or PHS -at the expense of having worse performance for short quenches [35].…”

Section: (D) Dotted Line]mentioning

confidence: 99%