INSPIRED BY TENSOR NETWORK SHI-JU RAN ICFO-INSTITUT DE CINCIES - - PowerPoint PPT Presentation
QUANTUM ENTANGLEMENT SIMULATORS INSPIRED BY TENSOR NETWORK SHI-JU RAN ICFO-INSTITUT DE CINCIES FOTNIQUES, THE BARCELONA INSTITUTE OF SCIENCE AND TECHNOLOGY November. 7th, 2017 @ Verona When we get to the very, very small world, we have a
SHI-JU RAN
ICFO-INSTITUT DE CIÈNCIES FOTÒNIQUES, THE BARCELONA INSTITUTE OF SCIENCE AND TECHNOLOGY
Pictures from internet only for academic use
When we get to the very, very small world, we have a lot of new things that would happen that represent completely new
—— Richard Feynman (1956)
A beautiful slide in a talk of Guifre Vidal
very difficult to simulated by classical computers.
the Hilbert space increases exponentially with the system size.
Photonic quantum simulator [A. Aspuru-Guzik, et al., Nat.
Quantum simulations with ultracold quantum gases,
for simulating different quantum problems; exponential speed-up than classical computers
Timeline of D-wave quantum computer A review article: M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013)
Computation block: superconducting circuit
systems of infinite size by few-body models.
Hamiltonian which is calculated by the ab-initial optimization principle (AOP) approach based on TN. Infinite 2D system Finite bulk Unit cell Entanglement bath sites Physical-bath interactions Physical interactions
Shi-Ju Ran, Angelo Piga, Cheng Peng, Gang Su, and Maciej Lewenstein, Phys. Rev. B 96, 155120 (2017)
the critical fields and exponents “Finite-size” effect: O(10-3)
Methodology Applications Contract & Truncate
Encoding
alization
nalization
Analytic study
states
fields
Numeric simulations
models
models
Outside physics
problems
learning
𝐷1 ↑> +𝐷2 ↓> →
𝑗=1 2
𝐷𝑗 |𝑡𝑗 >
𝑗,𝑘=1 2
𝐷𝑗𝑘 |𝑡𝑗 > |𝑡
𝑘 >
𝑗1…𝑗𝑂=1 2
𝐷𝑗1…𝑗𝑂 |𝑡1 > ⋯ |𝑡𝑂 >
𝐷1 ↑> +𝐷2 ↓> →
𝑗=1 2
𝐷𝑗 |𝑡𝑗 >
𝑗,𝑘=1 2
𝐷𝑗𝑘 |𝑡𝑗 > |𝑡
𝑘 >
𝑗1…𝑗𝑂=1 2
𝐷𝑗1…𝑗𝑂 |𝑡1 > ⋯ |𝑡𝑂 >
1D quantum states: MPS 2D quantum states: PEPS Multi-scale entanglement renormalization ansatz Memory cost increases in a polynomial way
=< 𝑗1𝑗2|𝑓−𝑗𝜐𝐼|𝑘1𝑘2 > j1 𝑗1 𝑘2 𝑗2
|𝜔0 >
A review article of area law: J. Eisert, M. Cramer , and M. B. Plenio, Rev. Mod. Phys. 82, 277 (2010).
Coarse-graining tensor renormalization group Corner-transfer matrix renormalization group Time-evolving block decimation algorithm
Coarse-graining tensor renormalization group Corner-transfer matrix renormalization group Time-evolving block decimation algorithm Our recent review article arXiv:1708.09213
Simplest local function that can be efficiently computed by classical computers —— contraction of local TN cluster 2) Minimal number of input parameters —— cell tensor
... ... ... ... ... . . . . . . ...
T
Scalar function
consistent local eigenvalue problems (Shi-Ju Ran, Phys. Rev. E 93, 053310)
TN encoding for 1D quantum (2D classical) systems The local eigenvalue problems that encodes the infinite TN
Shi-Ju Ran, Angelo Piga, Cheng Peng, Gang Su, and Maciej Lewenstein, Phys. Rev. B 96, 155120 (2017) Classical simulation for the physical-bath interactions Quantum simulators
Continuous MPS in imaginary time (arXiv:1608.06544) Transform to standard summation form
(Evolution form) The dimension of the bath sites determine the upper bound of the entanglement that can be captured between the finite bulk and the rest.
Quantum entanglement simulator can be easily realized in experiments： using super-conducting circuits or cold/ultra-cold atoms or ions
bath sites have simple physics: higher spins for spin models, Bosons/fermions for bosonic/fermionic models.
boundary terms (physical-bath interactions) can be expanded in the standard spin-pin interaction form. By choosing different matrix basis, the bath can be spins, bosons or fermions
physical-bath (PB) interactions Cluster Nearest-neighbor PB interactions Long-range PB interactions
Numeri Methods Density functional theory (ab-initial principle) Dynamic mean-field theory Ab-initial optimization principle of tensor network Effective models Tight binding model Single-impurity model Interacting few-body model
An input sample mapped from scalars (e.g., grey pixels) to normalized vectors
(E. Stoudenmire, and D. J. Schwab (2016))
Slightly entangled states are able to encode image classes
Ding Liu, Shi-Ju Ran, Peter Wittek, Cheng Peng, Raul Blázquez García, Gang Su, and Maciej Lewenstein, arXiv:1710.04833
Accuracy versus bond dimension (over-fitted with large bond dimensions) Accuracy with optimal bond dimensions
accuracy with fixed training samples from MNIST is nearly 100%.
(generalization ability) is more than 80%.
The inner product TN defines the fidelity The SVD of the top tensor defines the entanglement spectrum
The inner product TN defines the fidelity The SVD of the top tensor defines the entanglement spectrum
Distance (similarity)
from real-space renormalization Entanglement characterizes: quamtun complexity; the information gain from the measurement
Experimental/ numeric simulations
(quantum Hall state, super-conductors, black hole, etc.) Functional and controllable quantum devices (arXiv:1707.07838) Numeric algorithms beyond density functional theory for interacting electrons Quantum entanglement and image/information recovering Quantum
and visualization theory of image classes Non-linear dimensionality reduction of Hilbert space
Looking for the microscopic quantum models that mimics intelligences
TN-based quantum simulator TN-based machine learning Fidelity and data structure
Acknowledgement to ERC AdG OSYRIS (ERC-2013-AdG Grant No. 339106), the Spanish MINECO grants FOQUS (FIS2013-46768-P), FISICATEAMO (FIS2016-79508-P), and “Severo Ochoa” Programme (SEV-2015-0522), Catalan AGAUR SGR 874, Fundacio Cellex, EU FETPRO QUIC, CERCA Programme / Generalitat de Catalunya, Fundacio Catalunya - La Pedrera . Ignacio Cirac Program Chair.