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"1.4.1+1" diff --git a/Project.toml b/Project.toml new file mode 100644 index 0000000..991effa --- /dev/null +++ b/Project.toml @@ -0,0 +1,15 @@ +[deps] +ExampleJuggler = "3bbe58f8-ed81-4c4e-a134-03e85fcf4a1a" +ITensors = "9136182c-28ba-11e9-034c-db9fb085ebd5" +JuliaFormatter = "98e50ef6-434e-11e9-1051-2b60c6c9e899" +LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" +Markdown = "d6f4376e-aef5-505a-96c1-9c027394607a" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +Pluto = "c3e4b0f8-55cb-11ea-2926-15256bba5781" +PlutoSliderServer = "2fc8631c-6f24-4c5b-bca7-cbb509c42db4" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +Quantics = "87f76fb3-a40a-40c9-a63c-29fcfe7b7547" +QuanticsGrids = "634c7f73-3e90-4749-a1bd-001b8efc642d" +QuanticsTCI = "b11687fd-3a1c-4c41-97d0-998ab401d50e" +TCIITensorConversion = "9f0aa9f4-9415-4e6a-8795-331ebf40aa04" +TensorCrossInterpolation = "b261b2ec-6378-4871-b32e-9173bb050604" diff --git a/README.md b/README.md new file mode 100644 index 0000000..e977106 --- /dev/null +++ b/README.md @@ -0,0 +1,67 @@ +# T4APlutoExamples + +[![Export Pluto notebooks](https://github.com/tensor4all/T4APlutoExamples/actions/workflows/ExportPluto.yaml/badge.svg)](https://github.com/tensor4all/T4APlutoExamples/actions/workflows/ExportPluto.yaml) + +[![CI](https://github.com/tensor4all/T4APlutoExamples/actions/workflows/CI.yaml/badge.svg)](https://github.com/tensor4all/T4APlutoExamples/actions/workflows/CI.yaml) + +Website: https://tensor4all.org/T4APlutoExamples/pluto_notebooks/ + + +## Description + +[This repository](https://github.com/Tensor4All/T4APlutoExamples) contains tutorials for [Tensor4All group](https://tensor4all.org/) written in Julia with Pluto notebook. Pre-computed notebooks can be found at [this page](https://tensor4all.org/T4APlutoExamples/pluto_notebooks/). + +## How to open Pluto notebooks locally + +To open and run Pluto notebooks, you will need to install Julia. We recommend using [juliaup](https://github.com/JuliaLang/juliaup) to install Julia, following the instructions provided on the [official Julia website](https://julialang.org/downloads/). + +```sh +$ git clone https://github.com/tensor4all/T4APlutoExamples.git +$ cd T4APlutoExamples +$ julia + _ + _ _ _(_)_ | Documentation: https://docs.julialang.org + (_) | (_) (_) | + _ _ _| |_ __ _ | Type "?" for help, "]?" for Pkg help. + | | | | | | |/ _` | | + | | |_| | | | (_| | | Version 1.10.5 (2024-08-27) + _/ |\__'_|_|_|\__'_| | Official https://julialang.org/ release +|__/ | + +julia> using Pkg; Pkg.activate("."); Pkg.instantiate() +julia> Pkg.status() +Status `~/tensor4all/T4APlutoExamples/Project.toml` + [98e50ef6] JuliaFormatter v1.0.60 + [c3e4b0f8] Pluto v0.19.46 + [2fc8631c] PlutoSliderServer v0.3.31 +``` + +## Launch Pluto Notebook + +Let's run our Pluto notebook `pluto_notebooks/quantics1d.jl` for instance: + +```sh +$ julia --project + _ + _ _ _(_)_ | Documentation: https://docs.julialang.org + (_) | (_) (_) | + _ _ _| |_ __ _ | Type "?" for help, "]?" for Pkg help. + | | | | | | |/ _` | | + | | |_| | | | (_| | | Version 1.10.5 (2024-08-27) + _/ |\__'_|_|_|\__'_| | Official https://julialang.org/ release +|__/ | + +julia> using Pluto; Pluto.run(notebook="pluto_notebooks/quantics1d.jl") +``` + +This will launch a Pluto notebook server immediately. + +You can also run Pluto notebooks on the web using the `download` function. + +```julia-repl +julia> using Pluto; Pluto.run(notebook=download("https://raw.githubusercontent.com/tensor4all/T4APlutoExamples/refs/heads/main/pluto_notebooks/quantics1d.jl")) +``` + +## How to learn more about Pluto.jl + +See the official Pluto.jl tutorial [🔎 Basic Commands in Pluto](https://github.com/fonsp/Pluto.jl/wiki/%F0%9F%94%8E-Basic-Commands-in-Pluto) to learn more. diff --git a/assets/open_welcome.png b/assets/open_welcome.png new file mode 100644 index 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When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). +macro bind(def, element) + quote + local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end + local el = $(esc(element)) + global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) + el + end +end + +# ╔═╡ ca3996e3-7a32-41a9-9e3f-f5cb83f3fe5f +begin + using Markdown + using PlutoUI +end + +# ╔═╡ 89ca61fd-6bd0-483f-aad6-e3aa2e35046c +md""" + +To run this notebook `about_pluto.jl` locally, copy and paste the following into your Julia REPL: + +```julia +using Pkg; Pkg.activate(temp=true); Pkg.add("Pluto") +BASE_URL = "https://raw.githubusercontent.com/tensor4all/T4APlutoExamples/refs/heads/main/pluto_notebooks/" +notebook = "about_pluto.jl" +url = joinpath(BASE_URL, notebook) +using Pluto; Pluto.run(notebook=download(url)) +``` +""" + +# ╔═╡ 19b7809c-5a93-4963-b46a-360a332c476d +begin + img_main_menu=Resource("https://raw.githubusercontent.com/tensor4all/T4APlutoExamples/refs/heads/main/assets/open_welcome.png") + nothing +end + +# ╔═╡ 7b83cb86-f08a-41b0-b506-6cbe5393d3a3 +md""" +## How to open ALL Pluto notebooks locally + + +The following instructions are for EXPERT users familiar with `Julia` and `git.` + +Our tutorials are written in [Pluto.jl](https://plutojl.org/) notebook. To open all the Pluto notebooks locally, clone our [tutorial repository T4APlutoExamples](https://github.com/tensor4all/T4APlutoExamples) and navigate to directory `T4APlutoExamples`: + +```sh +$ git clone https://github.com/tensor4all/T4APlutoExamples.git +$ cd T4APlutoExamples +$ Manifest.toml Project.toml README.md pluto_notebooks scripts +``` + +Then we also need to install Pluto.jl. To do this, type `julia --project` in a terminal to open Julia REPL and run the following julia code: + +```julia-repl +$ julia --project + _ + _ _ _(_)_ | Documentation: https://docs.julialang.org + (_) | (_) (_) | + _ _ _| |_ __ _ | Type "?" for help, "]?" for Pkg help. + | | | | | | |/ _` | | + | | |_| | | | (_| | | Version 1.10.5 (2024-08-27) + _/ |\__'_|_|_|\__'_| | Official https://julialang.org/ release +|__/ | + +julia> using Pkg; Pkg.instantiate() + Activating project at `~/tensor4all/T4APlutoExamples` +``` + +Continued from previous REPL session, launch Pluto server locally via: + +```julia-repl +julia> using Pluto; Pluto.run() +[ Info: Loading... +┌ Info: +└ Opening http://localhost:1234/?secret=xxxxxx in your default browser... ~ have fun! +┌ Info: +│ Press Ctrl+C in this terminal to stop Pluto +└ +``` + +This will open http://localhost:1234/?secret=xxxxxx in your default browser and you will see the following main menu of Pluto notebook: + +$(img_main_menu) + +To open notebooks under `pluto_notebooks` directory, type `pluto_notebooks/welcome.jl` to text area under `Open a notebook` section. Then click `Open` button. +""" + +# ╔═╡ 00c93734-a3e0-4c45-a60a-93fafba51dc5 + + +# ╔═╡ ec419a4f-061c-4d67-a1b6-c123eb763132 +md""" +## Specifying Pluto notebook URL directly + +You can also open the notebook on the web directly. +""" + +# ╔═╡ 2cbff48a-a3ef-462d-a726-4c5ca85bb4c0 +md""" +## About PlutoUI.jl + +The drop down menu above is created using PlutoUI.jl. You'll notice that when the drop-down menu is updated from `welcome.jl' to `quantics1d.jl', the message "To run this notebook locally, run welcome.jl" should change to "To run this notebook locally, run quantics1d.jl". This interactive functionality is one of the key features of Pluto notebook. +""" + +# ╔═╡ 3766d4ae-2f5a-4515-82c2-b8984947efa4 +begin + ui_notebook = @bind notebook Select([ + "welcome.jl", + "quantics1d.jl", + "quantics2d.jl", + "quantics1d_advanced.jl", + "compress.jl", + "interfacingwithitensors.jl", + "qft.jl", + "about_pluto.jl", + "plots.jl", + ], default="welcome.jl") + nothing +end + +# ╔═╡ 07854cf9-da58-4c02-a54f-31a24d1f32b3 +md""" +Select file $(ui_notebook). + +To run this notebook $(notebook) locally, copy and paste the following into your Julia REPL: +""" + +# ╔═╡ cf22b220-b5d6-483f-a137-ff56d0216158 +begin + mdstr = """ + ```julia + using Pkg; Pkg.activate(temp=true); Pkg.add("Pluto") + BASE_URL = "https://raw.githubusercontent.com/tensor4all/T4APlutoExamples/refs/heads/main/pluto_notebooks/" + notebook = "$(notebook)" + url = joinpath(BASE_URL, notebook) + using Pluto; Pluto.run(notebook=download(url)) + ``` + """ + Markdown.parse(mdstr) +end + +# ╔═╡ 1286865c-875b-4b32-b27f-9796aab29b23 +md""" +## About Pluto.jl + +Since Pluto has a built-in package manager, packages are automatically installed when you use: + +```julia +using +# or +import +``` + +Your package environment is stored in the notebook file. If you open the notebook with a text editor you will find `PLUTO_PROJECT_TOML_CONTENTS` which corresponds to Project.toml and `PLUTO_MANIFEST_TOML_CONTENTS` which corresponds to Manifest.toml. When someone else opens your notebook with Pluto, the exact same package environment will be used and packages will work on their computer. + +See [Pluto’s built-in package management](https://plutojl.org/en/docs/packages/) to learn more. +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +Markdown = "d6f4376e-aef5-505a-96c1-9c027394607a" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" + +[compat] +PlutoUI = "~0.7.60" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.5" +manifest_format = "2.0" +project_hash = "be1c3711711942a0b3f42007b7d4abd4448cc38d" + +[[deps.AbstractPlutoDingetjes]] +deps = ["Pkg"] +git-tree-sha1 = "6e1d2a35f2f90a4bc7c2ed98079b2ba09c35b83a" +uuid = "6e696c72-6542-2067-7265-42206c756150" +version = "1.3.2" + +[[deps.ArgTools]] +uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" +version = "1.1.1" 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+ +[[deps.Random]] +deps = ["SHA"] +uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" + +[[deps.Reexport]] +git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b" +uuid = "189a3867-3050-52da-a836-e630ba90ab69" +version = "1.2.2" + +[[deps.SHA]] +uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce" +version = "0.7.0" + +[[deps.Serialization]] +uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b" + +[[deps.Sockets]] +uuid = "6462fe0b-24de-5631-8697-dd941f90decc" + +[[deps.SparseArrays]] +deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"] +uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" +version = "1.10.0" + +[[deps.Statistics]] +deps = ["LinearAlgebra", "SparseArrays"] +uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" +version = "1.10.0" + +[[deps.SuiteSparse_jll]] +deps = ["Artifacts", "Libdl", "libblastrampoline_jll"] +uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c" +version = "7.2.1+1" + +[[deps.TOML]] +deps = ["Dates"] +uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76" +version = 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\ No newline at end of file diff --git a/pluto_notebooks/compress.jl b/pluto_notebooks/compress.jl new file mode 100644 index 0000000..3530ef3 --- /dev/null +++ b/pluto_notebooks/compress.jl @@ -0,0 +1,1468 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# ╔═╡ a4c27a3b-be79-4972-9f4b-792f460c188b +begin + using Plots + + import TensorCrossInterpolation as TCI + using QuanticsTCI + import QuanticsGrids as QG +end + +# ╔═╡ 1cf43d12-22d4-43db-99cb-61b0bd8384d1 +md""" +To run this notebook compress.jl locally, copy and paste the following into your Julia REPL: + +```julia +using Pkg; Pkg.activate(temp=true); Pkg.add("Pluto") +BASE_URL = "https://raw.githubusercontent.com/tensor4all/T4APlutoExamples/refs/heads/main/pluto_notebooks/" +notebook = "compress.jl" +url = joinpath(BASE_URL, notebook) +using Pluto; Pluto.run(notebook=download(url)) +``` +""" + +# ╔═╡ e6946e55-1220-4c4a-8811-78a27dea6e39 +md""" +# Compressing exisiting data""" + +# ╔═╡ 0b38c4db-40f6-4129-99f0-16c138e5cebc +md""" +## TCI + +Let us demonstrate how to compress exisiting data by TCI. +First, we create a test dataset on a 3D grid.""" + +# ╔═╡ f65b2036-8b1c-4875-bd8d-d9d6a5967eff +tolerance = 1e-5 + +# ╔═╡ 4b6a17b2-502e-407a-bbab-f22e1c84604f +begin + # Replace this line with the dataset to be tested for compressibility. + grid_tt = range(-pi, pi; length = 200) + dataset_tt = [cos(x) + cos(y) + cos(z) for x in grid_tt, y in grid_tt, z in grid_tt] + size(dataset_tt) +end + +# ╔═╡ c348ba58-58d3-4b58-a512-45cf1c223dd5 +md""" +We now construct a TCI.""" + +# ╔═╡ 8c5ce2f9-0fc1-4bff-b24d-8fcee25733ca +errors_tt = let + # Construct TCI + tt, ranks, errors_tt = TCI.crossinterpolate2( + Float64, i -> dataset_tt[i...], collect(size(dataset_tt)), tolerance = tolerance) + + # Check error + ttdataset = [tt([i, j, k]) + for i in axes(grid_tt, 1), j in axes(grid_tt, 1), k in axes(grid_tt, 1)] + errors_tt = abs.(ttdataset .- dataset_tt) + println( + "TCI of the dataset_tt with tolerance $tolerance has link dimensions $(TCI.linkdims(tt)), " + * + "for a max error of $(maximum(errors_tt))." + ) + errors_tt +end; + +# ╔═╡ f3c6a898-c440-43c3-8bb4-2be1101d5fd4 +md""" +Let us plot the original data and the TCI error on a 2D cut.""" + +# ╔═╡ f52ef908-eb78-404d-98ab-461447368fce +let + # Original data + c1 = heatmap(dataset_tt[:, :, 1], aspect_ratio = 1) + title!("Original data") + + # TCI error + c2 = heatmap(log10.(abs.(errors_tt[:, :, 1])), aspect_ratio = 1) + title!("log10 of abs error of TCI") + + plot(c1, c2, size = (800, 500)) +end + +# ╔═╡ 2b274d67-bd87-4de9-9565-403ad8c49959 +md""" +## QTCI + +We now demonstrate how to compress existing data by QTCI.""" + +# ╔═╡ dc961a12-5d1e-488e-815a-d47e12defd86 +begin + # Number of bits + R = 8 + + # Replace with your dataset + grid_qtt = range(-pi, pi; length = 2^R + 1)[1:(end - 1)] # exclude the end point + dataset_qtt = [cos(x) + cos(y) + cos(z) for x in grid_qtt, y in grid_qtt, z in grid_qtt] + size(dataset_qtt) +end + +# ╔═╡ 90ca07b8-cd64-4bd0-99af-aa1389766fbc +md""" +### QuanticsTCI.jl +Let us first use `quanticscrossinterpolate` function in `QuanticsTCI.jl`.""" + +# ╔═╡ 421840b6-b506-46fd-937f-817af34c68ce +begin + # Perform QTCI + qtt, ranks, _ = quanticscrossinterpolate( + dataset_qtt, tolerance = tolerance, unfoldingscheme = :fused) +end + +# ╔═╡ cbd3a596-98dc-4d1a-9029-b7201da835bc +md""" +Below, we compute the error for the whole tensor, which may be too expensive for a large $\mathcal{R}$.""" + +# ╔═╡ d9a6f697-8fbc-44c2-a2d2-1f343757090c +begin + # Check error + qttdataset = [qtt([i, j, k]) + for i in axes(grid_qtt, 1), j in axes(grid_qtt, 1), + k in axes(grid_qtt, 1)] + qtterrors = abs.(qttdataset .- dataset_qtt) + println( + "Quantics TCI compression of the dataset with tolerance $tolerance has " * + "link dimensions $(TCI.linkdims(qtt.tci)), for a max error of $(maximum(qtterrors))." + ) +end + +# ╔═╡ db0507c0-cf9c-4f12-bcec-ab703974c522 +md""" +Again, let us plot the original data and the TCI error on a 2D cut.""" + +# ╔═╡ 9218abf3-28c6-4dfd-afdf-413b469abdc8 +let + # Original data + c1 = heatmap(qttdataset[:, :, 1], aspect_ratio = 1) + title!("Original data") + + c2 = heatmap(log10.(abs.(qtterrors[:, :, 1])), aspect_ratio = 1) + title!("log10 of abs error of QTCI") + + plot(c1, c2, size = (800, 500)) +end + +# ╔═╡ 62980e80-7283-4f91-a6fc-18cffb930f82 +md""" +### QuanticsGrids.jl + TensorCrossInterpolation.jl + +`QuanticsTCI.jl` is user-friendly, yet utilizing `QuanticsGrids.jl` directly provides greater flexibility.""" + +# ╔═╡ 4d02437a-b0f7-4c4e-a7d0-64bc5ee6cbd3 +function create_qgrid(R, qttdataset) + # 3D quantics grid with R bits and the fused reprensentation (default) + qgrid = QG.InherentDiscreteGrid{3}(R) + + # Function that returns the value of the dataset at the given quantics index + qf(qindex) = qttdataset[QG.quantics_to_grididx(qgrid, qindex)...] + + return qgrid, qf +end + +# ╔═╡ 8afc5a4d-5b82-44db-8d96-f105e05d36a0 +begin + qgrid, qf = create_qgrid(R, qttdataset) + + # Data at the quantics index [1, 1, ..., 1] = the index [1, 1, 1]. + qf(fill(1, R)) == qttdataset[1, 1, 1] +end + +# ╔═╡ 083f027f-1b5a-48a0-8993-24f2ff098332 +md""" +The `create_qgrid` function generates a 3D quantics grid and a closure (`qf`) for dataset access based on quantics indices, given a grid resolution (`R`) and a dataset (`qttdataset`). +This design reduces reliance on global variables, leading to faster function evalulations.""" + +# ╔═╡ cbe34f12-f0a5-4015-ba0e-5c2763685a56 +md""" +The effectiveness of TCI significantly depends on selecting appropriate initial pivots. +Optimal initial pivots are locations where the function intended for interpolation exhibits large absolute values.""" + +# ╔═╡ fe224fa0-8f49-43f3-923f-d18ef2124a60 +initialpivots, localdims = let + # Local dimensions + localdims = fill(8, R) + + # Generate initial pivots by maximainzing the aboslute value of the function from random points. + # This is a heuristic to find good initial pivots. + # The optimization is performed by single-sites updates. + ninitialpivots = 10 + initialpivots = [TCI.optfirstpivot(qf, localdims, [rand(1:d) for d in localdims]) + for _ in 1:ninitialpivots] + + for p in initialpivots + println("Initial pivot: $p $(qf(p))") + end + initialpivots, localdims +end + +# ╔═╡ 572d920b-8f5b-4b1e-a13e-3eb308766120 +let + # Perform (Q)TCI + qtt, _, _ = TCI.crossinterpolate2( + Float64, qf, localdims, initialpivots; tolerance = tolerance) + # Test error + qtt(initialpivots[1]) ≈ qf(initialpivots[1]) +end + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +QuanticsGrids = "634c7f73-3e90-4749-a1bd-001b8efc642d" +QuanticsTCI = "b11687fd-3a1c-4c41-97d0-998ab401d50e" +TensorCrossInterpolation = "b261b2ec-6378-4871-b32e-9173bb050604" + +[compat] +Plots = "~1.40.8" +QuanticsGrids = "~0.3.2" +QuanticsTCI = "~0.7.0" +TensorCrossInterpolation = "~0.9.12" 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b/pluto_notebooks/index.md new file mode 100644 index 0000000..ffe1462 --- /dev/null +++ b/pluto_notebooks/index.md @@ -0,0 +1,9 @@ +Please follow the instruction in the top cell of each notebook to run it locally. +- [Quantics TCI of univariate function](quantics1d.html) +- [Quantics TCI of univariate funciton (advanced topics)](quantics1d_advanced.html) +- [Compressing exisiting data](compress.html) +- [Interfacing with ITensors.jl](interfacingwithitensors.html) +- [Quantics TCI of multivariate function](quantics2d.html) +- [Quantum Fourier Transform](qft.html) +- [How to open ALL Pluto notebooks locally](about_pluto.html) +- [Introduction to Plots.jl](plots.html) diff --git a/pluto_notebooks/interfacingwithitensors.html b/pluto_notebooks/interfacingwithitensors.html new file mode 100644 index 0000000..2ec286c --- /dev/null +++ b/pluto_notebooks/interfacingwithitensors.html @@ -0,0 +1,20 @@ + + + + + + + + + + +

\ No newline at end of file diff --git a/pluto_notebooks/interfacingwithitensors.jl b/pluto_notebooks/interfacingwithitensors.jl new file mode 100644 index 0000000..5864e1b --- /dev/null +++ b/pluto_notebooks/interfacingwithitensors.jl @@ -0,0 +1,885 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# ╔═╡ 3a76d412-2c5b-4160-9f5d-2dc750fb44b3 +begin + import QuanticsGrids as QG + using QuanticsTCI: quanticscrossinterpolate + import TensorCrossInterpolation as TCI + using TCIITensorConversion + using ITensors + + B = 2^(-30) # global variable + function f(x) + return cos(x / B) * cos(x / (4 * sqrt(5) * B)) * exp(-x^2) + 2 * exp(-x) + end + + R = 40 # number of bits + xmin = 0.0 + xmax = 3.0 + qgrid = QG.DiscretizedGrid{1}(R, xmin, xmax; includeendpoint = false) + ci, ranks, errors = quanticscrossinterpolate( + Float64, f, qgrid; maxbonddim = 15) +end + +# ╔═╡ dd3b9f0b-4300-4856-adcd-120c71dff9cd +md""" +To run this notebook interfacingwithitensors.jl locally, copy and paste the following into your Julia REPL: + +```julia +using Pkg; Pkg.activate(temp=true); Pkg.add("Pluto") +BASE_URL = "https://raw.githubusercontent.com/tensor4all/T4APlutoExamples/refs/heads/main/pluto_notebooks/" +notebook = "interfacingwithitensors.jl" +url = joinpath(BASE_URL, notebook) +using Pluto; Pluto.run(notebook=download(url)) +``` +""" + +# ╔═╡ 33bbccb2-2df2-4e80-a7d9-cdc9391d72c6 +md""" +# Interfacing with ITensors.jl + +You can convert a TCI object to an ITensor MPS object using the `TCIITensorConversion.jl` library. This library provides conversions of tensor trains between `TensorCrossInterpolation.jl` and `ITensors.jl`. + +We first construct a TCI object: +""" + +# ╔═╡ 062b5b8e-56eb-4cfd-a045-336a99e8bb82 +md""" +One can create a tensor train object from the TCI object, and then convert it to an ITensor MPS: +""" + +# ╔═╡ 88dfda51-86f6-4331-8d0c-d12ab277a12a +begin + # Construct a TensorTrain object from the TensorCI2 object + tt = TCI.TensorTrain(ci.tci) + # Convert the TensorTrain object to an ITensor MPS object + M = ITensors.MPS(tt) +end + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +ITensors = "9136182c-28ba-11e9-034c-db9fb085ebd5" +QuanticsGrids = "634c7f73-3e90-4749-a1bd-001b8efc642d" +QuanticsTCI = "b11687fd-3a1c-4c41-97d0-998ab401d50e" +TCIITensorConversion = "9f0aa9f4-9415-4e6a-8795-331ebf40aa04" +TensorCrossInterpolation = "b261b2ec-6378-4871-b32e-9173bb050604" + +[compat] +ITensors = "~0.6.17" +QuanticsGrids = "~0.3.2" +QuanticsTCI = "~0.7.0" +TCIITensorConversion = "~0.1.4" +TensorCrossInterpolation = "~0.9.12" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.5" +manifest_format = "2.0" +project_hash = "608fc341f9395a1918480d4c093cfd396e36459f" + +[[deps.Accessors]] +deps = ["CompositionsBase", "ConstructionBase", "Dates", "InverseFunctions", "LinearAlgebra", "MacroTools", "Markdown", "Test"] +git-tree-sha1 = "f61b15be1d76846c0ce31d3fcfac5380ae53db6a" +uuid = "7d9f7c33-5ae7-4f3b-8dc6-eff91059b697" +version = "0.1.37" + + [deps.Accessors.extensions] + AccessorsAxisKeysExt = "AxisKeys" + AccessorsIntervalSetsExt = "IntervalSets" + AccessorsStaticArraysExt = "StaticArrays" + AccessorsStructArraysExt = "StructArrays" + AccessorsUnitfulExt = "Unitful" + + [deps.Accessors.weakdeps] + AxisKeys = "94b1ba4f-4ee9-5380-92f1-94cde586c3c5" + IntervalSets = "8197267c-284f-5f27-9208-e0e47529a953" 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\ No newline at end of file diff --git a/pluto_notebooks/plots.jl b/pluto_notebooks/plots.jl new file mode 100644 index 0000000..f4ff544 --- /dev/null +++ b/pluto_notebooks/plots.jl @@ -0,0 +1,1264 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# ╔═╡ a76d8151-fb1d-4dcb-aaf7-eb7f3d3c1f9e +begin + using Plots + using Plots.RecipesBase: @recipe, @shorthands + + using LaTeXStrings +end + +# ╔═╡ 8cabb3ac-bd6f-496c-84b2-bb536ba0ff54 +md""" +To run this notebook plots.jl locally, copy and paste the following into your Julia REPL: + +```julia +using Pkg; Pkg.activate(temp=true); Pkg.add("Pluto") +BASE_URL = "https://raw.githubusercontent.com/tensor4all/T4APlutoExamples/refs/heads/main/pluto_notebooks/" +notebook = "plots.jl" +url = joinpath(BASE_URL, notebook) +using Pluto; Pluto.run(notebook=download(url)) +``` +""" + +# ╔═╡ 679f9791-df49-47f6-b5e0-562737f265b5 +md""" +# Introduction to Plots.jl +This guide provides an introduction to [Plots.jl](https://docs.juliaplots.org/stable/), a powerful and flexible plotting package for Julia. +Plots.jl is a widely used Julia package for data visualization. +""" + +# ╔═╡ fe019b21-c774-49ff-981c-0487f0b07d42 +md""" +To plot a given function, we can pass it as an argument to the `plot` function. +""" + +# ╔═╡ e44c2429-fc7c-4609-b27c-4b527a7b8440 +plot(sin) + +# ╔═╡ 83905cd7-4548-482b-9514-3b9ef2d99e67 +md""" +Additionally, we can provide x-coordinates (`x`) and y-coordinates (`y`) as arguments to the `plot` function to create custom plots.""" + +# ╔═╡ 5744c00e-ff55-4cff-b2ed-7079c341175e +let + x = -π:0.1:π + y = sin.(x) + # line plot + plot(x, y, label="sin") +end + +# ╔═╡ b1c45986-eb23-42aa-ba27-8269686ad626 +md""" +We can draw multiple functions on a same plot pane using `plot!` function. +The following example demonstrates plotting the sine function $y = \sin(x)$ for $x \in [-\pi, \pi]$. +""" + +# ╔═╡ aeab03ea-5947-4088-b60e-d1b32e98d164 +let + x = -π:0.1:π + y1 = sin.(x) + y2 = cos.(x) + plt = plot() + # line plot + plot!(plt, x, y1, label="sin") + # scatter plot + plot!(plt, x, y2, label="cos", seriestype=:scatter) +end + +# ╔═╡ 9e90b55e-863a-4d6f-91d1-84e15ad1edd7 +md""" +Instead of specifying `seriestype=:scatter`, we can use the `scatter` or `scatter!` functions directly. +""" + +# ╔═╡ 5e630874-2ce5-4a8a-9cc0-c2474a9a6fa3 +let + x = 1:10 + y = rand(10) + plt = plot() + plot!(x, y) + scatter!(x, y, label="scatter", marker=:x) +end + +# ╔═╡ 25a968c0-4209-44e5-8e9a-ba580e0fbf4d +md""" +## Heatmap + +We can use the `heatmap` function to visualize a function on a two-dimensional domain. +""" + +# ╔═╡ f1871cd6-8d8c-4e38-920b-c3f145513b22 +let + x = 1:20 + y = 1:3 + + f(x, y) = 2x + y + heatmap(x, y, f) +end + +# ╔═╡ c8b63bfc-3ef3-41b5-9f9f-7b59a5c0cce2 +md""" +Of course, we can also pass the z-coordinates of the data (`z`) as the third argument to the `heatmap` function. +""" + +# ╔═╡ 86a17ea2-ce2c-44a5-ad69-b30b280dfb41 +let + x = 1:20 + y = 1:3 + + f(x, y) = 2x + y + @show size(f.(x', y)) + z = f.(x', y) + heatmap(x, y, z) +end + +# ╔═╡ a5c86153-8430-4336-a82f-c671b2c63eb5 +md""" +## Defining a custom seriestype + +We can define a custom seriestype by creating a new struct and defining a [recipe](https://docs.juliaplots.org/stable/recipes/#recipes) for it. +Here we define a custom seriestype `SemiLogy` that plots the y-axis on a logarithmic scale. +""" + +# ╔═╡ 0ece0753-f329-43e6-9a34-99eeb7345cf4 +begin + # defines mutable struct `SemiLogy` and sets shorthands + # `semilogy` and `semilogy!` + @userplot SemiLogy + @recipe function f(t::SemiLogy) + x = t.args[begin] + y = t.args[end] + ε = nextfloat(0.0) + + yscale := :log10 + # Adding ε avoids getting log10(0) from being -Inf + (x, ε .+ y) + end +end + +# ╔═╡ fa8f001c-e55b-457e-ae4e-80cf7aa1ffd7 +semilogy((-10:1:-7), 10.0 .^ (-10:1:-7)) + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" +Plots 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z1P88B!dAPa05cvfHqXY5nLIVj!m$F^#xi28T44jbhD%mPoZ%;9PTSJyc7-a;T zq$#SD%wLKj4 \ No newline at end of file diff --git a/pluto_notebooks/qft.jl b/pluto_notebooks/qft.jl new file mode 100644 index 0000000..16d5900 --- /dev/null +++ b/pluto_notebooks/qft.jl @@ -0,0 +1,2168 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# ╔═╡ 23d20fe2-152c-4368-808d-784ab72c507b +begin + using LaTeXStrings + using Plots + + import QuanticsGrids as QG + import TensorCrossInterpolation as TCI + using QuanticsTCI: quanticscrossinterpolate, quanticsfouriermpo +end + +# ╔═╡ 09aecf54-58c1-4e6e-abee-1e8574d9cf61 +using PlutoUI + +# ╔═╡ 0cc27419-c679-42e5-b63b-68da665c9d57 +begin + resource_qft1 = Resource("https://raw.githubusercontent.com/tensor4all/T4AJuliaTutorials/main/qft1.png", "alt"=>"qft1", "width"=>"30%") + + resource_qft2 = Resource("https://raw.githubusercontent.com/tensor4all/T4AJuliaTutorials/main/qft2.png", "alt"=>"qft contraction", "width"=>"60%") + nothing +end + +# ╔═╡ 4bbd8548-49cc-4abf-af40-909d15cc6c75 +begin + coeffs = [1.0, 1.0] + ϵs = [100.0, -50.0] + + _exp(x, ϵ) = exp(-ϵ * x) / (1 - exp(-ϵ)) + + fx(x) = sum(coeffs .* _exp.(x, ϵs)) +end + +# ╔═╡ 84e92a8c-289e-4831-88a1-f20d4d681aac +begin + plotx = range(0, 1; length = 1000) + + plot(plotx, fx.(plotx)) + xlabel!(L"x") + ylabel!(L"f(x)") +end + +# ╔═╡ 85718cda-191e-4445-bcca-0704efe5d6b1 +begin + R = 40 + xgrid = QG.DiscretizedGrid{1}(R, 0, 1) + + qtci, ranks, errors = quanticscrossinterpolate(Float64, fx, xgrid; tolerance = 1e-10) +end + +# ╔═╡ 1b129c10-2143-4235-95b2-0eb60885f7e0 +begin + # Construct QTT representation of T_{km} + fouriertt = quanticsfouriermpo(R; sign = 1.0, normalize = true) + + # Apply T_{km} to the QTT representation of f(x) + sitedims = [[2, 1] for _ in 1:R] + ftt = TCI.TensorTrain(qtci.tci) + hftt = TCI.contract(fouriertt, ftt; algorithm = :naive, tolerance = 1e-8) + + hftt *= 1 / sqrt(2)^R + + @show hftt +end + +# ╔═╡ 5f1dcbb3-1f0d-4263-8440-212f0b7f52b6 +begin + import TCIITensorConversion + using ITensors + import Quantics: fouriertransform, Quantics + + sites_m = [Index(2, "Qubit,m=$m") for m in 1:R] + sites_k = [Index(2, "Qubit,k=$k") for k in 1:R] + + fmps = MPS(ftt; sites = sites_m) + + # Apply T_{km} to the MPS representation of f(x) and reply the result by 1/sqrt(M) + # tag="m" is used to indicate that the MPS is in the "m" basis. + hfmps = (1 / sqrt(2)^R) * + fouriertransform(fmps; sign = 1, tag = "m", sitesdst = sites_k) +end + +# ╔═╡ 5c9be416-2eb0-4f7e-bc77-8a14c0cffd67 +md""" +To run this notebook qft.jl locally, copy and paste the following into your Julia REPL: + +```julia-repl +julia> using Pkg; Pkg.activate(temp=true); Pkg.add("Pluto") +julia> BASE_URL = "https://raw.githubusercontent.com/tensor4all/T4APlutoExamples/refs/heads/main/pluto_notebooks/" +julia> notebook = "qft.jl" +julia> url = joinpath(BASE_URL, notebook) +julia> using Pluto; Pluto.run(notebook=download(url)) +``` +""" + +# ╔═╡ 2089fb37-7a8d-46c2-b95c-a0feab70d250 +md""" +# Quantum Fourier Transform +""" + +# ╔═╡ ab2c70f2-a83e-454a-a183-12772af3eac4 +md""" +## 1D Fourier transform + +```math +%\newcommand{\hf}{\hat{f}} %Fourier transform of \bff +%\newcommand{\hF}{\hat{F}} +``` + +Consider a discrete function $f_m \in \mathbb{C}^M$, e.g. the +discretization, $f_m = f(x(m))$, of a one-dimensional function $f(x)$ on a grid $x(m)$. +Its discrete Fourier transform (DFT) is + +```math +\hat{f}_k = \sum_{m=0}^{M-1} T_{km} f_m , \qquad +T_{km} = \tfrac{1}{\sqrt{M}} e^{- i 2 \pi k \cdot m /M} . +``` + +For a quantics grid, $M = 2^\mathcal{R}$ is exponentially large and the (naive) DFT exponentially expensive to evaluate. +However, the QTT representation of $T$ is known to have a low-rank structure and can be represented as a tensor train with small bond dimensions. + +Thus, if the input function $f$ is given in the quantics representation as $(resource_qft1), + +then $\hat{f} = T f$ can be computed by efficiently contracting the tensor trains for $T$ and $f$ and recompressing the result: + + $(resource_qft2) + +Note that after the Fourier transform, the quantics indices $\sigma_1,\cdots,\sigma_\mathcal{R}$ are ordered in the inverse order of the input indices $\sigma'_1,\cdots,\sigma'_\mathcal{R}$. +This allows construction of the DFT operator with small bond dimensions.""" + +# ╔═╡ 81ef384b-b219-4214-973c-7ed8d892834a +md""" +We consider a function $f(x)$, which is the sum of exponential functions, defined on interval $[0,1]$: + +```math +f(x) = \sum_p \frac{c_p}{1 - e^{-\epsilon_p}} e^{-\epsilon_p x}. +``` + +Its Fourier transform is given by + +```math +\hat{f}_k = \int_0^1 dx \, f(x) e^{i \omega_k x} = - \sum_p \frac{c_p}{i\omega_k - \epsilon_p}. +``` + +for $k = 0, 1, \cdots $ and $\omega_k = 2\pi k$. + +If you are familiar with quantum field theory, you can think of $f(x)$ as a bosonic correlation function.""" + +# ╔═╡ 3bfec21e-ecf6-4628-a45f-8920eef79447 +md""" +First, we construct a QTT representation of the function $f(x)$.""" + +# ╔═╡ 02d6bed8-69a6-4b4c-852a-60d9fe52c4db +md""" +Second, we compute the Fourier transform of $f(x)$ using the QTT representation of $f(x)$ and the QTT representation of the DFT operator $T$: + +```math +\hat{f}_k = \int_0^1 dx \, f(x) e^{i \omega_k x} \approx \frac{1}{M} \sum_{m=0}^{M-1} f_m e^{i 2 \pi k m / M} = \frac{1}{\sqrt{M}} \sum_{m=0}^{M-1} T_{km} f_m. +``` + +for $k = 0, \ldots, M-1$ and $\omega_k = 2\pi k$. +This can be implemented as follows.""" + +# ╔═╡ 314c5b5a-5c3f-4c96-b004-02743bab093a +md""" +Let us compare the result with the exact Fourier transform of $f(x)$.""" + +# ╔═╡ a463bbb3-5a6c-4b09-a3d9-5cdb467c77c4 +begin + kgrid = QG.InherentDiscreteGrid{1}(R, 0) # 0, 1, ..., 2^R-1 + _expk(k, ϵ) = -1 / (2π * k * im - ϵ) + hfk(k) = sum(coeffs .* _expk.(k, ϵs)) # k = 0, 1, 2, ..., 2^R-1 +end + +# ╔═╡ 1cbf3d00-d1a2-40ee-af13-7d725bd0c195 +let + plotk = collect(0:300) + y = [hftt(reverse(QG.origcoord_to_quantics(kgrid, x))) for x in plotk] # Note: revert the order of the quantics indices + p1 = plot() + plot!(p1, plotk, real.(y), marker = :+, label = "QFT") + plot!(p1, plotk, real.(hfk.(plotk)), marker = :x, label = "Reference") + xlabel!(p1, L"k") + ylabel!(p1, L"\mathrm{Re}~\hat{f}(k)") + + p2 = plot() + plot!(p2, plotk, imag.(y), marker = :+, label = "QFT") + plot!(p2, plotk, imag.(hfk.(plotk)), marker = :x, label = "Reference") + xlabel!(L"k") + ylabel!(L"\mathrm{Im}~\hat{f}(k)") + + plot(p1, p2, size = (800, 500)) +end + +# ╔═╡ 327f847d-de5f-4e40-9888-450283f92c4e +md""" +The exponentially large quantics grid allows to compute the Fourier transform with high accuracy at high frequencies. +To check this, let us compare the results at high frequencies.""" + +# ╔═╡ b9f728f1-2265-4e0e-95f2-49c7b97d0421 +let + plotk = [10^n for n in 1:5] + @assert maximum(plotk) <= 2^R - 1 + y = [hftt(reverse(QG.origcoord_to_quantics(kgrid, x))) for x in plotk] # Note: revert the order of the quantics indices + + p1 = plot() + + plot!(p1, plotk, abs.(real.(y)), marker = :+, + label = "QFT", xscale = :log10, yscale = :log10) + plot!(p1, plotk, abs.(real.(hfk.(plotk))), marker = :x, + label = "Reference", xscale = :log10, yscale = :log10) + xlabel!(p1, L"k") + ylabel!(p1, L"\mathrm{Re}~\hat{f}(k)") + + p2 = plot() + + plot!(p2, plotk, abs.(imag.(y)), marker = :+, + label = "QFT", xscale = :log10, yscale = :log10) + plot!(p2, plotk, abs.(imag.(hfk.(plotk))), marker = :x, + label = "Reference", xscale = :log10, yscale = :log10) + xlabel!(p2, L"k") + ylabel!(p2, L"\mathrm{Im}~\hat{f}(k)") + + plot(p1, p2, size = (800, 500)) +end + +# ╔═╡ 89dd01d2-c9fc-4d69-a5b6-3303850bbe8c +md""" +You may use ITensors.jl to compute the Fourier transform of the function $f(x)$. +The following code explains how to do this.""" + +# ╔═╡ 77a9f7c9-36fe-4721-b6c3-7712930c3cf1 +# Evaluate Ψ for a given index +function _evaluate(Ψ::MPS, sites, index::Vector{Int}) + only(reduce(*, Ψ[n] * onehot(sites[n] => index[n]) for n in 1:length(Ψ))) +end + +# ╔═╡ 7348acb0-1e9c-493a-b192-facfd2d00109 +let + plotk = [10^n for n in 1:5] + @assert maximum(plotk) <= 2^R - 1 + y = [_evaluate(hfmps, reverse(sites_k), reverse(QG.origcoord_to_quantics(kgrid, x))) + for x in plotk] # Note: revert the order of the quantics indices + + p1 = plot() + plot!(p1, plotk, abs.(real.(y)), marker = :+, label = "QFT") + plot!(p1, plotk, abs.(real.(hfk.(plotk))), marker = :x, label = "Reference") + xlabel!(p1, L"k") + ylabel!(p1, L"\mathrm{Re}~\hat{f}(k)") + + p2 = plot() + plot!(p2, plotk, abs.(imag.(y)), marker = :+, label = "QFT") + plot!(p2, plotk, abs.(imag.(hfk.(plotk))), marker = :x, label = "Reference") + xlabel!(p2, L"k") + ylabel!(p2, L"\mathrm{Im}~\hat{f}(k)") + + plot(p1, p2, size = (800, 500)) +end + +# ╔═╡ 87937f1f-eb60-40ea-b0ac-9bef4ce98637 +md""" +## 2D Fourier transform + +We now consider a two-dimensional function $f(x, y) = \frac{1}{(1 - e^{-\epsilon})(1 - e^{-\epsilon'})} e^{-\epsilon x - \epsilon' y}$ defined on the interval $[0,1]^2$. + +Its Fourier transform is given by + +```math +\hat{f}_{kl} = \int_0^1 \int_0^1 dx dy \, f(x, y) e^{i \omega_k x + i\omega_l y} \approx \frac{1}{M^2} \sum_{m,n=0}^{M-1} f_{mn} e^{i 2 \pi (k m + l n) / M} = \frac{1}{M} \sum_{m,n=0}^{M-1} T_{km} T_{ln} f_{mn}. +``` + +The exact form of the Fourier transform is + +```math +\hat{f}_{kl} = \frac{1}{(i\omega_k - \epsilon) (i\omega_l - \epsilon')}. +``` + +for $k, l = 0, 1, \cdots $, $\omega_k = 2\pi k$ and $\omega_l = 2\pi l$. + +The 2D Fourier transform can be numerically computed in QTT format (with interleaved representation) in a straightforward way using Quantics.jl. +""" + +# ╔═╡ 0c1da36b-686c-4850-a0a0-ac4652d0b0ce +begin + ϵ = 1.0 + ϵprime = 2.0 + fxy(x, y) = _exp(x, ϵ) * _exp(y, ϵprime) + + # 2D quantics grid using interleaved unfolding scheme + xygrid = QG.DiscretizedGrid{2}(R, (0, 0), (1, 1); unfoldingscheme = :interleaved) + + # Resultant QTT representation of f(x, y) has bond dimension of 1. + qtci_xy, ranks_xy, errors_xy = quanticscrossinterpolate( + Float64, fxy, xygrid; tolerance = 1e-10) +end + +# ╔═╡ baabe2d4-d2c9-4932-af0a-b363854655ad +begin + # for discretizing `y` + sites_n = [Index(2, "Qubit,n=$n") for n in 1:R] + + sites_l = [Index(2, "Qubit,l=$l") for l in 1:R] + + sites_mn = collect(Iterators.flatten(zip(sites_m, sites_n))) + + fmps2 = MPS(TCI.TensorTrain(qtci_xy.tci); sites = sites_mn) + siteinds(fmps2) +end + +# ╔═╡ 35a93fe6-fa5e-4172-a7d2-228996cf9f5c +begin + # Fourier transform for x + tmp_ = (1 / sqrt(2)^R) * + fouriertransform(fmps2; sign = 1, tag = "m", sitesdst = sites_k, cutoff = 1e-20) + + # Fourier transform for y + hfmps2 = (1 / sqrt(2)^R) * + fouriertransform(tmp_; sign = 1, tag = "n", sitesdst = sites_l, cutoff = 1e-20) + + siteinds(hfmps2) +end + +# ╔═╡ 0fb125ca-eeec-4ada-ac62-12f45e486952 +md""" +For convinience, we swap the order of the indices. +""" + +# ╔═╡ 4fc4e652-c883-42b9-a049-79041773a584 +begin + # Convert to fused representation and swap the order of the indices + hfmps2_fused = MPS(reverse([hfmps2[2 * n - 1] * hfmps2[2 * n] for n in 1:R])) + + # From fused to interleaved representation + sites_kl = collect(Iterators.flatten(zip(sites_k, sites_l))) + hfmps2_reverse = Quantics.rearrange_siteinds(hfmps2_fused, [[x] for x in sites_kl]) + siteinds(hfmps2_reverse) +end + +# ╔═╡ 028094a0-16f7-4026-9c71-7bd6c31a1ce2 +begin + klgrid = QG.InherentDiscreteGrid{2}(R, (0, 0); unfoldingscheme = :interleaved) + + sparse1dgrid = collect(0:4) + + reconstdata = [_evaluate( + hfmps2_reverse, sites_kl, QG.origcoord_to_quantics(klgrid, (k, l))) + for k in sparse1dgrid, l in sparse1dgrid] + + hfkl(k::Integer, l::Integer) = _expk(k, ϵ) * _expk(l, ϵprime) + + exactdata = [hfkl(k, l) for k in sparse1dgrid, l in sparse1dgrid] + + c1 = heatmap(real.(exactdata)) + xlabel!(L"k") + ylabel!(L"l") + title!("Real part of Exact data") + + c2 = heatmap(real.(reconstdata)) + xlabel!(L"k") + ylabel!(L"l") + title!("Real part of Reconstructed data") + + c3 = heatmap(abs.(exactdata .- reconstdata)) + xlabel!(L"k") + ylabel!(L"l") + title!("Error") + + plot(c1, c2, c3, size = (1500, 400), layout = (1, 3)) +end + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +ITensors = "9136182c-28ba-11e9-034c-db9fb085ebd5" +LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +Quantics = "87f76fb3-a40a-40c9-a63c-29fcfe7b7547" +QuanticsGrids = "634c7f73-3e90-4749-a1bd-001b8efc642d" +QuanticsTCI = "b11687fd-3a1c-4c41-97d0-998ab401d50e" +TCIITensorConversion = "9f0aa9f4-9415-4e6a-8795-331ebf40aa04" +TensorCrossInterpolation = "b261b2ec-6378-4871-b32e-9173bb050604" + +[compat] +ITensors = "~0.6.17" +LaTeXStrings = "~1.3.1" +Plots = "~1.40.8" +PlutoUI = "~0.7.60" +Quantics = "~0.3.8" +QuanticsGrids = "~0.3.2" +QuanticsTCI = "~0.7.0" +TCIITensorConversion = "~0.1.4" +TensorCrossInterpolation = "~0.9.12" +""" + +# ╔═╡ 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\ No newline at end of file diff --git a/pluto_notebooks/quantics1d.jl b/pluto_notebooks/quantics1d.jl new file mode 100644 index 0000000..4b96a70 --- /dev/null +++ b/pluto_notebooks/quantics1d.jl @@ -0,0 +1,1680 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# ╔═╡ ce892f77-f0ee-4721-8976-9d4e217d1925 +begin + using Plots + gr() # Use GR backend for plotting + + import QuanticsGrids as QG + import TensorCrossInterpolation as TCI + + using QuanticsTCI: QuanticsTCI, quanticscrossinterpolate, integral +end + +# ╔═╡ 5c09f2e8-5e91-413f-8f21-407d13ed5b8b +md""" +To run this notebook locally, copy and paste the following into your Julia REPL: + +```julia +using Pkg; Pkg.activate(temp=true); Pkg.add("Pluto") +BASE_URL = "https://raw.githubusercontent.com/tensor4all/T4APlutoExamples/refs/heads/main/pluto_notebooks/" +notebook = "quantics1d.jl" +url = joinpath(BASE_URL, notebook) +using Pluto; Pluto.run(notebook=download(url)) +``` +""" + +# ╔═╡ f85ccc76-0793-41c6-acf5-23e621953b4b +md""" +# Quantics TCI of univariate function +""" + +# ╔═╡ a6c64c0d-6e01-4044-8f09-283d35f51518 +begin + # defines mutable struct `SemiLogy` and sets shorthands `semilogy` and `semilogy!` + @userplot SemiLogy + @recipe function f(t::SemiLogy) + x = t.args[begin] + y = t.args[end] + ε = nextfloat(0.0) + + yscale := :log10 + # Warning: Invalid negative or zero value 0.0 found at series index 16 for log10 based yscale + # prevent log10(0) from being -Inf + (x, ε .+ y) + end +end + +# ╔═╡ e241cb1b-d6f4-4017-bd9c-91269a7d44cb +md""" +## Example 1 + +The first example is taken from Fig. 1 in [Ritter2024](https://arxiv.org/abs/2303.11819). + +We are going to compute the integral $\mathrm{I}[f] = \int_0^{\ln 20} \mathrm{d}x f(x) $ of the function + +```math +f(x) = \cos\left(\frac{x}{B}\right) \cos\left(\frac{x}{4\sqrt{5}B}\right) e^{-x^2} + 2e^{-x}, +``` + +where $B = 2^{-30}$. +The integral evaluates to $\mathrm{I}[f] = 19/10 + O(e^{-1/(4B^2)})$. + +We first construct a QTT representation of the function $f(x)$ as follows:""" + +# ╔═╡ 783ffb2d-8b23-41a0-919b-2e51e0d0c8bd +begin + B = 2^(-30) # global variable + function f(x) + return cos(x / B) * cos(x / (4 * sqrt(5) * B)) * exp(-x^2) + 2 * exp(-x) + end + + println(f(0.2)) +end + +# ╔═╡ bfb37a67-322e-4e2a-ad7c-dc8a628ae44f +md""" +Let's examine the behaviour of $f(x)$. This function involves structure on widely different scales: rapid, incommensurate oscillations and a slowly decaying envelope. We'll use [PythonPlot.jl](https://github.com/JuliaPy/PythonPlot.jl) visualisation library which uses Python library [matplotlib](https://matplotlib.org/) behind the scenes. + +For small $x$ we have: +""" + +# ╔═╡ 0667a441-4318-4ff8-961d-bee751b0f5b1 +let + xs = LinRange(0, 2.0^(-23), 1000) + + plt = plot(title = "$(nameof(f))") + plot!(plt, xs, f.(xs), label = "$(nameof(f))", legend = true) + plt +end + +# ╔═╡ 1db28e87-516c-468d-aee2-77a108691054 +md""" +For $x \in (0, 3]$ we will get: +""" + +# ╔═╡ ac154938-3dfe-420f-a24a-43fb8b9a5060 +let + xs2 = LinRange(2.0^(-23), 3, 100000) + plt = plot(title = "$(nameof(f))") + plot!(plt, xs2, f.(xs2), label = "$(nameof(f))", legend = true) + plt +end + +# ╔═╡ 7d3f31de-a251-4ed4-99b4-380206595132 +md""" +### QTT representation + +We construct a QTT representation of this function on the domain $[0, \ln 20]$, discretized on a quantics grid of size $2^\mathcal{R}$ with $\mathcal{R} = 40$ bits: +""" + +# ╔═╡ fac80646-42ab-40f7-8233-c679cca2cf6b +begin + R = 40 # number of bits + xmin = 0.0 + xmax = log(20.0) + N = 2^R # size of the grid + # * Uniform grid (includeendpoint=false, default): + # -xmin, -xmin+dx, ...., -xmin + (2^R-1)*dx + # where dx = (xmax - xmin)/2^R. + # Note that the grid does not include the end point xmin. + # + # * Uniform grid (includeendpoint=true): + # -xmin, -xmin+dx, ...., xmin-dx, xmin, + # where dx = (xmax - xmin)/(2^R-1). + qgrid = QG.DiscretizedGrid{1}(R, xmin, xmax; includeendpoint = true) + ci, ranks, errors = quanticscrossinterpolate(Float64, f, qgrid; maxbonddim = 15) +end + +# ╔═╡ 575431fb-1cb5-4611-add7-471f7ae01dcd +length(ci.tci) + +# ╔═╡ e5282234-32be-4787-a33e-c2df02aefcaf +md""" +Here, we've created the object `ci` of type `QuanticsTensorCI2{Float64}`. This can be evaluated at an linear index $i$ ($1 \le i \le 2^\mathcal{R}$) as follows: +""" + +# ╔═╡ b48a3427-cd90-4d80-a97c-bde4b7da6716 +let + for i in [1, 2, 3, 2^R] # Linear indices + # restore original coordinate `x` from linear index `i` + x = QG.grididx_to_origcoord(qgrid, i) + println("x: $(x), i: $(i), tci: $(ci(i)), ref: $(f(x))") + end +end + +# ╔═╡ d3f8d6ce-92e0-4a2b-b904-18969197936c +md""" +We see that `ci(i)` approximates the original `f` at `x = QG.grididx_to_origcoord(qgrid, i)`. Let's plot them together. +""" + +# ╔═╡ b0f5b0e0-c4c3-4ead-8216-d5e04401ccc1 +let + maxindex = QG.origcoord_to_grididx(qgrid, 2.0^(-23)) + testindices = Int.(round.(LinRange(1, maxindex, 1000))) + + xs = [QG.grididx_to_origcoord(qgrid, i) for i in testindices] + ys = f.(xs) + yci = ci.(testindices) + + plt = plot(title = "$(nameof(f)) and TCI", xlabel = "x", ylabel = "y") + plot!(plt, xs, ys, label = "$(nameof(f))", legend = true) + plot!(plt, xs, yci, label = "tci", linestyle = :dash, alpha = 0.7, legend = true) + plt +end + +# ╔═╡ 7595b85c-7f02-4833-911e-87df232e4c11 +md""" +Above, one can see that the original function is interpolated very accurately. + +Let's plot of $x$ vs interpolation error $|f(x) - \mathrm{ci}(x)|$ for small $x$ +""" + +# ╔═╡ 70459274-2923-437b-9f09-9c9be792996e +let + maxindex = QG.origcoord_to_grididx(qgrid, 2.0^(-23)) + testindices = Int.(round.(LinRange(1, maxindex, 1000))) + + xs = [QG.grididx_to_origcoord(qgrid, i) for i in testindices] + ys = f.(xs) + yci = ci.(testindices) + plt = plot(title = "x vs interpolation error: $(nameof(f))", + xlabel = "x", ylabel = "interpolation error") + semilogy!(xs, abs.(ys .- yci), label = "log(|f(x) - ci(x)|)", yscale = :log10, + legend = :bottomright, ylim = (1e-16, 1e-7), yticks = 10.0 .^ collect(-16:1:-7)) + plt +end + +# ╔═╡ 3bdc28f2-5013-4285-a058-3b1bfc855467 +md""" +... and for all $x$:""" + +# ╔═╡ 3edf2304-4ac8-460f-8888-d52b8dfea0d6 +let + plt = plot(title = "x vs interpolation error: $(nameof(f))", + xlabel = "x", ylabel = "interpolation error") + + testindices = Int.(round.(LinRange(1, 2^R, 1000))) + xs = [QG.grididx_to_origcoord(qgrid, i) for i in testindices] + ys = f.(xs) + yci = ci.(testindices) + semilogy!(xs, abs.(ys .- yci), label = "log(|f(x) - ci(x)|)", legend = true, + ylim = (1e-16, 1e-6), yticks = 10.0 .^ collect(-16:1:-6)) + plt +end + +# ╔═╡ e93fcdcb-6573-4c31-9a0b-c32e1f91ebd7 +md""" +The function is approximated with an accuracy $\approx 10^{-7}$ over the entire domain. + +We are now ready to compute the integral $\mathrm{I}[f] = \int_0^{\ln 20} \mathrm{d}x f(x) \simeq 19/10$ using the QTT representation of $f(x)$.""" + +# ╔═╡ e9fdf643-3e46-414e-9f8b-7746d91d0f7f +integral(ci), 19 / 10 + +# ╔═╡ edfe0936-17a8-416d-92ad-6c7c9d897fd2 +md""" +`integral(ci)` is equivalent to calling `QuanticsTCI.sum(ci)` and multiplying the result by the interval length divided by $2^\mathcal{R}$.""" + +# ╔═╡ c9365566-6df1-4a16-8a31-a515fe98e0d6 +sum(ci) * (log(20) - 0) / 2^R, 19 / 10 + +# ╔═╡ 1ef93487-1cac-4da4-912e-f3a69022a853 +md""" +### About `ci::QuanticsTensorCI2{Float64}` + +Let's dive into the `ci` object: +""" + +# ╔═╡ 6bbdc9e0-5ff1-436f-a19c-3e7cb89a824d +println(typeof(ci)) + +# ╔═╡ 4996289b-e733-43aa-9a90-ab3143a7d6ec +md""" +As we've seen before, `ci` is an object of `QuanticsTensorCI2{Float64}` in `QuanticsTCI.jl`, which is a thin wrapper of `TensorCI2{Float64}` in `TensorCrossInterpolation.jl`. +The undering object of `TensorCI2{Float64}` type can be accessed as `ci.tci`. This will be useful for obtaining more detailed information on the TCI results. + +For instance, `ci.tci.maxsamplevalue` is an estimate of the abosolute maximum value of the function, and `ci.tci.pivoterrors` stores the error as function of the bond dimension computed by prrLU. +In the following figure, we plot the normalized error vs. bond dimension, showing an exponential decay. +""" + +# ╔═╡ a426a6ad-75d1-46ec-b05a-8e51301ac545 +begin + # Plot error vs bond dimension obtained by prrLU + plt = plot(title = "normalized error vs. bond dimension: $(nameof(f))", + xlabel = "Bond dimension", ylabel = "Normalization error") + semilogy!(1:length(ci.tci.pivoterrors), ci.tci.pivoterrors ./ ci.tci.maxsamplevalue, + marker = :x, ylim = (1e-8, 10), yticks = (10.0 .^ (-10:1:0)), legend = false) + plt +end + +# ╔═╡ bee885a0-3170-4410-a783-3c221e29ab71 +md""" +### Function evaluations + +Our TCI algorithm does not call elements of the entire tensor, but constructs the TT (Tensor Train) from some elements chosen adaptively. On which points $x \in [0, 3]$ was the function evaluated to construct a QTT representation of the function $f(x)$? Let's find out. One can retrieve the information on the function evaluations as follows. +""" + +# ╔═╡ 6d75cc75-8170-498f-8520-a44917b7fd6d +begin + # Dict{Float64,Float64} + # key: `x` + # value: function value at `x` + evaluated = QuanticsTCI.cachedata(ci) +end + +# ╔═╡ eda97da8-f49f-43d9-857a-ff5bee58be16 +md""" +Let's plot `f` and the evaluated points together. +""" + +# ╔═╡ 5ce76560-4fe2-49e6-ac79-62d48100b5c1 +let + f̂(x) = ci(QG.origcoord_to_quantics(qgrid, x)) + xs = LinRange(0, 2.0^(-23), 1000) + + xs_evaluated = collect(keys(evaluated)) + fs_evaluated = [evaluated[x] for x in xs_evaluated] + + plt = plot( + title = "$(nameof(f)) and TCI", xlabel = "x", ylabel = "y", xlim = (0, maximum(xs))) + plot!(plt, xs, f.(xs), label = "$(nameof(f))") + scatter!(plt, xs_evaluated, fs_evaluated, marker = :x, label = "evaluated points") + plt +end + +# ╔═╡ 2ae41928-e443-44ac-80a8-616e37277da5 +md""" +## Example 2 + +We now consider the function: + +```math +\newcommand{\sinc}{\mathrm{sinc}} +\begin{align} +f(x) &= \sinc(x)+3e^{-0.3(x-4)^2}\sinc(x-4) \nonumber\\ +&\quad - \cos(4x)^2-2\sinc(x+10)e^{-0.6(x+9)} + 4 \cos(2x) e^{-|x+5|}\nonumber \\ +&\quad +\frac{6}{x-11}+ \sqrt{(|x|)}\arctan(x/15).\nonumber +\end{align} +``` + +One can construct a QTT representation of this function on the domain $[-10, 10)$ using a quantics grid of size $2^\mathcal{R}$ ($\mathcal{R}=20$): +""" + +# ╔═╡ 7407ef57-02ee-4a28-8557-d258486f2918 +# Function of interest +function oscillation_fn(x) + return ( + sinc(x) + 3 * exp(-0.3 * (x - 4)^2) * sinc(x - 4) - cos(4 * x)^2 - + 2 * sinc(x + 10) * exp(-0.6 * (x + 9)) + 4 * cos(2 * x) * exp(-abs(x + 5)) + + 6 * 1 / (x - 11) + sqrt(abs(x)) * atan(x / 15)) +end + +# ╔═╡ 68cac44c-41a6-498d-9957-dd3da91679c8 +let + R = 20 # number of bits + N = 2^R # size of the grid + + qgrid = QG.DiscretizedGrid{1}(R, -10, 10; includeendpoint = false) + + # Convert to quantics format and sweep + ci, ranks, errors = quanticscrossinterpolate( + Float64, oscillation_fn, qgrid; maxbonddim = 15) +end + +# ╔═╡ db6a1232-8fc0-4ebd-838b-5905d469ae11 +for i in [1, 2, 2^R] # Linear indices + x = QG.grididx_to_origcoord(qgrid, i) + println("x: $(x), tci: $(ci(i)), ref: $(oscillation_fn(x))") +end + +# ╔═╡ 4a53fd00-1320-4f87-8c92-ddcce7f954c3 +md""" +Above, one can see that the original function is interpolated very accurately. The function `grididx_to_origcoord` transforms a linear index to a coordinate point $x$ in the original domain ($-10 \le x < 10$). + +In the following figure, we plot the normalized error vs. bond dimension, showing an exponential decay. +""" + +# ╔═╡ 1d102f52-822f-43a8-bad3-2416e8c79aa8 +let + # Plot error vs bond dimension obtained by prrLU + plt = plot(xlabel = "Bond dimension", ylabel = "Normalization error", + title = "normalized error vs. bond dimension") + semilogy!(1:length(ci.tci.pivoterrors), ci.tci.pivoterrors ./ ci.tci.maxsamplevalue, + marker = :x, ylim = (1e-8, 10), yticks = (10.0 .^ (-10:1:0)), legend = false) + plt +end + +# ╔═╡ 86a2c2ce-928e-4f21-9128-b33e143f1c2e +md""" +## Example 3 + +### Control the error of the TCI by a tolerance + +We interpolate the same function as in Example 2, but this time we use a tolerance to control the error of the TCI. The tolerance is a positive number that determines the maximum error of the TCI, which is scaled by an estimate of the abosolute maximum of the function. +The TCI algorithm will adaptively increase the bond dimension until the error is below the tolerance.""" + +# ╔═╡ fb59b981-e550-4763-8c2e-5797c383aeb4 +begin + tol = 1e-8 # Tolerance for the error + + # Convert to quantics format and sweep + ci_tol, ranks_tol, errors_tol = quanticscrossinterpolate( + Float64, oscillation_fn, qgrid; + tolerance = tol, + normalizeerror = true, # Normalize the error by the maximum sample value, + verbosity = 1, loginterval = 1 # Log the error every `loginterval` iterations + ) +end + +# ╔═╡ 6c7ac83c-f9e1-4999-883e-913e37f0d03d +println("Max abs sampled value is $(ci_tol.tci.maxsamplevalue)") + +# ╔═╡ ab49ca09-97d3-4223-891f-966df3309739 +errors_tol ./ ci_tol.tci.maxsamplevalue + +# ╔═╡ f4c3409b-a1c4-4db6-9c31-7559178c6b03 +md""" +### Estimate the error of the TCI +Wait! +Since we did not sample the function over the entire domain, we do not know the true error of the TCI. +In theory, we can estimate the error of the TCI by comparing the function values at the sampled points with the TCI values at the same points. +But, it is not practical to compare the function values with the TCI values at all points in the domain. +The function `estimatetrueerror` in `TensorCrossInterpolation.jl` provides a good estimate of the error of the TCI. +The algorithm finds indices (points) where the error is large by a randomized global search algorithm starting with a set of random initial points.""" + +# ╔═╡ 4c6c4cc7-7240-45cc-9500-51a28da3a690 +pivoterror_global = TCI.estimatetrueerror( + TCI.TensorTrain(ci.tci), ci.quanticsfunction; nsearch = 100) # Results are sorted in descending order of the error + +# ╔═╡ d64ff0f1-794d-4e53-878a-45b81a8707aa +md""" +Now, you can see the error estimate of the TCI is below the tolerance of $10^{-8}$ (or close to it).""" + +# ╔═╡ 8945822d-c359-4ed2-a56a-8e08ae38aaf9 +begin + println("The largest error found is $(pivoterror_global[1][2]) and the corresponding pivot is $(pivoterror_global[1][1]).") + println("The tolerance used is $(tol * ci_tol.tci.maxsamplevalue).") +end + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +QuanticsGrids = "634c7f73-3e90-4749-a1bd-001b8efc642d" +QuanticsTCI = "b11687fd-3a1c-4c41-97d0-998ab401d50e" +TensorCrossInterpolation = "b261b2ec-6378-4871-b32e-9173bb050604" + +[compat] +Plots = "~1.40.8" +QuanticsGrids = "~0.3.2" +QuanticsTCI = "~0.7.0" +TensorCrossInterpolation = "~0.9.12" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.5" +manifest_format = "2.0" +project_hash = "d534049ad302d8463279b933d07c8ff6b6a75970" + +[[deps.Adapt]] +deps = ["LinearAlgebra", 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diff --git a/pluto_notebooks/quantics1d_advanced.jl b/pluto_notebooks/quantics1d_advanced.jl new file mode 100644 index 0000000..be2b3b4 --- /dev/null +++ b/pluto_notebooks/quantics1d_advanced.jl @@ -0,0 +1,1433 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# ╔═╡ 84193dab-f772-447b-8c8a-bc46d5a2d71d +begin + import QuanticsGrids as QG + import TensorCrossInterpolation as TCI + + using Plots +end + +# ╔═╡ d4733947-56dd-43c4-8938-266b01769e1c +md""" +To run this notebook quantics1d_advanced.jl locally, copy and paste the following into your Julia REPL: + +```julia +using Pkg; Pkg.activate(temp=true); Pkg.add("Pluto") +BASE_URL = "https://raw.githubusercontent.com/tensor4all/T4APlutoExamples/refs/heads/main/pluto_notebooks/" +notebook = "quantics1d_advanced.jl" +url = joinpath(BASE_URL, notebook) +using Pluto; Pluto.run(notebook=download(url)) +``` +""" + +# ╔═╡ 7f6d4f2d-18cb-4b15-98c2-285840155b76 +md""" +# Quantics TCI of univariate funciton (advanced topics) +""" + +# ╔═╡ 066d2ac8-a9a1-4d98-a31e-bab5566d1958 +md""" +## Example 1 (continuation) + +### Performance tips + +Let's recall again the function $f(x)$ from the [previous page](./quantics1d.ipynb). + +```math +f(x) = \cos\left(\frac{x}{B}\right) \cos\left(\frac{x}{4\sqrt{5}B}\right) e^{-x^2} + 2e^{-x}, +``` + +where $B = 2^{-30}$. + +In the [previous page](./quantics1d.ipynb), we implemented $f(x)$ as +""" + +# ╔═╡ c9d45bbc-c23f-427e-8bf8-5ea51d539d23 +md""" +```julia +B = 2^(-30) # global variable +# This is simple, but not recommended +function f(x) + return cos(x/B) * cos(x/(4*sqrt(5)*B)) * exp(-x^2) + 2 * exp(-x) +end +``` +""" + +# ╔═╡ 9a065cdc-5cfd-4b24-a529-20422628b47b +md""" +However, since the implementation treats `B` as a untyped global variables, Julia compiler won't generate efficient code. See [Performance tips](https://docs.julialang.org/en/v1/manual/performance-tips/#Avoid-untyped-global-variables) at the official Julia documentatoin to learn more. In Julia, this can be written as below: +""" + +# ╔═╡ f3653d4d-283f-4758-8b57-90e34a6ed53c +begin + # Define callable struct + Base.@kwdef struct Ritter2024 <: Function + B::Float64 = 2^(-30) + end + + # Make Ritter2024 be "callable" object. + function (obj::Ritter2024)(x) + B = obj.B + return cos(x / B) * cos(x / (4 * sqrt(5) * B)) * exp(-x^2) + 2 * exp(-x) + end + + f = Ritter2024() + nothing +end + +# ╔═╡ 9fe50058-6f68-4d63-93e2-0ee209cf3f0f +md""" +Such \"callable\" objects are sometimes called \"functors.\" See [Function like objects](https://docs.julialang.org/en/v1/manual/methods/#Function-like-objects) at the official Julia documentation to learn more. +""" + +# ╔═╡ 0c9eac91-99c7-4613-ba04-99a8bb512ae6 +md""" +### The inside work of `quanticscrossinterpolate` + +Let us see the inside work of the `quanticscrossinterpolate` function in `QuanticsTCI.jl` by implementing the previous example using `QuanticsGrids.jl` and `TensorCrossInterpolation.jl`. + +We first define a grid as in the previous example: +""" + +# ╔═╡ 36063722-609e-4acd-8340-5d86a4397ec2 +begin + R = 40 # number of bits + xmin = 0.0 + xmax = log(20) + qgrid = QG.DiscretizedGrid{1}(R, xmin, xmax; includeendpoint = false) +end + +# ╔═╡ 44353d0b-ee68-4d17-bc95-5ae002c6c0da +md""" +We now define a function that takes a quantics index, which is named `qf`. +""" + +# ╔═╡ c9490dba-9814-4354-8d18-630dcb2d006e +begin + localdims = fill(2, R) # Sizes of local indices + qf(q) = f(QG.quantics_to_origcoord(qgrid, q)) + cf = TCI.CachedFunction{Float64}(qf, localdims) +end + +# ╔═╡ 18b5d059-98ee-42d8-b6e4-6348a857ca8e +md""" +Here, we've defined a function `qf` that accepts quantics `q` as an argument. Then we wrap `qf` using `TCI.CachedFunction`. +The function `QuanticsGrids.quantics_to_origcoord` converts a quantics index to a point (in this example, of type Float64) in the original coordinate system. +`TCI.CachedFunction` caches function evaluations; +`cf` caches the pair of input and output that are used during constructing a QTT representation. +This is useful if a function evaluation takes more than 100 ns or if you want to record function evaluations. + +Choosing good initial pivots is critical for numerical stability. +In the following code, we generate initial pivots by finding local maxima of $|f(x)|$. The function `optfirstpivot` maximizes the given function `qf` using single-index updates (zero-temperature Monte Carlo). +""" + +# ╔═╡ cb628558-99dc-4fb3-b636-e939fa1895f6 +begin + nrandominitpivot = 5 + + # random initial pivot + initialpivots = [TCI.optfirstpivot(qf, localdims, [rand(1:d) for d in localdims]) + for _ in 1:nrandominitpivot] + + qf.(initialpivots) # Function values at initial pivots +end + +# ╔═╡ a7fe45e4-81ea-4e0d-95c9-e8f170a85df2 +md""" +We are ready to construct a QTT representation of `cf` via `TCI.crossinterpolate2`: +""" + +# ╔═╡ 87cbea37-c332-497c-a73c-31f73f510a88 +ci, ranks, errors = TCI.crossinterpolate2( + Float64, cf, localdims, initialpivots; maxbonddim = 15) + +# ╔═╡ 9cf65a3f-3822-4592-ad76-c0d16f167593 +md""" +The integral of $f(x)$ can be computed by summing all the elements in the QTT representation and multiplying by the interval length divided by $2^\mathcal{R}$.""" + +# ╔═╡ b4c129b6-d233-41ce-ac83-81083db8ece8 +TCI.sum(ci) * (log(20) - 0) / 2^R, 19 / 10 + +# ╔═╡ a9136ba7-83b0-4b0d-9961-1f71e57b1ae7 +md""" +You can retrieve the results of the function evaluations during the TCI construction as follows. +""" + +# ╔═╡ c3fd86f9-4ca4-412a-b302-d37945f620fc +begin + cache = TCI.cachedata(cf) # Dict{Vector{Int},Float64} + + _qs = collect(keys(cache)) # quantics indices + _xs = QG.quantics_to_origcoord.(Ref(qgrid), _qs) # original coordinates + xs_evaluated = sort(_xs) +end + +# ╔═╡ 0fa89b30-4171-4ec7-80a3-09ac26084063 +md""" +The object `xs_evaluated` stores points what we wanted to know. + +We expect `ci::TensorCrossInterpolation.TensorCI2{Float64}` approximates `cf`(and `f`): +Just in case, we will check `qf` approximates `f` and `qf` and `cf` functions output the same result: +""" + +# ╔═╡ 2fc85737-cb8b-45d9-92a3-5d3316ecf7a8 +begin + x = 0.2 + # convert `x` in the original coordinate system to the corresponding `q` of quantics + q = QG.origcoord_to_quantics(qgrid, x) + println("f(x) = $(f(x)), qf(q) = $(qf(q)), cf(q) = $(cf(q)), ci(q) = $(ci(q))") +end + +# ╔═╡ bb62d10a-0d23-464f-891e-cd5e22db900f +@show length(TCI.cachedata(cf)) + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +QuanticsGrids = "634c7f73-3e90-4749-a1bd-001b8efc642d" +TensorCrossInterpolation = "b261b2ec-6378-4871-b32e-9173bb050604" + +[compat] +Plots = "~1.40.8" +QuanticsGrids = "~0.3.2" +TensorCrossInterpolation = "~0.9.12" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.10.5" +manifest_format = "2.0" +project_hash = "b97e5a5f39ec0933ef29369781a2464f4bbe73f8" + +[[deps.Adapt]] +deps = ["LinearAlgebra", "Requires"] +git-tree-sha1 = "6a55b747d1812e699320963ffde36f1ebdda4099" +uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e" +version = "4.0.4" +weakdeps = ["StaticArrays"] + + [deps.Adapt.extensions] + AdaptStaticArraysExt = 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\ No newline at end of file diff --git a/pluto_notebooks/quantics2d.jl b/pluto_notebooks/quantics2d.jl new file mode 100644 index 0000000..1fd0e88 --- /dev/null +++ b/pluto_notebooks/quantics2d.jl @@ -0,0 +1,1452 @@ +### A Pluto.jl notebook ### +# v0.19.46 + +using Markdown +using InteractiveUtils + +# ╔═╡ d85608d8-93ff-4875-b187-15467bb35765 +begin + using Printf + + using LaTeXStrings + using Plots + gr() # Use GR backend for plotting + + import TensorCrossInterpolation as TCI + import QuanticsGrids: DiscretizedGrid, origcoord_to_quantics, origcoord_to_grididx, + grididx_to_origcoord + using QuanticsTCI +end + +# ╔═╡ 13ed9ed3-9979-4e5d-a73e-334a01631242 +md""" +```julia +using Pkg; Pkg.activate(temp=true); Pkg.add("Pluto") +BASE_URL = "https://raw.githubusercontent.com/tensor4all/T4APlutoExamples/refs/heads/main/pluto_notebooks/" +notebook = "quantics2d.jl" +url = joinpath(BASE_URL, notebook) +using Pluto; Pluto.run(notebook=download(url)) +``` +""" + +# ╔═╡ 5412cb9c-a4c6-47be-be1d-dd8e269925da +md""" +# Quantics TCI of multivariate function +""" + +# ╔═╡ 404dabef-0813-4623-84d3-1fe8dfa4b481 +md""" +## Artificial function with widely different length scales + +As in the univariate case, we first demonstrate the multivariate case on an artificial function with widely different length scales: +""" + +# ╔═╡ d3c8602d-9093-4944-98c4-dbd6789d2957 +function f(x, y) + (exp(-0.4 * (x^2 + y^2)) + 1 + sin(x * y) * exp(-x^2) + + cos(3 * x * y) * exp(-y^2) + cos(x + y)) + + 0.05 * cos(1 / 0.001 * (0.2 * x - 0.4 * y)) + + 0.0005 * cos(1 / 0.0001 * (-0.2 * x + 0.7 * y)) + + 1e-5 * cos(1 / 1e-7 * (20 * x)) +end + +# ╔═╡ 6189c4b4-c513-42f1-a564-3195dcec4564 +md""" +To construct a 2D quantics grid, put `2` in the type parameter of `DiscretizedGrid`, and use tuples to specify lower and upper limits in each dimension:""" + +# ╔═╡ a4b7df6c-c912-43e5-ab9b-76159e3add9d +begin + R = 40 + grid = DiscretizedGrid{2}(R, (-5, -5), (5, 5)) +end + +# ╔═╡ a7247226-a54c-4769-a101-e31283b7e4a8 +md""" +To illustrate the different length scales, we show a series of progressively smaller parts of the function domain in the following.""" + +# ╔═╡ de55a1e0-66f3-4b8b-a227-17562e65e2f0 +begin + function myplotheatmap!( + plt, f, xlim::Tuple, ylim::Tuple; xlim_box = nothing, ylim_box = nothing) + x = LinRange(xlim..., 400) + y = LinRange(ylim..., 400) + s = heatmap!(plt, x, y, f) + + if !isnothing(xlim_box) && !isnothing(ylim_box) + plot!( + plt, + [xlim_box[1], xlim_box[2], xlim_box[2], xlim_box[1], xlim_box[1]], + [ylim_box[1], ylim_box[1], ylim_box[2], ylim_box[2], ylim_box[1]], + color = "lightgreen", lw = 2, label = "" + ) + end + xlabel!(L"$x$") + ylabel!(L"$y$") + plt + end + + function myplotheatmap( + func, xlim::Tuple, ylim::Tuple; xlim_box = nothing, ylim_box = nothing) + plt = plot(xlim = xlim, ylim = ylim, aspect_ratio = :equal, + xlabel = L"$x$", ylabel = L"$y$") + myplotheatmap!(plt, func, xlim, ylim; xlim_box = xlim_box, ylim_box = ylim_box) + end +end + +# ╔═╡ da85bc0e-6162-4eac-9169-08a9f2855c3c +myplotheatmap(f, (-5, 5), (-5, 5), xlim_box = (0.25, 1.75), ylim_box = (1.25, 2.75)) + +# ╔═╡ 968836b2-a1a2-4130-b7f6-d459aaa27c05 +myplotheatmap(f, (0.25, 1.75), (1.25, 2.75), xlim_box = (0.94, 1.0), ylim_box = (1.84, 1.9)) + +# ╔═╡ 84fab71c-4d08-46e9-8f45-0654c28664b9 +myplotheatmap(f, (0.94, 1.0), (1.84, 1.9), xlim_box = (0.97, 0.97 + 1e-7), + ylim_box = (1.88, 1.88 + 1e-7)) + +# ╔═╡ 93d8a0c4-86df-4009-93e9-42baa11e0c14 +let + xs = LinRange(0.97, 0.97 + 1e-7, 400) + ys = LinRange(1.88, 1.88 + 1e-7, 400) + + (hm, cb) = let + # currently GR backend does not support colorbar_ticks + # https://github.com/JuliaPlots/Plots.jl/issues/3560 + # we manually create a colorbar using with vertical heatmap. + n = 100 + colors = cgrad(:inferno, n, categorical = false) + hm = heatmap( + xs, ys, f, aspect_ratio = :equal, + color = colors, colorbar = false, + xticks = ([0.97, 0.97 + 0.9e-7], ["0.97", "0.97+1e-7"]), + yticks = ([1.88, 1.88 + 1e-7], ["1.88", "1.88" * "\n" * "+1e-7"]) + ) + + m = minimum(f.(xs, ys')) + M = maximum(f.(xs, ys')) + _yy = _xx = range(0, 1, n) + cb = heatmap( + _xx, _yy, (x, y) -> y, + ticks = false, + ratio = 20, + legend = false, + fillcolor = colors, + lims = (0, 1), + framestyle = :box + ) + mstr = @sprintf "%.7f" m + Mstr = @sprintf "%.7f" M + annotate!(cb, 3, 0, text(mstr, 8)) + annotate!(cb, 3, 1, text(Mstr, 8)) + hm, cb + end + + plot(hm, cb) +end + +# ╔═╡ d9a8e202-2f32-497c-a977-51cfbb538e1d +md""" +We can now obtain a QTT for `f` in the same way as in the 1D case:""" + +# ╔═╡ 2b34bc61-05a3-413c-9746-e7e67076ee13 +begin + # Construct 2D quantics + qtci, ranks, errors = quanticscrossinterpolate(Float64, f, grid) + χ = 1:length(qtci.tci.pivoterrors) + plot(χ, nextfloat(0.0) .+ qtci.tci.pivoterrors ./ qtci.tci.maxsamplevalue, + xlabel = L"\chi", ylabel = "Normalized error", yscale = :log10, + ylims = (1e-8, 1e1), yticks = (10.0 .^ (-8:1:1)), legend = false) +end + +# ╔═╡ 8fbee7e2-15f7-4f2f-8229-f678a71c7eed +md""" +Checking the error on the same slices as before, we see that the approximation is accurate everywhere:""" + +# ╔═╡ d68328cd-ce39-4c55-b89a-b6d45f9a90e3 +# Function that evaluates log10 of the interplation error at (x, y) +function errflog10(x, y) + i = origcoord_to_grididx(grid, (x, y)) + log10(nextfloat(0.0) + abs(f(grididx_to_origcoord(grid, i)...) - qtci(i))) +end + +# ╔═╡ 2e41e949-5cb2-42f2-afea-09d1737cf708 +let + ε = 1e-10 + myplotheatmap(errflog10, (-5, 5 - ε), (-5, 5 - ε), + xlim_box = (0.25, 1.75), ylim_box = (1.25, 2.75)) +end + +# ╔═╡ 04d603a0-e42f-439d-81cf-e7d3ce68ab16 +myplotheatmap( + errflog10, (0.25, 1.75), (1.25, 2.75), xlim_box = (0.94, 1.0), ylim_box = (1.84, 1.9)) + +# ╔═╡ f3fa3855-58cf-4b84-aa89-807dec97524b +myplotheatmap(errflog10, (0.94, 1.0), (1.84, 1.9), + xlim_box = (0.97, 0.97 + 1e-7), ylim_box = (1.88, 1.88 + 1e-7)) + +# ╔═╡ 7fb33773-e2c9-45e1-8615-2183a38eb077 +let + xs = LinRange(0.97, 0.97 + 1e-7, 400) + ys = 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