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Pluto notebook for finding a polynomial for a quantum circuit model
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| ### A Pluto.jl notebook ### | |
| # v0.19.9 | |
| using Markdown | |
| using InteractiveUtils | |
| # ╔═╡ 195eb152-17e7-11ed-32ab-ab200fb170af | |
| using Yao, YaoPlots, Polynomials | |
| # ╔═╡ f6f10336-205c-4581-9418-de6fdd9b6418 | |
| using Mods, Primes | |
| # ╔═╡ f84bc62b-dc1b-4682-afe3-be01537d1ef8 | |
| md"## Problem definition" | |
| # ╔═╡ 6cbbaf4a-804d-4a50-960d-0b9726359dcd | |
| function generate_circuit(n::Int, x::T; depth=2) where T | |
| c = chain(vcat([vcat( | |
| [control(n, i, i+1=>X) for i=1:n-1], # entangler | |
| [put(n, i=>matblock([one(T) one(T); one(T) x]; tag="var")) for i=1:n]) # gate [1 1; 1 x] on every qubit | |
| for k=1:depth]...)) | |
| end | |
| # ╔═╡ 26365575-9176-44b6-8526-4ebde3f5fc62 | |
| function circuit_run(n::Int, x::T) where T | |
| reg = ArrayReg(ones(T, 1<<n)) # a 10 qubit state initialized to ones. | |
| reg = apply(reg, generate_circuit(n, x)) | |
| return expect(repeat(n, X), reg) # get the expectation value of operator XXXXXX. | |
| end | |
| # ╔═╡ 729ffdc0-9bb5-4bb8-9b3c-83bc7517199a | |
| md"## Polynomial algebra" | |
| # ╔═╡ eecda948-45c8-4b23-bf5f-cf692d6212ad | |
| Polynomial([1, -1]) | |
| # ╔═╡ dc1130a2-1951-4404-b2da-a3aa358a1287 | |
| circuit2 = generate_circuit(5, Polynomial([0, 1])); | |
| # ╔═╡ b9b328c4-0c46-47f2-9654-077114d50a2e | |
| md"The matrix representation" | |
| # ╔═╡ 5bda2c47-81f8-4b34-ba2d-45c8aa5f96cc | |
| mat(Polynomial{Int64,:x}, circuit2) | |
| # ╔═╡ ebf613bf-0434-4f3d-b9fa-7e975d13551a | |
| # ╠═╡ show_logs = false | |
| circuit_run(5, Polynomial([0, 1])) | |
| # ╔═╡ 6330de9d-8a59-42af-80d1-c95b5d75139c | |
| md"## Finite Field algebra" | |
| # ╔═╡ 2fa2b501-fbed-474e-9cce-eebd8ab604a8 | |
| circuit = generate_circuit(5, Mod{17}(Int32(4))); | |
| # ╔═╡ 62664a48-8e73-4bbd-8d5c-512b602fa495 | |
| vizcircuit(circuit) | |
| # ╔═╡ b8dafabd-9f31-40c1-8b40-cf36a97da07e | |
| mat(Mod{17,Int32}, circuit) | |
| # ╔═╡ 41814391-a9bc-4969-80a9-14b2ec88d9fc | |
| begin | |
| # this calls into the Chinese remainder theorem | |
| function improved_counting(ys::AbstractArray...) | |
| map(yi->improved_counting(yi...), zip(ys...)) | |
| end | |
| improved_counting(ys::Mod...) = Mods.CRT(ys...) | |
| end | |
| # ╔═╡ 02c1b7c3-a613-43e9-9d2e-7b48f47ae724 | |
| # iterate over finite fields over largest several prime numbers until convergence | |
| # `f` is a function. Given an input data type `T`, `f(T)` returns the function value. | |
| # `TI` is an integer type as the data type. | |
| # `max_iter` is the number of prime numbers to evaluate. | |
| function big_integer_solve(f, ::Type{TI}, max_iter::Int=100) where TI | |
| N = typemax(TI) | |
| local res, respre, YS | |
| for k = 1:max_iter | |
| N = prevprime(N-TI(1)) | |
| @debug "iteration $k, computing on GP($(N)) ..." | |
| T = Mods.Mod{N,TI} | |
| rk = f(T) | |
| if max_iter==1 | |
| # if the iteration number if 1, return the value directly. | |
| return map(x->BigInt.(Mods.value.(x)), rk) # needs test | |
| end | |
| if k != 1 | |
| # call Chinese remainder theorem to improve the counting of target value | |
| push!.(YS, rk) | |
| res = map(x->improved_counting(x...), YS) | |
| all(respre .== res) && return res | |
| respre = res | |
| else # k=1 | |
| YS = reshape([Any[] for i=1:length(rk)], size(rk)) | |
| push!.(YS, rk) | |
| respre = map(x->BigInt.(value.(x)), rk) | |
| end | |
| end | |
| @warn "result is potentially inconsistent." | |
| return res | |
| end | |
| # ╔═╡ dc87507a-6586-4bc8-ac82-a8cdf6577a72 | |
| # `f` is a function that evaluates | |
| function polynomial_fit(f; maxorder, max_iter=100) | |
| result = big_integer_solve(Int32, max_iter) do T | |
| xs = 0:maxorder | |
| ys = [f(T(x)) for x in xs] # compute `f(x)` for different `x`s, on type `T`. | |
| res = fill(T[], size(ys[1])) # contraction result can be a tensor | |
| for ci in 1:length(ys[1]) | |
| A = zeros(T, maxorder+1, maxorder+1) | |
| for j=1:maxorder+1, i=1:maxorder+1 | |
| A[j,i] = T(xs[j])^(i-1) | |
| end | |
| # gaussian elimination | |
| res[ci] = A \ T.(getindex.(ys, Ref(ci))) | |
| end | |
| res | |
| end | |
| map(Polynomial, result) | |
| end | |
| # ╔═╡ 907b28e8-8a55-46b9-bd18-d5df1e646ba7 | |
| # ╠═╡ show_logs = false | |
| polynomial_fit(x->circuit_run(5, x); maxorder=14) | |
| # ╔═╡ e5f959ee-5f4f-4a52-a4a5-30e11abbb0e4 | |
| md"note: you need to decide the maximum order manually." | |
| # ╔═╡ 00000000-0000-0000-0000-000000000001 | |
| PLUTO_PROJECT_TOML_CONTENTS = """ | |
| [deps] | |
| Mods = "7475f97c-0381-53b1-977b-4c60186c8d62" | |
| Polynomials = "f27b6e38-b328-58d1-80ce-0feddd5e7a45" | |
| Primes = "27ebfcd6-29c5-5fa9-bf4b-fb8fc14df3ae" | |
| Yao = "5872b779-8223-5990-8dd0-5abbb0748c8c" | |
| YaoPlots = "32cfe2d9-419e-45f2-8191-2267705d8dbc" | |
| [compat] | |
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| YaoPlots = "~0.8.0" | |
| """ | |
| # ╔═╡ 00000000-0000-0000-0000-000000000002 | |
| PLUTO_MANIFEST_TOML_CONTENTS = """ | |
| # This file is machine-generated - editing it directly is not advised | |
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| uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3" | |
| version = "1.3.4+4" | |
| [[deps.Xorg_libXrender_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] | |
| git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96" | |
| uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa" | |
| version = "0.9.10+4" | |
| [[deps.Xorg_libpthread_stubs_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb" | |
| uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74" | |
| version = "0.1.0+3" | |
| [[deps.Xorg_libxcb_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"] | |
| git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6" | |
| uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b" | |
| version = "1.13.0+3" | |
| [[deps.Xorg_xtrans_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845" | |
| uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10" | |
| version = "1.4.0+3" | |
| [[deps.Yao]] | |
| deps = ["BitBasis", "LinearAlgebra", "LuxurySparse", "Reexport", "YaoAPI", "YaoArrayRegister", "YaoBlocks", "YaoSym"] | |
| git-tree-sha1 = "c23c27ba3186a6558ff8bc3f4b7bc16e5af76c1e" | |
| uuid = "5872b779-8223-5990-8dd0-5abbb0748c8c" | |
| version = "0.8.2" | |
| [[deps.YaoAPI]] | |
| git-tree-sha1 = "3244c75eb3fbae41e3a3fb622d1808da1ce83b78" | |
| uuid = "0843a435-28de-4971-9e8b-a9641b2983a8" | |
| version = "0.4.1" | |
| [[deps.YaoArrayRegister]] | |
| deps = ["Adapt", "BitBasis", "LegibleLambdas", "LinearAlgebra", "LuxurySparse", "MLStyle", "Random", "SparseArrays", "StaticArrays", "StatsBase", "TupleTools", "YaoAPI"] | |
| git-tree-sha1 = "81ef9b5f217d960c5663091176631de5b6c3f7ec" | |
| uuid = "e600142f-9330-5003-8abb-0ebd767abc51" | |
| version = "0.9.1" | |
| [[deps.YaoBlocks]] | |
| deps = ["BitBasis", "CacheServers", "ChainRulesCore", "ExponentialUtilities", "InteractiveUtils", "LegibleLambdas", "LinearAlgebra", "LuxurySparse", "MLStyle", "Random", "SparseArrays", "StaticArrays", "StatsBase", "TupleTools", "YaoAPI", "YaoArrayRegister"] | |
| git-tree-sha1 = "3a09ed20b1113f094d024e2f48f3789e5e22dd40" | |
| uuid = "418bc28f-b43b-5e0b-a6e7-61bbc1a2c1df" | |
| version = "0.13.2" | |
| [[deps.YaoPlots]] | |
| deps = ["Luxor", "Yao"] | |
| git-tree-sha1 = "a7a3a460d1bc9f8e28ef336f3d870a959851c530" | |
| uuid = "32cfe2d9-419e-45f2-8191-2267705d8dbc" | |
| version = "0.8.0" | |
| [[deps.YaoSym]] | |
| deps = ["BitBasis", "LinearAlgebra", "LuxurySparse", "Requires", "SparseArrays", "YaoArrayRegister", "YaoBlocks"] | |
| git-tree-sha1 = "3518fa5611d252971771f4f4c015341f04367c89" | |
| uuid = "3b27209a-d3d6-11e9-3c0f-41eb92b2cb9d" | |
| version = "0.6.0" | |
| [[deps.Zlib_jll]] | |
| deps = ["Libdl"] | |
| uuid = "83775a58-1f1d-513f-b197-d71354ab007a" | |
| version = "1.2.12+3" | |
| [[deps.Zstd_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927" | |
| uuid = "3161d3a3-bdf6-5164-811a-617609db77b4" | |
| version = "1.5.2+0" | |
| [[deps.gdk_pixbuf_jll]] | |
| deps = ["Artifacts", "Glib_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pkg", "Xorg_libX11_jll", "libpng_jll"] | |
| git-tree-sha1 = "c23323cd30d60941f8c68419a70905d9bdd92808" | |
| uuid = "da03df04-f53b-5353-a52f-6a8b0620ced0" | |
| version = "2.42.6+1" | |
| [[deps.libaom_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4" | |
| uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b" | |
| version = "3.4.0+0" | |
| [[deps.libass_jll]] | |
| deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] | |
| git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47" | |
| uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0" | |
| version = "0.15.1+0" | |
| [[deps.libblastrampoline_jll]] | |
| deps = ["Artifacts", "Libdl", "OpenBLAS_jll"] | |
| uuid = "8e850b90-86db-534c-a0d3-1478176c7d93" | |
| version = "5.1.1+0" | |
| [[deps.libfdk_aac_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55" | |
| uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" | |
| version = "2.0.2+0" | |
| [[deps.libpng_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] | |
| git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c" | |
| uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" | |
| version = "1.6.38+0" | |
| [[deps.libvorbis_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] | |
| git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" | |
| uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" | |
| version = "1.3.7+1" | |
| [[deps.nghttp2_jll]] | |
| deps = ["Artifacts", "Libdl"] | |
| uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" | |
| version = "1.47.0+0" | |
| [[deps.p7zip_jll]] | |
| deps = ["Artifacts", "Libdl"] | |
| uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" | |
| version = "17.4.0+0" | |
| [[deps.x264_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" | |
| uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" | |
| version = "2021.5.5+0" | |
| [[deps.x265_jll]] | |
| deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
| git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" | |
| uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" | |
| version = "3.5.0+0" | |
| """ | |
| # ╔═╡ Cell order: | |
| # ╠═195eb152-17e7-11ed-32ab-ab200fb170af | |
| # ╟─f84bc62b-dc1b-4682-afe3-be01537d1ef8 | |
| # ╠═26365575-9176-44b6-8526-4ebde3f5fc62 | |
| # ╠═6cbbaf4a-804d-4a50-960d-0b9726359dcd | |
| # ╟─729ffdc0-9bb5-4bb8-9b3c-83bc7517199a | |
| # ╠═62664a48-8e73-4bbd-8d5c-512b602fa495 | |
| # ╠═eecda948-45c8-4b23-bf5f-cf692d6212ad | |
| # ╠═dc1130a2-1951-4404-b2da-a3aa358a1287 | |
| # ╟─b9b328c4-0c46-47f2-9654-077114d50a2e | |
| # ╠═5bda2c47-81f8-4b34-ba2d-45c8aa5f96cc | |
| # ╠═ebf613bf-0434-4f3d-b9fa-7e975d13551a | |
| # ╟─6330de9d-8a59-42af-80d1-c95b5d75139c | |
| # ╠═f6f10336-205c-4581-9418-de6fdd9b6418 | |
| # ╠═2fa2b501-fbed-474e-9cce-eebd8ab604a8 | |
| # ╠═b8dafabd-9f31-40c1-8b40-cf36a97da07e | |
| # ╠═dc87507a-6586-4bc8-ac82-a8cdf6577a72 | |
| # ╠═02c1b7c3-a613-43e9-9d2e-7b48f47ae724 | |
| # ╠═41814391-a9bc-4969-80a9-14b2ec88d9fc | |
| # ╠═907b28e8-8a55-46b9-bd18-d5df1e646ba7 | |
| # ╟─e5f959ee-5f4f-4a52-a4a5-30e11abbb0e4 | |
| # ╟─00000000-0000-0000-0000-000000000001 | |
| # ╟─00000000-0000-0000-0000-000000000002 |
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