GM3D/2212.12372 H5C check.ipynb

## 2212.12372 H5C check.ipynb
{
  "cells": [
    {
      "metadata": {
        "trusted": true
      },
      "id": "7f622568",
      "cell_type": "code",
      "source": "using PolynomialRings",
      "execution_count": 5,
      "outputs": []
    },
    {
      "metadata": {
        "trusted": true
      },
      "id": "1dfb9214",
      "cell_type": "code",
      "source": "# since all the quantities in the hamlitonian are commuting, we can do the caluculation in a commutative polynomial ring.\n\nR = @ring! ℚ[σ₁, σ₂, σ₃, σ₄, σ₅, σ₆, σ₇, σ₈, σ₉, σ₁₀]\npaulis = (σ₁, σ₂, σ₃, σ₄, σ₅, σ₆, σ₇, σ₈, σ₉, σ₁₀)",
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 8,
          "data": {
            "text/plain": "(σ₁, σ₂, σ₃, σ₄, σ₅, σ₆, σ₇, σ₈, σ₉, σ₁₀)"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "# this function renders all squares of paulis into 1.\n\nfunction square_of_pauli_is_one(f)\n    for z in (σ₁, σ₂, σ₃, σ₄, σ₅, σ₆, σ₇, σ₈, σ₉, σ₁₀)\n        f = divrem(f, z^2-1)[2]\n    end\n    return f\nend\n\nsquare_of_pauli_is_one(σ₇^4 + 1)\n# x^4 + 1 = x^4 - 1 + 2 = (x^2 - 1) * (x^2 + 1) + 2 = 2 mod (x^2 - 1)",
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 9,
          "data": {
            "text/plain": "2//1"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "id": "cd782540",
      "cell_type": "code",
      "source": "# n = 3 case gives the correct result.\nx1, x2, x3 = (σ₁-1)/2, (σ₂-1)/2, (σ₃-1)/2\nh31 = (x1 -4*x2 -3*x3)^2\nh32 = (-2*x1 +x2 + 2*x3 + 4)^2\nh33 = (2*x1 + 2*x2 + 4)^2\nh34 = (3*x1 -2*x2 +4*x3 + 2)^2\nHC30 = h31 + h32 + h33 + h34\nHC3 = square_of_pauli_is_one(HC30)\n# HM3 matches with (S51)",
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 12,
          "data": {
            "text/plain": "-4//1*σ₁*σ₂ + 5//2*σ₁*σ₃ + 3//1*σ₂*σ₃ + -3//2*σ₁ + -7//2*σ₂ + -4//1*σ₃ + 87//2"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "id": "0c3169a8",
      "cell_type": "code",
      "source": "# n = 5.\n# definition of x_i is read from table S2.\nx1, x2, x3, x4, x5 = (1-σ₁)/2, (σ₂-1)/2, (σ₃-1)/2, (σ₄-1)/2, (σ₅-1)/2",
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 13,
          "data": {
            "text/plain": "(-1//2*σ₁ + 1//2, 1//2*σ₂ + -1//2, 1//2*σ₃ + -1//2, 1//2*σ₄ + -1//2, 1//2*σ₅ + -1//2)"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "id": "0375efbb",
      "cell_type": "code",
      "source": "h51 = (6x1 -8x2 + 2x3 -4x4 -4x5 + 2)^2\nh52 = (-4x1 -3x2 + 11x3 -5x4 -3x5 +4)^2\nh53 = (6x1 + 6x2 + 3x3 -0*x4 -3x5 + 9)^2\nh54 = (4x1 -2x2 +0*x3 +12x4 + 4x5 + 8)^2\nh55 = (-2x1 + 2x2 -6x3 -2x4 +x5)^2\nh56 = (-3x1 + 5x2 -3x3 + 4x4 -17x5 + 8)^2",
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 14,
          "data": {
            "text/plain": "9//4*σ₁^2 + 15//2*σ₁*σ₂ + 25//4*σ₂^2 + -9//2*σ₁*σ₃ + -15//2*σ₂*σ₃ + 9//4*σ₃^2 + 6//1*σ₁*σ₄ + 10//1*σ₂*σ₄ + -6//1*σ₃*σ₄ + 4//1*σ₄^2 + -51//2*σ₁*σ₅ + -85//2*σ₂*σ₅ + 51//2*σ₃*σ₅ + -34//1*σ₄*σ₅ + 289//4*σ₅^2 + 36//1*σ₁ + 60//1*σ₂ + -36//1*σ₃ + 48//1*σ₄ + -204//1*σ₅ + 144//1"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "id": "e5abee6b",
      "cell_type": "code",
      "source": "HC50 = h51 + h52 + h53 + h54 + h55 + h56\nHC5 = square_of_pauli_is_one(HC50)",
      "execution_count": 15,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 15,
          "data": {
            "text/plain": "27//2*σ₁*σ₂ + -7//2*σ₁*σ₃ + -29//1*σ₂*σ₃ + -18//1*σ₁*σ₄ + 39//2*σ₂*σ₄ + -63//2*σ₃*σ₄ + -35//2*σ₁*σ₅ + -34//1*σ₂*σ₅ + -5//2*σ₃*σ₅ + 9//2*σ₄*σ₅ + -91//1*σ₁ + 9//1*σ₂ + 28//1*σ₃ + 23//1*σ₄ + -543//2*σ₅ + 630//1"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "# n = 10\nsigns = (1, 1, -1, -1, 1, -1, 1, -1, 1, -1)\nxs = signs.*[(1-z)/2 for z in paulis]\nD₁₀ = [0 0 3 0 0 0 3 0 -3 -3;0 0 2 0 4 -4 0 4 -2 4; -3 0 0 0 0 0 -3 0 0 0; 1 2 1 4 -4 -2 -2 0 -1 0; 2 0 2 -2 0 0 1 -1 0 4; 0 0 -3 -3 0 0 0 0 -3 3; -3 3 -1 0 1 2 1 2 -2 -1; 0 -2 0 1 2 -1 1 -3 3 -3; 0 -2 -2 0 -2 0 0 0 2 2; 2 -2 0 -2 0 2 -2 2 0 0; 0 -2 -2 0 1 3 1 -2 -2 -1]",
      "execution_count": 16,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 16,
          "data": {
            "text/plain": "11×10 Matrix{Int64}:\n  0   0   3   0   0   0   3   0  -3  -3\n  0   0   2   0   4  -4   0   4  -2   4\n -3   0   0   0   0   0  -3   0   0   0\n  1   2   1   4  -4  -2  -2   0  -1   0\n  2   0   2  -2   0   0   1  -1   0   4\n  0   0  -3  -3   0   0   0   0  -3   3\n -3   3  -1   0   1   2   1   2  -2  -1\n  0  -2   0   1   2  -1   1  -3   3  -3\n  0  -2  -2   0  -2   0   0   0   2   2\n  2  -2   0  -2   0   2  -2   2   0   0\n  0  -2  -2   0   1   3   1  -2  -2  -1"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "t̄₁₀ = [3 4 0 1 2 3 2 3 2 2 0]'",
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 17,
          "data": {
            "text/plain": "11×1 adjoint(::Matrix{Int64}) with eltype Int64:\n 3\n 4\n 0\n 1\n 2\n 3\n 2\n 3\n 2\n 2\n 0"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "s0 = D₁₀*xs + t̄₁₀",
      "execution_count": 18,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 18,
          "data": {
            "text/plain": "11×1 Matrix{@ring(ℚ[σ₁,σ₂,σ₃,σ₄,σ₅,σ₆,σ₇,σ₈,σ₉,σ₁₀])}:\n 3//2*σ₃ + -3//2*σ₇ + 3//2*σ₉ + -3//2*σ₁₀ + 3//1\n σ₃ + -2//1*σ₅ + -2//1*σ₆ + 2//1*σ₈ + σ₉ + 2//1*σ₁₀ + 2//1\n 3//2*σ₁ + 3//2*σ₇ + -3//1\n -1//2*σ₁ + -σ₂ + 1//2*σ₃ + 2//1*σ₄ + 2//1*σ₅ + -σ₆ + σ₇ + 1//2*σ₉ + -5//2\n -σ₁ + σ₃ + -σ₄ + -1//2*σ₇ + -1//2*σ₈ + 2//1*σ₁₀ + 2//1\n -3//2*σ₃ + -3//2*σ₄ + 3//2*σ₉ + 3//2*σ₁₀ + 3//1\n 3//2*σ₁ + -3//2*σ₂ + -1//2*σ₃ + -1//2*σ₅ + σ₆ + -1//2*σ₇ + σ₈ + σ₉ + -1//2*σ₁₀ + 1//1\n σ₂ + 1//2*σ₄ + -σ₅ + -1//2*σ₆ + -1//2*σ₇ + -3//2*σ₈ + -3//2*σ₉ + -3//2*σ₁₀ + 8//1\n σ₂ + -σ₃ + σ₅ + -σ₉ + σ₁₀ + 1//1\n -σ₁ + σ₂ + -σ₄ + σ₆ + σ₇ + σ₈\n σ₂ + -σ₃ + -1//2*σ₅ + 3//2*σ₆ + -1//2*σ₇ + -σ₈ + σ₉ + -1//2*σ₁₀"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "HC0_10 = sum([s*s for s in s0])",
      "execution_count": 19,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 19,
          "data": {
            "text/plain": "27//4*σ₁^2 + -11//2*σ₁*σ₂ + 29//4*σ₂^2 + -4//1*σ₁*σ₃ + -7//2*σ₂*σ₃ + 9//1*σ₃^2 + 2//1*σ₁*σ₄ + -5//1*σ₂*σ₄ + 9//2*σ₃*σ₄ + 17//2*σ₄^2 + -7//2*σ₁*σ₅ + -7//2*σ₂*σ₅ + -5//2*σ₃*σ₅ + 7//1*σ₄*σ₅ + 21//2*σ₅^2 + 2//1*σ₁*σ₆ + 3//1*σ₂*σ₆ + -9//1*σ₃*σ₆ + -13//2*σ₄*σ₆ + 5//2*σ₅*σ₆ + 19//2*σ₆^2 + σ₁*σ₇ + -1//2*σ₂*σ₇ + -3//1*σ₃*σ₇ + 5//2*σ₄*σ₇ + 6//1*σ₅*σ₇ + -2//1*σ₆*σ₇ + 15//2*σ₇^2 + 2//1*σ₁*σ₈ + -6//1*σ₂*σ₈ + 4//1*σ₃*σ₈ + -5//2*σ₄*σ₈ + -5//1*σ₅*σ₈ + -11//2*σ₆*σ₈ + 4//1*σ₇*σ₈ + 19//2*σ₈^2 + 5//2*σ₁*σ₉ + -7//1*σ₂*σ₉ + 3//2*σ₃*σ₉ + -4//1*σ₄*σ₉ + -3//1*σ₅*σ₉ + 3//2*σ₆*σ₉ + -4//1*σ₇*σ₉ + 17//2*σ₈*σ₉ + 11//1*σ₉^2 + -11//2*σ₁*σ₁₀ + -1//2*σ₂*σ₁₀ + -3//2*σ₃*σ₁₀ + -10//1*σ₄*σ₁₀ + -2//1*σ₅*σ₁₀ + -9//1*σ₆*σ₁₀ + 5//1*σ₇*σ₁₀ + 21//2*σ₈*σ₁₀ + 9//2*σ₉*σ₁₀ + 65//4*σ₁₀^2 + -15//2*σ₁ + 20//1*σ₂ + 5//2*σ₃ + -15//1*σ₄ + -33//1*σ₅ + -9//1*σ₆ + -34//1*σ₇ + -16//1*σ₈ + -9//2*σ₉ + -7//1*σ₁₀ + 429//4"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "HC_10 = square_of_pauli_is_one(HC0_10)",
      "execution_count": 20,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 20,
          "data": {
            "text/plain": "-11//2*σ₁*σ₂ + -4//1*σ₁*σ₃ + -7//2*σ₂*σ₃ + 2//1*σ₁*σ₄ + -5//1*σ₂*σ₄ + 9//2*σ₃*σ₄ + -7//2*σ₁*σ₅ + -7//2*σ₂*σ₅ + -5//2*σ₃*σ₅ + 7//1*σ₄*σ₅ + 2//1*σ₁*σ₆ + 3//1*σ₂*σ₆ + -9//1*σ₃*σ₆ + -13//2*σ₄*σ₆ + 5//2*σ₅*σ₆ + σ₁*σ₇ + -1//2*σ₂*σ₇ + -3//1*σ₃*σ₇ + 5//2*σ₄*σ₇ + 6//1*σ₅*σ₇ + -2//1*σ₆*σ₇ + 2//1*σ₁*σ₈ + -6//1*σ₂*σ₈ + 4//1*σ₃*σ₈ + -5//2*σ₄*σ₈ + -5//1*σ₅*σ₈ + -11//2*σ₆*σ₈ + 4//1*σ₇*σ₈ + 5//2*σ₁*σ₉ + -7//1*σ₂*σ₉ + 3//2*σ₃*σ₉ + -4//1*σ₄*σ₉ + -3//1*σ₅*σ₉ + 3//2*σ₆*σ₉ + -4//1*σ₇*σ₉ + 17//2*σ₈*σ₉ + -11//2*σ₁*σ₁₀ + -1//2*σ₂*σ₁₀ + -3//2*σ₃*σ₁₀ + -10//1*σ₄*σ₁₀ + -2//1*σ₅*σ₁₀ + -9//1*σ₆*σ₁₀ + 5//1*σ₇*σ₁₀ + 21//2*σ₈*σ₁₀ + 9//2*σ₉*σ₁₀ + -15//2*σ₁ + 20//1*σ₂ + 5//2*σ₃ + -15//1*σ₄ + -33//1*σ₅ + -9//1*σ₆ + -34//1*σ₇ + -16//1*σ₈ + -9//2*σ₉ + -7//1*σ₁₀ + 203//1"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "",
      "execution_count": null,
      "outputs": []
    }
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    "kernelspec": {
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      "display_name": "Julia 1.8.3",
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    "gist": {
      "id": "a401e5a09b543f11d54ca593e9a08ade",
      "data": {
        "description": "Documents/Papers/2212.12372 H5C check.ipynb",
        "public": true
      }
    },
    "_draft": {
      "nbviewer_url": "https://gist.github.com/a401e5a09b543f11d54ca593e9a08ade"
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	{
	"cells": [
	{
	"metadata": {
	"trusted": true
	},
	"id": "7f622568",
	"cell_type": "code",
	"source": "using PolynomialRings",
	"execution_count": 5,
	"outputs": []
	},
	{
	"metadata": {
	"trusted": true
	},
	"id": "1dfb9214",
	"cell_type": "code",
	"source": "# since all the quantities in the hamlitonian are commuting, we can do the caluculation in a commutative polynomial ring.\n\nR = @ring! ℚ[σ₁, σ₂, σ₃, σ₄, σ₅, σ₆, σ₇, σ₈, σ₉, σ₁₀]\npaulis = (σ₁, σ₂, σ₃, σ₄, σ₅, σ₆, σ₇, σ₈, σ₉, σ₁₀)",
	"execution_count": 8,
	"outputs": [
	{
	"output_type": "execute_result",
	"execution_count": 8,
	"data": {
	"text/plain": "(σ₁, σ₂, σ₃, σ₄, σ₅, σ₆, σ₇, σ₈, σ₉, σ₁₀)"
	},
	"metadata": {}
	}
	]
	},
	{
	"metadata": {
	"trusted": true
	},
	"cell_type": "code",
	"source": "# this function renders all squares of paulis into 1.\n\nfunction square_of_pauli_is_one(f)\n for z in (σ₁, σ₂, σ₃, σ₄, σ₅, σ₆, σ₇, σ₈, σ₉, σ₁₀)\n f = divrem(f, z^2-1)[2]\n end\n return f\nend\n\nsquare_of_pauli_is_one(σ₇^4 + 1)\n# x^4 + 1 = x^4 - 1 + 2 = (x^2 - 1) * (x^2 + 1) + 2 = 2 mod (x^2 - 1)",
	"execution_count": 9,
	"outputs": [
	{
	"output_type": "execute_result",
	"execution_count": 9,
	"data": {
	"text/plain": "2//1"
	},
	"metadata": {}
	}
	]
	},
	{
	"metadata": {
	"trusted": true
	},
	"id": "cd782540",
	"cell_type": "code",
	"source": "# n = 3 case gives the correct result.\nx1, x2, x3 = (σ₁-1)/2, (σ₂-1)/2, (σ₃-1)/2\nh31 = (x1 -4x2 -3x3)^2\nh32 = (-2x1 +x2 + 2x3 + 4)^2\nh33 = (2x1 + 2x2 + 4)^2\nh34 = (3x1 -2x2 +4*x3 + 2)^2\nHC30 = h31 + h32 + h33 + h34\nHC3 = square_of_pauli_is_one(HC30)\n# HM3 matches with (S51)",
	"execution_count": 12,
	"outputs": [
	{
	"output_type": "execute_result",
	"execution_count": 12,
	"data": {
	"text/plain": "-4//1σ₁σ₂ + 5//2σ₁σ₃ + 3//1σ₂σ₃ + -3//2σ₁ + -7//2σ₂ + -4//1*σ₃ + 87//2"
	},
	"metadata": {}
	}
	]
	},
	{
	"metadata": {
	"trusted": true
	},
	"id": "0c3169a8",
	"cell_type": "code",
	"source": "# n = 5.\n# definition of x_i is read from table S2.\nx1, x2, x3, x4, x5 = (1-σ₁)/2, (σ₂-1)/2, (σ₃-1)/2, (σ₄-1)/2, (σ₅-1)/2",
	"execution_count": 13,
	"outputs": [
	{
	"output_type": "execute_result",
	"execution_count": 13,
	"data": {
	"text/plain": "(-1//2σ₁ + 1//2, 1//2σ₂ + -1//2, 1//2σ₃ + -1//2, 1//2σ₄ + -1//2, 1//2*σ₅ + -1//2)"
	},
	"metadata": {}
	}
	]
	},
	{
	"metadata": {
	"trusted": true
	},
	"id": "0375efbb",
	"cell_type": "code",
	"source": "h51 = (6x1 -8x2 + 2x3 -4x4 -4x5 + 2)^2\nh52 = (-4x1 -3x2 + 11x3 -5x4 -3x5 +4)^2\nh53 = (6x1 + 6x2 + 3x3 -0x4 -3x5 + 9)^2\nh54 = (4x1 -2x2 +0x3 +12x4 + 4x5 + 8)^2\nh55 = (-2x1 + 2x2 -6x3 -2x4 +x5)^2\nh56 = (-3x1 + 5x2 -3x3 + 4x4 -17x5 + 8)^2",
	"execution_count": 14,
	"outputs": [
	{
	"output_type": "execute_result",
	"execution_count": 14,
	"data": {
	"text/plain": "9//4σ₁^2 + 15//2σ₁σ₂ + 25//4σ₂^2 + -9//2σ₁σ₃ + -15//2σ₂σ₃ + 9//4σ₃^2 + 6//1σ₁σ₄ + 10//1σ₂σ₄ + -6//1σ₃σ₄ + 4//1σ₄^2 + -51//2σ₁σ₅ + -85//2σ₂σ₅ + 51//2σ₃σ₅ + -34//1σ₄σ₅ + 289//4σ₅^2 + 36//1σ₁ + 60//1σ₂ + -36//1σ₃ + 48//1σ₄ + -204//1σ₅ + 144//1"
	},
	"metadata": {}
	}
	]
	},
	{
	"metadata": {
	"trusted": true
	},
	"id": "e5abee6b",
	"cell_type": "code",
	"source": "HC50 = h51 + h52 + h53 + h54 + h55 + h56\nHC5 = square_of_pauli_is_one(HC50)",
	"execution_count": 15,
	"outputs": [
	{
	"output_type": "execute_result",
	"execution_count": 15,
	"data": {
	"text/plain": "27//2σ₁σ₂ + -7//2σ₁σ₃ + -29//1σ₂σ₃ + -18//1σ₁σ₄ + 39//2σ₂σ₄ + -63//2σ₃σ₄ + -35//2σ₁σ₅ + -34//1σ₂σ₅ + -5//2σ₃σ₅ + 9//2σ₄σ₅ + -91//1σ₁ + 9//1σ₂ + 28//1σ₃ + 23//1σ₄ + -543//2*σ₅ + 630//1"
	},
	"metadata": {}
	}
	]
	},
	{
	"metadata": {
	"trusted": true
	},
	"cell_type": "code",
	"source": "# n = 10\nsigns = (1, 1, -1, -1, 1, -1, 1, -1, 1, -1)\nxs = signs.*[(1-z)/2 for z in paulis]\nD₁₀ = [0 0 3 0 0 0 3 0 -3 -3;0 0 2 0 4 -4 0 4 -2 4; -3 0 0 0 0 0 -3 0 0 0; 1 2 1 4 -4 -2 -2 0 -1 0; 2 0 2 -2 0 0 1 -1 0 4; 0 0 -3 -3 0 0 0 0 -3 3; -3 3 -1 0 1 2 1 2 -2 -1; 0 -2 0 1 2 -1 1 -3 3 -3; 0 -2 -2 0 -2 0 0 0 2 2; 2 -2 0 -2 0 2 -2 2 0 0; 0 -2 -2 0 1 3 1 -2 -2 -1]",
	"execution_count": 16,
	"outputs": [
	{
	"output_type": "execute_result",
	"execution_count": 16,
	"data": {
	"text/plain": "11×10 Matrix{Int64}:\n 0 0 3 0 0 0 3 0 -3 -3\n 0 0 2 0 4 -4 0 4 -2 4\n -3 0 0 0 0 0 -3 0 0 0\n 1 2 1 4 -4 -2 -2 0 -1 0\n 2 0 2 -2 0 0 1 -1 0 4\n 0 0 -3 -3 0 0 0 0 -3 3\n -3 3 -1 0 1 2 1 2 -2 -1\n 0 -2 0 1 2 -1 1 -3 3 -3\n 0 -2 -2 0 -2 0 0 0 2 2\n 2 -2 0 -2 0 2 -2 2 0 0\n 0 -2 -2 0 1 3 1 -2 -2 -1"
	},
	"metadata": {}
	}
	]
	},
	{
	"metadata": {
	"trusted": true
	},
	"cell_type": "code",
	"source": "t̄₁₀ = [3 4 0 1 2 3 2 3 2 2 0]'",
	"execution_count": 17,
	"outputs": [
	{
	"output_type": "execute_result",
	"execution_count": 17,
	"data": {
	"text/plain": "11×1 adjoint(::Matrix{Int64}) with eltype Int64:\n 3\n 4\n 0\n 1\n 2\n 3\n 2\n 3\n 2\n 2\n 0"
	},
	"metadata": {}
	}
	]
	},
	{
	"metadata": {
	"trusted": true
	},
	"cell_type": "code",
	"source": "s0 = D₁₀*xs + t̄₁₀",
	"execution_count": 18,
	"outputs": [
	{
	"output_type": "execute_result",
	"execution_count": 18,
	"data": {
	"text/plain": "11×1 Matrix{@ring(ℚ[σ₁,σ₂,σ₃,σ₄,σ₅,σ₆,σ₇,σ₈,σ₉,σ₁₀])}:\n 3//2σ₃ + -3//2σ₇ + 3//2σ₉ + -3//2σ₁₀ + 3//1\n σ₃ + -2//1σ₅ + -2//1σ₆ + 2//1σ₈ + σ₉ + 2//1σ₁₀ + 2//1\n 3//2σ₁ + 3//2σ₇ + -3//1\n -1//2σ₁ + -σ₂ + 1//2σ₃ + 2//1σ₄ + 2//1σ₅ + -σ₆ + σ₇ + 1//2σ₉ + -5//2\n -σ₁ + σ₃ + -σ₄ + -1//2σ₇ + -1//2σ₈ + 2//1σ₁₀ + 2//1\n -3//2σ₃ + -3//2σ₄ + 3//2σ₉ + 3//2σ₁₀ + 3//1\n 3//2σ₁ + -3//2σ₂ + -1//2σ₃ + -1//2σ₅ + σ₆ + -1//2σ₇ + σ₈ + σ₉ + -1//2σ₁₀ + 1//1\n σ₂ + 1//2σ₄ + -σ₅ + -1//2σ₆ + -1//2σ₇ + -3//2σ₈ + -3//2σ₉ + -3//2σ₁₀ + 8//1\n σ₂ + -σ₃ + σ₅ + -σ₉ + σ₁₀ + 1//1\n -σ₁ + σ₂ + -σ₄ + σ₆ + σ₇ + σ₈\n σ₂ + -σ₃ + -1//2σ₅ + 3//2σ₆ + -1//2σ₇ + -σ₈ + σ₉ + -1//2σ₁₀"
	},
	"metadata": {}
	}
	]
	},
	{
	"metadata": {
	"trusted": true
	},
	"cell_type": "code",
	"source": "HC0_10 = sum([s*s for s in s0])",
	"execution_count": 19,
	"outputs": [
	{
	"output_type": "execute_result",
	"execution_count": 19,
	"data": {
	"text/plain": "27//4σ₁^2 + -11//2σ₁σ₂ + 29//4σ₂^2 + -4//1σ₁σ₃ + -7//2σ₂σ₃ + 9//1σ₃^2 + 2//1σ₁σ₄ + -5//1σ₂σ₄ + 9//2σ₃σ₄ + 17//2σ₄^2 + -7//2σ₁σ₅ + -7//2σ₂σ₅ + -5//2σ₃σ₅ + 7//1σ₄σ₅ + 21//2σ₅^2 + 2//1σ₁σ₆ + 3//1σ₂σ₆ + -9//1σ₃σ₆ + -13//2σ₄σ₆ + 5//2σ₅σ₆ + 19//2σ₆^2 + σ₁σ₇ + -1//2σ₂σ₇ + -3//1σ₃σ₇ + 5//2σ₄σ₇ + 6//1σ₅σ₇ + -2//1σ₆σ₇ + 15//2σ₇^2 + 2//1σ₁σ₈ + -6//1σ₂σ₈ + 4//1σ₃σ₈ + -5//2σ₄σ₈ + -5//1σ₅σ₈ + -11//2σ₆σ₈ + 4//1σ₇σ₈ + 19//2σ₈^2 + 5//2σ₁σ₉ + -7//1σ₂σ₉ + 3//2σ₃σ₉ + -4//1σ₄σ₉ + -3//1σ₅σ₉ + 3//2σ₆σ₉ + -4//1σ₇σ₉ + 17//2σ₈σ₉ + 11//1σ₉^2 + -11//2σ₁σ₁₀ + -1//2σ₂σ₁₀ + -3//2σ₃σ₁₀ + -10//1σ₄σ₁₀ + -2//1σ₅σ₁₀ + -9//1σ₆σ₁₀ + 5//1σ₇σ₁₀ + 21//2σ₈σ₁₀ + 9//2σ₉σ₁₀ + 65//4σ₁₀^2 + -15//2σ₁ + 20//1σ₂ + 5//2σ₃ + -15//1σ₄ + -33//1σ₅ + -9//1σ₆ + -34//1σ₇ + -16//1σ₈ + -9//2σ₉ + -7//1*σ₁₀ + 429//4"
	},
	"metadata": {}
	}
	]
	},
	{
	"metadata": {
	"trusted": true
	},
	"cell_type": "code",
	"source": "HC_10 = square_of_pauli_is_one(HC0_10)",
	"execution_count": 20,
	"outputs": [
	{
	"output_type": "execute_result",
	"execution_count": 20,
	"data": {
	"text/plain": "-11//2σ₁σ₂ + -4//1σ₁σ₃ + -7//2σ₂σ₃ + 2//1σ₁σ₄ + -5//1σ₂σ₄ + 9//2σ₃σ₄ + -7//2σ₁σ₅ + -7//2σ₂σ₅ + -5//2σ₃σ₅ + 7//1σ₄σ₅ + 2//1σ₁σ₆ + 3//1σ₂σ₆ + -9//1σ₃σ₆ + -13//2σ₄σ₆ + 5//2σ₅σ₆ + σ₁σ₇ + -1//2σ₂σ₇ + -3//1σ₃σ₇ + 5//2σ₄σ₇ + 6//1σ₅σ₇ + -2//1σ₆σ₇ + 2//1σ₁σ₈ + -6//1σ₂σ₈ + 4//1σ₃σ₈ + -5//2σ₄σ₈ + -5//1σ₅σ₈ + -11//2σ₆σ₈ + 4//1σ₇σ₈ + 5//2σ₁σ₉ + -7//1σ₂σ₉ + 3//2σ₃σ₉ + -4//1σ₄σ₉ + -3//1σ₅σ₉ + 3//2σ₆σ₉ + -4//1σ₇σ₉ + 17//2σ₈σ₉ + -11//2σ₁σ₁₀ + -1//2σ₂σ₁₀ + -3//2σ₃σ₁₀ + -10//1σ₄σ₁₀ + -2//1σ₅σ₁₀ + -9//1σ₆σ₁₀ + 5//1σ₇σ₁₀ + 21//2σ₈σ₁₀ + 9//2σ₉σ₁₀ + -15//2σ₁ + 20//1σ₂ + 5//2σ₃ + -15//1σ₄ + -33//1σ₅ + -9//1σ₆ + -34//1σ₇ + -16//1σ₈ + -9//2σ₉ + -7//1*σ₁₀ + 203//1"
	},
	"metadata": {}
	}
	]
	},
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	"cell_type": "code",
	"source": "",
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