ldpc 0.7.0

A toolbox for classical and quantum LDPC codes.
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159

use crate::css::Css;
use crate::codes::LinearCode;
use sparse_bin_mat::{BinNum, SparseBinMat, SparseBinVec};

// This implement a variation of the method
// introduced in https://arxiv.org/abs/0903.5256
// to compute logical operator generators of a CSS
// code from the linear codes.
pub(super) fn from_linear_codes(x_code: &LinearCode, z_code: &LinearCode) -> Css<SparseBinMat> {
    Logicals::new(x_code, z_code).compute()
}

struct Logicals {
    raw_x_generators: Vec<SparseBinVec>,
    raw_z_generators: Vec<SparseBinVec>,
    x_logicals: Vec<Vec<usize>>,
    z_logicals: Vec<Vec<usize>>,
    length: usize,
}

impl Logicals {
    fn new(x_code: &LinearCode, z_code: &LinearCode) -> Self {
        Self {
            raw_x_generators: Self::to_generator_vec(z_code),
            raw_z_generators: Self::to_generator_vec(x_code),
            x_logicals: Vec::new(),
            z_logicals: Vec::new(),
            length: x_code.len(),
        }
    }

    fn to_generator_vec(code: &LinearCode) -> Vec<SparseBinVec> {
        code.generator_matrix()
            .rows()
            .map(|row| row.to_vec())
            .collect()
    }

    fn compute(mut self) -> Css<SparseBinMat> {
        while let Some(x_generator) = self.raw_x_generators.pop() {
            if let Some(z_generator) = self.find_anticommuting_z_generator(&x_generator) {
                self.update_remaining_generators(&x_generator, &z_generator);
                self.push_logicals(x_generator, z_generator);
            }
        }
        Css {
            x: SparseBinMat::new(self.length, self.x_logicals),
            z: SparseBinMat::new(self.length, self.z_logicals),
        }
    }

    fn anticommute(x_generator: &SparseBinVec, z_generator: &SparseBinVec) -> bool {
        x_generator.dot_with(z_generator).unwrap() == BinNum::one()
    }

    fn find_anticommuting_z_generator(
        &mut self,
        x_generator: &SparseBinVec,
    ) -> Option<SparseBinVec> {
        self.raw_z_generators
            .iter()
            .position(|z_generator| Self::anticommute(x_generator, z_generator))
            .map(|position| self.raw_z_generators.swap_remove(position))
    }

    fn update_remaining_generators(
        &mut self,
        x_generator: &SparseBinVec,
        z_generator: &SparseBinVec,
    ) {
        self.raw_z_generators
            .iter_mut()
            .filter(|gen| Self::anticommute(x_generator, gen))
            .for_each(|gen| *gen = z_generator + gen);
        self.raw_x_generators
            .iter_mut()
            .filter(|gen| Self::anticommute(gen, z_generator))
            .for_each(|gen| *gen = x_generator + gen);
    }

    fn push_logicals(&mut self, x_generator: SparseBinVec, z_generator: SparseBinVec) {
        self.x_logicals.push(x_generator.to_positions_vec());
        self.z_logicals.push(z_generator.to_positions_vec());
    }
}

#[cfg(test)]
mod test {
    use super::*;
    use rand::thread_rng;

    #[test]
    fn steane_code() {
        let hamming = LinearCode::hamming_code();
        let logicals = from_linear_codes(&hamming, &hamming);
        assert_commutations(
            logicals,
            Css {
                x: hamming.parity_check_matrix(),
                z: hamming.parity_check_matrix(),
            },
        )
    }

    #[test]
    fn shor_code() {
        let x_code = LinearCode::from_parity_check_matrix(SparseBinMat::new(
            9,
            vec![
                vec![0, 1],
                vec![1, 2],
                vec![3, 4],
                vec![4, 5],
                vec![6, 7],
                vec![7, 8],
            ],
        ));
        let z_code = LinearCode::from_parity_check_matrix(SparseBinMat::new(
            9,
            vec![vec![0, 1, 2, 3, 4, 5], vec![3, 4, 5, 6, 7, 8]],
        ));
        let logicals = from_linear_codes(&x_code, &z_code);
        assert_commutations(logicals, Css {x: x_code.parity_check_matrix(), z: z_code.parity_check_matrix()});
    }

    #[test]
    fn random_hypergraph_product() {
        let code = LinearCode::random_regular_code()
            .num_bits(25)
            .num_checks(15)
            .bit_degree(3)
            .check_degree(5)
            .sample_with(&mut thread_rng())
            .unwrap();
        let logicals = from_linear_codes(&code, &code);
        assert_commutations(logicals, Css {x: code.parity_check_matrix(), z: code.parity_check_matrix()});
    }

    fn assert_commutations(logicals: Css<SparseBinMat>, par_matrices: Css<&SparseBinMat>) {
        assert_logicals_commute_with_stabilizers(&logicals.x, par_matrices.x);
        assert_logicals_commute_with_stabilizers(&logicals.z, par_matrices.z);
        assert_anticommuting_logical_pairs(&logicals.x, &logicals.z);
    }

    fn assert_logicals_commute_with_stabilizers(
        logicals: &SparseBinMat,
        stabilizers: &SparseBinMat,
    ) -> bool {
        (logicals * &stabilizers.transposed()).is_zero()
    }

    fn assert_anticommuting_logical_pairs(
        x_logicals: &SparseBinMat,
        z_logicals: &SparseBinMat,
    ) -> bool {
        (x_logicals * &z_logicals.transposed())
            == SparseBinMat::identity(x_logicals.number_of_rows())
    }
}