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3 Description of the algorithm
 3.1 Elementary version
 3.2 Some more details
 3.3 Empirical estimate of the success probability

3 Description of the algorithm

3.1 Elementary version

3.1-1 What it does?

In the simplest possible terms, we are given a pair of matrices P and Q with orthogonal rows, PQ^T=0. The matrices have entries in a finite field F=\mathop{\rm GF}(q), where q is a power of a prime. The goal is to find the smallest weight of a non-zero vector c over the same field F, such that c be orthogonal with the rows of P, Pc^T=0, and linearly independent from the rows of Q.

3.1-2 The algorithm

We first construct a generator matrix G whose rows form a basis of the F-linear space of all vectors orthogonal to the rows of P. At each step, a random permutation S is generated and applied to the columns of G. Then, Gauss' elimination with back substitution renders the resulting matrix to the reduced row echelon form, after which the inverse permutation S^{-1} is applied to the columns. Rows of the resulting matrix G_S that are linearly independent from the rows of Q are considered as candidates for the minimum weight vectors. Thus, after N steps, we are getting an upper bound on the distance which is improving with increasing N.

3.1-3 Intuition

The intuition is that each row of G_S is guaranteed to contain at least rank(G_S)-1 zeros. Thus, we are sampling mostly lower-weight vectors from the linear space orthogonal to the rows of P. Further, it is easy to see that any vector obtained this way is irreducible [DKP15], i.e., it cannot be decomposed into a pair of zero-syndrome vectors with non-overlapping supports.

Furthermore, the eventual convergence is guaranteed. Indeed, if c is a minimum-weight codeword of weight d, consider a permutation S which places one position from its support into the 1st column, and the remaining positions into the last d-1 columns. Vector c being the lowest-weight non-trivial vector, no pivot column may be in the block of last d-1 columns. This guarantees that vector c is obtained as the first row of G_S. (This argument is adapted to degenerate quantum codes from [CGLN21]).

3.1-4 CSS version of the algorithm

The described version of the algorithm is implemented in the function DistRandCSS (4.1). It applies to the case of Calderbank-Shor-Steane (CSS) codes, where the matrices P=H_X and Q=H_Z are called the CSS generator matrices, and the computed minimum weight is the distance d_Z of the code. The number of columns n is the block length of the code, and it encodes k qudits, where k=n-rank(H_X)-rank(H_Z). To completely characterize the code, we also need the distance d_X which can be obtained by calling the same function with the two matrices interchanged. The conventional code distance d is the minimum of d_X and d_Z. Parameters of such a q-ary CSS code are commonly denoted as [[n,k,(d_X,d_Z)]]_q, or simply [[n,k,d]]_q as for a general q-ary stabilizer code.

3.1-5 Generic version of the algorithm

CSS codes are a subclass of general F-linear stabilizer codes which are specified by a single stabilizer generator matrix H=(A|B) written in terms of two blocks of n columns each. The orthogonality condition is given in a symplectic form,

A B^T-B A^T=0,

or, equivalently, as orthogonality between the rows of H and the symplectic-dual matrix \tilde H=(B|-A). Non-trivial vectors in the code must be orthogonal to the rows of P=\tilde H and linearly independent from the rows of Q=H. The difference with the CSS version of the algorithm is that we must minimize the symplectic weight of c=(a|b), given by the number of positions i, 1\le i\le n, such that either a_i or b_i (or both) be non-zero.

The parameters of such a code are denoted as [[n,k,d]]_q, where k=n-rankH is the number of encoded qudits, and d is the minimal symplectic weight of a non-trivial vector in the code. It is easy to check that a CSS code can also be represented in terms of a single stabilizer generator matrix. Namely, for a CSS code with generators H_X and H_Z, the stabilizer generator matrix has a block-diagonal form, H=diag(H_X,H_Z).

A version of the algorithm for general F-linear stabilizer codes is implemented in the function DistRandStab (4.1).

Important Notice: In general, here one could use most general permutations of 2n columns, or restricted permutations of n two-column blocks preserving the pair structure of the matrix. While the latter method would be much faster, there is no guarantee that every vector would be found. As a result, we decided to use general permutations of 2n columns.

3.2 Some more details

3.2-1 Quantum stabilizer codes

Representation of quantum codes in terms of linear spaces is just a convenient map. In the case q=2 (qubits), the details can be found, e.g., in the book of Nielsen and Chuang, [NC00]. Further details on the theory of stabilizer quantum error correcting codes based on qubits can be found in the Caltech Ph.D. thesis of Daniel Gottesman [Got97] and in the definitive 1997 paper by Calderbank, Rains, Shor, and Sloane [CRSS98]. Theory of stabilizer quantum codes based on qudits (q-state quantum systems) was developed by Ashikhmin and Knill [AK01] (prime fields with q prime) and by Ketkar, Klappenecker, Kumar, & Sarvepalli [KKKS06] (extension fields with q a non-trivial power of a prime).

In the binary case (more generally, when q is a prime), F-linear codes coincide with additive codes. The linear codes [e.g., over \mathop{\rm GF}(4) in the binary case [CRSS98]] is a different construction which assumes an additional symmetry. A brief summary of F-linear quantum codes [where F=\mathop{\rm GF}(q) with q=p^m, m>1 a non-trivial power of a prime] can be found in the introduction of Ref. [ZP20]. The construction is equivalent to a more physical approach in terms of a lifted Pauli group suggested by Gottesman [Got14].

3.2-2 The algorithm

Case of classical linear codes

The algorithm 3.1-2 is closely related to the algorithm for finding minimum-weight codewords in a classical linear code as presented by Leon [Leo88], and a related family of information set (IS) decoding algorithms [Kru89] [CG90].

Consider a classical linear q-ary code [n,k,d]_q encoding k symbols into n, specified by a generator matrix G of rank k. Using Gauss' algorithm and column permutations, the generator matrix can be rendered into a systematic form, G=(I|A), where the two blocks are I, the size-k identity matrix, and a k by n-k matrix A. In such a representation, the first k positions are called the information set of the code (since the corresponding symbols are transmitted directly) and the remaining n-k symbols provide the redundancy. Any k linearly-independent columns of G can be chosen as the information set, which defines the systematic form of G up to a permutation of the rows of A.

The IS algorithm and the original performance bounds [Leo88] [Kru89] [CG90] are based on the observation that for a long random code a set of k+\Delta randomly selected columns, with \Delta of order one, are likely to contain an information set. ISs are (approximately) in one-to-one correspondence with the column permutations, and a random IS can thus be generated as a set of pivot columns in the Gauss' algorithm after a random column permutation. Thus, if there is a codeword c of weight d, the probability to find it among the rows of reduced-row-echelon form G_S after a column permutation S can be estimated as that for a randomly selected set of k columns to hit exactly one non-zero position in c.

The statistics of ISs is more complicated in other ensembles of random codes, e.g., in linear low-density parity-check (LDPC) codes where the check matrix H (of rank n-k and with rows orthogonal to those of G) is additionally required to be sparse. Nevertheless, a provable bound can be obtained for a related covering set (CS) algorithm where a randomly selected set of s\ge k-1 positions of a putative codeword are set to be zero, and the remaining positions are constructed with the help of linear algebra. In this case, the optimal choice [DKP17] is to take s\approx n(1-\theta), where \theta is the erasure threshold of the family of the codes under consideration. Since \theta\ge R (here R=k/n is the code rate), here more zeros must be selected, and the complexity would grow (assuming the distance d remains the same, which is usually not the case for LDPC codes).

Note however that rows of G_S other than the last are not expected to contain as many zeros (e.g., the first row is only guaranteed to have k-1 zeros), so it is possible that the performance of the IS algorithm on classical LDPC codes is actually closer to that on random codes as estimated by Leon [Leo88].

Case of quantum CSS codes

In the case of a random CSS code (with matrices P and Q selected randomly, with the only requirement being the orthogonality between the rows of P and Q), the performance of the algorithm 3.1-2 can be estimated as that of the CS algorithm, in terms of the erasure threshold of a linear code with the parity matrix P, see [DKP17].

Unfortunately, such an estimate fails dramatically in the case of quantum LDPC codes, where rows of P and Q have weights bounded by some constant w. This is a reasonable requirement since the corresponding quantum operators (supported on w qudits) have to actually be measured frequently as a part of the operation of the code, and it is reasonable to expect that the measurement accuracy goes down (exponentially) quickly as w is increased. Then, the linear code orthogonal to the rows of P has the distance \le w (the minimal weight of the rows of Q), and the corresponding erasure threshold is exactly zero. In other words, there is a finite probability that a randomly selected w symbols contain a vector orthogonal to the rows of P (and such a vector would likely have nothing to do with non-trivial quantum codewords which must be linearly independent from the rows of Q).

On the other hand, for every permutation S in the algorithm 3.1-2, the matrix G_S contains exactly k=n-rank(P)-rank(Q) rows orthogonal to rows of P and linearly independent from rows of Q (with columns properly permuted). These vectors contain at least s zeros, where [1-\theta_*(P,Q)] n\le s\le n-rank(Q), where \theta_*(P,Q) is the erasure threshold for Z-like codewords in the quantum CSS code with H_X=P and H_Z=Q.

What is it that we do not understand?

What missing is an understanding of the statistics of the ISs of interest, namely, the ISs that overlap with a minimum-weight codeword in one (or a few) positions.

Second, we know that a given column permutation S leads to the unique information set, and that every information set can be obtained by a suitably chosen column permutation. However, there is no guarantee that the resulting information sets have equal probabilities. In fact, it is easy to construct small matrices where different information sets are obtained from different numbers of column permutations (and thus have different probabilities). It is not clear whether some of the ISs may have vanishingly small probabilities in the limit of large codes; in such a case the algorithm may take an excessively long time to converge.

3.3 Empirical estimate of the success probability

The probability to find a codeword after N rounds of the algorithm can be estimated empirically, by counting the number of times each codeword of the minimum weight was discovered. We expect the probability P(c) to discover a given codeword c to depend only on its (symplectic) weight wgt(c), with the probability a monotonously decreasing function of the weight. If, after N steps, codewords c_1, c_2, \ldots , c_m of the same (minimal) weight w are discovered n_1, n_2, \ldots , n_m times, respectively, we can estimate the corresponding Poisson parameter as

\lambda_w =\frac{1}{N m}\sum_{i=1}^m n_i.

Then, the probability that a codeword c_0 of the true minimal weight d < w be not discovered after N steps can be upper bounded as (the inequalities tend to saturate and become equalities in the limit of small \lambda_w)

P_{\rm fail} < (1-\lambda_w)^N < e^{-N\lambda_w}=\exp\left(-m^{-1}\sum_{i=1}^m n_i\right)\equiv \exp(-\langle n\rangle).

Thus, the probability to fail is decreasing as an exponent of the parameter \langle n\rangle, the average number of times a minimum-weight codeword has been found.

The hypothesis about all P(c_i) being equal to \lambda_w is testable, e.g., if one considers the distribution of the ratios x_i=n_i/N, where N=\sum_{i=1}^m n_i is the total number of codewords found. These quantities sum up to one and are distributed according to multinomial distribution[Ste53]. Further, under our assumption of all P(c_i) being equal, we also expect the outcome probabilities in the multinomial distribution to be all equal, \pi_i=1/m, 1\le i\le m.

This hypothesis can be tested using Pearson's \chi^2 test. Namely, in the limit where the total number of observations N diverges, the quantity

X^2=\sum_{i=1}^m \frac{(n_i-N \pi_i)^2}{ N\pi_i}= N^{-1}\sum_{i=1}^m \frac{n_i^2}{\pi_i}-N \stackrel{\pi_i=1/m}\to\frac{m}{N}\sum_{i=1}^m n_i^2-N,

is expected to be distributed according to the \chi^2_{m-1} distribution with m-1 parameters, see [CL54] [Cra99].

In practice, we can approximate with the \chi^2_{m-1} distribution as long as the total N be large compared to the number m of the codewords found (i.e., the average \langle n\rangle must be large, which is the same condition as needed for confidence in the result.)

With debug[4] set (binary value 8) in DistRandCSS and DistRandStab (4.1), whenever more than one minimum-weight vector is found, the quantity X^2 is computed and output along with the average number of times \langle n\rangle a minimum-weight codeword has been found. However, no attempt is made to analyze the corresponding value or calculate the likelihood of the null hypothesis that the codewords be equiprobable.

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