<P/>
<Subsection Label="Subsection_TheGoal">
<Heading>What it does?</Heading>
<P/>
In the simplest possible terms, we are given a pair of matrices <Math>P</Math>
and <Math>Q</Math> with orthogonal rows, <Math>PQ^T=0</Math>. The matrices have entries in
a finite field <Math>F=\mathop{\rm GF}(q)</Math>, where <Math>q</Math> is a power of a prime. The goal
is to find the smallest weight of a non-zero vector <Math>c</Math> over the same
field <Math>F</Math>, such that <Math>c</Math> be orthogonal with the rows of <Math>P</Math>, <Math>Pc^T=0</Math>,
and linearly independent from the rows of <Math>Q</Math>.
<P/>
</Subsection>
<P/>
We first construct a generator matrix <Math>G</Math> whose rows form
a basis of the <Math>F</Math>-linear space of all vectors orthogonal to the rows
of <Math>P</Math>. At each step, a random permutation <Math>S</Math> is generated and
applied to the columns of <Math>G</Math>. Then, Gauss' elimination with back
substitution renders the resulting matrix to the reduced
row echelon form, after which the inverse permutation <Math>S^{-1}</Math> is
applied to the columns. Rows of the resulting matrix <Math>G_S</Math> that are
linearly independent from the rows of <Math>Q</Math> are considered as candidates
for the minimum weight vectors. Thus, after <Math>N</Math> steps, we are getting
an upper bound on the distance which is improving with increasing <Math>N</Math>.
<P/>
</Subsection>
<P/>
The intuition is that each row of <Math>G_S</Math> is guaranteed to contain at
least <Code>rank</Code><Math>(G_S)-1</Math> zeros. Thus, we are sampling mostly
lower-weight vectors from the linear space orthogonal to the rows of
<Math>P</Math>. Further, it is easy to see that any vector obtained this way is
<Emph>irreducible</Emph> <Cite Key="Dumer-Kovalev-Pryadko-bnd-2015"/>, i.e., it
cannot be decomposed into a pair of zero-syndrome vectors with
non-overlapping supports.
<P/>
Furthermore, the eventual convergence is guaranteed. Indeed, if <Math>c</Math>
is a minimum-weight codeword of weight <Math>d</Math>, consider a permutation <Math>S</Math>
which places one position from its support into the 1st column, and
the remaining positions into the last <Math>d-1</Math> columns. Vector <Math>c</Math> being
the lowest-weight non-trivial vector, no pivot column may be in the
block of last <Math>d-1</Math> columns. This guarantees that vector <Math>c</Math> is
obtained as the first row of <Math>G_S</Math>.
(This argument is adapted to degenerate quantum codes from
<Cite Key="Cuellar-etal-2020"/>).
<P/>
</Subsection>
<Subsection Label="Subsection_AlgorithmCSS">
<Heading>CSS version of the algorithm</Heading>
<P/>
The described version of the algorithm is implemented in the function <Code>DistRandCSS</Code>
(<Ref Sect="Section_DistanceFunctions"/>). It applies to the case of
Calderbank-Shor-Steane (CSS) codes, where the matrices <Math>P=H_X</Math> and
<Math>Q=H_Z</Math> are called the CSS generator matrices, and the computed
minimum weight is the distance <Math>d_Z</Math> of the code. The number of
columns <Math>n</Math> is the block length of the code, and it encodes <Math>k</Math>
qudits, where <Math>k=n-</Math><Code>rank</Code><Math>(H_X)-</Math><Code>rank</Code><Math>(H_Z)</Math>. To completely
characterize the code, we also need the distance <Math>d_X</Math> which can be
obtained by calling the same function with the two matrices interchanged.
The conventional code distance <Math>d</Math> is the minimum of <Math>d_X</Math> and <Math>d_Z</Math>.
Parameters of such a <Math>q</Math>-ary CSS code are commonly denoted as
<Math>[[n,k,(d_X,d_Z)]]_q</Math>, or simply <Math>[[n,k,d]]_q</Math> as for a general
<Math>q</Math>-ary stabilizer code.
<P/>
</Subsection>
<Subsection Label="Subsection_AlgorithmGeneric">
<Heading>Generic version of the algorithm</Heading>
<P/>
CSS codes are a subclass of general <Math>F</Math>-linear stabilizer codes which
are specified by a single stabilizer generator matrix <Math>H=(A|B)</Math>
written in terms of two blocks of <Math>n</Math> columns each. The orthogonality
condition is given in a symplectic form, <Display> A B^T-B A^T=0, </Display> or,
equivalently, as orthogonality between the rows of <Math>H</Math> and the
symplectic-dual matrix <Math>\tilde H=(B|-A)</Math>. Non-trivial vectors in the code must be
orthogonal to the rows of <Math>P=\tilde H</Math> and linearly independent from
the rows of <Math>Q=H</Math>. The difference with the CSS version of the
algorithm is that we must minimize the <Emph>symplectic</Emph> weight of
<Math>c=(a|b)</Math>, given by the number of positions <Math>i</Math>, <Math>1\le i\le n</Math>, such
that either <Math>a_i</Math> or <Math>b_i</Math> (or both) be non-zero.
<P/>
The parameters of such a code are denoted as <Math>[[n,k,d]]_q</Math>, where
<Math>k=n-</Math><Code>rank</Code><Math>H</Math> is the number of encoded qudits, and <Math>d</Math> 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 <Math>H_X</Math> and <Math>H_Z</Math>, the stabilizer generator matrix has a
block-diagonal form, <Math>H=</Math><Code>diag</Code><Math>(H_X,H_Z)</Math>.
<P/>
A version of the algorithm for general <Math>F</Math>-linear stabilizer codes is
implemented in the function <Code>DistRandStab</Code> (<Ref Sect="Section_DistanceFunctions"/>).
<P/>
<Emph>Important Notice</Emph>: In general, here one could use most general
permutations of <Math>2n</Math> columns, or restricted permutations of <Math>n</Math>
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 <Math>2n</Math> columns.
<P/>
</Subsection>
</Section>
<Section Label="Chapter_Description_Section_Some_more_details">
<Heading>Some more details</Heading>
<P/>
Representation of quantum codes in terms of linear spaces
is just a convenient map. In the case <Math>q=2</Math> (qubits), the details can
be found, e.g., in the book of Nielsen and Chuang, <Cite
Key="Nielsen-book"/>. 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 <Cite Key="gottesman-thesis"/>
and in the definitive 1997 paper by Calderbank, Rains, Shor, and
Sloane <Cite Key="Calderbank-1997"/>. Theory of stabilizer quantum
codes based on qudits (<Math>q</Math>-state quantum systems)
was developed by Ashikhmin and Knill
<Cite Key="Ashikhmin-Knill-2001"/>
(prime fields with <Math>q</Math> prime) and by Ketkar, Klappenecker, Kumar, & Sarvepalli <Cite
Key="Ketkar-Klappenecker-Kumar-Sarvepalli-2006"/> (extension fields
with <Math>q</Math> a non-trivial power of a prime).
<P/>
In the binary case (more generally, when <Math>q</Math> is a prime), <Math>F</Math>-linear
codes coincide with <Emph>additive</Emph> codes. The <Emph>linear</Emph> codes [e.g.,
over <Math>\mathop{\rm GF}(4)</Math> in the binary case <Cite Key="Calderbank-1997"/>] is a
different construction which assumes an additional symmetry. A brief
summary of <Math>F</Math>-linear quantum codes [where <Math>F=\mathop{\rm GF}(q)</Math> with <Math>q=p^m</Math>,
<Math>m>1</Math> a non-trivial power of a prime] can be found in the introduction
of Ref. <Cite Key="Zeng-Pryadko-hprod-2020"/>. The construction is
equivalent to a more physical approach in terms of a lifted Pauli
group suggested by Gottesman <Cite
Key="Gottesman-prime-power-2014"/>.
<P/>
</Subsection>
<P/>
<Alt Only="LaTeX"><![CDATA[
\noindent\underline{%
]]></Alt>
Case of <Emph>classical linear codes</Emph>
<Alt Only="LaTeX"><![CDATA[
}
]]></Alt>
<P/>
The algorithm <Ref Subsect="Subsection_TheProcedure"/> is closely
related to the algorithm for finding minimum-weight codewords in a
classical linear code as presented by Leon <Cite Key="Leon-1988"/>,
and a related family of <Emph>information set</Emph> (IS) decoding algorithms <Cite
Key="Kruk-1989"/> <Cite Key="Coffey-Goodman-1990"/>.
<P/>
Consider a classical linear <Math>q</Math>-ary code <Math>[n,k,d]_q</Math> encoding <Math>k</Math>
symbols into <Math>n</Math>, specified by a generator matrix <Math>G</Math> of rank <Math>k</Math>.
Using Gauss' algorithm and column permutations, the generator matrix
can be rendered into a <Emph>systematic form</Emph>, <Math>G=(I|A)</Math>, where the two
blocks are <Math>I</Math>, the size-<Math>k</Math> identity matrix, and a <Math>k</Math> by <Math>n-k</Math>
matrix <Math>A</Math>. In such a representation, the first <Math>k</Math> positions are
called the information set of the code (since the corresponding
symbols are transmitted directly) and the remaining <Math>n-k</Math> symbols
provide the redundancy. Any <Math>k</Math> linearly-independent columns of <Math>G</Math>
can be chosen as the information set, which defines the systematic
form of <Math>G</Math> up to a permutation of the rows of <Math>A</Math>.
<P/>
The IS algorithm and the original performance bounds
<Cite Key="Leon-1988"/>
<Cite Key="Kruk-1989"/>
<Cite Key="Coffey-Goodman-1990"/>
are based on the observation that for a long random code a set of
<Math>k+\Delta</Math> randomly selected columns, with <Math>\Delta</Math> 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 <Emph>pivot</Emph> columns in the Gauss'
algorithm after a random column permutation. Thus, if there is a
codeword <Math>c</Math> of weight <Math>d</Math>, the probability to find it among the rows
of reduced-row-echelon form <Math>G_S</Math> after a column permutation <Math>S</Math> can
be estimated as that for a randomly selected set of <Math>k</Math> columns to hit
exactly one non-zero position in <Math>c</Math>.
<P/>
The statistics of ISs is more complicated in other ensembles of random
codes, e.g., in linear <Emph>low-density parity-check</Emph> (LDPC) codes where
the check matrix <Math>H</Math> (of rank <Math>n-k</Math> and with rows orthogonal to those
of <Math>G</Math>) is additionally required to be sparse. Nevertheless, a
provable bound can be obtained for a related <Emph>covering set</Emph> (CS)
algorithm where a randomly selected set of <Math>s\ge k-1</Math> 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 <Cite Key="Dumer-Kovalev-Pryadko-IEEE-2017"/> is to
take <Math>s\approx n(1-\theta)</Math>, where <Math>\theta </Math> is the erasure threshold
of the family of the codes under consideration. Since <Math>\theta\ge R</Math>
(here <Math>R=k/n</Math> is the code rate), here more zeros must be selected, and
the complexity would grow (assuming the distance <Math>d</Math> remains the same,
which is usually <Emph>not</Emph> the case for LDPC codes).
<P/>
Note however that rows of <Math>G_S</Math> other than the last are not expected
to contain as many zeros (e.g., the first row is only guaranteed to
have <Math>k-1</Math> zeros), so it is <Emph>possible</Emph> that the performance of the
IS algorithm on classical LDPC codes is actually closer to that on
random codes as estimated by Leon <Cite Key="Leon-1988"/>.
<P/>
<Alt Only="LaTeX"><![CDATA[
\medskip\noindent\underline{%
]]></Alt>
Case of <Emph>quantum CSS codes</Emph>
<Alt Only="LaTeX"><![CDATA[
}
]]></Alt>
<P/>
In the case of a random CSS code (with matrices <Math>P</Math> and <Math>Q</Math> selected
randomly, with the only requirement being the orthogonality between
the rows of <Math>P</Math> and <Math>Q</Math>), the performance of the algorithm <Ref
Subsect="Subsection_TheProcedure"/> can be estimated as that of the CS
algorithm, in terms of the erasure threshold of a linear code with the
parity matrix <Math>P</Math>, see <Cite Key="Dumer-Kovalev-Pryadko-IEEE-2017"/>.
<P/>
Unfortunately, such an estimate fails dramatically in the case of
<Emph>quantum LDPC codes</Emph>, where rows of <Math>P</Math> and <Math>Q</Math> have weights bounded
by some constant <Math>w</Math>. This is a reasonable requirement since the
corresponding quantum operators (supported on <Math>w</Math> 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 <Math>w</Math> is increased. Then, the
linear code orthogonal to the rows of <Math>P</Math> has the distance <Math>\le w</Math>
(the minimal weight of the rows of <Math>Q</Math>), and the corresponding erasure
threshold is exactly zero. In other words, there is a finite
probability that a randomly selected <Math>w</Math> symbols contain a vector
orthogonal to the rows of <Math>P</Math> (and such a vector would likely have nothing to
do with non-trivial <Emph>quantum</Emph> codewords which must be linearly
independent from the rows of <Math>Q</Math>).
<P/>
On the other hand, for every permutation <Math>S</Math> in the algorithm <Ref
Subsect="Subsection_TheProcedure"/>, the matrix <Math>G_S</Math> contains exactly
<Math>k=n-</Math><Code>rank</Code><Math>(P)-</Math><Code>rank</Code><Math>(Q)</Math> rows orthogonal to rows of <Math>P</Math> and
linearly independent from rows of <Math>Q</Math> (with columns properly
permuted). These vectors contain at least <Math>s</Math> zeros, where
<Math>[1-\theta_*(P,Q)] n\le s\le n-</Math><Code>rank</Code><Math>(Q)</Math>, where <Math>\theta_*(P,Q)</Math> is
the erasure threshold for <Math>Z</Math>-like codewords in the quantum CSS code
with <Math>H_X=P</Math> and <Math>H_Z=Q</Math>.
<P/>
<Alt Only="LaTeX"><![CDATA[
\medskip\noindent\underline{%
]]></Alt>
<Emph>What is it that we do not understand?</Emph>
<Alt Only="LaTeX"><![CDATA[
}
]]></Alt>
<P/>
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.
<P/>
Second, we know that a given column permutation <Math>S</Math> 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 <Emph>different</Emph> 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.
<P/>
</Subsection>
</Section>
<Section Label="Section_Empirical">
<Heading>Empirical estimate of the success probability</Heading>
<P/>
The probability to find a codeword after <Math>N</Math> rounds of the algorithm
can be estimated empirically, by counting the number of times each
codeword of the minimum weight was discovered. We <Emph>expect</Emph> the
probability <Math>P(c)</Math> to discover a given codeword <Math>c</Math> to depend only on
its (symplectic) weight <Code>wgt</Code><Math>(c)</Math>, with the probability a
monotonously decreasing function of the weight. If, after <Math>N</Math> steps,
codewords <Math>c_1</Math>, <Math>c_2</Math>, <Math>\ldots</Math> , <Math>c_m</Math> of the same (minimal) weight <Math>w</Math>
are discovered <Math>n_1</Math>, <Math>n_2</Math>, <Math>\ldots</Math> , <Math>n_m</Math> times, respectively, we can
estimate the corresponding Poisson parameter as
<Display> \lambda_w =\frac{1}{N m}\sum_{i=1}^m n_i. </Display>
<P/>
Then, the probability that a codeword <Math>c_0</Math> of the true minimal weight
<Math> d < w </Math> be <Emph>not</Emph> discovered after <Math>N</Math> steps can be upper
bounded as (the inequalities tend to saturate and become equalities in
the limit of small <Math>\lambda_w</Math>) <Display> 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). </Display> Thus, the probability to fail is
decreasing as an exponent of the parameter <Math>\langle n\rangle</Math>, the
<Emph>average number of times a minimum-weight codeword has been found.</Emph>
<P/>
The hypothesis about all <Math>P(c_i)</Math> being equal to <Math>\lambda_w</Math> is
testable, e.g., if one considers the distribution of the ratios
<Math>x_i=n_i/N</Math>, where <Math>N=\sum_{i=1}^m n_i</Math> is the total number of
codewords found. These quantities sum up to one and are distributed
according to multinomial distribution<Cite
Key="Steel-1953"/>. Further, under our assumption
of all <Math>P(c_i)</Math> being equal, we also expect the outcome probabilities
in the multinomial distribution to be all equal, <Math>\pi_i=1/m</Math>, <Math>1\le
i\le m</Math>.
<P/>
This hypothesis can be tested using Pearson's test. Namely,
in the limit where the total number of observations <Math>N</Math> diverges, the
quantity <Display> 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, </Display> is expected to be
distributed according to the <Math>\chi^2_{m-1}</Math> distribution with <Math>m-1</Math>
parameters, see <Cite Key="Chernoff-Lehmann-1954"/> <Cite
Key="Cramer-book-1999"/>.
<P/>
In practice, we can approximate with the <Math>\chi^2_{m-1}</Math> distribution
as long as the total <Math>N</Math> be large compared to the number <Math>m</Math> of the
codewords found (i.e., the average <Math>\langle n\rangle</Math> must be large,
which is the same condition as needed for confidence in the result.)
<P/>
With <Code>debug[4]</Code> set (binary value 8) in <Code>DistRandCSS</Code> and
<Code>DistRandStab</Code> (<Ref Sect="Section_DistanceFunctions"/>), whenever more
than one minimum-weight vector is found, the quantity <Math>X^2</Math> is
computed and output along with the average number of times <Math>\langle
n\rangle</Math> 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.
<P/>
</Section>
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.
