Best Known Quantum Codes

An \([[n, k]]\) quantum stabiliser code \(Q\) is said to be a best known \([[n, k]]\) quantum code (BKQC) if \(C\) has the highest minimum weight among all known \([[n, k]]\) quantum codes. The acronym QECC (Quantum Error Correcting Code) will be used to more easily distinguish from the best known linear codes database (BKLC).

Magma currently has a database for binary quantum codes, though it should be noted that these codes are considered to be over the alphabet \(GF(4)\), not \(GF(2)\). The database for codes over \(GF(4)\) currently contains constructions of all best known quantum codes of length \(35\). This includes self-dual quantum codes up to length \(35\), which are stored in the database as dimension \(0\) quantum codes.

Quantum codes of length up to \(12\) are optimal, in the sense that their minimum weights meet the upper bound. Thus the user has access to \(665\) best-known binary quantum codes.

The Magma QECC database uses the tables of bounds and constructions compiled by Markus Grassl (Karlsruhe), available online at [Grassl, n.d.], which are based on the results in [Calderbank et al., 1998]. Good codes have also been contributed by Eric Rains and Zlatko Varbanov.

The user can display the method used to construct a particular QECC code through use of a verbose mode, triggered by the verbose flag BestCode. When it is set to true, all of the functions in this section will output the steps involved in each code they construct.

QECC(F, n, k): FldFin, RngIntElt, RngIntElt -> CodeQuantum, BoolElt

BKQC(F, n, k): FldFin, RngIntElt, RngIntElt -> CodeQuantum, BoolElt

BestKnownQuantumCode(F, n, k): FldFin, RngIntElt, RngIntElt -> CodeQuantum, BoolElt

Given a finite field \(F\), and positive integers \(n\) and \(k\) such that \(k \le n\), return an \([[n,k]]\) quantum code over \(F\) which has the largest minimum weight among all known \([[n,k]]\) quantum codes. A second boolean return value signals whether or not the desired code exists in the database.

The database currently exists for \(GF(4)\) (which are in fact binary quantum codes) up to length \(35\).

Example: QECC Egs (ex-4ef878)

The weight distribution of a small best known quantum code is calculated, verifying its minimum weight. Note that the larger the dimension of a quantum code, the easier it is to calculate its weight distribution.

> F<w> := GF(4);
> Q := QECC(F,25,16);
> Q:Minimal;
[[25, 16, 3]] Quantum code over GF(2^2)
> time WD_S, WD_N, WD := WeightDistribution(Q);
Time: 0.010
> WD_S;
[ <0, 1>, <1, 2>, <2, 1>, <14, 4>, <15, 16>, <16, 38>, <17, 79>, <18, 126>, <19,
129>, <20, 77>, <21, 27>, <22, 9>, <23, 3> ]
> WD_N;
[ <0, 1>, <1, 2>, <2, 1>, <3, 399>, <4, 6527>, <5, 75363>, <6, 707543>, <7,
5404369>, <8, 34084490>, <9, 180107319>, <10, 804255370>, <11, 3052443894>, <12,
9883860222>, <13, 27348684334>, <14, 64649758926>, <15, 130286413858>, <16,
222912028997>, <17, 321704696752>, <18, 387985433701>, <19, 385943417035>, <20,
310898936275>, <21, 197566276671>, <22, 95232787563>, <23, 32688613821>, <24,
7109768160>, <25, 735493959> ]
> WD;
[ <3, 399>, <4, 6527>, <5, 75363>, <6, 707543>, <7, 5404369>, <8, 34084490>, <9,
180107319>, <10, 804255370>, <11, 3052443894>, <12, 9883860222>, <13,
27348684334>, <14, 64649758922>, <15, 130286413842>, <16, 222912028959>, <17,
321704696673>, <18, 387985433575>, <19, 385943416906>, <20, 310898936198>, <21,
197566276644>, <22, 95232787554>, <23, 32688613818>, <24, 7109768160>, <25,
735493959> ]

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So the \([[25,16]]\) code is impure, and has a minimum distance of \(3\).

Example: QECC Self Dual (ex-aece57)

Unlike linear codes, dimension \(0\) quantum codes are non-trivial and are the subject of much study. These are the self-dual quantum codes, which form a special subclass of quantum stabilizer codes. It can be seen that a length \(n\) self-dual code is described by a \(n\times n\) generator matrix, an indication of the non-triviality of its structure.

> F<w> := GF(4);
> C := QECC(GF(4),8, 0);
> C;
[[8, 0, 4]] self-dual Quantum code over GF(2^2), stabilised by:
[  1   0   0   1   0   1   1   0]
[  w   0   0   w   0   w   w   0]
[  0   1   0   1   0   1   0   1]
[  0   w   0   w   0   w   0   w]
[  0   0   1   1   0   0   1   1]
[  0   0   w   w   0   0   w   w]
[  0   0   0   0   1   1   1   1]
[  0   0   0   0   w   w   w   w]

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Example: QECC Verbose (ex-e5b5cc)

The verbose flag BestCode will show the method by which the best code is constructed in the database. In this example the construction of a \([[25, 11, 4]]\) quantum code is described.

> SetVerbose("BestCode",true);
> F<w> := GF(4);
> Q := QECC(F,25,11);
Construction of a [[ 25 , 11 , 4 ]] Quantum Code:
[1]:  [[40, 30, 4]] Quantum code over GF(2^2)
        QuasiCyclicCode of length 40 stacked to height 2 with generating
        polynomials: 1,  w^2*x^4 + w*x^3 + w^2*x^2 + w*x + w^2,  x^4 + w^2*x^2 +
        w^2*x + w^2,  x^4 + w*x^3 + x^2,  w*x^4 + x^3 + w^2*x,  w*x^4 + x^3 +
        x^2 + x,  w^2*x^4 + x^2 + w*x,  x^4 + w^2*x^3 + w^2*x^2 + x + 1,  w,
        x^4 + w^2*x^3 + x^2 + w^2*x + 1,  w*x^4 + x^2 + x + 1,  w*x^4 + w^2*x^3
        + w*x^2,  w^2*x^4 + w*x^3 + x,  w^2*x^4 + w*x^3 + w*x^2 + w*x,  x^4 +
        w*x^2 + w^2*x,  w*x^4 + x^3 + x^2 + w*x + w
[2]:  [[21, 11, 4]] Quantum code over GF(2^2)
        Shortening of [1] at { 2, 3, 4, 6, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19,
        21, 24, 28, 34, 37 }
[3]:  [[25, 11, 4]] Quantum code over GF(2^2)
        ExtendCode [2] by 4
> Q:Minimal;
[[25, 11, 4]] Quantum code over GF(2^2)

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