A hybrid code can simultaneously encode classical and quantum information into quantum digits such that the information is protected against errors when transmitted through a quantum channel. It is shown that a hybrid code has the remarkable feature that it can detect more errors than a comparable quantum code that is able to encode the classical and quantum information. Weight enumerators are introduced for hybrid codes that allow to characterize the minimum distance of hybrid codes. Surprisingly, the weight enumerators for hybrid codes do not obey the usual MacWilliams identity.
Recently, the field of quantum error-correcting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum error-correction. In this paper we give a brief exposition of the theory of quantum stabilizer codes. We review the stabilizer formalism of quantum codes, establish the connection between classical codes and stabilizer codes and the main methods for constructing quantum codes from classical codes. In addition to the expository part, we include new results that cannot be found elsewhere. Specifically, after reviewing some important bounds for quantum codes, we prove the nonexistence of pure perfect quantum stabilizer codes with minimum distance greater than 3. Finally, we illustrate the general methods of constructing quantum codes from classical codes by explicitly constructing two new families of quantum codes and conclude by showing how to construct new quantum codes by shortening.
We prove a conjecture of Yu, Lai, and Oh: their strengthened quantum Hamming bounds, a family that interpolates between Hamming-type bounds and the Singleton bound, hold for all quantum codes of dimension
K>1, including impure and nonadditive codes. The result covers every integer local dimension
p≥2, not necessarily a prime power, every integer
t≥1, both distances
2t+1 and
2t+2, and every interpolation parameter
e∈{0,…,t}. The case
e=0 implies the ordinary quantum Hamming bound for general codes, recently proved by Zhang and Chen. Yu, Lai, and Oh proved their bounds for pure codes and for general codes of distance three or four. More generally, for every integer
p≥2 and every quantum code of length
n, dimension
K>1 and distance
d≥3, we prove
Kf(0)≤pnEf(X), where
X∼Bin(n,1−p−2), for every real polynomial
f of degree less than
d that is nonnegative on
{0,…,n} and positive at zero. For pure codes this inequality is implicit in the argument of Yu, Lai, and Oh. The proof uses the linear-programming method with quantum weight enumerators.