A Note on Quantum Hamming Bound

Quantum stabilizer codes are a known class of quantum codes that can protect quantum information against noise and decoherence. Stabilizer codes can be constructed from self-orthogonal or dualcontaining classical codes, see for example [3,8,11] and references therein. It is desirable to study upper and lower bounds on the minimum distance of classical and quantum codes, so the computer search on the code parameters can be minimized. It is a well-known fact that Singleton and Hamming bounds hold for classical codes [10]. Also, upper and lower bounds on the achievable minimum distance of quantum stabilizer codes are needed. Perhaps the simplest upper bound is the quantum Singleton bound, also known as the Knill-Laflamme bound. The binary version of the quantum Singleton bound was first proved by Knill and Laflamme in [12], see also [1,2], and later generalized by Rains using weight enumerators in [16].

Theorem 1 (Quantum Singleton Bound). An ((n, K, d)) q stabilizer code with K > 1 satisfies K ≤ q n-2d+2 .

(

Codes which meet the quantum Singleton bound with equality are called quantum MDS codes. If we assume that K = q k , then this bound can be stated as k ≤ n -2d + 2. In [11] It has been shown that these codes cannot be indefinitely long and showed that the maximal length of a q-ary quantum MDS codes is upper bounded by 2q 2 -2. This could probably be tightened to q 2 + 2. It would be interesting to find quantum MDS codes of length greater than q 2 + 2 since it would disprove the MDS conjecture for classical codes [10]. A related open question is regarding the construction of codes with lengths between q and q 2 -1. At the moment there are no analytical methods for constructing a quantum MDS code of arbitrary length in this range (see [9] for some numerical results).

Another important bound for quantum codes is the quantum Hamming bound. The quantum Hamming bound states (see [6,7]) that: Theorem 2 (Quantum Hamming Bound). Any pure ((n, K, d)) q stabilizer code satisfies

The previous quantum Hamming bound holds only for nondegenerate (pure) quantum codes. However, the degenerate (impure) quantum codes are particularly interesting class of quantum codes because they can pack more quantum information. In addition, the errors of small weights do not need active error correction strategies.

So far no degenerate quantum code has been found that beats this bound. Gottesman showed that impure single and double error-correcting binary quantum codes cannot beat the quantum Hamming bound [8]. It is proved in [11] that Hamming bound holds for quantum stabilizer codes with distance d = 3.

In general, does Hamming bound exist for any distance d in ((n, K, d)) q stabilizer codes? This has been an open question for a decade. In this note we prove Hamming bound for double errorcorrecting stabilizer codes with distance d = 5 and also give a sketch to prove it for general distance d.

There have been several approaches to prove bounds on the quantum code parameters. In [1] Ashikhmin and Litsyn derived many bounds for quantum codes by extending a novel method originally introduced by Delsarte [5] for classical codes. Using this method they proved the binary versions of Theorems 1,2. We use this method to show that the Hamming bound holds for all double error-correcting quantum codes. See [11] for a similar result for single error-correcting codes. But first we need Theorem 3 and the Krawtchouk polynomial of degree j in the variable x,

Theorem 3. Let Q be an ((n, K, d)) q stabilizer code of dimension K > 1. Suppose that S is a nonempty subset of {0, . . . , d -1} and N = {0, . . . , n}. Let

be a polynomial satisfying the conditions i) f x > 0 for all x in S, and

Proof. See [11].

We demonstrate usefulness of the previous Theorem by showing that quantum Hamming bound holds for impure codes when d = 5.

Lemma 4 (Quantum Hamming Bound). An ((n, K, 5)) q stabilizer code with K > 1 satisfies

, where f x = ( e j=0 K j (x)) 2 , S = {0, 1, . . . , 4} and N={0,1,. . . ,n}. Calculating f (x) and f x gives us

and,

Clearly f x > 0 for all x ∈ S . Also, f (x) ≤ 0 for all x ∈ N \S since the binomial coefficients for the negative values are zero. The Hamming bound is given by

So, there are four different comparisons where f (0)/f 0 ≥ f (x)/f x , for x = 1, 2, 3, 4. We find a lower bound for n that holds for all values of q. From Lemmas 5,6,7, and 8, shown below, for n ≥ 7 it follows that

While the above method is a general method to prove Hamming bound for impure quantum codes, the number of terms increases with a large minimum distance. It becomes difficult to find the true bound using this method. However, one can derive more consequences from Theorem 3; see, for instance, [1,2,13,15]. Lemma 5. The inequality f (0)/f 0 ≥ f (1)/f 1 holds for n ≥ 6 and q ≥ 2.

in the left side (n-1) approximates to (n-2). Also, in the right side (n-2) and (n-1) approximate to (n). So,

divide both sides by (q 2 -1) 2 (q 2 -1) 2 and approximate

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