Abstract. Quantum stabilizer states over Fm can be represented as self-dual additive codes over F m 2. These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operation correspond to equivalence classes of codes. We have previously used this fact to classify self-dual additive codes over F4. In this paper we classify selfdual additive codes over F9, F16, and F25. Assuming that the classical MDS conjecture holds, we are able to classify all self-dual additive MDS codes over F9 by using an extension technique. We prove that the minimum distance of a self-dual additive code is related to the minimum vertex degree in the associated graph orbit. Circulant graph codes are introduced, and a computer search reveals that this set contains many strong codes. We show that some of these codes have highly regular graph representations. 1.
Abstract. Quantum stabilizer states over Fm can be represented as self-dual additive codes over F m 2. These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operation correspond to equivalence classes of codes. We have previously used this fact to classify self-dual additive codes over F4. In this paper we classify selfdual additive codes over F9, F16, and F25. Assuming that the classical MDS conjecture holds, we are able to classify all self-dual additive MDS codes over F9 by using an extension technique. We prove that the minimum distance of a self-dual additive code is related to the minimum vertex degree in the associated graph orbit. Circulant graph codes are introduced, and a computer search reveals that this set contains many strong codes. We show that some of these codes have highly regular graph representations. 1.
Quantum stabilizer states over Fm can be represented as self-dual additive codes over Fm2. These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operationcorrespond to equivalence classes of codes. We have previously used this fact to classify self-dual additive codes over F4. In this paper we classify selfdual additive codes over F9, F16, and F25. Assuming that the classical MDSconjecture holds, we are able to classify all self-dual additive MDS codes over F9 by using an extension technique. We prove that the minimum distanceof a self-dual additive code is related to the minimum vertex degree in theassociated graph orbit. Circulant graph codes are introduced, and a computersearch reveals that this set contains many strong codes. We show that some ofthese codes have highly regular graph representations.
Additive codes over GF(9) that are self-dual with respect to the Hermitian trace inner product have a natural application in quantum information theory, where they correspond to ternary quantum error-correcting codes. However, these codes have so far received far less interest from coding theorists than self-dual additive codes over GF(4), which correspond to binary quantum codes. Self-dual additive codes over GF(9) have been classified up to length 8, and in this paper we extend the complete classification to codes of length 9 and 10. The classification is obtained by using a new algorithm that combines two graph representations of self-dual additive codes. The search space is first reduced by the fact that every code can be mapped to a weighted graph, and a different graph is then introduced that transforms the problem of code equivalence into a problem of graph isomorphism. By an extension technique, we are able to classify all optimal codes of length 11 and 12. There are 56,005,876 (11,3^11,5) codes and 6493 (12,3^12,6) codes. We also find the smallest codes with trivial automorphism group.
We study objects that can be represented as graphs, error-correcting codes, quantum states, or Boolean functions. It is known that self-dual additive codes, which can also be interpreted as quantum states, can be represented as graphs, and that two codes are equivalent when the corresponding graphs are equivalent with respect to local complementation (LC). We give classifications of such codes. Circulant graph codes are introduced, and it is shown that some of these codes have highly regular graph representations. We show that the orbit of a bipartite graph under edge local complementation (ELC) corresponds to the equivalence class of a binary linear code. We classify ELC orbits, give a new method for classifying binary linear codes, and show that the information sets and the minimum distance of a code can be derived from the corresponding ELC orbit. Self-dual additive codes over GF(4) can also be interpreted as quadratic Boolean functions. In this context we define PARIHN, peak-to-average power ratio with respect to the {I, H, N}n transform, and prove that PARIHN equals the size of the maximum independent set over the associated LC orbit of graphs. We define the aperiodic propagation criteria (APC) of a Boolean function, and show that it is related to the minimum distance of a self-dual additive code over GF(4), and to the degree of entanglement in the associated quantum state. We give a generalization to non-quadratic Boolean functions, and relate APC to known cryptographic criteria. Interlace polynomials encode many properties of the LC and ELC orbits of a graph. We enumerate interlace polynomials and circle graphs, show that there exist non-unimodal interlace polynomials, and relate properties of interlace polynomials to properties of codes and quantum states. We define self-dual bent functions, and provide constructions and classifications.
A short introduction to quantum error correction is given, and it is shown that zero-dimensional quantum codes can be represented as self-dual additive codes over GF(4) and also as graphs. We show that graphs representing several such codes with high minimum distance can be described as nested regular graphs having minimum regular vertex degree and containing long cycles. Two graphs correspond to equivalent quantum codes if they are related by a sequence of local complementations. We use this operation to generate orbits of graphs, and thus classify all inequivalent self-dual additive codes over GF(4) of length up to 12, where previously only all codes of length up to 9 were known. We show that these codes can be interpreted as quadratic Boolean functions, and we define non-quadratic quantum codes, corresponding to Boolean functions of higher degree. We look at various cryptographic properties of Boolean functions, in particular the propagation criteria. The new aperiodic propagation criterion (APC) and the APC distance are then defined. We show that the distance of a zero-dimensional quantum code is equal to the APC distance of the corresponding Boolean function. Orbits of Boolean functions with respect to the {I,H,N}^n transform set are generated. We also study the peak-to-average power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove that PAR_IHN of a quadratic Boolean function is related to the size of the maximum independent set over the corresponding orbit of graphs. A construction technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It is finally shown that both PAR_IHN and APC distance can be interpreted as partial entanglement measures.
We prove that the quantum code distance is NP-hard to approximate within an additive error of cN, for some constant c>0, where N is the number of qubits. Our reductions are deterministic. This improves the previous square-root additive gap to Ω(N) and resolves the explicitly stated linear-gap question of Kapshikar and Kundu. Our result holds for CSS codes with identical X- and Z-check spaces, and with a constant rate and constant relative distance. For every fixed λ>1, there is a constant c>0 such that hardness still holds even when every nonidentity stabilizer has weight greater than λ times the quantum distance. We also improve the hardness gap of graph state distance on N vertices of Grigorescu, Jha, and Samperton from cube-root to Ω(N), resolving their explicitly stated open question. Both hardness results are asymptotically optimal since both distances are at most N. Our graph state distance hardness result holds for balanced bipartite graphs with a binary adjacency matrix that is its own inverse (mod 2).
Our main technique for both hardness bounds above is classical: we show how to convert any code C of length m into a self-dual code A(C) of length N=Θ(m) while exactly doubling the original coset metric. The conversion is deterministic and efficient. We call it the metric self-dual completion of C. It comes with a linear embedding τ:F2m↪F2N. The embedding doubles all Hamming distances between vectors in F2m and all pairwise distances between corresponding cosets. The embedding also guarantees that all codewords of A(C) of weight at most 2m are exactly τ(C).
It is established in literature that finding stabilizer quantum error correcting codes (QECCS) is the same as finding self-dual additive codes over the finite field F4 under the Hermitian trace inner product. Additionally, every self-dual additive code can be represented by a graph adjacency matrix. Many selfdual additive codes are constructed from circulant graphs. We introduce new graph code constructions: the Toeplitz Graph, the Multidimensional Toeplitz Graph (MDT), and the Generalized Toeplitz (GT) graph constructions. We consider some of the properties of the Toeplitz and Multidimensional Toeplitz Graphs and compare the Circulant and GT code constructions.