SUBMITTED TO 2002 IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY, LAUSANNE 1 Graphs, Quadratic Forms, and Quantum Codes Markus Grassl, Andreas Klappenecker, and Martin R¨otteler Extended Abstract of a paper submitted to ISIT 2002, Lausanne, Switzerland Abstract — We show that any stabilizer code over a finite field is equivalent to a graphical quantum code. Further- more we prove that a graphical quantum code over a finite field is a stabilizer code. The technique used in the proof establishes a new connection between quantum codes and quadratic forms. We provide some simple examples to illus- trate our results. Keywords — Graphs, quadratic forms, quantum error- correcting codes. I. Graphical Quantum Codes Let A be the additive group of a finite field F p m. Denote by H the complex vector space C α of dimension α = |A|. Let B be an orthonormal basis of H ⊗n consisting of basis vectors |y〉 labeled by elements of the group A n . Let K ∼ = A k and N ∼ = A n be subgroups of A k+n such that A k+n = K × N . Following the definition of Schlingemann and Werner in [1], a graphical quantum code is an α k -dimensional sub- space Q of H ⊗n , which is spanned by the vectors |x〉 = 1 √ α n  y∈N  k+n  i,j=1 i<j χ(z i ,z j ) Γ ij  |y〉, (1) where x ∈ K and z = x + y ∈ K × N ∼ = A k+n . The coefficients on the right hand side are given by the values of a non-degenerate symmetric bicharacter χ on A × A. The exponents Γ ij are given by the adjacency matrix Γ of a weighted undirected graph with integral weights, Γ ij ∈ Z. As (1) is independent of the diagonal elements Γ ii , we can assume without loss of generality that the graph has no loops. In [1] the authors raised the question whether or not every stabilizer code is equivalent to a graphical quantum code. Our main result gives an affirmative answer to this question: Theorem 1: Any stabilizer code over the alphabet A = F p m is equivalent to a graphical quantum code. Conversely, any graphical quantum code over A is a stabilizer code. In the sequel, we will prove this theorem. First, we show that any graphical code over an extension field F p m can be reformulated as a graphical code over the prime field F p . Then we compute the stabilizer associated with a graphical M. Grassl and M. R¨otteler are with the Institut f¨ ur Algorith- men und Kognitive Systeme (IAKS), Arbeitsgruppe Quantum Com- puting, Prof. Th. Beth, Fakult¨at f¨ ur Informatik, Universit¨ at Karls- ruhe, Am Fasanengarten 5, 76 128 Karlsruhe, Germany (e-mail: {grassl,roettele}@ira.uka.de). A. Klappenecker is with the Department of Computer Science, Texas A&M University, College Station, TX 77843-3112, USA (e- mail: klappi@cs.tamu.edu). code, followed by the construction of a graphical represen- tation of a stabilizer code. We conclude by giving examples which illustrate both directions of our main theorem. Lemma 1: Any symmetric bicharacter χ over the abelian group A ∼ = F m p can be written as χ(h, g) = exp  2πi p b(h, g)  , (2) where b is a symmetric bilinear form over F p , i.e., b(h, g)= h t Mg where M is a symmetric matrix over F p . Proof: For fixed h ∈ A, the mapping g → χ(h, g) is a character of A. Any character ζ of A can be written as ζ (g) = exp(2πi/p·h t g) where h t g denotes the inner product of the group element g identified with a vector in F m p and the vector h ∈ F m p . As χ(h 1 + h 2 ,g)= χ(h 1 ,g)χ(h 2 ,g) and the group A is (non-canonically) isomorphic to its character group A ∗ , the bicharacter χ can be written as χ(h, g) = exp  2πi p (Mh) t g  , where M is an m × m matrix over F p . Symmetry of the bicharacter implies symmetry of M . Using this lemma, eq. (1) can be rewritten as |x〉 = 1 √ α n  y∈N  k+n  i,j=1 i<j exp(2πi/p · (z t i Mz j )) Γij  |y〉 = 1 √ α n  y∈N exp  2πi p q(v)   |y〉. (3) Here we identify x + y ∈ F k+n p m with v =(v i ) ∈ F m(k+n) p . Furthermore, q is the quadratic form q(v) := m(k+n)  i,j=1 i<j Γ ′ ij v i v j (4) on F m(k+n) p defined by the symmetric matrix Γ ′ := Γ ⊗ M . Hence the states (1) of the graphical quantum code Q can be expressed in the form |x〉 = 1 √ α n  y∈N ζ (q(x + y))|y〉, (5) where ζ is a non-trivial additive character of F p and q is the quadratic form (4) on F m(k+n) p . We will take advantage of this presentation in the following sections.