Coding Analog of Superadditivity Using Entanglement-Assisted Quantum Tensor Product Codes Over $\mathbb {F}_{p^k}$

Abstract

We provide a procedure to construct entanglement-assisted Calderbank-Shor-Steane (CSS) codes over qudits from the parity check matrices of two classical codes over F q , where q = p k , p is prime, and k is a positive integer. The construction procedure involves the proposed Euclidean Gram-Schmidt orthogonalization algorithm, followed by a procedure to extend the quantum operators to obtain stabilizers of the code. Using this construction, we provide a construction of entanglement-assisted tensor product codes from classical tensor product codes over F q . We further show that a nonzero rate entanglement-assisted tensor product code can be obtained from a classical tensor product code whose component codes yield zero rate entanglement-assisted CSS codes. We view this result as the coding analog of superadditivity.INDEX TERMS Entanglement-assisted (EA) codes, entanglement-assisted Calderbank-Shor-Steane (CSS) code, extension procedure, quantum error correction, tensor product codes (TPCs). Engineering uantum Transactions on IEEENadkarni and Garani: Coding Analog of Superadditivity Using Entanglement-Assisted Quantum Tensor Product Codes

Encoding of Nonbinary Entanglement-Unassisted and Assisted Stabilizer Codes

“…Let α and p(x) = x 2 + x + 2 be the primitive element and the primitive polynomial of F 9 , respectively. In Table 2, we provide the ternary representation [b 1 b 0 ], the field trace, and the representation [Tr p k /p (αβ) Tr p k /p (β)] of the elements of F 9 , useful for representing X (9) (•) and Z (9) (•) operators in terms of X (3) (•) and Z (3) (•) operators using equations (4) and (5), respectively, provided in Table 3. (9) (β) and Z (9) (β) (β ∈ F 9 ).…”

“…In Figure 3, we provide the encoding circuit architecture 9 to encode the one qudit state, i.e., the two subqudit state |φ into the codeword |ψ using DFT 3 , DFT −1 3 , ADD 3 , and M 2 gates. Practically, these one and two subqudit gates correspond to one or two qudit gates whose operator form is obtained using equations ( 4) and (5).…”

“…To illustrate our framework, we consider d = 9 (p = 3, k = 2), where the quantum states and operators are represented based on F 9 with primitive element α and primitive polynomial p(x) = x 2 + x + 2. Let R be a subgroup of G ⊗3 9 (n = 3) generated by the following minimal generators: R 1 = X (9) (α) ⊗ Z (9) (α 6 ) ⊗ X (9) (α),…”

“…the generators in G provided in equations ( 37)-( 40) are extended by one subqudit as follows: (9) (α) ⊗ Z (9) (α 6 ) ⊗ X (9) (α) Z (9) (α 6 ) ⊗ X (9) (α) ⊗ Z (9) (α 6 ) X (9) (1) ⊗ Z (9) (α 5 ) ⊗ X (9) (1) Z (9) (α 5 ) ⊗ X (9) (α 4 ) ⊗ Z (9) (α 5 ) Original qudits ⊗ ⊗ ⊗ ⊗ X (3) (2), Z (3) (1), I 3 , I 3 .…”

“…Harnessing the power of entanglement, information processing tasks like teleportation, super dense coding [7], superadditivity of quantum channel capacities [8], and the construction of non-zero rate quantum codes from zero rate quantum codes [9] that do not have any classical analog can be achieved. Using pre-shared entangled states between the encoder and decoder, we recently proposed a general framework for constructing EA stabilizer codes over qudits of dimension d = p k from first principles in [10].…”