Short Quantum Accumulate Codes with High Rate and Multiple Error Corrections Capability

This paper proposes short but high rate quantum codes, derived from the classical accumulate codes, having capability of correcting multiple errors simultaneously. The high coding rate is obtained from k = 2 with minimum blocklength n, but still satisfying both Singletone and Hamming bounds to have error correction capability of t = 2 qubits, We perform an intensive computer search to find the best matrix of parity check that match with the accumulator matrix to provide zero simplectic inner product (SIP) mod 2. We found that it is difficult to obtain a perfect parity check matrix that provide zero diagonal element in SIP to guarantee completely unique syndromes. However, since it may be difficult, we also present in this paper a theoretical quantum word error rate (QWER), when some syndromes do not have unique values to identify the quantum errors of X, Z, and Y. The results in this paper are expected to provide groundbreaking foundation on the development of high rate quantum codes with $k\geq 2$, especially for short blocklength.