|Fidelity of Finite Length Quantum Codes in Qubit Erasure Channel
|Alexei Ashikhmin, Bell Labs, Nokia, United States
|Q.3: Quantum Error Correcting Codes and Decoding
|Quantum Systems, Codes, and Information
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|We consider the probability of decoding error, equivalently the fidelity, of finite length quantum codes assuming quantum information transmission via qubit erasure channel. First, we obtain an analytical expression on this probability via combinatorial invariants of quantum codes. Next, we derive a generalization of the MacWilliams identities for those invariants and use them to formulate a linear programming optimization problem, which allows us to obtain lower bounds on the number of unrecoverable sets of $i$ erasures and the probability of decoding error. Our example shows that the obtain bounds can be very tight. Next, we derive upper (achievability) bounds on the probability of decoding error for finite length stabilizer and CSS codes. CSS codes have many advantages for practical applications. Our results show that CSS codes of small lengths, like 100 qubits, visibly lose in performance to generic stabilizer codes.