Authors: Masataka Ohta
Errors on computations depend on their arguments and are, in general, different arguments by arguments. Though Shor used a simple error model that a qubit is disturbed/decohered locally by its environment state only, he overlooked a fact that an initial environment state around a qubit is output from a QEC (Quantum Error Correction) encoder and is affected by and entangled with all the argument qubits used to compute the qubit, which makes the errors depend on the argument qubits. As linear superposition for quantum parallelism keeps the errors different, usual QEC schemes just do not work. It is actually demonstrated that Shor code fails to correct a single qubit error if an input qubit to an encoder is entangled with an external qubit. Quantum block codes are not helpful to correct errors on qubits of a block entangled with qubits outside of the block. Though it may be possible to construct an improved QEC circuit for N quantum parallel computations to correct N different errors, detections and corrections of N different errors require O(N) information, which makes hardware complexity of the circuit O(N), which is no better than classical N parallel computations with N parallel hardware, which means quantum supremacy with QEC is denied.
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