Conservative Quantum Computing

Conservation laws limit the accuracy of physical implementations of elementary quantum logic gates. If the computational basis is represented by a component of spin and physical implementations obey the angular momentum conservation law, any physically realizable unitary operators with size less than n qubits cannot implement the controlled-NOT gate within the error probability 1/(4n^2), where the size is defined as the total number of the computational qubits and the ancilla qubits. An analogous limit for bosonic ancillae is also obtained to show that the lower bound of the error probability is inversely proportional to the average number of photons. Any set of universal gates inevitably obeys a related limitation with error probability O(1/n^2)$. To circumvent the above or related limitations yielded by conservation laws, it is recommended that the computational basis should be chosen as the one commuting with the additively conserved quantities.