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Many people say entanglement is essential. For example, the answer to this question seems to say so Is entanglement necessary for quantum computation?.

I understand that quantum encryption protocols are more secure if the qubits are not in a separable state. Otherwise, the qubits are confined in a subspace of dimension less than the overall Hilbert space.

I can understand that the immense dimension, $2^n$, of the Hilbert space, in which entangled states are inevitable, is the source of quantum computing power. I can understand many algorithms may not work if the entangled states are taken away from the Hilbert space. But is entanglement essential to quantum algorithms? Is it an essential source of quantum speedup? Deutsch's algorithm seems to offer a counter-example. In it, the two qubits are separable throughout the steps. In particular, the second qubit stays as $|-\rangle$ throughout the steps.

glS
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Yuan John Jiang
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When people talk about a speed-up, they (usually) mean that the problem they're trying to solve has different inputs, which can be parametrised by their size $n$. The best classical algorithm has a certain asymptotic behaviour as a function of $n$. A quantum algorithm has a speed-up if its scaling is better (it grows more slowly). Your example of Deutsch's algorithm is then a moot point - there is no with-size scaling because there are no instances of different sizes.

DaftWullie
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