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I think that ahelwer's answer touches on some the ways that we think about the complexity of algorithms. However — given that we don't literally have "oracles" in the real world which we wish to query, you might wonder why we would worry about query complexity, or the idea of oracles at all. I will try to give some perspective on this, and in ...


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Another way to think about this: To simulate what goes on in a quantum computer we have to do a lot of matrix math using $(2^N \times 2^N)$ matrices$^1$, and the action of (most) of the clifford gates can be actually be accomplished by applying some non-linear, low complexity matrix operation instead of a matrix multiplication. For example, the Pauli-X gate,...


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One of the simulators in Microsoft Quantum Development Kit is Toffoli simulator which seems to do exactly what you want. It supports a limited set of primitive gates (X, CNOT and Toffoli gates, as well as other gates when their effect is X or identity), measurements in the computational basis and DumpMachine to output the state of the simulator. It is a ...


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It is important to realize that the depolarizing and dephasing channel (and pretty much any other noise model for that matter) do not represent unitary operations. This means there is no unitary operation (that takes qubit states to qubit states) corresponding to these channels. Rather, channels are more general: they map density operators to density ...


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To my mind, this theorem is not very well stated in this form, if taken out of context. Where it says "phase gates", this may be misleading. It means specifically just $S=\sqrt{Z}$ and not what I think of as a phase gate, which can have an arbitrary phase (but they have very specifically introduced their terminology about 3 pages earlier). This is a key ...


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