Sanchayan Dutta's answer correctly points out that $|1\rangle-|1\rangle$ doesn't actually arise in your example problem, but it doesn't answer the question in the title: what is $|1\rangle-|1\rangle$? ...

If you initialize three qubits to $|0\rangle$, apply a Hadamard gate to each, then measure each in the computational basis, the result will be an independent coin flip for each bit: that is, any of ...

All of your math is correct. Both of the states that you calculated are entangled states. An entangled state is one in which there exist measurable properties of subsystems that are correlated. In (1),...

The tensor product of two objects with $m$ and $n$ components is an object with $mn$ components that consists of the pairwise products of the components of the inputs. The Kronecker product and the $v ... View answer 2 votes Your intuition is correct if you think of the superpositions as classical probability distributions, which is how they are usually described in popularizations. What's unique to quantum mechanics is ... View answer 2 votes As DaftWullie said in a comment,$\log m$is most likely a shorthand for$\lceil\log_2 m\rceil$. Traditionally in computer science, one looks only at the asymptotic complexity of algorithms. ... View answer Accepted answer 2 votes In Shor's (quantum) algorithm, you compute the modular exponent only a small number of times—far smaller than$N$. The algorithm is probabilistic, and if you're lucky you may compute it only once. ... View answer 2 votes Grover's algorithm is a Circuit SAT solver that finds a satisfying assignment in around$2^{n/2}$evaluations of the circuit, where$n$is the number of inputs. You can build a circuit that takes a ... View answer 2 votes What makes quantum wave mechanics different from classical is that the wave is defined over a configuration space with a huge number of dimensions. In nonrelativistic undergraduate quantum mechanics (... View answer 1 votes Your question talks about the theoretical capabilities of theoretical computers. Anyone who says that quantum computers won't replace classical computers is probably talking about the actual ... View answer Accepted answer 1 votes Where does that single evaluation of$f(x)$actually occur? Is it in the construction of$U_f$?$U_f$is an implementation of$f$in quantum gates. The evaluation of$f$occurs in the course of ... View answer 1 votes Grover's algorithm is a (quantum-)circuit-SAT solver. I suppose it could also be a literal black box solver, but it would only work with black boxes that don't decohere your entangled input state, and ... View answer 1 votes From the question title, it sounds like you're interested in brute-force password cracking. There is a quantum algorithm for this that outperforms brute force, in principle. It's called Grover's ... View answer 0 votes Grover's algorithm is a CIRCUIT-SAT solver. Given a circuit with$n$boolean inputs, it finds a satisfying input in$O(2^{n/2})\$ evaluations of the circuit in the worst case, which is interesting ...