Just storing the amplitudes requires mindboggling amounts of computer memory. “Quantumcomputing For The Very Curious.” Quantam.country. March 18, 2019. Can you think of other heuristics that might help you guess an answer? Summing all this up: if we measure a qubit with state α∣0⟩+β∣1⟩\alpha |0\rangle+\beta|1\rangleα∣0⟩+β∣1⟩ in the computational basis, then the outcome is a classical bit: either 000, with probability ∣α∣2|\alpha|^2∣α∣2, or 111, with probability ∣β∣2|\beta|^2∣β∣2. Alright, let’s review what we’ve learnt. Note that it of course acts linearly on superpositions of computational basis states, as we expect for a quantum gate. explaining how to use quantum computers to efficiently simulate quantum field theories. I must admit that I do. Yet the evidence so far suggests our universe does allow such universal machines. QED. One purpose is, as explained earlier, to help strengthen your memory, so your new knowledge is consolidated in your long-term memory. I said that the CNOT leaves the control qubit alone, and modifies the target qubit. Let’s suppose we have three qubits, for instance, and computational basis states such as ∣000⟩,∣001⟩|000\rangle, |001\rangle∣000⟩,∣001⟩, and so on. And so the NOT and Hadamard gates correspond to the special cases where U=XU = XU=X and U=HU = HU=H, respectively. People will say a state like 0.6∣0⟩+0.8∣1⟩0.6|0\rangle+0.8|1\rangle0.6∣0⟩+0.8∣1⟩ is a superposition of ∣0⟩|0\rangle∣0⟩ and ∣1⟩|1\rangle∣1⟩. You might enjoy it, and feel as though you’re learning about basketball. So suppose we have a vector ∣ψ⟩|\psi\rangle∣ψ⟩ which can be written in component form as: ∣ψ⟩=[ab⋮z].|\psi\rangle = \left[ \begin{array}{c} a \\ b \\ \vdots \\ z \end{array} \right].∣ψ⟩=⎣⎢⎢⎢⎡ab⋮z⎦⎥⎥⎥⎤. What makes the Hadamard gate interesting as a quantum gate? That perhaps sounds strange! Having extolled the virtues of spaced-repetition testing, let’s try another question: This question is similar to an earlier question: “How many dimensions does the state space of a qubit have?” It may seem inefficient to have such similar questions, but it helps build fluency with the material when you have the “same” information encoded into memory in multiple ways, triggering off different associations. It was an amazing leap of imagination, and the foundation of modern computing. With that said, it’s different from those forms too. But it’s not quite the simplest possible quantum circuit. \end{aligned}∣x,y⟩→∣x,y⊕x⟩.. So let’s come back to our opening question: will aliens have computers? In order to argue that his machine could imitate any algorithmic process, Turing considered what operations a human mathematician could perform when carrying out an algorithm. By contrast, Shor’s quantum factoring algorithm would make factoring into a comparatively easy task, if large-scale quantum computers can be built. And, we hope, that may enable us to solve some computational problems faster. Here I present a very personal viewpoint, focusing on my own path to quantum entanglement and then on to applications. You might wonder if there’s some quantum gate which takes two bits xxx and yyy as input, in the computational basis, ∣x,y⟩|x,y\rangle∣x,y⟩, and then outputs a single qubit ∣x∧y⟩|x \wedge y\rangle∣x∧y⟩, where x∧yx \wedge yx∧y is just the logical AND of the bits xxx and yyy. Note that your answer here may be the same as your answer to the previous exercise, if you answered that exercise using just a single Toffoli gate and no other quantum gates. Another complication is that when you talk about an “efficient simulation” in computation you mean the time and space overhead isn’t too large. Yet the book’s goal—to understand nature in computational terms—is one widely shared by the quantum computing community.
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