Last night’s insomnia rabbit hole: Do mirrors flip images left-right? Why not up-down? Our brains seem to have a bias towards certain rotations, probably due to the symmetry and typical motion of the human body. And this can help explain why special relativity is so counterintuitive. (1/7) 🧵🧪⚛️

The tool to do this is just the Hadamard gate again. This uses interference at a beamsplitter to turn two packets into one whose direction depends on whether the packets are in-phase or out-of-phase. (12/15)

Here is the same thing, but now simulated as a quantum wave function for all four possible sets of outputs. The initial state – a superposition of two wave packets could be obtained using the Hadamard gate (a beamsplitter) shown above. (7/15)

Classically, we have to plug in both inputs separately to find out, using the box twice. But if the box operates with qubits, perhaps we can somehow put our input qubit in as a superposition of 0 and 1 (generated using a Hadamard gate as shown here), and get the answer in one shot. (4/15)

Here’s the evolution of the wave function for the CNOT operation. Two particles collide when their positions are equal – when a blob hits the dashed diagonal line. At the end, the two lower blobs are swapped. We just need to reposition the blobs and apply a phase gate to fix the phases. (11/14)

By placing the barrier just off from the midpoint between the packets, a particle incoming from the left results in two in-phase packets, and from the right, in two out-of-phase packets. The reverse is also true: the Hadamard gate translates phase information into position of the particle. (7/14)

Now the Hadamard gate, which turns a single packet into a superposition of two packets. This is just a 50/50 beamsplitter – a narrow barrier with 50% reflection and 50% transmission. (6/14)

Next a phase gate. Here one of the packets passes through a region of attractive potential (dashed red line), shifting its phase, as shown by the color scale. The phase shift could be anything, but here it’s 180 deg., making this a Z gate. (5/14)