Nice to have you back john! some things that come to mind on the topic: 1. Seeing the definition of 2-qubit operator , is the inner product still a 2x2 matrix, instead of scalar? (because its still sandwiched between 2 states) 2. looking at your illustration of identity matrix in the two qubit case, can i say that for the identity operator, the whole thing "factorizes" into two separate parts (just |i> and just |k>) while in the general unitary evolution, |i> state can influence the future of |k> and vice-versa? 3. So in general, U and U dagger are different than U tilde and U tilde dagger? Are there even more special cases where U = U tilde or U dagger = U tilde? 4. is the "unitarity in both time and space" in any way inspired by how special relativity treats space and time coordinates?
Hey! All great questions. 1. Indeed it would end up being an operator :) you do similar calculations when performing a partial trace for example. 2. Yes! Precisely, the 2 qubit gate would entangle the qubits, but applying the adjoint of the unitary would unwind this entanglement, giving us back what ever our initial state was. 3. The swap matrix is precisely a matrix like this! 4. I believe there is some philosophical motivation like that, but I like to think about it in the sense that, 2 qubit gates (if you forget about tiny details like what the matrix elements are) look very similar whether you evolve in time or "evolve in space". It's just 2 legs in, 2 legs out.
Great question. In my eyes this channel does the "path to equilibrium" and describing equilibrium. Unitary circuits are just a "new" type of dynamics to throw into that puzzle :). And they can be connected to things like driven systems. So overall, I won't be going too much into quantum computing unless it's useful :).
You can think of it in a few ways. One neat way is to rewrite it as: S exp(I J ZZ), where this J is just a shift of J_z. So it can be thought of as a swap times an evolution generated by a ZZ interaction!
Nice to have you back john! some things that come to mind on the topic:
1. Seeing the definition of 2-qubit operator , is the inner product still a 2x2 matrix, instead of scalar?
(because its still sandwiched between 2 states)
2. looking at your illustration of identity matrix in the two qubit case, can i say that for the identity operator, the whole thing "factorizes" into two separate parts (just |i> and just |k>) while in the general unitary evolution, |i> state can influence the future of |k> and vice-versa?
3. So in general, U and U dagger are different than U tilde and U tilde dagger?
Are there even more special cases where U = U tilde or U dagger = U tilde?
4. is the "unitarity in both time and space" in any way inspired by how special relativity treats space and time coordinates?
Hey! All great questions.
1. Indeed it would end up being an operator :) you do similar calculations when performing a partial trace for example.
2. Yes! Precisely, the 2 qubit gate would entangle the qubits, but applying the adjoint of the unitary would unwind this entanglement, giving us back what ever our initial state was.
3. The swap matrix is precisely a matrix like this!
4. I believe there is some philosophical motivation like that, but I like to think about it in the sense that, 2 qubit gates (if you forget about tiny details like what the matrix elements are) look very similar whether you evolve in time or "evolve in space". It's just 2 legs in, 2 legs out.
@@JonathonRiddell As a follow up to question 1; I don't see how is still an operator. Shouldn't it be a matrix element not an operator?
Yes taking it with all 4 is a scalar / a matrix element.
by the way, your previous vids seem to be a lot about stat mech (especially equilibriation), are you pivoting more to the quantum computing side now?
Great question. In my eyes this channel does the "path to equilibrium" and describing equilibrium. Unitary circuits are just a "new" type of dynamics to throw into that puzzle :). And they can be connected to things like driven systems.
So overall, I won't be going too much into quantum computing unless it's useful :).
How should the quantity (pi/4,pi/4,Jz) be thought about? Can I think about it like a regular unit vector like we usually do for spin rotations?
You can think of it in a few ways. One neat way is to rewrite it as:
S exp(I J ZZ), where this J is just a shift of J_z. So it can be thought of as a swap times an evolution generated by a ZZ interaction!
(Maybe I misunderstand your question)
Do I have a British accent yet?
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Dang I’m stupid 🤣
Hi! Waiting for some valuable response on my problem related simulation, I have sent through Mail.