Notes (Quantum Computing)

# Notes (Quantum Computing) These are only meant as a personal note and not intended as formal study. Beyond Unitary Evolution --- When you look at what quantum algorithm does you always end up rotation on a block sphere in 3d space or 2d space for simplicity. But why is any operation/transformation on a quantum state represented by rotation on the 3d bloch sphere? <center> <figure align="center"> ![blochsphere](https://hackmd.io/_uploads/S198t8Tjel.png) <figcaption>Figure 1: Bloch sphere representation</figcaption> </figure> </center> The reason is quite elegant. According to postulate of quantum mechanics, a quantum state in closed system follows unitary evolution which preserves it's norm and inner product. These unitary matrix acting on a quantum state are a part of SU(n) group, which is a group of special unitary complex matrices of size n*n with determinant 1. Eg: SU(2) is a group of 2*2 unitary complex matrix with determinant 1 Another special group is SO(n) which is a group of special orthogonal matrices which correspond to rotation in n*n space. An interesting property that we get is that "SU(2) is double cover of SO(3) which represent rotation in 3d space". Meaning for {U,-U} in SU(2) maps to one R in SO(3). **(Spinors if you know what I mean)** Another intresting property is that "SU(2) can be decomposed as sequence of rotations(Rx,Ry,Rz) with a phase in 2x2 complex hilbert space". **"So it's a map from 2d complex hilbert space to real 3d space"** So moving on, my uneazyness stemmed from this limitation that every quantum operations are just sequence of rotations in a bloch sphere. This to me is a profound limitation that bounds our freedom to design any quantum algorithm. This may be the reason why it's so hard and tricky, because it's all straight forward rotations. Okay, so we must find a way out. A way out of bloch sphere, essentially way out of SU(2) or any unitary operation. What would that be? Short answer is non unitary operation that too on closed system. See, open quantum system evolves through non unitary evolution but not closed. If we introduce non unitary evolution on closed quantum system we're essentially rejecting one postulate of quantum mechanics. I believe there's a video lecture by Scott Aaronson on how non unitary evolution leads to CTC (Closed time curves), maybe similar to Gödel rotating universe. Nevertheless, what I initially wanted to do was to entangle two qubits, decompose unitary transformation U to two non unitaries T1,T2 such that when I apply T1 on the first quantum state there's some sort of phase kickback and T2 operates in maybe reverse of T1 on the second qubit. Makes no sense right? Short form what I wanted to do was to break unitary into non unitaries such that when these non unitaries were applied to entangled states the overall operation would still be unitary. But I quickly found out that it's not possible for the overall operation to be unitary without the non unitaries themselves be unitaries thus a contradiction. It's like jumping from a plane without parachute. **Notably, Given's decomposition allows us to decompose any dxd unitary matrix into d(d-1)/2 two level unitary matrices**. If I ignore the assumption that overall operation is unitary and leave it as non unitary transformation on closed system then I'd need to fight 100 year old highly tested quantum mechanics. To think I'd win would be utterly naive. A strong and easy counter argument to reason from would be to reason from the truth that hilbert space is not real. Doesn't even have to be an hibert space, we could cook some other space up and project it to the physical space. So, technically going beyond hilbert space would be bs. PS: A good resource for visualizing how hilbert space and projective space are connected: https://youtu.be/KEzmw6cKOlU?si=1Y-sEj4JoiKVEjJD <hr> It's absolutely perplexing to find out that conjugate variable in heisenberg uncertainity principle are fourier transform pairs. It's wild how deeply fourier transform takes part in defining fundamental nature of our reality. <hr> # Does relative phase change due to relative change in time provide a way for quantum speedup in interference based algorithm? Don't know. Most likely not. Probably bs. But will be fun to figure out theoretical bounds on this one. <hr> The naive wish to simulate the entire universe. Even simulating a single cell feels impossible at least for a few hundred years. Will humanity prevail?