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quantum-kittens
GitHub Repository: quantum-kittens/platypus
 Path: blob/main/notebooks/summer-school/2021/lec3.1.ipynb
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Kernel: Python 3

Noise in Quantum Computers (part 1)

In this lecture, Zlatko discusses the different types of noise involved in real quantum computers. He considers the simple example of the X gate and compares the results of the ideal and real experiments. To reach noiseless computers, we need to discuss types of quantum errors. In this lecture, Zlatko discusses common coherent error using ideal and noisy gates. We conclude that coherent error will build up in a quadratic way and will affect the performance, irrespective of the circuit depth.

  • Download the lecturer's notes here

  • Download the lecturer's slides here

FAQ

What's the definition of coherent noise ? It can de described by a unitary, such as the XϵX_\epsilon .
Does the quadratic build up of coherent noise negate quadratic time speedups from something like Grover's algorithm? Or is it not so naively related? It’s quadratic in the error amplitude, which is different from quadratic and the number of qubits. If you assume that only one keep it in your many qubit register has noise, then it’s probably not so bad, but if you assume that every qubit has the same amount of noise, or similar level of noise, then the amount of total noise will scale also exponentially, as we will see in the second part of this lecture, especially in the case of state preparation errors, covered in some of the bonus material.
Is it safe to say that the PMF of Bernoulli is a Gaussian? Bernoulli is for only 2 outcomes, Binomial limits to Gaussian in the limit of large N.
So in theory the more samples/shots we have the more precise measurement because the result has biggest probability?Is there a reason to use as little shots as possible? Yes, ideally you take an infinite number of shots, but each shot takes time, so the more shots, the longer you have to wait. Also the error on the gates may drift over time so you may want to avoid that.
Are "involuntary measurements" represented by noisy gates? Incoherent noise discuss about it.
Are all quantum computation errors linear ? Or can they be non linear as well? Operations on quantum states - even errors - can usually be described by linear operations on the state - these can be non-linear on expectation values
Suppose that from time to time the state ∣1⟩|1\rangle decays into state ∣0⟩|0\rangle, but very rarely the opposite happens (the phenomenon can be described as spontaneous emission), is this kind of error representable with a noisy gate? It doesn't seem linear. The average evolution of the state (density matrix) is deterministic, smooth, and linear in the density matrix. On a shot by shot (quantum monte carlo trajectory) the evolution is not linear indeed. On average the Bloch vector will just shrink.
We leave the factor of two in the definition of the rotation because the actual angle by which the Bloch sphere will rotate is the angle of theta. What does it mean? Why should we keep the factor two? What's the relation between theta/2 in Rx(theta) and the expectation values of X,Y, Z?[slice 18, Visualize: Evolution on the Bloch sphere] The factor of two there is needed if you want the angle theta to be the angle by which the Bloch vector of rotates. You can see it because the difference in the eigenvalues of any of the Pauli operators is two, so it’s not normalized to one. In other words, the plus state rotates at +1, and the # minus state rotates at -1, so the difference between the two states is rotating at two times
In slide 18, How can you quickly find expectations along X, Y and Z without much calculation? Use the fact that RX(theta)R_X(theta) is the following (see image) and that acted on ∣0⟩|0\rangle or ∣1⟩|1\rangle gives ∣0⟩|0\rangle and X flips them. Then use the expectation value for X and Y. The other way is to use the density matrix and write it in terms of Paulis and then use the orthonormality of Paulis but that requires a little more density matrix familiarity.
How does σy apply to the y-axis of the sphere? If you write out the action of a rotation around sigma_y , denotes RY(theta)R_Y(theta), you will see that it does not touch ⟨Y⟩\langle Y\rangle for any state, but will swap between ⟨X⟩\langle X\rangle and ⟨Z⟩\langle Z\rangle only at a sinusoidal rate.