Don't Hit the Ground | PennyLane Challenges

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Challenge statement

This challenge is included in the QHack 2023 Flashback Badge Challenge event.

Preparing a fiducial state, usually denoted by $\lvert 0 \rangle,$ is the first step before carrying out any quantum computations. For most quantum computers, this is a straightforward process (although sometimes energy and time consuming). We need to bring the quantum device to almost absolute zero so that it relaxes to its ground state —the state of minimal energy— which is our choice of fiducial state.

Why does this happen? Quantum systems are never really isolated, so they will exchange energy with their environment. The net effect is that any quantum properties of the system's state, i.e. superpositions and entanglement, are lost after some time.

How do we model this energy exchange at finite temperature? The Generalized Amplitude Damping channel provides a good approximation. Suppose $\gamma$ is the photon loss rate at zero temperature, and $p$ is the probability that a qubit emits a photon to the finite-temperature environment (the system will also absorb photons with probability $1-p$ ). We can approximate the interaction with the environment for a duration $t$ via the circuit below.

That is, we compose many Generalized Amplitude Damping channels with infinitesimal noise parameters $\gamma\Delta t$ and de-excitation probability $p$ . A shorter step $\Delta t$ gives a more precise calculation, but we will need more Generalized Amplitude Damping channels to model the same duration $T$ .

We would like to know how quickly a system can relax to its fiducial state, given some photon loss rate $\gamma$ and emission probability $p$ . Assuming that the system is in the initial state

\lvert + \rangle = \frac{1}{\sqrt{2}}\lvert 0\rangle + \frac{1}{\sqrt{2}}\lvert 1\rangle,

your task is to estimate the relaxation half-life, which is the time at which we obtain the outcome $\lvert 1 \rangle$ with probability 1/4 (the measurement is performed in the computational basis).

Challenge code

You must complete the half_life function to calculate the time $T$ at which the probability of measuring $\lvert 1 \rangle$ becomes 1/4.

Input

As input to this problem, you are given:

gamma (float): The zero-temperature photon loss rate.
p (float): The de-excitation probability due to temperature effects

Output

This code will output a float equal to your estimate of the relaxation half-life. Note that you may require the step and iterations of your circuit to actually reach the half-life.

Test cases

The following public test cases are available to you. Note that there are additional hidden test cases that we use to verify that your code is valid in full generality.

test_input: [0.1,0.92]
expected_output: 9.05
test_input: [0.2,0.83]
expected_output: 7.09

If your solution matches the correct one within the given tolerance specified in check (in this case it's a 2e-1 relative error tolerance), the output will be "Success!". Otherwise, you will receive an "Incorrect" prompt.

Good luck!