# Log-normal State Preparation

## Background

Many quantum algorithms rely on initializing qubits in a specific state. The promised speedup of the algorithm depends on the ability to prepare the quantum state efficiently. The challenge of preparing a quantum state is an example of a broader use case of compiling isometrics into specific quantum circuit and it is known that, in general, an exponential number of gates, O(2n), are needed.

One of the popular distributions is the log-normal distribution, used in many places such as in the Black-Scholes model option pricing formula. The log-normal distribution is given by:

$f(x) = \frac{1}{x\sigma\sqrt{2\pi}}exp^{-\frac{(\ln(x)-\mu)^2}{2\sigma^2}}$

## The Problem

In this assignment, you are tasked with preparing a log-normal distribution with μ = 0 and σ = 0.1.

## Conditions

• You may use up to 10 qubits and split them between state and auxiliary qubits as you like.
• You may choose any mapping between the state vector domain D to the distribution domain (i.e. the discretization) as long as it is continuous.
• The target gates set is a general single-qubit rotation ’u’ as well as ’CX’
• The target distribution error should be err < 10−2 , where the metric is the L2 norm and values outside the state-vector domain are assumed to be zero:
$err$
$= \left( \int_{-\infty}^\infty \left( p(x) - f(x)\right) ^2 dx\right) ^{0.5}$
$=\left( \sum\limits_{i=0}^{2^n-1} \displaystyle\int_{x \isin D_i} \Big( \frac{p_i}{|D_i|} - f(x)\Big) ^2dx + \int_{x\notin D} f(x)^2dx\right) ^{0.5}$

Where pi is the probability of the ’i’ state (i.e p3 corresponds to |11⟩ ) and n is the number of state qubits.

For example: Suppose you have two qubits prepared with the state $\frac{1}{3}$ |00$\rangle$ +$\frac{2}{3}$ |01$\rangle$+ $\frac{2}{3}$ |10$\rangle$ + 0|11$\rangle$ and choose the discretized domain as (0.1, 1, 1.5,2 ,3), then the error will be:

$err^2$
$= \displaystyle\int_{0.1}^1 \Big( \frac{\frac{1}{3}^2}{0.9} - f(x)\Big) ^2 dx +$
$\displaystyle\int_{1}^{1.5} \Big( \frac{\frac{2}{3}^2}{0.5} - f(x)\Big) ^2 dx +$
$\displaystyle\int_{1.5}^2 \Big( \frac{\frac{2}{3}^2}{0.5} - f(x)\Big) ^2 dx +$
$\displaystyle\int_{2}^3 \Big( \frac{0^2}{ 1} - f(x)\Big) ^2 dx +$
$\displaystyle\int_{0}^{0.1} f(x)^2 dx +$
$\displaystyle\int_{3}^\infty f(x)^2 dx$

The first four terms calculate the error over the three discretized domain (0.1 to 1, 1 to 1.5, 1.5 to 2, and 2 to 3) and the last two terms calculate the error over the range from 0 to 0.1 and then from 3 to infinity, where there is no estimation.

For example Python code that calculates L2 distance, see our Competition support site

## Success metric

The winning submission will be the solution that meets the above conditions and has the shortest circuit depth.

When submitting, please make sure you note:

• What is the discretization that you used?
• Which qubits are to be measured for the log-normal distribution and which qubits were auxiliary qubits? Use no more than 10 qubits in total.

## Questions?

Post your questions or see what others have asked on our Competition support site.

We look forward to reviewing your solution. Here are some things to know:

You are welcome to submit multiple solutions. If you submit something today, and find an improvement later on, you're welcome to submit the new solution as well.
Each problem requires slightly different information. Please make sure you carefully review the problem instructions.
The description of the approach is very helpful to the judges in understanding your work. Please include as much detail as you wish. We're interested in how you approached this problem, how your solution works, how long it took you to develop it, and whether there were particular problems that you overcame. We'd love to hear as much as you want to tell us. You can submit this in the 'description' field or include it as the 'Additional file (optional)' attachment.
The QASM code is how we can verify the 2-qubit gate count and/or circuit depth. It should include only CX and single-qubit unitary gates. This note on our Discourse support site might help.
If you created the circuit in something other than QASM, please submit the source code as well.
You will receive an email confirming the submission within a few minutes.
Submissions are closed. the results are here. Thank you!
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*QASM Code