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:

Where p_i 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: