What should we add?

Recently a novel efficient algorithm was proposed for preparation of uniform quantum superposition states, which was published in Quantum Information Processing journal. The algorithm tries to solve the problem of preparation of a uniform superposition state of the form

$$ | \psi> = \frac{1}{\sqrt{M}} \sum_{j=0}^{M-1} |j>, $$

where $M$ denotes the number of distinct states in the superposition state and $2 ≤ M ≤ 2^n$ . They also show that the superposition state can be efficiently prepared, using a deterministic approach, with a gate complexity and circuit depth of only $O(log_2 M)$ for all $M$. This demonstrates an exponential reduction in gate complexity in comparison with other existing deterministic approaches (including qiskit's) in the literature for the general case of this problem. Another advantage of the proposed approach is that it requires only $n =\lceil log 2 M \rceil$ qubits. Furthermore, neither ancilla qubits nor any quantum gates with multiple controls are needed in their approach for creating the uniform superposition state $|\psi>$.

Below I have provided the implementation of this algorithm.

import numpy as np
from qiskit import QuantumCircuit , QuantumRegister
""" Input: A positive integer M with 2 < M < 2^n, M not equal to 2^r for any natural number r.
Output: A quantum circuit that creates the uniform superposition state: 
$\frac{1}{\sqrt{M}} \sum_{j=0}^{M-1} \ket{j} $. Number of qubits: Using n = np.ceil(np.log2(M)), creates the uniform superposition state with the least number of qubits. 
"""
def uniform_superposition(M, n):
    N = [int(x) for x in list(np.binary_repr(M))][::-1]
    k = len(N)
    L = [index for (index ,item) in enumerate(N) if item ==1] #Locations of '1's
    qreg = QuantumRegister(n, 'q')
    quantum_circuit = QuantumCircuit(qreg)
    quantum_circuit.x(qreg[L[1:k]])
    Mcurrent = 2**(L[0])
    theta = -2*np.arccos(np.sqrt(Mcurrent/M))
    if L[0]>0: #if M is even
        quantum_circuit.h(qreg[0:L[0]])
    quantum_circuit.ry(theta , qreg[L[1]])
    quantum_circuit.ch(qreg[L[1]], qreg[L[0]:L[1]], ctrl_state='0')
    for m in range(1,len(L) -1):
        theta = -2*np.arccos(np.sqrt (2**L[m]/ (M- Mcurrent)))
        quantum_circuit.cry(theta , qreg[L[m]], qreg[L[m+1]], ctrl_state='0')
        quantum_circuit.ch(qreg[L[m+1]], qreg[L[m]:L[m+1]], ctrl_state='0')
        Mcurrent = Mcurrent + 2**(L[m])
    return quantum_circuit

If there's interest, it could be added to the circuit library, since it already some state preparation logic example.

The research paper used for creating this algorithm.
-- [An efficient quantum algorithm for preparation of uniform quantum superposition states][1]
[1]: Shukla, A., Vedula, P. An efficient quantum algorithm for preparation of uniform quantum superposition states. Quantum Inf Process 23, 38 (2024). https://doi.org/10.1007/s11128-024-04258-4