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Uniform Quantum Superposition States

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Uniform quantum superposition states are a fundamental concept in quantum mechanics, representing a state where a quantum system exists in a linear combination of multiple basis states, with each basis state contributing equally to the overall superposition.

Definition

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In the context of an -qubit system, a uniform quantum superposition state is defined as: Here, represents the computational basis states of the -qubit system, and is the total number of distinct states in the superposition. The normalization factor ensures that the total probability of finding the system in one of the basis states is equal to 1.

Importance in Quantum Computation

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Uniform superposition states play a crucial role in quantum computation algorithms. They are often utilized as initial states or intermediate states during quantum computations. The ability to efficiently prepare uniform superposition states is essential for the implementation of various quantum algorithms (e.g., Grover's algorithm, Quantum Fourier Transform), as it impacts the overall efficiency and success of quantum computations.

Preparation of uniform quantum superposition states when

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For an -qubit system, Hadamard gates acting on each of the qubits (each initialized to the ) can be used to prepare uniform quantum superposition states when is of the form . In this case case with qubits, the combined Hadamard gate is expressed as the tensor product of Hadamard gates:

The resulting uniform quantum superposition state is then: This generalizes the preparation of uniform quantum states using Hadamard gates for any [1].

Measurement of this uniform quantum state results in a random random state between and .

Examples:

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Example 1:

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For a system with qubit, the Hadamard gate is applied to the single qubit:

Applying to yields the uniform quantum superposition state:

Example 2:

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For a system with qubits, the combined Hadamard gate is the tensor product of two Hadamard gates:

Mathematically, this is expressed as:

Applying to , yields the superposition states with equal weights.

Preparation of uniform quantum superposition states in the general case,

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An efficient and deterministic approach for preparing the superposition state with a gate complexity and circuit depth of only for all was presented in [2]. This approach requires only qubits. Importantly, neither ancilla qubits nor any quantum gates with multiple controls are needed in this approach for creating the uniform superposition state .

References

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  1. ^ Nielsen, Michael A.; Chuang, Isaac (2010). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 978-1-10700-217-3. OCLC 43641333.
  2. ^ Alok Shukla and Prakash Vedula (2024). "An efficient quantum algorithm for preparation of uniform quantum superposition states". Quantum Information Processing. 23:38 (1): 1–32. arXiv:2306.11747. doi:10.1007/s11128-024-04258-4.

Sources

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