[Paper Review] Creating superpositions that correspond to efficiently integrable probability distributions

Lov K. Grover, Terry Rudolph|ArXiv.org|Aug 15, 2002

Quantum Computing Algorithms and ArchitectureComputer Science244 citations

TL;DR

This paper presents an efficient quantum algorithm to prepare a superposition state corresponding to a discretized, efficiently integrable probability distribution—such as log-concave distributions—by recursively subdividing regions and applying controlled rotations based on classically computed cumulative probabilities. The key contribution is a method to coherently prepare quantum states with amplitudes proportional to the square roots of probabilities, enabling quantum interference and speedups in subsequent processing.

ABSTRACT

We give a simple and efficient process for generating a quantum superposition of states which form a discrete approximation of any efficiently integrable (such as log concave) probability density functions.

Motivation & Objective

• To develop a method for efficiently preparing a quantum superposition state |ψ⟩ = Σ√pᵢ|i⟩ corresponding to a discretized probability distribution {pᵢ}.
• To address whether such states can be created efficiently when the underlying distribution is efficiently integrable, such as log-concave distributions.
• To enable quantum advantage by allowing coherent manipulation of probability distributions through unitary operations, beyond classical sampling.
• To explore applications in quantum search with non-uniform priors and in estimating Fourier components via amplitude estimation.
• To demonstrate that quantum computation can coherently access classical probability distributions without requiring classical randomness as input.

Proposed method

• The method recursively subdivides the domain of a probability distribution into finer regions, starting from a coarse m-qubit state representing region probabilities.
• For each region i, the algorithm classically computes f(i) = ∫_{left half of i} p(x)dx / ∫_{region i} p(x)dx, the conditional probability of being in the left half.
• An ancilla register is prepared in a state |θᵢ⟩ where θᵢ = arccos(√f(i)), using a quantum circuit that coherently computes f(i).
• A controlled rotation of angle θᵢ is applied to an additional qubit, entangling it with the region state to produce √pᵢ⁽ᵐ⁾|i⟩(cosθᵢ|0⟩ + sinθᵢ|1⟩).
• The ancilla register is uncomputed to disentangle it, leaving the state in the desired form with increased qubit count.
• The process is iterated until n qubits are used, yielding a superposition over N = 2ⁿ states with amplitudes √pᵢ.

Experimental results

Research questions

• RQ1 Can a quantum computer efficiently prepare a superposition state |ψ⟩ = Σ√pᵢ|i⟩ for a discretized probability distribution {pᵢ} when the cumulative distribution function is efficiently integrable?
• RQ2 Is it possible to coherently generate such a superposition using only quantum operations, even if the classical integration algorithm is probabilistic?
• RQ3 How can non-uniform priors be incorporated into quantum search algorithms using this method?
• RQ4 What quantum advantages arise from preparing such a superposition compared to classical sampling?
• RQ5 Can this method be extended to multivariate probability distributions?

Key findings

• The algorithm efficiently prepares a quantum superposition state corresponding to a discretized, efficiently integrable probability distribution using recursive subdivision and controlled rotations.
• The method works for all log-concave distributions—including Gaussian, exponential, and Poisson—due to their efficiently computable cumulative distribution functions.
• The use of ancilla qubits to coherently compute and apply rotation angles ensures that probabilistic classical integration algorithms can be simulated coherently on a quantum computer.
• The resulting superposition enables quantum interference effects not accessible via classical sampling, such as through Walsh-Hadamard transformations that generate non-log-concave output distributions.
• The method supports applications in amplitude estimation and quantum search with non-uniform priors, potentially enabling quadratic speedups over classical counterparts.
• The construction is scalable and maintains efficiency even when the number of states N = 2ⁿ grows exponentially, provided the cumulative distribution is efficiently integrable.