[Paper Review] A Quantum Optics Argument for the #P-hardness of a Class of Multidimensional Integrals

Peter P. Rohde, Dominic W. Berry+2 more|arXiv (Cornell University)|Jul 18, 2016

Optical Network Technologies3 citations

TL;DR

This paper establishes that a class of multidimensional integrals arising in quantum optics is #P-hard by proving their computational equivalence to matrix permanents—known to be #P-hard—using a quantum optical formalism. By expressing BosonSampling output amplitudes in phase-space via characteristic functions, the authors show that evaluating these integrals is as hard as computing permanents, providing a novel quantum physics-inspired approach to computational complexity theory.

ABSTRACT

Matrix permanents arise naturally in the context of linear optical networks fed with nonclassical states of light. In this letter we tie the computational complexity of a class of multi-dimensional integrals to the permanents of large matrices using a simple quantum optics argument. In this way we prove that evaluating integrals in this class is extbf{\#P}-hard. Our work provides a new approach for using methods from quantum physics to prove statements in computer science.

Motivation & Objective

• To establish the computational complexity of a specific class of multidimensional integrals arising in quantum optical systems.
• To demonstrate that evaluating these integrals is #P-hard by leveraging the known #P-hardness of matrix permanents.
• To provide a new method for analyzing computational complexity using tools from quantum optics, particularly phase-space representations.
• To show that the equivalence between permanent-based and integral-based formalisms in BosonSampling enables cross-fertilization between quantum physics and computational complexity theory.
• To illustrate that special cases (e.g., permutation matrices) yield classically efficient integrals, confirming consistency with known permanent complexity results.

Proposed method

• Formalize the output amplitudes of linear optical networks using quantum optical characteristic functions in phase space.
• Express the output state as a multidimensional integral over complex amplitudes α, using the Wigner function and Gaussian state representations.
• Derive an explicit integral expression for the probability amplitude of a given photon-number configuration, equivalent to the permanent-based formalism.
• Establish mathematical equivalence between the integral form and the matrix permanent via symmetrization and Gaussian integration techniques.
• Use the known #P-hardness of permanent computation to infer the #P-hardness of the corresponding integral evaluation.
• Analyze special cases (e.g., permutation matrices) to show separability of the integral, enabling efficient classical evaluation.

Experimental results

Research questions

• RQ1 Is the evaluation of the class of multidimensional integrals arising in quantum optical phase-space formalism computationally hard?
• RQ2 Can the #P-hardness of matrix permanents be transferred to a corresponding class of integrals through physical equivalence in quantum optics?
• RQ3 How does the structure of the integral formalism reflect the computational complexity of the underlying permanent?
• RQ4 Are there identifiable structural features in the integrals (e.g., separability) that correspond to classically tractable cases of the permanent?
• RQ5 Can quantum optical tools be systematically used to derive complexity-theoretic results about mathematical integrals?

Key findings

• The class of multidimensional integrals derived from the phase-space representation of BosonSampling output amplitudes is #P-hard to compute in the worst case.
• The integral formalism is mathematically equivalent to the permanent-based formalism, and since permanents are #P-hard, so are these integrals.
• For permutation matrices, the integral becomes separable and evaluates exactly to 1, confirming that such cases are classically efficient—consistent with the fact that perm(σ) = 1 for all σ.
• The integral expression for the output probability amplitude is given by a multidimensional Gaussian integral involving products of |α_j|² and phase-space variables.
• The equivalence between the permanent and the integral formalism is exact and holds for all input states and unitary transformations in the linear optical network.
• This work demonstrates that quantum optics tools can be used as a novel framework to prove complexity-theoretic statements about mathematical problems.