The Computational Complexity of Linear Optics (1011.3245v1)

Abstract: We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.

• The paper demonstrates that if a classical algorithm could sample from BosonSampling, the polynomial hierarchy would collapse.
• It employs matrix analysis and random matrix theory to link the hardness of the permanent function to classical simulation limits.
• The results underscore quantum supremacy and set vital benchmarks for verifying sampling problems in quantum computing research.

The Computational Complexity of Linear Optics

Introduction

The paper "The Computational Complexity of Linear Optics" by Scott Aaronson and Alex Arkhipov investigates the intersection of theoretical computer science and quantum optics. It tackles the feasibility of simulating certain quantum processes with classical computations, particularly focusing on linear-optical elements and their computational implications. This essay summarizes the paper's core contributions, methodologies, implications, and its potential avenues for future research in AI and quantum computation.

Overview and Main Contributions

At its core, the paper proposes and explores the BosonSampling problem, characterized by generating identical photons, passing them through a linear-optical network, and performing nonadaptive photon counting in various modes. The authors assert that, under reasonable complexity assumptions, no classical algorithm can efficiently simulate this model. Key contributions include:

1. Polynomial Hierarchy Collapse: The paper shows that if a classical polynomial-time algorithm could sample from the same distribution as a linear-optical network, it would result in the collapse of the polynomial hierarchy to the third level. This extends the work on quantum supremacy by Shor, demonstrating inherent computational hardness in quantum systems beyond factoring.
2. Approximate Simulation Hardness: Even approximate or noisy classical simulations of such quantum systems imply significant complexity-class collapses, contingent on two conjectures: the Permanent-of-Gaussians Conjecture and the Permanent Anti-Concentration Conjecture.
3. The Permanent Function and #P-Hardness: The authors rigorously connect the problem to the permanent function of matrices, a known #P-complete problem, establishing that approximating the permanent for matrices with Gaussian entries is computationally intractable for classical machines.

Methodologies

To establish these results, the authors employ a multi-faceted approach:

1. Complexity Theory Techniques: They leverage reductions and classical simulation cases to bridge the gap between linear-optical processes and computational hardness results.
2. Matrix Analysis and Random Matrix Theory: Utilizing properties of Haar-random unitary matrices and their submatrices, the authors provide probabilistic bounds and concentration inequalities for the permanent of Gaussian matrices.
3. Interdisciplinary Methods: The paper interweaves quantum physical properties with deep theoretical results from computer science, demonstrating the feasibility of linking physical processes with computational complexity outcomes.

Numerical and Bold Claims

The central numerical results that underscore their arguments include:

1. Probability Estimates: Rigorous bounds show that the probability distribution from sampling problems associated with linear-optics is far from trivial, emphasizing #P-hardness-related distributions.
2. Error Bounds: The combination of permanent-related conjectures and anti-concentration bounds ensures that even approximate classical simulations retain unacceptable errors under classical computation constraints.

Implications for AI and Quantum Computing

The implications of this research are profound:

1. Quantum Supremacy: The work provides strong evidence that certain quantum computations enjoy a clear, provable advantage over classical counterparts, reinforcing the concept of quantum supremacy.
2. Sampling Problem Frameworks: It introduces a robust framework for exploring the hardness of sampling problems, which could provide insights for other complex systems in AI and computational physics.
3. Classical Simulation Barriers: By demonstrating intrinsic barriers to classical simulations, the paper outlines vital checks needed for validating quantum versus classical computational experiments, particularly in noisy environments.

Future Directions

The paper opens several avenues for future exploration:

1. Broader Quantum Systems: Extending the results to other quantum models and understanding the implications on more general computational paradigms.
2. Algorithmic Improvements and Error Correction: Investigating refined algorithms and sophisticated error-correction techniques to bring theoretical results closer to practical quantum computations.
3. Verification Protocols: Developing more robust interactive protocols or sampling verifications that can be feasibly implemented with experimental setups.

Conclusion

Aaronson and Arkhipov's paper rigorously articulates the computational complexity inherent in linear optics, suggesting that quantum computers, even of rudimentary forms, potentially outperform classical machines in specific sampling tasks. By tying the problem to the mathematical hardness of the permanent function and establishing robust conjectures, this work forms a cornerstone in demonstrating quantum computational superiority, promising substantial impacts on both theoretical and experimental fronts in computer science and physics.