Average-case complexity versus approximate simulation of commuting quantum computations (1504.07999v2)

Abstract: We use the class of commuting quantum computations known as IQP (Instantaneous Quantum Polynomial time) to strengthen the conjecture that quantum computers are hard to simulate classically. We show that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error. One conjecture relates to the hardness of estimating the complex-temperature partition function for random instances of the Ising model; the other concerns approximating the number of zeroes of random low-degree polynomials. We observe that both conjectures can be shown to be valid in the setting of worst-case complexity. We arrive at these conjectures by deriving spin-based generalisations of the Boson Sampling problem that avoid the so-called permanent anticoncentration conjecture.

• The paper demonstrates that under plausible average-case hardness conjectures, IQP quantum computations are intractable to approximate even with constant additive error.
• The paper links the difficulty of IQP simulation to complex-temperature Ising model partition functions and the counting of zeroes in low-degree polynomials.
• The paper warns that efficient classical approximate simulations could collapse the Polynomial Hierarchy, underscoring a strong quantum advantage.

Average-case Complexity Versus Approximate Simulation of Commuting Quantum Computations

This paper by Bremner, Montanaro, and Shepherd explores the complexities associated with simulating quantum computations, specifically focusing on the class of quantum computations known as Instantaneous Quantum Polytime (IQP). The central thesis rests on the challenge of classically simulating quantum computers, emphasizing that quantum computations, particularly those involving IQP, could intrinsically be hard to simulate by classical means, assuming certain average-case hardness conjectures hold true.

Key Contributions

1. IQP and Complexity Conjectures: The authors investigate the complexity of simulating IQP computations, which are described as non-universal but still potentially classically inaccessible. They establish that if either of two plausible average-case hardness conjectures is valid, then IQP computations are challenging to simulate classically, even with a constant additive error. This strengthens the belief in the computational advantage of quantum over classical systems.
2. Complex-temperature Ising Model and Polynomial Zero Conjectures: The paper associates IQP hardness with average-case problems in two domains:
   • Approximating the complex-temperature partition function for random instances of the Ising model.
   • Counting the number of zeroes of random low-degree polynomials.
3. Average-case Hardness and Conjectures: The authors derive conjectures based on random-self-reducibility and anticoncentration phenomena, proposing that certain Ising model instances and polynomial evaluations are hard on average. These conjectures bridge the established worst-case complexity results with average-case recognition, reinforcing the computational difficulty of approximating IQP circuits.
4. Approximate Simulations: Through their analysis, they show that if efficient classical algorithms exist for such simulations (up to a specified additive error), it would imply unexpected complexity-theoretic collapses, such as the collapse of the Polynomial Hierarchy to a lower level—a highly unlikely scenario under contemporary understanding.

Implications and Future Directions

The implications of this research are notable in both theoretical and practical realms. The findings suggest that while outright proving quantum supremacy remains difficult due to the absence of conclusive lower bounds on classical computing, conditional empirical evidence based on complexity-theoretic conjectures could demonstrate such a claim. Practically, this reinforces the necessity to develop and refine quantum architectures capable of implementing IQP circuits efficiently, to empirically test the boundaries of classical simulability.

Moreover, the theoretical discourse opened by this paper presents fertile ground for future research. Establishing formal proofs for the proposed conjectures remains a pressing priority. Simultaneously, the elucidation of IQP circuit dynamics could reveal new pathways to quantum advantage in terms of specific computational tasks, particularly those rooted in statistical physics and combinatorial challenges.

This discourse on IQP fills an essential gap in understanding the capabilities and limitations of quantum computers, fostering advances toward a more defined landscape where quantum advantage can be leveraged for practical computational problems.