Hardness of classically sampling quantum chemistry circuits

Abstract: Significant advances have been made in the study of quantum advantage both in theory and experiment, although these have mostly been limited to artificial setups. In this work, we extend the scope to address quantum advantage in tasks relevant to chemistry and physics. Specifically, we consider the unitary cluster Jastrow (UCJ) ansatz-a variant of the unitary coupled cluster ansatz, which is widely used to solve the electronic structure problem on quantum computers-to show that sampling from the output distributions of quantum circuits implementing the UCJ ansatz is likely to be classically hard. More specifically, we show that there exist UCJ circuits for which classical simulation of sampling cannot be performed in polynomial time, under a reasonable complexity-theoretical assumption that the polynomial hierarchy does not collapse. Our main contribution is to show that a class of UCJ circuits can be used to perform arbitrary instantaneous quantum polynomial-time (IQP) computations, which are already known to be classically hard to simulate under the same complexity assumption. As a side result, we also show that UCJ equipped with post-selection can generate the class post-BQP. Our demonstration, worst-case nonsimulatability of UCJ, would potentially imply quantum advantage in quantum algorithms for chemistry and physics using unitary coupled cluster type ansatzes, such as the variational quantum eigensolver and quantum-selected configuration interaction.

Hardness of Classically Sampling Quantum Chemistry Circuits

The paper "Hardness of Classically Sampling Quantum Chemistry Circuits" presents a nuanced contribution to the domain of quantum advantage, specifically focusing on tasks relevant to quantum chemistry and physics. The authors investigate the difficulty of classically simulating the sampling of quantum circuits that employ the unitary cluster Jastrow (UCJ) ansatz. This ansatz, a variant of the unitary coupled cluster widely used in quantum electronic structure problems, is shown to be likely classically hard to simulate under reasonable complexity-theoretical assumptions.

Summary of Main Insights

The core argument of the paper revolves around establishing the classical intractability of sampling from the output distributions of quantum circuits that implement the UCJ ansatz. By concentrating on the 1-UCJ form—a single layer unitary cluster Jastrow—they demonstrate that the worst-case complexity scenario suggests sampling from these circuits cannot be performed in polynomial time, given the assumption that the polynomial hierarchy does not collapse.

The paper leverages the complexity of instantaneous quantum polynomial-time (IQP) computations, known to be classically hard to simulate, which provides a foundational comparison. The primary result indicates that specific UCJ circuits can emulate any IQP circuit capable of executing universal quantum computation. Thus, the impossibility of weak classical simulation of these UCJ circuits emerges directly from established results concerning IQP complexity. In particular:

• Post-Selection Capabilities: The study extends to showcase that UCJ equipped with post-selection can generate post-BQP computations, aligning with the classical hardness of post-IQP.
• Complexity Hierarchy: The authors detail a complexity hierarchy, demonstrating that 1-UCJ$_{\rm JW}$ (where JW denotes the Jordan-Wigner transformation applied to the ansatz) includes IQP, thus inheriting the unsimulability features.

Practical and Theoretical Implications

This exploration into UCJ circuits has significant implications for quantum algorithms applied in chemistry and physics. Notably, methodologies such as the variational quantum eigensolver and quantum-selected configuration interaction stand to benefit by potentially demonstrating quantum advantage through these circuits.

From a theoretical perspective, the work presents a compelling argument for the expanded applicability of quantum advantage concepts to realistic tasks beyond contrived setups. The use of UCJ circuits could bolster the practical adoption of quantum computing in industry-specific problems where electronic structure calculations are pertinent.

Future Directions

The paper suggests several avenues for further exploration:

1. Average-Case Complexity: Investigating whether the average-case realizations of UCJ circuits exhibit the same level of complexity would help refine boundaries of quantum-classical computational demarcation.

2. Noisy Implementations: Given near-term quantum technologies are characterized by noise, assessing the performance and advantage under noisy conditions remains crucial for real-world applicability.

3. Expectation Value Estimation: While sampling is shown to be hard, the classical simulatability of expectation value tasks using UCJ circuits is an open question, akin to issues with IQP circuits.

Overall, the findings underscore the potential of expanding quantum advantage to circuits with direct applicability in quantum chemistry, encouraging broader focus on such practically relevant quantum workloads.