Fermionic-Adapted Shadow Tomography Efficiently Calculates Multiple Dynamical Correlation Functions

Dynamical correlation functions, which describe how a system responds to external influences, present a significant challenge for classical computation, yet understanding them is crucial in many areas of physics. Taehee Ko, Mancheon Han, and Sangkook Choi from the School of Computational Sciences at the Korea Institute for Advanced Study present a new approach to calculating these functions efficiently using quantum computers. Their work introduces Fermionic-Adapted Shadow Tomography (FAST), a framework that reformulates dynamical correlation functions to work with shadow tomography techniques, dramatically reducing the computational resources needed. The team demonstrates that FAST protocols require fewer measurements and less complex quantum circuits than existing methods, potentially offering a substantial improvement in the ability to simulate and understand complex quantum systems.

Simulating these functions presents a significant challenge for classical computers, but this work leverages quantum computation to overcome those limitations. This new framework improves efficiency by reducing the number of measurements needed to characterize a quantum system’s behavior.

Traditional methods often require measuring each pair of observable properties individually, a process that becomes increasingly demanding as the system grows in complexity. FAST protocols, however, can estimate multiple dynamical correlation functions simultaneously, requiring fewer quantum circuits and reducing the overall computational burden. Across a range of scenarios, the protocols demonstrate a reduction in the number of circuits needed by a factor of one or two, representing a substantial improvement in efficiency.

Reducing Measurement Overhead in Quantum Algorithms

Quantum computing research focuses on efficiently estimating observable properties, particularly within variational quantum algorithms and quantum simulation. A key challenge is minimizing the number of measurements needed to achieve accurate results, a significant bottleneck on current quantum hardware. Researchers are exploring various techniques to address this, including methods to improve sampling efficiency and reduce computational cost. Shadow tomography is a powerful technique that allows for efficient estimation of quantum states using a relatively small number of measurements. Adaptive sampling dynamically adjusts measurement strategies to focus on the most important regions of the quantum state.

Researchers are also investigating simultaneous measurement, where multiple observable properties are measured in a single quantum circuit, and partial tomography, which estimates only the relevant parts of the quantum state. Optimising the mapping of quantum information onto qubits is also crucial, with adaptive and ternary tree mappings offering potential improvements. Improving accuracy also requires addressing errors and noise in quantum computations. Error mitigation techniques reduce the impact of noise, while adaptive algorithms dynamically adjust the quantum circuit during optimisation. Researchers are also exploring improved methods for approximating time evolution in quantum simulations and developing more efficient quantum linear system solvers. These protocols reformulate these functions to be compatible with shadow tomography techniques, enabling more streamlined calculations than previously possible. Results demonstrate that FAST protocols enhance sample efficiency and reduce the number of measurement circuits needed by a factor of one or two, across a range of scenarios, when compared to existing methods. The team achieved this by developing novel representations of correlation functions that are directly compatible with quantum measurement.

They found ways to express these functions in terms of probabilities obtainable from quantum circuits, allowing for a more streamlined calculation. This involved reformulating the mathematical expressions to align with the capabilities of shadow tomography, a technique that efficiently characterizes quantum states through a series of measurements. These advancements have significant implications for fields such as materials science and quantum chemistry, potentially leading to the discovery of new materials with tailored properties and advancements in quantum technologies. The team’s approach achieves polynomial sample complexity, requiring at most two-copy measurements with uncontrolled Hamiltonian simulation, and offers broader applicability than some alternative techniques.

Further optimisation is possible by considering the choice of fermion-to-qubit mapping. Future research directions include investigating the potential optimality of the protocols and exploring extensions to chained measurement strategies, potentially reducing circuit numbers and enabling constant-depth execution with different mappings. Tailoring measurement protocols to specific mappings, such as the Bravyi-Kitaev and ternary tree mappings, could also lead to even more efficient simulations on quantum devices.