Estimating Outcome Probabilities of Quantum Circuits through Quasiprobabilities
The paper under review introduces an innovative methodology for estimating the outcome probabilities of quantum circuits, leveraging quasiprobability representations and Monte Carlo sampling techniques. The primary focus is on handling the non-classical aspects inherent to quantum computation by using a measure of negativity, defined through the 1-norm of the quasiprobability. The results presented in the paper underscore a significant association between negativity and non-classical resources in quantum computation.
The authors set forth a framework for estimating outcome probabilities that are typically challenging to compute efficiently using classical methods, due to the inherent exponential complexity of direct calculations involving the Born rule. Notably, for many classes of quantum circuits, such as stabilizer circuits or fermionic linear optics, probability calculations can be efficiently simulated using classical computation approaches like those delineated in the Gottesman-Knill theorem.
This paper offers a broader methodology using quasiprobability representations, a mathematical tool historically employed in physics to simulate quantum processes. These quasiprobability distributions are typically real-valued but may take on negative values, which cannot be directly interpreted as probabilities. The central tenet of this work is an unbiased estimator that remains efficient so long as the negativity of the system grows at most polynomially with circuit size.
One of the paper's key contributions is tying the efficiency of the estimation method to a measure of negativity, expressing the total negativity of a circuit as a "currency" in quantum computing. For classes of quantum circuits where total negativity is polynomially bounded, the estimation can be performed efficiently, offering a poly-precision approximation of the quantum probabilities involved.
A critical insight provided is the simulation of quantum trajectories and the evaluation of circuit negativity under a Monte Carlo sampling approach to the quasiprobability distributions. Additionally, detailed protocols are introduced for sampling from a modified distribution of these trajectories, with efficient implementation nuanced by the structure of the circuits and gates involved.
Practically, the methods offer a simulation advantage beyond classical limitations by demonstrating efficiency in simulating systems where negativity is the resource, identifying methods for leveraging predominantly nonnegative circuit elements effectively. The paper proposes alternative estimators that exploit symmetries in the quantum gates' operations and regrouping of unitary operations to enhance calculation efficiency.
This work's implications extend to understanding how quasiprobability representations can be crafted or selected to balance the representation of negative and nonnegative operations within circuits. The authors argue that a critical task for researchers is identifying and utilizing quasiprobability representations in which crucial operations are represented nonnegatively, thus reducing computational overhead and expanding the classes of simulatable operations.
In conclusion, the research presented offers a valuable perspective on using negativity as a quantifiable resource in quantum computation, providing a robust mechanism for estimating outcome probabilities across specific classes of quantum processes. This advancement broadens the horizon for classical simulation of quantum circuits, allowing researchers to tackle previously intractable problems with greater optimism and effectiveness. Future work will likely explore extending these frameworks to more complex and varied quantum systems, further elucidating the potential of quasiprobability sampling in quantum computing applications.