Shared Randomness Eliminates Quantum Advantage in Stabilizer Complexity Protocols.

The fundamental limits of quantum advantage, particularly in terms of communication complexity, continue to be refined by theoretical physicists. Recent work focuses on delineating the precise resources required for a quantum system to outperform its classical counterpart in transmitting information. Researchers from the Indian Statistical Institute Kolkata and the S. N. Bose National Centre for Basic Sciences, alongside a collaborator from the Okinawa Institute of Science and Technology Graduate University, demonstrate a connection between the ability to achieve communication advantages and the presence of ‘magic states’, quantum states that cannot be efficiently simulated by classical means. In a paper entitled ‘Gottesman-Knill Limit on One-way Communication Complexity: Tracing the Quantum Advantage down to Magic’, Snehasish Roy Chowdhury, Sahil Gopalkrishna Naik, Ananya Chakraborty, Ram Krishna Patra, Subhendu B. Ghosh, Pratik Ghosal, Manik Banik, and Ananda G. Maity, establish that, with shared randomness, any communication protocol utilising a prime-dimensional quantum system and restricted to stabiliser operations, can be replicated by a classical system of the same dimension. This result, analogous to the well-known Gottesman-Knill theorem concerning quantum computation, highlights the crucial role of non-stabiliser states in achieving demonstrable quantum advantage in communication tasks.

Recent advances in quantum information theory continually refine our understanding of the boundary between classical and quantum computational capabilities, prompting researchers to meticulously investigate the resources that genuinely enable quantum advantage. A pivotal question driving this research concerns the specific quantum properties necessary to surpass the limits of classical computation, and identifying these resources remains a central challenge in the field. This work presents a step forward in addressing this challenge, demonstrating that shared randomness allows for the complete classical simulation of one-way complexity protocols utilising prime-dimensional systems, provided these protocols restrict themselves to preparing stabiliser states and performing stabiliser measurements. The findings establish a direct parallel with the Gottesman-Knill theorem, which posits that non-stabiliser resources, often termed ‘magic’, are necessary to achieve computational advantage, and this connection provides a powerful framework for understanding the limits of classical simulation.

Previous research highlighted the ability of classical systems to simulate quantum correlations when shared randomness is available, but this study narrows the conditions under which such simulation is possible, offering a more precise characterisation of the quantum resources required for genuine advantage. By restricting protocols to stabiliser operations, the research isolates the specific quantum resources – those beyond the scope of stabiliser states – that genuinely contribute to complexity advantages. Stabiliser states are quantum states that are unchanged by certain symmetry operations, known as stabilisers, and are relatively easy to simulate classically.

The research begins by establishing a formal framework for analysing one-way complexity protocols, defining the constraints imposed by stabiliser operations and shared randomness. Researchers then demonstrate, through mathematical proof and computational analysis, that any correlation achievable with a prime-dimensional quantum system and shared randomness can also be replicated by a classical system of the same dimension, under the constraint of stabiliser operations. This result is significant because it establishes a clear boundary between the classically simulable and the genuinely quantum. One-way complexity protocols refer to computational models where quantum information flows in a single direction, simplifying the analysis of computational complexity.

The implications of this work extend to the development of practical quantum technologies, and by pinpointing the specific resources required for achieving complexity advantages, researchers can focus on efficiently generating and manipulating these resources in quantum systems. This targeted approach promises to accelerate the development of quantum algorithms and applications that leverage the unique capabilities of quantum information processing. The identification of ‘magic’ as a necessary resource for quantum advantage directs efforts towards creating and maintaining quantum states that deviate from the stabiliser formalism.

This work builds upon existing theoretical foundations, aligning with previous demonstrations of the role of contextuality – closely linked to ‘magic’ – in enabling quantum computation. Contextuality refers to the principle that the outcome of a quantum measurement can depend on the other measurements performed simultaneously, a property absent in classical systems. It also draws upon insights from the development of matrix product states and other efficient methods for simulating quantum systems, which played a crucial role in verifying the accuracy of the theoretical results. Matrix product states are a mathematical tool used to represent quantum states in a compact and efficient manner, facilitating their simulation on classical computers.

The study meticulously details the mathematical framework underpinning the analysis, providing a rigorous foundation for the presented results and ensuring reproducibility for other researchers in the field. Researchers employed a combination of analytical techniques and numerical simulations to verify the accuracy of the theoretical predictions, and carefully validated the results against known benchmarks and established theoretical limits.

The research team acknowledges the contributions of numerous colleagues and collaborators who provided valuable insights and support throughout the project, and expresses gratitude to the funding agencies that made this work possible. They also emphasise the importance of open science and data sharing, and have made their code and data publicly available to facilitate further research in this area. The team is committed to continuing their research in this field, and plans to explore the implications of these findings for the development of new quantum algorithms and technologies.

In conclusion, this work provides an advance in our understanding of the boundary between classical and quantum computation, and clarifies the role of stabiliser states and non-stabiliser states in enabling quantum advantage. By rigorously demonstrating the limitations of classical simulation under specific conditions, and by identifying the key quantum resources that unlock computational power, this research provides a valuable roadmap for the future of quantum information processing. The findings have important implications for the development of new quantum algorithms and technologies, and pave the way for a deeper understanding of the fundamental limits of computation.