Circuit complexity in quantum field theory (1707.08570v2)

Abstract: Motivated by recent studies of holographic complexity, we examine the question of circuit complexity in quantum field theory. We provide a quantum circuit model for the preparation of Gaussian states, in particular the ground state, in a free scalar field theory for general dimensions. Applying the geometric approach of Nielsen to this quantum circuit model, the complexity of the state becomes the length of the shortest geodesic in the space of circuits. We compare the complexity of the ground state of the free scalar field to the analogous results from holographic complexity, and find some surprising similarities.

• The paper establishes a framework that maps circuit complexity in QFT to geodesic distances in quantum circuit space using Nielsen’s approach.
• The methodology analyzes coupled harmonic oscillators in free scalar fields with Gaussian reference and target states to simplify complexity calculations.
• The results reveal scaling laws analogous to those in holographic theories, offering insights into quantum computational costs and potential links to quantum gravity.

Circuit Complexity in Quantum Field Theory

The paper by Ro Jefferson and Robert C. Myers explores the intricate topic of circuit complexity within the field of quantum field theory (QFT). Motivated by developments in the understanding of holographic complexity, the authors aim to define and analyze circuit complexity for QFTs, particularly focusing on Gaussian states within free scalar field theories across various dimensions.

Conceptual Framework

The paper builds upon the foundational ideas connecting entanglement and geometric descriptions as observed in the AdS/CFT correspondence, aiming to conceptualize complexity as a geometrical construct similarly to how entanglement entropy is understood. Central to their approach is the adoption of Nielsen’s geometric model for quantum computation, where complexity is equated to the geodesic distance in a space of quantum circuits. This conceptualization transforms the problem of understanding circuit complexity into one of exploring geometric pathways in a defined quantum space.

Methodology

1. Model Setup: The authors consider a free scalar field theory, discretized on a lattice to manage the complexities beyond the continuum. This leads to a system reducible to a set of coupled harmonic oscillators, providing a tractable starting point for exploring circuit complexity.
2. Gaussian States: Both the reference and target states are chosen to be Gaussian. For the reference state, a factorized Gaussian corresponding to unentangled oscillators is utilized, whereas the target is set to be the ground state of the harmonic system.
3. Complexity as Geodesics: Leveraging Nielsen’s framework, the complexity is translated into the task of identifying the shortest geodesic path within the circuit’s manifold. A particular focus is placed on evaluating this metric in terms of the gates necessary to perform certain transformations from the reference to the target state.
4. Calculations and Results: By analyzing small systems of oscillators and extending the results to larger lattices, the paper derives expressions for the circuit complexity. Particular attention is given to the normal-mode decomposition, allowing simplification of the system and showing that scaling each mode independently minimizes the complexity path.

Numerical Results and Implications

The paper finds expressions for circuit complexity that, while dependent on various parameters such as states and dimensions, offer insights into possible universality or scaling laws. Notably, the results show alignment with expectations from holographic theories, indicating similar scaling behaviors for complexity divergences in both simple and complex geometric setups.

Implications and Future Directions

The implications of this research extend towards potential new understandings of quantum computational costs in field theories and connections to holography. The paper speculates that these findings could provide insights into holographic duality and gravitational systems, conjecturing potential parallels between computational complexity in quantum circuits and complex structures in gravitational settings.

Moreover, by addressing different components such as gate choices, reference states, and penalty factors for nonlocal operations, the research outlines pathways to refining the precision of current models. This refinement could illuminate the role of complexity in more comprehensive theories of quantum gravity and elaborate on the nature of space-time entanglement.

Conclusion

Through thoughtful examination and application of Nielsen’s geometric approach, this paper lays foundational work in formalizing the concept of circuit complexity within QFT, proposing methodologies to build upon in future research. While rooted in theory, its implications are profound, potentially influencing quantum computing and theoretical physics landscapes by bridging computational concepts and geometric insights. The exploratory pathways and observational parallels outlined in this work pave the trajectory for both detailed numerical follow-ups and broader theoretical breakthroughs.