Entropy scaling and simulability by Matrix Product States (0705.0292v2)

Abstract: We investigate the relation between the scaling of block entropies and the efficient simulability by Matrix Product States (MPS), and clarify the connection both for von Neumann and Renyi entropies (see Table I). Most notably, even states obeying a strict area law for the von Neumann entropy are not necessarily approximable by MPS. We apply these results to illustrate that quantum computers might outperform classical computers in simulating the time evolution of quantum systems, even for completely translational invariant systems subject to a time independent Hamiltonian.

• The paper establishes that a strict area law in von Neumann entropy alone does not guarantee efficient MPS simulation of quantum states.
• It demonstrates that logarithmic scaling of Rényi entropies for α<1 supports MPS approximability, whereas faster growth for α>1 or linear von Neumann entropy hinders it.
• The paper highlights that these insights have direct implications for quantum simulations, indicating classical limitations and potential quantum computing advantages.

Entropy Scaling and Simulability by Matrix Product States

This paper, authored by Norbert Schuch, Michael M. Wolf, Frank Verstraete, and J. Ignacio Cirac, explores the critical investigation of the relationship between entropy scaling and the simulability of quantum states using Matrix Product States (MPS). While MPS are recognized as a robust framework for approximating the ground states of one-dimensional quantum systems, the nuanced criteria governing their efficient applicability remain an open question. The authors provide a comprehensive analysis that contributes nuanced insights into when quantum states may be efficiently approximated by MPS, particularly focusing on the relationship with entropy scaling.

Key Results and Analysis

1. Entropy Scaling and Approximability: It is a common belief that for a quantum state to be efficiently simulable by MPS, it should adhere to an area law in terms of its von Neumann entropy. However, this work challenges such assumptions by evaluating both von Neumann and Rényi entropies. The authors demonstrate through their investigations that a straightforward area-law compliance is insufficient for guaranteeing approximability by MPS. They provide evidence that even some states adhering strictly to the area law for von Neumann entropy resist approximation by MPS.
2. Analysis of Entropy Measures:
   • The paper finds that a logarithmic scaling of Rényi entropies $S_\alpha$, for $\alpha < 1$, ensures approximability by MPS. Conversely, if these entropies grow faster than logarithmically for $\alpha > 1$, efficient approximation by MPS is deemed infeasible.
   • Notably, linear growth of von Neumann entropy is similarly shown to preclude the possibility of efficient approximability by MPS.
   • The results underscore that relying solely on von Neumann entropy to benchmark approximability can be misleading due to its asymptotic nature and the continuity inequality.
3. Implications for Quantum Simulations: The authors extend their theoretical findings to propose that quantum computers could outperform classical computers in specific simulations. The results imply that classical simulations, especially concerning the temporal evolution of certain quantum systems, might be inherently limited and unable to replicate the efficiency possible with quantum computing methods. The paper notes that this holds even when simulations are for translationally invariant systems under time-independent Hamiltonians.
4. Numerical and Theoretical Demonstrations: The paper provides both numerical examples and theoretical constructs, demonstrating particular states that withstand efficient MPS approximation while fulfilling certain entropy conditions. This includes states exhibiting constant von Neumann entropy, yet challenging to approximate, as well as other test states with varied entropy scaling curves.

Theoretical and Practical Implications

The research holds substantial implications for both theoretical quantum physics and practical computational strategies. The challenges identified in entropy-based characterizations of MPS suitability suggest a need for more refined criteria that consider other metrics, such as smooth Rényi entropies and the role of non-local correlations. Practically, acknowledging these limitations could drive innovations in quantum simulation methodologies, potentially leading to more accurate and scalable quantum computation frameworks.

Future Directions

The work opens several avenues for further exploration:

• Extending findings to higher-dimensional systems and verifying their utility beyond one-dimensional setups.
• Investigating the role of other entropy measures or quantum information metrics in determining the approximability of quantum states by MPS.
• Enhancing the theoretical framework governing MPS-related computation, particularly concerning its scalability and integration into diverse quantum computing architectures.

In summary, this paper provides a significant leap in understanding the nuanced relationship between entropy scaling and the simulability of quantum states with MPS, questioning established assumptions and suggesting new directions for future research.