Sharp complexity phase transitions generated by entanglement

Abstract

Entanglement is one of the physical properties of quantum systems responsible for the computational hardness of simulating quantum systems. But while the runtime of specific algorithms, notably tensor network algorithms, explicitly depends on the amount of entanglement in the system, it is unknown whether this connection runs deeper and entanglement can also cause inherent, algorithm-independent complexity. In this work, we quantitatively connect the entanglement present in certain quantum systems to the computational complexity of simulating those systems. Moreover, we completely characterize the entanglement and complexity as a function of a system parameter. Specifically, we consider the task of simulating single-qubit measurements of $k$--regular graph states on $n$ qubits. We show that, as the regularity parameter is increased from $1$ to $n-1$, there is a sharp transition from an easy regime with low entanglement to a hard regime with high entanglement at $k=3$, and a transition back to easy and low entanglement at $k=n-3$. As a key technical result, we prove a duality for the simulation complexity of regular graph states between low and high regularity.

References in corpus (6)

• Multi-party entanglement in graph states
• Universal resources for measurement-based quantum computation
• Wigner function negativity and contextuality in quantum computation on rebits
• Universal quantum computation with little entanglement
• Classical simulation versus universality in measurement based quantum computation
• Entanglement and local information access for graph states

Cited by in corpus (7)

• Benchmarking highly entangled states on a 60-atom analog quantum simulator
• Error-resilience Phase Transitions in Encoding-Decoding Quantum Circuits
• Compilation of algorithm-specific graph states for quantum circuits
• Achieving the volume-law entropy regime with random-sign Dicke states
• The Foliage Partition: An Easy-to-Compute LC-Invariant for Graph States
• Computable and noncomputable in the quantum domain: statements and conjectures
• Quantum Complexity in Rule-Based Constrained Many-Body Models: Scars, Fragmentation, and Chaos