Estimating Entanglement Entropy of Quantum States with Bounded Stabilizer Dimension

Core Concepts

Any pseudoentangled state ensemble with a gap of t bits of entropy requires Ω(t) non-Clifford gates to prepare.

Abstract

The content discusses the relationship between pseudoentanglement and non-Clifford complexity in quantum computing. The key highlights are: The authors present an algorithm that can efficiently estimate the entanglement entropy across any bipartition of qubits for quantum states with bounded stabilizer dimension. Specifically, if a state |ψ⟩ has stabilizer dimension at least n-k, the algorithm outputs upper and lower bounds on the entanglement entropy that are within k bits of the true value. As a corollary, the authors show that any family of Clifford circuits that produces a pseudoentangled ensemble {|Ψk⟩, |Φk⟩}k with entropy gap f(n) vs. g(n) satisfying f(n) - g(n) ≥ t must use Ω(t) auxiliary non-Clifford single-qubit gates. This matches the lower bound on non-Clifford gates needed for pseudorandom states, showing that pseudoentanglement is as computationally expensive as pseudorandomness in terms of non-Clifford complexity. The authors also discuss a concurrent work by Gu, Oliveiro, and Leone that obtains related results on the relationship between magic and entanglement.

Stats

Any n-qubit state |ψ⟩ that is stabilized by at least 2^(n-t) Pauli operators requires Ω(t) non-Clifford gates to prepare.

Quotes

"Any pseudoentangled state ensemble with a gap of t bits of entropy requires Ω(t) non-Clifford gates to prepare." "Pseudoentangled states can be instantiated in polynomial time and logarithmic depth, assuming the existence of quantum-secure one-way functions."

Key Insights Distilled From

Pseudoentanglement Ain't Cheap

by Sabee Grewal... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.00126.pdf

Deeper Inquiries

How can the techniques developed in this work be extended to other measures of quantum state complexity beyond circuit depth, such as space complexity or query complexity

The techniques developed in this work can be extended to other measures of quantum state complexity beyond circuit depth by considering different aspects of the quantum state's properties. For example, space complexity in quantum computing refers to the amount of memory or qubits required to represent a quantum state or perform a quantum computation. By analyzing the entanglement entropy across different partitions of qubits, similar to the approach taken in this work, one could potentially derive bounds on the space complexity of quantum states. Additionally, query complexity in quantum computing relates to the number of queries needed to solve a specific computational problem. By investigating the entanglement properties of quantum states in the context of query complexity tasks, one could potentially establish connections between the entanglement structure of states and the efficiency of quantum algorithms in terms of query complexity. This extension would involve analyzing how the entanglement entropy across different partitions of qubits influences the number of queries required to perform certain quantum computations. Overall, the techniques developed in this work, which focus on estimating entanglement entropy across qubit partitions, can be adapted and applied to explore various aspects of quantum state complexity, including space complexity and query complexity, providing insights into the relationships between entanglement properties and computational efficiency in quantum algorithms.

What are the implications of the connection between pseudoentanglement and non-Clifford complexity for the AdS/CFT correspondence and the complexity of the holographic dictionary

The connection between pseudoentanglement and non-Clifford complexity has significant implications for the AdS/CFT correspondence and the complexity of the holographic dictionary. In the context of the AdS/CFT correspondence, which relates quantum gravity in anti-de Sitter space to quantum field theory on its boundary, understanding the non-Clifford complexity of pseudoentangled states can provide insights into the computational resources required to model and analyze holographic dualities. The complexity of the holographic dictionary, which maps between bulk gravitational theories and boundary quantum field theories, can be influenced by the non-Clifford gates needed to prepare pseudoentangled states. If constructing pseudoentangled states with optimal gaps requires a linear number of non-Clifford gates, as shown in the work, this could imply that certain aspects of the holographic dictionary are computationally challenging or resource-intensive. Moreover, the connection between pseudoentanglement and non-Clifford complexity may shed light on the computational properties of quantum systems that exhibit holographic dualities. Understanding the lower bounds on non-Clifford resources for pseudoentangled states could provide valuable insights into the computational aspects of quantum gravity theories and their correspondence with quantum field theories, enhancing our understanding of the holographic principle and its implications for quantum information science.

Are there other cryptographic objects in quantum computing, beyond pseudorandomness and pseudoentanglement, that exhibit similar lower bounds on non-Clifford resources

In addition to pseudorandomness and pseudoentanglement, there are other cryptographic objects in quantum computing that exhibit similar lower bounds on non-Clifford resources. One such cryptographic object is pseudorandom quantum states, which are ensembles of quantum states that cannot be distinguished from truly random states by any polynomial-time adversary. Similar to pseudoentangled states, pseudorandom quantum states may require a certain number of non-Clifford gates to prepare, as demonstrated in previous works. By analyzing the non-Clifford complexity of pseudorandom quantum states, researchers can establish lower bounds on the resources needed to generate these states efficiently. This can have implications for quantum cryptography, quantum secure communication, and quantum information processing, where the security and efficiency of cryptographic protocols rely on the computational hardness of certain quantum tasks. Overall, exploring the relationship between non-Clifford complexity and cryptographic objects beyond pseudorandomness and pseudoentanglement can provide valuable insights into the computational foundations of quantum cryptography and the security of quantum communication protocols.