[Paper Review] Quantum information science and complex quantum systems

Michael A. Nielsen|ArXiv.org|Oct 1, 2002

Quantum Computing Algorithms and Architecture13 references3 citations

TL;DR

This paper argues that quantum information science provides a powerful scientific framework for studying complex quantum systems by leveraging concepts like logical depth, entanglement, and dynamic strength measures. It demonstrates that quantum dynamics can be fungible resources for universal quantum computation and introduces a metric-based strength measure that enables lower bounds on computational complexity, offering new tools to analyze emergent quantum phenomena.

ABSTRACT

What makes quantum information science a science? This paper explores the idea that quantum information science may offer a powerful approach to the study of complex quantum systems.

Motivation & Objective

• To establish quantum information science as a fundamental scientific discipline, not merely engineering, by framing it as a tool for understanding complex quantum systems.
• To address the intrinsic scientific value of quantum information science by linking it to the study of emergent phenomena in many-body quantum systems.
• To develop quantitative measures—particularly dynamic strength measures—capable of characterizing the complexity and computational power of quantum operations.
• To show that quantum dynamical operations are fungible, enabling universal quantum computation using any entangling two-qudit gate and local operations.
• To connect the theory of quantum computation with broader many-body physics by using quantum information concepts to probe complex quantum behavior.

Proposed method

• Using Bennett's concept of logical depth to define complexity as the runtime of a near-optimal program generating a system, distinguishing simple, random, and complex systems.
• Applying this framework to quantum systems, showing that factoring large integers is logically deep under classical computation but not under quantum computation due to Shor's algorithm.
• Defining a metric-based strength measure for unitary operations as the minimal distance to local unitaries, quantifying non-locality and computational power.
• Introducing three axiomatic properties—chaining, stability, and locality—for strength measures to ensure consistency and physical relevance.
• Constructing a constructive, near-optimal algorithm to simulate a CNOT gate using any entangling two-qudit unitary and local gates, proving dynamical fungibility.
• Using the strength measure to derive lower bounds on circuit depth, linking dynamic complexity to computational complexity.

Experimental results

Research questions

• RQ1 What makes quantum information science a science rather than just engineering, and how can it provide intrinsic scientific insight into complex quantum systems?
• RQ2 Can the concept of logical depth be adapted to quantum systems to quantify their complexity, especially in the context of quantum algorithms like Shor’s?
• RQ3 Are all entangling quantum dynamical operations equally powerful for universal quantum computation, and can this be shown constructively?
• RQ4 How can a strength measure for unitary operations be defined such that it reflects both physical and computational properties, including non-locality and computational complexity?
• RQ5 Can such a strength measure be used to derive lower bounds on the number of gates required to implement a given quantum operation?

Key findings

• The list of prime factors of large integers is logically deep under classical computation (implying high complexity) but not under quantum computation, due to Shor’s algorithm, demonstrating a quantum advantage in complexity.
• Quantum dynamical operations are fungible: any entangling two-qudit unitary, combined with local operations, can be used to perform universal quantum computation, as shown by a constructive algorithm in Bremner et al. [9].
• The strength measure defined as the minimal distance to local unitaries satisfies key physical properties: chaining, stability, and locality, making it a robust and meaningful quantifier of non-locality.
• A strength measure satisfying these axioms enables the derivation of lower bounds on circuit depth: if the strength of a unitary scales superpolynomially, then so must the number of CNOT gates required to implement it.
• The framework provides a new theoretical bridge between quantum information science and many-body physics, allowing the study of complex quantum systems through entanglement and dynamic complexity.
• The results suggest that quantum information concepts—especially those related to entanglement and dynamic strength—can serve as overarching theories to understand emergent behavior in complex quantum systems.