The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes (1607.05256v1)

Abstract: These are lecture notes from a weeklong course in quantum complexity theory taught at the Bellairs Research Institute in Barbados, February 21-25, 2016. The focus is quantum circuit complexity---i.e., the minimum number of gates needed to prepare a given quantum state or apply a given unitary transformation---as a unifying theme tying together several topics of recent interest in the field. Those topics include the power of quantum proofs and advice states; how to construct quantum money schemes secure against counterfeiting; and the role of complexity in the black-hole information paradox and the AdS/CFT correspondence (through connections made by Harlow-Hayden, Susskind, and others). The course was taught to a mixed audience of theoretical computer scientists and quantum gravity / string theorists, and starts out with a crash course on quantum information and computation in general.

• The paper explores how quantum complexity in preparing states and applying unitary transformations drives advances in quantum money and cryptography.
• It investigates the computational challenges of executing quantum algorithms, including Grover’s search, to optimize quantum operations.
• The work extends these insights to black hole physics by analyzing the difficulty of decoding Hawking radiation and resolving the information paradox.

The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes

The paper, documented through a series of lectures at the 28th McGill Invitational Workshop on Computational Complexity, embarks on an exploration of the intersection between quantum computing theory and quantum gravity. This intersection is portrayed through the examination of computational complexity in preparing quantum states and applying unitary transformations, relating these concepts to the nuanced domains of quantum money and black hole information theory.

Central to the discourse is the notion of quantum complexity, which has traditionally been associated with decision problems solvable in quantum polynomial time. However, this work posits a shift in perspective by addressing the intricacies involved in creating specific quantum states or executing particular transformations, suggesting new applications and theoretical developments in areas like quantum cryptography and cosmology.

Quantum Complexity and State Preparation

A significant focus is placed on the intricacies of preparing quantum states and applying quantum transformations. Questions explored include the complexity involved and the number of operations required, especially considering universal gate sets. The paper discusses the potential of quantum states as a resource in quantum computations and information systems, probing the technical limits of such resources.

The discourse expands into quantum money as an application — an intriguing concept proposed by Wiesner and further refined by Bennett et al., establishing a system where quantum states are used to create unforgeable currency. This quantum money paradigm leverages the no-cloning theorem, illustrating a practical consequence of quantum mechanics that could revolutionize how authenticity is managed in digital systems.

The research points to several quantum algorithms and complexities, such as the Grover’s search algorithm and the challenge of defining approximate unitary transformations. These algorithms serve as foundational components in addressing quantum state preparation and manipulation within polynomial constraints.

Black Hole Information and Quantum Gravity

An enthralling component of the workshop’s lectures is the extension of quantum complexity ideas to black holes, particularly addressing the twin challenges of the black hole information paradox and the firewall problem. The work navigates through concepts posited by Hawking and later by Harlow and Hayden, questioning information loss and unitary evolution in black holes. The complexity of reconstructing the Hawking radiation into its original state when swallowed by a black hole serves as a fertile ground for theoretical exploration.

By examining these physicist-conceived paradoxes through the computational lens, the research bridges a conceptual gap: it suggests that the challenges in decoding Hawking radiation might be inherently related to quantum complexity. More precisely, this decoding task's difficulty ties into some of the hardest problems in quantum computing, akin to solving instances of the hidden subgroup problem over non-abelian groups.

Implications and Future Directions

This multi-dimensional exploration of computational complexity thrusts open numerous avenues for future research. The lecture series speculates on the broader implications for AI and quantum information systems, proposing a deeper understanding of the theoretical underpinnings that bind quantum states to cosmological phenomena. This creates a unique tapestry where technological advancements could influence theoretical physics, raising questions about the nature of computation beyond classical paradigms.

Furthermore, the engagement with quantum gates, state complexity, and unitary applications underscores upcoming challenges in quantum circuit optimization and simplification. The insights into possible oracle separations and the hypothesized class differences in quantum computational frameworks like $\mathsf{QMA}$ and $\mathsf{QCMA}$ indicate there remains much to learn about the foundational structures that make quantum computation distinctly potent.

In conclusion, the lectures encapsulate a comprehensive journey from quantum states to black hole physics, enkindling a vibrant discussion around the quantum mechanical narrative that underlies many apparently disparate fields. The quantum complexity of states and transformations, far from being a niche computational curiosity, emerges as a pivotal theme, connecting complex algorithms, quantum financial systems, and the mysteries of the cosmos.