Quantum circuit lower bounds in the magic hierarchy

Abstract: We introduce the magic hierarchy, a quantum circuit model that alternates between arbitrary-sized Clifford circuits and constant-depth circuits with two-qubit gates ($\textsf{QNC}^0$). This model unifies existing circuit models, such as $\textsf{QAC}^0_f$ and models with adaptive intermediate measurements. Despite its generality, we are able to prove nontrivial lower bounds. We prove new lower bounds in the first level of the hierarchy, showing that certain explicit quantum states cannot be approximately prepared by circuits consisting of a Clifford circuit followed by $\textsf{QNC}^0$. These states include ground states of some topologically ordered Hamiltonians and nonstabilizer quantum codes. Our techniques exploit the rigid structure of stabilizer codes and introduce an infectiousness property: if even a single state in a high distance code can be approximately prepared by one of these circuits, then the entire subspace must lie close to a perturbed stabilizer code. We also show that proving state preparation lower bounds beyond a certain level of the hierarchy would imply classical circuit lower bounds beyond the reach of current techniques in complexity theory. More broadly, our techniques go beyond lightcone-based methods and highlight how the magic hierarchy provides a natural framework for connecting circuit complexity, condensed matter, and Hamiltonian complexity.

Overview of Quantum Circuit Lower Bounds in the Magic Hierarchy

The paper introduces the concept of the magic hierarchy, a quantum circuit model designed to integrate existing circuit models, notably those utilizing large-scale Clifford circuits and constant-depth circuits with two-qubit gates, denoted as $\textsf{QNC}^0$. This framework provides a nuanced approach to the study of quantum circuits, establishing new lower bounds and offering insights into complex quantum states that cannot be generated using these models.

Key Contributions and Results

1. Magic Hierarchy Definition: The magic hierarchy classifies quantum circuits based on the number of alternations between Clifford circuits and $\textsf{QNC}^0$ circuits. The hierarchy is structured such that higher levels indicate higher complexity in terms of circuit entanglement and nonstabilizerness properties.

2. Lower Bound Proofs: The paper explores the first level of the magic hierarchy to demonstrate lower bounds. It shows that certain quantum states, including ground states of topologically ordered Hamiltonians and nonstabilizer quantum codes, cannot be prepared using circuits from the first level of the magic hierarchy, or $\altCQ[1]$. This is a seminal contribution as it suggests limitations within this circuit model, pushing for the development of new techniques beyond lightcone-based methods.

3. Mutual Information Technique: A detailed method involving mutual information is used to derive lower bounds. It states that quantum states, characterized by discrete mutual information values among subsets of qubits, cannot be approximated if this mutual information does not align with integer values. This technique provides a rich framework for analyzing correlations within quantum states.

4. Implications of Infectiousness: The research introduces the concept of infectiousness for codes within quantum circuits. It establishes that if a single quantum state from a reliable code can be approximately generated using $\altCQ[1]$ circuits, then the entire code must exhibit behavior akin to perturbed stabilizer codes.

5. Connections and Comparisons: The work delineates how the magic hierarchy relates to other circuit models such as those using Fanout gates and adaptive intermediate measurements. It establishes equivalence up to constant factors in complexity measures like $T$-depth, providing a comprehensive analysis of circuit limitations and capabilities.

Implications and Future Work

The results significantly enhance the understanding of quantum circuit capabilities, particularly highlighting the constraints of the first level of the magic hierarchy model. The techniques provided in the paper underscore the challenges in preparing intricate quantum states without increasing circuit depth or complexity.

There are several exciting avenues for future exploration stemming from this paper. Firstly, understanding whether there exists a strict hierarchy in the computational power at different levels could unravel deeper insights into quantum complexity. Additionally, connecting these theoretical results with practical implementations on quantum hardware could realize novel approaches in state preparation and quantum error correction.

Moreover, the connections drawn between the magic hierarchy and classical circuit lower bounds present profound implications for complexity theory, suggesting that substantial breakthroughs in quantum circuit complexity might also lead to advances in classical computational limits.

Open Questions: The paper posits numerous open questions, one of which examines the possibility of achieving quantum advantage using magic hierarchy models—particularly at higher levels. Answering these questions could potentially redefine the boundaries of quantum computing capabilities.

In summary, this paper fosters new paths in quantum circuit research, forging robust connections between quantum and classical computational complexity, and encouraging further exploration into advanced quantum algorithms and states.