Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus (1902.03178v6)

Abstract: We present a completely new approach to quantum circuit optimisation, based on the ZX-calculus. We first interpret quantum circuits as ZX-diagrams, which provide a flexible, lower-level language for describing quantum computations graphically. Then, using the rules of the ZX-calculus, we give a simplification strategy for ZX-diagrams based on the two graph transformations of local complementation and pivoting and show that the resulting reduced diagram can be transformed back into a quantum circuit. While little is known about extracting circuits from arbitrary ZX-diagrams, we show that the underlying graph of our simplified ZX-diagram always has a graph-theoretic property called generalised flow, which in turn yields a deterministic circuit extraction procedure. For Clifford circuits, this extraction procedure yields a new normal form that is both asymptotically optimal in size and gives a new, smaller upper bound on gate depth for nearest-neighbour architectures. For Clifford+T and more general circuits, our technique enables us to to `see around' gates that obstruct the Clifford structure and produce smaller circuits than naive 'cut-and-resynthesise' methods.

• The paper presents a deterministic simplification strategy using ZX-calculus and graph transformations to reduce circuit complexity.
• It outlines a method for circuit extraction from simplified ZX-diagrams that ensures generalized flow and optimizes gate count and depth for Clifford circuits.
• Experimental results show the approach outperforms traditional methods, paving the way for more resource-efficient quantum algorithm design.

Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus

In "Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus," the authors introduce a novel approach to quantum circuit optimization leveraging the ZX-calculus. This method, grounded in graph theory, simplifies quantum circuits by representing them as ZX-diagrams, which are more flexible and expressive than traditional circuit representations. The core contribution is a deterministic simplification strategy for these ZX-diagrams by employing the graph transformations of local complementation and pivoting.

Methodology and Results

The paper demonstrates that the inherently flexible ZX-diagrams, derived from quantum circuits, can be systematically simplified using the ZX-calculus. This novel approach, unlike existing techniques that focus on gate substitution and phase polynomial optimization, uses graph transformations to achieve simplification. The simplification process involves manipulating ZX-diagrams through specific rules that remove certain interior spiders by performing graph operations and substituting them with simpler components.

One of the paper's highlights includes establishing a sound procedure for extracting quantum circuits from simplified ZX-diagrams. This procedure ensures that the reduced diagram possesses a graph-theoretic property called the generalised flow, facilitating deterministic extraction. The authors provide a thorough walkthrough, demonstrating this process with a running example in a Jupyter Notebook format, showcasing the technique's superiority in optimizing quantum circuits that feature Clifford circuits.

For Clifford circuits, the authors introduce a new normal form, which optimizes the size and gate depth efficiently. This normal form yields an asymptotically optimal gate count and a smaller upper bound on gate depth for specific architectures. These insights have been proven useful through experimental evidence, where the proposed method outshines traditional optimization techniques.

Implications and Future Work

The implications of this work are far-reaching in the field of quantum computing. By offering a foundational framework for simplifying ZX-diagrams, this technique enables circuit optimizations that surpass conventional methods, especially in reducing the count of non-Clifford gates such as T gates. The ability to manipulate ZX-diagrams facilitates optimizations that are not achievable with other techniques, thus providing a new pathway for developing efficient quantum algorithms.

The paper's approach suggests several promising future directions. Extending these graph-theoretic simplifications to more general classes of diagrams could unlock deeper circuit optimizations, potentially leading to breakthroughs in the development of quantum algorithms that require fewer resources. Furthermore, adapting the circuit extraction strategies to varied qubit topologies opens up possibilities for optimization on constrained architectures, which is crucial for practical applications on current quantum hardware.

Conclusion

This paper presents a significant theoretical advancement in quantum circuit optimization, providing a robust method to simplify circuits using the ZX-calculus and graph theory. The innovative use of local complementation and pivoting operations marks a departure from typical optimization tasks, enabling complex reductions beyond the reach of traditional approaches. As the field of quantum computing evolves, the insights from this paper are poised to play a pivotal role in designing more efficient and practical quantum circuits, propelling future research and applications in quantum technology.