# Categorical Quantum Mechanics and FTQC

Nov. 2023

Massachusetts Institute of Mathematics

Categorical quantum mechanics and FTQC are two related topics that explore the foundations and applications of quantum computation using mathematical and computational tools. In this article, I will explain the main concepts and results of these fields, and how they can help us achieve the ultimate goal of building a large-scale and reliable quantum computer.

Categorical quantum mechanics is the study of quantum processes and their composition using monoidal category theory, a branch of mathematics that deals with abstract structures and their transformations. A quantum process is a physical operation that transforms a quantum system, such as a qubit, from one state to another. For example, applying a quantum gate, such as a Hadamard or a CNOT, is a quantum process. A quantum system can also be composed of several subsystems, such as a pair of entangled qubits, and quantum processes can act on them in parallel or sequentially. For example, performing a quantum teleportation protocol is a quantum process that involves both parallel and sequential composition of quantum processes.

Monoidal category theory provides a general framework to describe and reason about quantum processes and their composition. A monoidal category consists of objects, which represent quantum systems, and morphisms, which represent quantum processes. The morphisms can be composed in two ways: sequentially, by connecting the output of one morphism to the input of another, and in parallel, by taking the tensor product of two morphisms. A monoidal category also has a special morphism called the identity, which does nothing to the object it acts on, and a special object called the unit, which represents the trivial quantum system. A monoidal category can also have additional features, such as a dagger, which reverses the direction of a morphism, or a compact structure, which allows us to distinguish between inputs and outputs of a morphism. These features capture some of the essential properties of quantum mechanics, such as the reversibility of quantum processes, the entanglement of quantum systems, and the process-state duality of quantum information.

One of the advantages of using monoidal category theory to study quantum mechanics is that it allows us to use a graphical language, called string diagrams, to represent and manipulate quantum processes and their composition. A string diagram is a pictorial representation of a morphism, where the objects are drawn as wires and the morphisms are drawn as boxes or symbols attached to the wires. The composition of morphisms is represented by connecting the wires, either vertically for sequential composition or horizontally for parallel composition. The identity morphism is represented by a straight wire, and the unit object is represented by an empty space. The dagger of a morphism is represented by flipping the diagram vertically, and the compact structure is represented by bending the wires. String diagrams provide a convenient and intuitive way to visualize and calculate quantum processes, and can often simplify the algebraic expressions that would otherwise be cumbersome to write and manipulate.

Categorical quantum mechanics has been used to develop various models and calculi for quantum computation, such as the ZX-calculus, the ZW-calculus, and the GHZ/W-calculus. These calculi are based on different types of algebras that encode the basic quantum operations and principles, such as quantum gates, measurements, superposition, and complementarity. These algebras can be represented as special types of morphisms in a monoidal category, such as Frobenius algebras, Hopf algebras, or complementary observables. These calculi can be used to reason about quantum protocols, circuits, algorithms, and information in a categorical and diagrammatic way, and can often lead to new insights and simplifications.

FTQC, or fault-tolerant quantum computation, is the goal of building a quantum computer that can perform large and complex quantum computations without being affected by the errors and noise that inevitably occur in the physical implementation of quantum systems and processes. Quantum systems and processes are very sensitive to external disturbances, such as thermal fluctuations, electromagnetic interference, or measurement errors, which can corrupt the quantum information and cause the computation to fail. To overcome this challenge, we need to develop methods and techniques to protect and correct the quantum information from the errors and noise, and to ensure that the computation can proceed reliably and accurately.

One of the main methods to achieve FTQC is to use quantum error correction (QEC) codes, which are schemes that encode a logical qubit, which represents the quantum information we want to store and manipulate, into a larger number of physical qubits, which represent the actual quantum devices we use to implement the quantum system. The encoding is done in such a way that the logical qubit can be recovered from the physical qubits even if some of them are corrupted by errors or noise, by using certain recovery operations that can detect and correct the errors. A QEC code can be characterized by several parameters, such as the number of physical qubits required, the types and rates of errors that can be corrected, the types and costs of operations that can be performed on the encoded qubits, and the resulting logical error rate that affects the computation.

QEC codes can be designed and analyzed using categorical quantum mechanics, by using the framework of dagger symmetric monoidal categories. A QEC code can be seen as a special type of morphism that encodes a logical qubit into a physical qubit, and a recovery operation can be seen as another type of morphism that decodes a physical qubit into a logical qubit. The composition of these morphisms can be represented by string diagrams, and the properties and performance of the QEC code can be studied by using the algebraic and graphical techniques of categorical quantum mechanics. For example, the notion of distance, which measures the number of errors that a QEC code can correct, can be defined in terms of the orthogonality of the encoded states, which can be expressed by using the dagger and the compact structure of the category. Similarly, the notion of transversality, which measures the ease of performing logical operations on the encoded qubits, can be defined in terms of the commutativity of the encoding and the logical operations, which can be expressed by using the symmetry and the coherence of the category.

There are many types and families of QEC codes that have been proposed and studied using categorical quantum mechanics, such as stabilizer codes, topological codes, subsystem codes, subsystem surface codes, and quantum low-density parity-check codes. These codes have different advantages and disadvantages in terms of the parameters mentioned above, and can be suitable for different types of quantum hardware and architectures. Some of these codes have been implemented and tested on existing quantum devices, and have shown promising results in reducing the error rates and increasing the coherence times of the qubits. However, there are still many challenges and open problems in developing and optimizing QEC codes for FTQC, such as finding the optimal trade-off between the overhead and the performance, designing efficient and scalable encoding and decoding circuits, and adapting the codes to the specific features and constraints of the quantum hardware.

In conclusion, categorical quantum mechanics and FTQC are two important and active areas of research in quantum computation, which aim to understand and exploit the power and potential of quantum processes and systems. By using the mathematical and computational tools of monoidal category theory and string diagrams, we can develop and analyze various models and methods for quantum computation, and design and implement reliable and robust quantum computers that can solve hard and complex problems that are beyond the reach of classical computers.

New York General Group