Related papers: Picturing Counting Reductions with the ZH-Calculus
The ZX-calculus is a powerful framework for reasoning in quantum computing. It provides in particular a compact representation of matrices of interests. A peculiar property of the ZX-calculus is the absence of a formal sum allowing the…
Recent completeness results on the ZX-Calculus used a third-party language, namely the ZW-Calculus. As a consequence, these proofs are elegant, but sadly non-constructive. We address this issue in the following. To do so, we first describe…
From Feynman diagrams to tensor networks, diagrammatic representations of computations in quantum mechanics have catalysed progress in physics. These diagrams represent the underlying mathematical operations and aid physical interpretation,…
Graphical languages, like quantum circuits or ZX-calculus, have been successfully designed to represent (memoryless) quantum computations acting on a finite number of qubits. Meanwhile, delayed traces have been used as a graphical way to…
Graphical calculi are vital tools for representing and reasoning about quantum circuits and processes. Some are not only graphically intuitive but also logically complete. The best known of these is the ZX-calculus, which is an industry…
The ZX, ZW and ZH calculi are all graphical calculi for reasoning about pure state qubit quantum mechanics. All of these languages use certain diagrammatic decorations, called !-boxes and phase variables, to indicate not just one diagram…
This note describes how the the scalable ZXH calculus can be used to represent in a compact way the quantum gates that are diagonal in the computational basis. This includes controlled and multi-controlled Z gates, their generalizations,…
Quantum Error-Correcting Codes (QECCs) play a crucial role in enhancing the robustness of quantum computing and communication systems against errors. Within the realm of QECCs, stabilizer codes, and specifically graph codes, stand out for…
The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, yields insight into the hierarchy of complexity classes. In particular, the theory produces algebraic tests for a…
The computational cost of counting the number of solutions satisfying a Boolean formula, which is a problem instance of #SAT, has proven subtle to quantify. Even when finding individual satisfying solutions is computationally easy (e.g.…
Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate…
Quantum computations are easily represented in the graphical notation known as the ZX-calculus, a.k.a. the red-green calculus. We demonstrate its use in reasoning about measurement-based quantum computing, where the graphical syntax…
Quantum computing is an emerging technology in which quantum mechanical properties are suitably utilized to perform certain compute-intensive operations faster than classical computers. Quantum algorithms are designed as a combination of…
We study the problem of counting the number of homomorphisms from an input graph $G$ to a fixed (quantum) graph $\bar{H}$ in any finite field of prime order $\mathbb{Z}_p$. The subproblem with graph $H$ was introduced by Faben and Jerrum…
ZX-Calculus is a versatile graphical language for quantum computation equipped with an equational theory. Getting inspiration from Geometry of Interaction, in this paper we propose a token-machine-based asynchronous model of both pure…
The ZX-calculus is a graphical language for quantum processes with built-in rewrite rules. The rewrite rules allow equalities to be derived entirely graphically, leading to the question of completeness: can any equality that is derivable…
Mapping a quantum algorithm to any practical large-scale quantum computer will require a sequence of compilations and optimizations. At the level of fault-tolerant encoding, one likely requirement of this process is the translation into a…
Boolean Satisfiability (SAT) problems are critical in fields such as artificial intelligence and cryptography, where efficient solutions are essential. Conventional probabilistic solvers often encounter scalability issues due to complex…
Reducing the number of non-Clifford quantum gates present in a circuit is an important task for efficiently implementing quantum computations, especially in the fault-tolerant regime. We present a new method for reducing the number of…
Diagrammatic techniques for reasoning about monoidal categories provide an intuitive understanding of the symmetries and connections of interacting computational processes. In the context of categorical quantum mechanics, Coecke and…