Mapping fermionic systems to qubits on a quantum computer is often the first step for algorithms in quantum chemistry and condensed matter physics. However, it is difficult to reconcile the many different approaches that have been proposed, such as those based on binary matrices, ternary trees, and stabilizer codes. This challenge is further exacerbated by the many ways to describe them -- transformation of Majorana operators, action on Fock states, encoder circuits, and stabilizers of local encodings -- making it challenging to know when the mappings are equivalent. In this work, we present a graphical framework for fermion-to-qubit mappings that streamlines and unifies various representations through the ZX-calculus. To start, we present the correspondence between linear encodings of the Fock basis and phase-free ZX-diagrams. The commutation rules of scalable ZX-calculus allows us to convert the fermionic operators to Pauli operators under any linear encoding. Next, we give a translation from ternary tree mappings to scalable ZX-diagrams, which not only directly represents the encoder map as a CNOT circuit, but also retains the same structure as the tree. Consequently, we graphically prove that ternary tree transformations are equivalent to linear encodings, a recent result by Chiew et al. The scalable ZX representation moreover enables us to construct an algorithm to directly compute the binary matrix for any ternary tree mapping. Lastly, we present the graphical representation of local fermion-to-qubit encodings. Its encoder ZX-diagram has the same connectivity as the interaction graph of the fermionic Hamiltonian and also allows us to easily identify stabilizers of the encoding.
Mapping fermionic systems to qubits on a quantum computer is often the first step for algorithms in quantum chemistry and condensed matter physics. However, it is difficult to reconcile the many different approaches that have been proposed, such as those based on binary matrices, ternary trees, and stabilizer codes. This challenge is further exacerbated by the many ways to describe them -- transformation of Majorana operators, action on Fock states, encoder circuits, and stabilizers of local encodings -- making it challenging to know when the mappings are equivalent. In this work, we present a graphical framework for fermion-to-qubit mappings that streamlines and unifies various representations through the ZX-calculus. To start, we present the correspondence between linear encodings of the Fock basis and phase-free ZX-diagrams. The commutation rules of scalable ZX-calculus allows us to convert the fermionic operators to Pauli operators under any linear encoding. Next, we give a translation from ternary tree mappings to scalable ZX-diagrams, which not only directly represents the encoder map as a CNOT circuit, but also retains the same structure as the tree. Consequently, we graphically prove that ternary tree transformations are equivalent to linear encodings, a recent result by Chiew et al. The scalable ZX representation moreover enables us to construct an algorithm to directly compute the binary matrix for any ternary tree mapping. Lastly, we present the graphical representation of local fermion-to-qubit encodings. Its encoder ZX-diagram has the same connectivity as the interaction graph of the fermionic Hamiltonian and also allows us to easily identify stabilizers of the encoding.
We consider two approaches to designing fermion-qubit mappings: (1) ternary tree transformations, which use Pauli representations of the Majorana operators that correspond to root-to-leaf paths of a tree graph and (2) linear encodings of the Fock basis, such as the Jordan-Wigner and Bravyi-Kitaev transformations, which store linear binary transformations of the fermionic occupation number vectors in the computational basis of qubits. These approaches have emerged as distinct concepts, with little notational consistency between them. In this paper we propose a universal description of fermion-qubit mappings, which reveals the relationship between ternary tree transformations and linear encodings. Using our notation, we show that every product-preserving ternary tree transformation is equivalent to a linear encoding of the Fock basis.
We introduce the Scalable ZX-calculus (SZX-calculus for short), a formal and compact graphical language for the design and verification of quantum computations. The SZX-calculus is an extension of the ZX-calculus, a powerful framework that captures graphically the fundamental properties of quantum mechanics through its complete set of rewrite rules. The ZX-calculus is, however, a low level language, with each wire representing a single qubit. This limits its ability to handle large and elaborate quantum evolutions. We extend the ZX-calculus to registers of qubits and allow compact representation of sub-diagrams via binary matrices. We show soundness and completeness of the SZX-calculus and provide two examples of applications, for graph states and error correcting codes.
While the quantum circuit model remains the dominant framework for representing and executing quantum algorithms, it is not always the most optimal framework for representing the quantum circuit’s deeper structure. In a standard circuit diagram, computation is shown mainly as a time ordered sequence of gates acting on qubit wires. ZX calculus, by contrast, represents the same process using two families of generators, the Z spiders and the X spiders, together with wires and Hadamard structure, so that algebraic identities, entanglement structure, and rewrite based simplifications become visible directly in the diagram itself. ZX calculus is not another visual representation for the same quantum circuit we have been using, but a graphical calculus in which diagrams carry both computational meaning and transformation rules. This is also the reason for the name ZX ; the formalism is built around the interaction between Z -type and X -type observables, phases, and basis changes. By turning matrix manipulations into diagrammatic rewrites, ZX calculus reduces much of the redundancies and abstract nature usually associated with quantum computation and provides a more structural language for understanding, optimizing, and verifying quantum processes.
Many stabilizer quantum error-correcting codes are built from a finite pattern repeated across space or time, such as lattice codes, translation-invariant graph states, and quantum convolutional codes. Ordinary stabilizer ZX-diagrams capture only finite truncations of such systems, obscuring the repeated structure that defines them. We introduce the delayed stabilizer ZX-calculus, a finite graphical language for these infinite, translation-invariant processes. It extends the odd-prime-dimensional stabilizer ZX-calculus with a single new generator, the delay, which feeds data from one time step to the next. We equip the calculus with two semantics. In the first semantics, we interpret the behaviour of a delayed ZX-diagram as an equivalence class of sequences of quantum channels; where two sequences are identified if they have the same information content. We show that the behaviour of a delayed ZX-diagram uniquely determines an infinite stabilizer group. In the second semantics, we interpret the delay as a formal variable, encoding the translation-invariant families of Pauli operators as generating functions. This allows us to represent a delayed ZX-diagram in terms of a tableau of generating functions, from which the infinite stabilizer group can be recovered. Finally, we give a complete axiomatization of the delayed stabilizer ZX-calculus, featuring generalised Euler decomposition and colour change rules. Using generalised forms of local complementation and pivoting, we reduce every diagram to a unique normal form. This establishes soundness, universality, and completeness for the generating tableau semantics.
We present a quantum compilation algorithm that maps Clifford encoders, encoding maps for stabilizer quantum codes, to a unique graphical representation in the ZX calculus. Specifically, we develop a canonical form in the ZX calculus and prove canonicity as well as efficient reducibility of any Clifford encoder into the canonical form. The diagrams produced by our compiler visualize information propagation and entanglement structure of the encoder, revealing properties that may be obscured in the circuit or stabilizer-tableau representation. Consequently, our canonical representation may be an informative technique for the design of new stabilizer quantum codes via graph theory analysis.
We introduce the Spin-ZX calculus as an elevation of Penrose's diagrams and associated binor calculus to the level of a formal diagrammatic language. The power of doing so is illustrated by the variety of scientific areas we apply it to: permutational quantum computing, quantum machine learning, condensed matter physics, and quantum gravity. Respectively, we analyse permutational computing transition amplitudes, evaluate barren plateaus for SU(2) symmetric ansätze, study properties of AKLT states, and derive the minimum quantised volume in loop quantum gravity.
Our starting point is the mixed-dimensional ZX calculus, a purely diagrammatic language that has been proven to be complete for finite-dimensional Hilbert spaces. That is, any equation that can be derived in the Hilbert space formalism, can also be derived in the mixed-dimensional ZX calculus. We embed the Spin-ZX calculus inside the mixed-dimensional ZX calculus, rendering it a quantum information flavoured diagrammatic language for the quantum theory of angular momentum, i.e. SU(2) representation theory. We diagrammatically derive the fundamental spin coupling objects - such as Clebsch-Gordan coefficients, symmetrising mappings between qubits and spin spaces, and spin Hamiltonians - under this embedding.
Our results establish the Spin-ZX calculus as a powerful tool for representing and computing with SU(2) systems graphically, offering new insights into foundational relationships and paving the way for new diagrammatic algorithms for theoretical physics.