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arXiv:2103.16081 · Quantum

A Graphical Calculus for Quantum Computing with Multiple Qudits using Generalized Clifford Algebras

Robert Lin

Published Nov 17, 2025• 0 citations
arXivHTMLPDF

Abstract

In this work, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, building off the algebraic framework developed in our prior work. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many properties of the graphical calculus using purely algebraic methods, including a novel algebraic proof of a Yang-Baxter equation and a construction of a corresponding braid group representation. Our algebraic proof, which applies to arbitrary qudit dimension, also enables a resolution of an open problem of Cobanera and Ortiz on the construction of self-dual braid group representations for even qudit dimension. We also derive several new identities for the braid elements, which are key to our proofs. Furthermore, we demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. Additionally, in terms of quantum computation, we demonstrate that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators. In fact, these braid elements are almost Clifford gates, for they normalize the generalized Pauli group up to an extra factor ζ , which is an appropriate square root of a primitive root of unity.

DOI
10.22331/q-2025-11-17-1913

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In this work, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, building off the algebraic framework developed in our prior work. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many properties of the graphical calculus using purely algebraic methods, including a novel algebraic proof of a Yang-Baxter equation and a construction of a corresponding braid group representation. Our algebraic proof, which applies to arbitrary qudit dimension, also enables a resolution of an open problem of Cobanera and Ortiz on the construction of self-dual braid group representations for even qudit dimension. We also derive several new identities for the braid elements, which are key to our proofs. Furthermore, we demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. Additionally, in terms of quantum computation, we demonstrate that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators. In fact, these braid elements are almost Clifford gates, for they normalize the generalized Pauli group up to an extra factor ζ , which is an appropriate square root of a primitive root of unity.
In this dissertation, we investigate algebraic structures in finite-dimensional Hilbert spaces, as concerns quantum computation and quantum information, as well as these structures' applications to lattices. On the quantum computation side, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies these axioms. Subsequently, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, using the algebraic framework developed. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many algebraic identities, including a novel algebraic proof of a Yang-Baxter equation. We also derive a new identity for the braid elements, which is key to our proofs. We then use the Yang-Baxter equation proof to resolve an open question of Cobanera and Ortiz. We demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. In addition, we show how to explicitly compute various vector states in an efficient manner using algebraic methods. On the quantum information side, we introduce a new decomposition of quantum channels acting on group algebras, which we term Kraus-like operator decompositions (Kraus-like decompositions for short). An important motivation for this new decomposition is a general nonexistence result that we show for Kraus operator decompositions for quantum channels in this setting. We show that the notion of convex Kraus-like operator decompositions (in which the coefficients in the sum decomposition are nonnegative and satisfy a sum rule) that are induced by the irreducible characters of a finite group is equivalent to the notion of a conditionally negative-definite length when the length is a class function. For a general finite group G, we prove a stability condition which shows that if the semigroup associated with a length has a convex Kraus-like operator decomposition for all t>0 small enough, then it has a convex Kraus-like operator decomposition for all time t>0. Using the stability condition, we show that for a general finite group, conditional negativity of the length function is equivalent to a set of semidefinite linear constraints on the length function. By a result of Schoenberg, our result implies that in the group algebra setting, a semigroup P_t induced by a length function which is a class function is a quantum channel for all t greater than or equal to 0 if and only if it possesses a convex Kraus-like operator decomposition for all t>0. Finally, motivated by the importance of lattice problems in quantum cryptography, we extend the algebraic framework for multi-qudit representations of generalized Clifford algebras to d-dimensional lattices mod P. We show that under suitable number-theoretic conditions, the subalgebra induced by a lattice has trivial center. Under the trivial center constraint, we construct for pairs of lattice vectors satisfying an algebraic constraint a unitary operator based on the product of generalized Clifford algebra generators associated to each lattice vector.
In this article, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies these axioms. Strengths of our algebraic framework include the minimality of its assumptions, and the readiness by which one may give an explicit construction satisfying these assumptions. In terms of applications, this algebraic framework provides a solid foundation which opens the way for developing a graphical calculus for multi-qudit representations of generalized Clifford algebras using purely algebraic methods, which is addressed in a follow-up paper.
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Zd{\mathbb{Z}}_{d}Zd​ parafermions are exotic non-Abelian quasiparticles generalizing Majorana fermions, which correspond to the case d=2d=2d=2. In contrast to Majorana fermions, braiding of parafermions with d>2d>2d>2 allows one to perform an entangling gate. This has spurred interest in parafermions, and a variety of condensed matter systems have been proposed as potential hosts for them. In this work, we study the computational power of braiding parafermions more systematically. We make no assumptions on the underlying physical model but derive all our results from the algebraical relations that define parafermions. We find a family of 2d2d2d representations of the braid group that are compatible with these relations. The braiding operators derived this way reproduce those derived previously from physical grounds as special cases. We show that if a ddd-level qudit is encoded in the fusion space of four parafermions, braiding of these four parafermions allows one to generate the entire single-qudit Clifford group (up to phases), for any ddd. If ddd is odd, then we show that in fact the entire many-qudit Clifford group can be generated.
Abstract Unitary fusion categories formalise the algebraic theory of topological quantum computation. These categories come naturally enriched in a subcategory of the category of Hilbert spaces, and by looking at this subcategory, one can identify a collection of generators for implementing quantum computation. We represent such generators for the Fibonacci and Ising models, namely the encoding of qubits and the associated braid group representations, with the ZX-calculus and show that in both cases, the Yang–Baxter equation is directly connected to an important rule in the complete ZX-calculus known as the P-rule, which enables one to interchange the phase gates defined with respect to complementary bases. In the Ising case, this reduces to a familiar rule relating two distinct Euler decompositions of the Hadamard gate as <?CDATA π/2\pi/2π/2?> π / 2 Z- and X-phase gates, whereas in the Fibonacci case, we give a previously unconsidered exact solution of the P-rule involving the Golden ratio. We demonstrate the utility of these representations by giving graphical derivations of the single-qubit braid equations for Fibonacci anyons and the single- and two-qubit braid equations for Ising anyons. We furthermore present a fully graphical procedure for simulating and simplifying braids with the ZX-representation of Fibonacci anyons.
Qudits are the natural generalization of qubits, that is, the basic unit of information in quantum computation. This generalization allows for quantum states to be represented with the same cyclic nature present in qubits, while also permitting the basic unit of information to be stored in d-level systems. In this document, we are concerned with studying universality over qudits on arbitrary dimension d. The question of single-qudit universality of a set of operators can be reduced to studying the adjoint representation of the set and determining whether or not its generated group is finite. Thus, we set out to find universal sets of operators in quantum computation over qudits, for any possible value of d. Moreover, we focus on finding “Clifford + T-like” universal sets of operators in arbitrary dimensions, in the hopes of maintaining a strong analog to the qubits case and its most studied universal set. We accomplish this goal by studying how the Clifford group behaves on arbitrary dimensions, and how this behavior relates to the typical Clifford group in prime dimensions. Furthermore, we find that the Clifford group on arbitrary dimension splits as the typical Clifford group on each of the coprime factors of d, which leads to a fundamental difference between the study of universality on prime power dimensions and arbitrary dimensions containing co-prime factors. Then, we focus on studying the structure of universal sets in dimension d, for any d containing at least two co-prime factors. Thus, after studying which T-like gates drive the single-qudit Clifford group to universality on prime power dimensions; that is, guaranteeing the irreducibility of the adjoint representation and an infinite generated group, we combine these two approaches to obtain “Clifford + T-like” universal sets on all possible dimensions.