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Unitary States and Gates of Three-Qubit Quantum Systems

Mars B. Gabbassov, Tolybay Z. Kuanov, Berik I. Tuleuov, Turganbay K. Yermagambetov

Published Oct 20, 2026• 0 citations
Publisher

Abstract

Pure states of three-qubit quantum systems are represented by eight-dimensional complex vectors, whereas their unitary transformations are described by $8 \times 8$ complex matrices. Motivated by this correspondence, we extend our algebraic model for two-qubit systems to eight-dimensional spaces endowed with commutative and associative multiplication. To each eight-dimensional number we assign a matrix and prove that the resulting correspondence is a homomorphic bijection. We derive conditions under which the associated matrix is unitary and hence represents a three-qubit quantum gate. This leads to a specific algebraic notion of a unitary state: a normalized state whose multiplication matrix is unitary. We construct 16,384 basis spaces; within each fixed space, multiplication generates a commuting family of quantum gates. For any two unitary states in the class considered, we give an explicit transition-gate construction within one of these algebraic families. For real components, the unitary states in a representative basis space form a union of thirty-two four-dimensional manifolds in \(\mathbb{R}^{8}\). The construction therefore yields new structured classes of three-qubit unitary operators and invariant state sets, while its extension to larger multiqubit systems remains open.

DOI
10.20944/preprints202610.1882.v1

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Pure states of three-qubit quantum systems are represented by eight-dimensional complex vectors, whereas their unitary transformations are described by 8×88 \times 88×8 complex matrices. Motivated by this correspondence, we extend our algebraic model for two-qubit systems to eight-dimensional spaces endowed with commutative and associative multiplication. To each eight-dimensional number we assign a matrix and prove that the resulting correspondence is a homomorphic bijection. We derive conditions under which the associated matrix is unitary and hence represents a three-qubit quantum gate. This leads to a specific algebraic notion of a unitary state: a normalized state whose multiplication matrix is unitary. We construct 16,384 basis spaces; within each fixed space, multiplication generates a commuting family of quantum gates. For any two unitary states in the class considered, we give an explicit transition-gate construction within one of these algebraic families. For real components, the unitary states in a representative basis space form a union of thirty-two four-dimensional manifolds in (\mathbb{R}^{8}). The construction therefore yields new structured classes of three-qubit unitary operators and invariant state sets, while its extension to larger multiqubit systems remains open.
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This work is devoted to the construction and study of commutative gates for a two-qubit quantum system. Using four-dimensional algebra developed by the Kazakh mathematician Abenov M.M. all groups of commutative gates have been constructed, and among all states of a two-qubit quantum system, unitary states with which a specific gate is connected have been identified. An explicit type of gate is described that transfers a quantum system from one unitary state to another unitary state. The proposed approach opens up new possibilities for the design of quantum algorithms not only for two-qubit quantum systems, but also for nnn-qubit quantum systems.
Quantum simulations of many-body systems using 2-qubit Yang-Baxter gates offer a benchmark for quantum hardware. This can be extended to the higher dimensional case with nnn-qubit generalisations of Yang-Baxter gates called nnn-simplex operators. Such multi-qubit gates potentially lead to shallower and more efficient quantum circuits as well. Finding them amounts to identifying unitary solutions of the nnn-simplex equations, the building blocks of higher dimensional integrable systems. These are a set of highly non-linear and over determined system of equations making it notoriously hard to solve even when the local Hilbert spaces are spanned by qubits. We systematically overcome this for higher simplex operators constructed using two methods: from Clifford algebras and by lifting Yang-Baxter operators. The n=3n=3n=3 or the tetrahedron case is analyzed in detail. For the qubit case our methods produce 13 inequivalent families of unitary tetrahedron operators. 12 of these families are obtained by appending the 5 unitary families of 4 by 4 constant Yang-Baxter operators of Dye-Hietarinta, with a single qubit operator. As applications, universal sets of single, two and three qubit gates are realized using such unitary tetrahedron operators. The ideas presented in this work can be naturally extended to the higher simplex cases.
Abstract We present two results on multiqubit Werner states, defined to be those states that are invariant under the collective action of any given single-qubit unitary that acts simultaneously on all the qubits. Motivated by the desire to characterize entanglement properties of Werner states, we construct a basis for the real linear vector space of Werner invariant Hermitian operators on the Hilbert space of pure states; it follows that any mixed Werner state can be written as a mixture of these basis operators with unique coefficients. Continuing a study of ‘polygon diagram’ Werner states constructed in earlier work, with a goal to connect diagrams to entanglement properties, we consider a family of multiqubit states that generalize the singlet, and show that their 2-qubit reduced density matrices are separable.
The evolution of quantum states serves as good fundamental studies in understanding the quantum information systems which finally lead to the research on quantum computation. To carry out such a study, mathematical tools such as the Lie group and their associated Lie algebra is of great importance. In this study, the Lie algebra of su(8) is represented in a tensor product operation between three Pauli matrices. This can be realized by constructing the generalized Gell-Mann matrices and comparing them to the Pauli bases. It is shown that there is a one-to-one correlation of the Gell-Mann matrices with the Pauli basis which resembled the change of coordinates. Together with the commutator relations and the frequency analysis of the structure constant via the algebra, the Lie bracket operation will be highlighted providing insight into relating quantum circuit model with Lie Algebra. These are particularly useful when dealing with three-qubit quantum circuit problems which involve quantum gates that is derived from the SU(8) Lie group.
We start with the simplest quantum system (a two-level system, i.e., a qubit) and discuss a one-to-one mapping of the quantum state in a two-dimensional Hilbert space to a vector in an eight dimensional probability space (probability simplex). We then show how the usual transformations of the quantum state, specifically the Hadamard gate and the single-qubit phase gate, can be accomplished with appropriate transformations of the mapped vector in the probability simplex. One key defining feature of both the mapping to the simplex and the transformations in the simplex is that they are not linear. These results show that both the initial state and the time evolution of a qubit can be fully captured in an eight dimensional probability simplex (or equivalently using three classical probabilistic bits). We then discuss multi-partite quantum systems and their mapping to the probability simplex. Here, the key tool is the identical tensor product structure of combining multiple quantum systems as well as multiple probability spaces. Specifically, we explicitly show how to implement an analog of the two-qubit controlled-not (CNOT) gate in the simplex. We leave it an open problem how much the quantum dynamics of NNN qubits can be captured in a probability simplex with 3N3N3N classical probabilistic bits. Finally, we also discuss the equivalent of the Schrodinger's equation for the wavefunction (in a Hilbert space of arbitrary dimension), which dictates the time evolution of the vectors in the simplex.