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 $n$-qubit quantum systems.
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 n-qubit quantum systems.
Pure states of two-qubit quantum systems are described by a four-dimensional vector of complex numbers, and unitary operators transferring a two-qubit quantum system from one state to another have the form of a 4 × 4 matrix with complex elements. This fact brings to mind the idea of studying the spaces of four-dimensional numbers with complex components. Moreover, the results obtained by the authors for four-dimensional numbers with real components inspire some optimism. In this paper we construct four-dimensional spaces of complex numbers by analogy with four-dimensional spaces of real numbers. Each four-dimensional number is mapped to a matrix formed from its components and it is proved that the constructed mapping is a bijection and a homomorphism. In the space of fourdimensional numbers, of the eight basis elements, half are real and half are imaginary. The presence of such symmetry distinguish these spaces from the space of quaternions, in which one basis element is real and the rest are imaginary. The symmetry of the basis numbers makes these spaces a natural generalization of one-dimensional and two-dimensional (complex) algebra. The conditions under which the corresponding matrices are gates for two-qubit quantum systems are defined. The notion of unitary state of a two-qubit quantum system is introduced, to which various gates from commutative groups of gates correspond. It is shown that any gate of a unitary state transforms a unitary state into a unitary state and a non-unitary state into a non-unitary state. Almost all gates used in the construction of quantum circuits, in particular H, SWAP, CX, CY, CZ, have the same properties. The problem of searching for a gate that transfers a quantum system from one unitary state to another unitary state has been solved. Thus, with the help of fourdimensional spaces of complex numbers it was possible to construct whole classes of two-qubit gates, which opens new possibilities for the construction of quantum algorithms. The results obtained have important theoretical and practical implications for quantum computing.
Pure states of three-qubit quantum systems are represented by eight-dimensional complex vectors, whereas their unitary transformations are described by 8×8 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 R8. 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.
The goal of finding a commutative group of gates from the set of unitary matrices is to simplify the process of quantum computing. If a set of gates is commutative, then we can apply the gates in any order without affecting the final outcome. This is important because quantum systems are very sensitive to errors and decoherence, and any disruption to the system can cause errors in the computation.By having a commutative group of gates, we can simplify the process of designing quantum algorithms and reduce the risk of errors. This is particularly important for solving complex computational problems that cannot be solved by classical computers.
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 n-qubit generalisations of Yang-Baxter gates called n-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 n-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=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.
Quantum information and quantum computation are emerging research areas based on the properties of quantum resources, such as superposition and entanglement. In the quantum gate array version, the use of convenient and proper gates is essential. While these gates adopt theoretically convenient forms to reproduce computational algorithms, their design and feasibility depend on specific quantum systems and physical resources used in their setup. These gates should be based on systems driven by physical interactions ruled by a quantum Hamiltonian. Then, the gate design is restricted to the properties and the limitations imposed by the interactions and the physical elements involved. This work shows how anisotropic Heisenberg-Ising interactions, written in a non-local basis, allow the reproduction of quantum computer operations based on unitary processes. We show that gates can be generated by a pulse sequence of driven magnetic fields. This fact states alternative techniques in quantum gate design for magnetic systems. A brief final discussion around associated fault tolerant extensions to the current work is included.
The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length codewords by △ z √-1 produces stabilizer states, that are N = 2 over Z4. We show that exponentiating these Z4-valued quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form N + 1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL(2, N). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary 2-design. The Kerdock design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary 2-designs on encoded qubits, i.e., to construct logical unitary 2-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to 16 qubits.