In this paper, we give a characterization of generalized quantum gates.We also show that many important operators are generalized quantumgates, moreover, some of these operators can be represented as theconvex combination of only two unitary operators. Our results answerwhat kinds of operations a duality quantum computing admits. Wepoint out that the set of all generalized quantum gates coincides withthe set of all restricted allowable generalized quantum gates. Thus,our results are also valid for restricted allowable generalized quantumgates
In this paper, we give a characterization of generalized quantum gates.We also show that many important operators are generalized quantumgates, moreover, some of these operators can be represented as theconvex combination of only two unitary operators. Our results answerwhat kinds of operations a duality quantum computing admits. Wepoint out that the set of all generalized quantum gates coincides withthe set of all restricted allowable generalized quantum gates. Thus,our results are also valid for restricted allowable generalized quantumgates
An allowable generalized quantum gate(introduced by Long,Liu and Wang)has the form of U 1 0 d kk k c U -= ∑,where Uk's are unitary operators on a Hilbert space H and 1 0 1 d kkc -= |∑|≤and 1 k |c|≤(0≤k≤d-1).In this work we consider a kind of AGQGs,called restricted allowable generalized quantum gates(RAGQGs),satisfying 1 0 0 1 d kk c -= ∑||≤.Some properties of the set RAGQG(H)of all RAGQGs on H are established.Especially,we prove that the extreme points of RAGQG(H)are exactly unitary operators on H and that B(H)=R+RAGQG(H).
In this paper, we give the most general duality gates, or generalized quantum gates in duality quantum computers. Here we show by explicit construction that a n-bit duality quantum computer with d slits can be simulated perfectly with an ordinary quantum computer with n qubits and one auxiliary qudit. Using this model, we give the most general form of duality gates which is of the form Σ d- 1 i = 0 p i U i , and the p i 's are complex numbers with module less or equal to 1 and constrained by |Σ i p i |≤1.
Quantum computing with qudits, quantum systems with d>2 levels, offers a powerful extension beyond qubits, expanding the computational possibilities of quantum systems, allowing the simplification of the implementation of several algorithms and, possibly, providing a foundation for optimised error correction. In this work, we propose a novel formulation of qudit gates that is universally applicable for any number of levels d, without restrictions on the dimensionality. By extending the mathematical framework of quantum gates to arbitrary dimensions, we derive explicit gate operations that form a universal set for quantum computation on qudits of any size. We demonstrate the validity of our approach through the implementation of the Quantum Fourier Transform (QFT) for arbitrary d, verifying both the correctness and utility of our generalized gates. This novel methodology broadens the design space for quantum algorithms and fault-tolerant architectures, paving the way for advancements in qudit-based quantum computing.
Kliuchnikov, Maslov, and Mosca proved in 2012 that a 2×2 unitary matrix V can be exactly represented by a single-qubit Clifford+ T circuit if and only if the entries of V belong to the ring Z[1/2,i] . Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+ T circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+ T circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number-theoretic characterizations for certain restricted Clifford+ T circuits by considering unitary matrices over subrings of Z[1/2,i] . We focus on the subrings Z[1/2] , Z[1/2] , Z[1/i2] , and Z[1/2,i] , and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates {X,CX,CCX} with an analogue of the Hadamard gate and an optional phase gate.
In this paper, we firstly briefly review the duality quantum computer. Distinctly, the generalized quantum gates, the basic evolution operators in a duality quantum computer are no longer unitary, and they can be expressed in terms of linear combinations of unitary operators. All linear bounded operators can be realized in a duality quantum computer, and unitary operators are just the extreme points of the set of generalized quantum gates. A d-slits duality quantum computer can be realized in an ordinary quantum computer with an additional qudit using the duality quantum computing mode. Duality quantum computer provides flexibility and clear physical picture in designing quantum algorithms, serving as a useful bridge between quantum and classical algorithms. In this review, we will show that duality quantum computer can simulate quantum systems more efficiently than ordinary quantum computers by providing descriptions of the recent efficient quantum simulation algorithms of Childs et al [Quantum Information & Computation, 12(11-12): 901-924 (2012)] for the fast simulation of quantum systems with a sparse Hamiltonian, and the quantum simulation algorithm by Berry et al [Phys. Rev. Lett. 114, 090502 (2015)], which provides exponential improvement in precision for simulating systems with a sparse Hamiltonian.
As problems in computer science become increasingly more computationally intense, researchers have begun to examine other methods of computation. One of the largest new models of computation is the quantum computer. Based on the concepts of quantum mechanics, quantum computers process bits of quantum information, or qubits, using quantum gates. So far some impressive results have been obtained using quantum computers on current problems (e.g. search problems). Many quantum algorithms make use of the Hadamard gate, and a few use a more general form of that gate. In this paper we present a generalized quantum gate that can be shown to encapsulate the standard gates used in quantum algorithms and quantum information processing: Pauli X, Y, and Z, Hadamard, T phase shift, S phase shift, and identity. This generalized gate may also lead to new, interesting, and useful gates for quantum computation.