Duality quantum computer and the efficient quantum simulations

Wei, Shi-Jie, Gui‐Lu Long

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.

Long, Gui-Lu, Liu . +1 more

Shi-Jie Wei, Gui‐Lu Long

Practical quantum systems are open systems due to interactions with their environment. Understanding the evolution of open systems dynamics is important for quantum noise processes , designing quantum error correcting codes, and performing simulations of open quantum systems. Here we proposed an efficient quantum algorithm for simulating the evolution of an open quantum system on a duality quantum computer. In contrast to unitary evolution in a usual quantum computer, the evolution operator in a duality quantum computer is a linear combination of unitary operators. In this duality algorithm, the time evolution of open quantum system is realized by using Kraus operators which is naturally realized in duality quantum computing. Compared to the Lloyd's quantum algorithm [Science.273, 1073(1996)] , the dependence on the dimension of the open quantum system in our algorithm is decreased. Moreover, our algorithm uses a truncated Taylor series of the evolution operators, exponentially improving the performance on the precision compared with existing quantum simulation algorithms with unitary evolution operations.

pXiangyu Kong, Gui Lu Longp

Shi-Jie Wei, Dong Ruan, Gui‐Lu Long

Because of inevitable coupling with the environment, nearly all practical quantum systems are open system, where the evolution is not necessarily unitary. In this paper, we propose a duality quantum algorithm for simulating Hamiltonian evolution of an open quantum system. In contrast to unitary evolution in a usual quantum computer, the evolution operator in a duality quantum computer is a linear combination of unitary operators. In this duality quantum algorithm, the time evolution of the open quantum system is realized by using Kraus operators which is naturally implemented in duality quantum computer. This duality quantum algorithm has two distinct advantages compared to existing quantum simulation algorithms with unitary evolution operations. Firstly, the query complexity of the algorithm is O(d(3)) in contrast to O(d(4)) in existing unitary simulation algorithm, where d is the dimension of the open quantum system. Secondly, By using a truncated Taylor series of the evolution operators, this duality quantum algorithm provides an exponential improvement in precision compared with previous unitary simulation algorithm.

Guilu Long, Yang Liu