This note concerns the appearance of a dynamical phase in realizations of geometric phase shift gates. This dynamical phase is a global phase dependent upon the input state and is therefore irrelevant to the gate operation. Thus, the geometric nature of geometric phase shift gates is established.
實現可集成的量子計算的關鍵步驟是實現保真度足夠高的一組普適量子邏輯門 .最近幾年發展的幾何量子計算使用幾何位相來實現量子邏輯門 ,其特點是利用幾何位相的整體幾何性質來避免某些局域的無規噪聲的影響 ,從而實現較高保真度的量子門 .文章先簡要介紹常規幾何量子邏輯門的概念 ,然后重點介紹最近提出的非常規幾何量子計算 :量子計算中使用的邏輯門的總位相既包含有幾何位相 ,又包含有動力學位相 ,但它僅依賴于一些幾何特征 .而且 ,對于任意的量子位輸入態 ,在量子門操作過程中積累的位相要么是零 ,要么是僅依賴幾何特征的位相 . \nThe physical implementation of a universal set of high-fidelity quantum gates is a key step in quantum computation. Geometric quantum computation is a promising scenario to achieve high-fidelity quantum gates. In this scheme, quantum gates are realized based on geometric phases, which are dependent only on some global geometric features. In this Communication, we present an overview of geometric quantum computation, especially the unconventional geometric quantum computation recently proposed by us. In the latter scheme, the total phase accumulated in a gate operation consists of both geometric and dynamic components, but for any input state it still only depends on the global geometric features.
Geometric phases arise naturally in a variety of quantum systems with observable consequences. They also arise in quantum computations when dressed states are used in gating operations. Here we show how they arise in these gating operations and how one may take advantage of the dressed states producing them. Specifically, we show that for a given, but arbitrary Hamiltonian, and at an arbitrary time $\ensuremath{\tau}$, there always exists a set of dressed states such that a given gate operation can be performed by the Hamiltonian up to a phase $\ensuremath{\varphi}$. The phase is a sum of a dynamical phase and a geometric phase. We illustrate the dressed phase for several systems.
We propose a new class of unconventional geometric gates involving nonzero dynamic phases, and elucidate that geometric quantum computation can be implemented by using these gates. Comparing with the conventional geometric gate operation, in which the dynamic phase shift must be removed or avoided, the gates proposed here may be operated more simply. We illustrate in detail that unconventional nontrivial two-qubit geometric gates with built-in fault-tolerant geometric features can be implemented in real physical systems.
Recently, it has been proposed to perform quantum computation through the Berry phase (adiabatic cyclic geometric phase) shift with NMR. This geometric quantum gate will hopefully be fault tolerant to certain types of error because of the geometric property of the Berry phase. Here we give a scheme to realize the NMR C-NOT gate through the Aharonov-Anandan phase (non-adiabatic cyclic phase) shift on the dynamic phase free evolution loop. In our scheme, the gate is run non-adiabatically, thus it is less affected by the decoherence. Moreover, in the scheme we have chosen the zero dynamic phase time evolution loop in obtaining the geometric phase shift; we need not perform any extra operation to cancel the dynamic phase.
A single-loop scenario is proposed to realize nonadiabatic geometric quantum computation. Conventionally, a so-called multiloop approach is used to remove the dynamical phase accumulated in the operation process for geometric quantum gates. More intriguingly, we here illustrate in detail how to use a special single-loop method to remove the dynamical phase and thus to construct a set of universal quantum gates based on the nonadiabatic geometric phase shift. The present scheme is applicable to NMR systems and may be feasible in other physical systems.