Composite pulses in NMR as nonadiabatic geometric quantum gates

Abstract

We show that some composite pulses widely employed in nuclear magnetic resonance experiments are regarded as non-adiabatic geometric quantum gates with Aharanov-Anandan phases. Thus, we reveal the presence of a fundamental issue on quantum mechanics behind a traditional technique. To examine the robustness of such composite pulses against fluctuations, we present a simple noise model in a two-level system. Then, we find that the composite pulses possesses purely geometrical nature even under a certain type of fluctuations.PACS numbers: 03.65. Vf, 82.56.Jn, Geometric phases have been attracting a lot of attention from the view point of the foundation of quantum mechanics and mathematical physics [1,2,3,4]. Recently, their application to quantum information processing is spotlighted [5,6], because they are expected to be robust against noise. However, the robustness of a geometric quantum gate (GQG), which is a quantum gate only using geometric phases, is not completely verified. Various examinations on this issue have been reported [7,8,9,10,11,12]. Blais and Tremblay [7] claimed that no advantage of the GQGs exists compared to the corresponding quantum gates with dynamical phases, while Zhu and Zanardi [8] showed that their non-adiabatic GQGs are robust against fluctuations in control parameters.In this paper, we show that some composite pulses widely employed in nuclear magnetic resonance (NMR) [13,14] to accomplish reliable operations is regarded as non-adiabatic GQGs based on an Aharonov-Anandan (AA) phase [15], and propose a simple noise model in a two-level system. Then, we classify fluctuations in terms of the robustness of the GQGs.An AA phase appears under non-adiabatic cyclic time evolution of a quantum system [15]. We note that the generalization to the non-cyclic case is given in Ref. [3,16]. Let us write the Bloch vector at t (0 ≤ t ≤ 1) as n(t)(∈ R 3 ). We denote a state vector given n(t) as |n(t) (∈ C 2 ). Namely, n(t) = n(t)|σ|n(t) , where σ = t (σ x , σ y , σ z ). The symbol t means the transposition of a vector. Time evolution is described by the Schrödinger equation with the Hamiltonian H(t). Note that |n(t)| = 1. Hereafter, we denote n(0) as n. We take the natural unit system in which = 1. Suppose that |n(1) = e iγ |n (γ ∈ R): n(1) = n. The AA phase γ g is defined as [15] γ g = γ − γ d ,(1) * Present address: CCSE, Japan Atomic Energy Agency, 6-9-3 Higashi-Ueno, Tokyo 110-0015, Japan and CREST(JST), 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan where( 2) is a dynamical phase. Next, suppose n + and n − are two Bloch vectors satisfying (a) n + · n − = −1 (i.e., n + |n − = 0) and (b) n ± (1) = n ± (i.e., there exist γ ± ∈ R such that |n ± (1) = e iγ± |n ± . An arbitrary quantum state |n is expressed by |n = a + |n + + a − |n − , where a ± = n ± |n . We call n ± basis Bloch vector corresponding to H(t). The initial state |n is transformed into the final state |n(1) = a + e iγ+ |n + + a − e iγ− |n − . Thus, the time evolution operator U at t = 1 generated by H(t) (t ∈ [0, 1]) is rewritten as U =...

Geometric Quantum Gates, Composite Pulses, and Trotter–Suzuki Formulas

“…Recently, a geometric quantum gate (GQG in short), which is a quantum gate only with geometric phases, is spotlighted in quantum information processing [5,6], because they are expected to be robust against noise. Although its robustness has not yet been generally confirmed [7,8,9,10,11,12], some GQG's are robust against certain types of fluctuations [13].On the other hand, composite rf-pulses are extensively employed in NMR [14,15], which are robust against systematic errors of the system. Note that rf-pulses are means for controlling spin states and have direct correspondence to quantum gates.…”

“…In order to proceed our discussion, we first review an AA phase that appears under non-adiabatic cyclic time evolution of a quantum system [19] and a GQG with it for a single qubit [13]. The qubit state |n(t) (∈ C 2 ) at t (∈ [0, T ]) corresponds to the Bloch vector…”

Geometric quantum gates in liquid-state NMR based on a cancellation of dynamical phases

“…We assume that the pulse duration is infinitely short for simplicity. It should be noted that this operation does not generate a dynamical phase since the y-axis is perpendicular to both k 1 and k 2 [4,28].…”

“…In particular, the importance of its application to quantum information processing has been increasing recently. A promising way to achieve this is to employ geometric phases (or, more generally, nonAbelian holonomies) [1,2], because geometric phases are expected to be robust against noise and decoherence under a proper condition [3,4]. A large number of studies for applying their potential robustness to quantum computing have been done, e,g., phase-shift gates with Berry phases [5], nonadiabatic geometric quantum gates [6,7,8,9,10,11,12,13], holonomic quantum computing [14,15,16,17,18,19,20,21], quantum gates with noncyclic geometric phases [22], and so on.…”

“…In this paper, we combine Zhu and Wang's approach with Jones et al's one, employing an Aharonov-Anandan phase for fast gate operation and a spin echo technique for dynamical phase cancellation, and demonstrate one-qubit gates with a commercial liquid-state nuclear magnetic resonance (NMR) system. In many experiments of nonadiabatic geometric quantum gates [11,12,13], the gate operations in which the dynamical phase is arranged to vanish [4,28] have been adopted. In the present paper, we show that we may have another option for physical realization of geometric quantum gates.…”