In NMR experiments and quantum computation, many pulse (quantum gate) sequences called the composite pulses, were developed to suppress one of two dominant errors; a pulse length error and an off-resonance error. We describe, in this paper, a general prescription to design a single-qubit concatenated composite pulse (CCCP) that is robust against two types of errors simultaneously. To this end, we introduce a new property, which is satisfied by some composite pulses and is sufficient to obtain a CCCP. Then we introduce a general method to design CCCPs with shorter execution time and less number of pulses. DRAFT taneously, although they are restricted within null operation and π-rotation (NOT gate) of a nuclear spin. In order to realize a composite pulse which is robust against these two errors simultaneously and without any restriction, we designed a ConCatenated Composite Pulse (CCCP) by concatenating CORPSE and SCROFULOUS. 16) The CCCP reported previously 16) consists of 3 composite pulses, each of which consists of 3 elementary pulses. Consequently, this CCCP is made of 9 pulses in total. Although this CCCP is robust against both PLE and ORE, its execution time is considerably longer than the corresponding elementary pulse. CCCPs made of less number of elementary pulses are certainly desirable from a viewpoint of decoherence suppression.Establishing general prescription to design a CCCP and its improvement are the subjects of this paper. Some composite pulses have interesting property, which we call the residualerror-preserving (REP) property. Employing two types of composite pulses with mutually exclusive REP properties is essential to design a successful CCCP robust against both PLE and ORE. By using this method, we obtain many CCCPs systematically. Moreover, we further

We propose a simple formalism to design unitary gates robust against given systematic errors. This formalism generalizes our previous observation [Y. Kondo and M. Bando, J. Phys. Soc. Jpn. 80, 054002 (2011)] that vanishing dynamical phase in some composite gates is essential to suppress pulse-length errors. By employing our formalism, we derive a new composite unitary gate which can be seen as a concatenation of two known composite unitary operations. The obtained unitary gate has high fidelity over a wider range of error strengths compared to existing composite gates.

Unitary operations acting on a quantum system must be robust against systematic errors in control parameters for reliable quantum computing. Composite pulse technique in nuclear magnetic resonance realizes such a robust operation by employing a sequence of possibly poor-quality pulses. In this study, we demonstrate that two kinds of composite pulses-one compensates for a pulse length error in a one-qubit system and the other compensates for a J -coupling error in a two-qubit system-have a vanishing dynamical phase and thereby can be seen as geometric quantum gates, which implement unitary gates by the holonomy associated with dynamics of cyclic vectors defined in the text.

We show that all geometric quantum gates (GQG's in short), which are quantum gates only with geometric phases, are robust against control field strength errors. As examples of this observation, we show (1) how robust composite rf-pulses in NMR are geometrically constructed and (2) a composite rf-pulse based on Trotter-Suzuki Formulas is a GQG.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, 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. Most of composite rf-pulses in NMR are designed with the knowledge of initial states, and thus it is often not replaceable with simple pulses. However, there are fully compensating composite rf-pulses that are replaceable with simple pulses without further modifications of other pulses, and thus are compatible in use for quantum computation, as demonstrated in ion traps [16] and Josephson junctions [17] as well as in NMR [18].In this letter, we discuss the relation between fully compensating composite quantum gates which is robust against control field strength errors and non-adiabatic GQG's with Aharanov-Anandan (AA) phases [19]. Let us define an ideal single-qubit operationwhere we take the natural unit system in which = 1. m is a unit vector (∈ R 3 ), while σ is a standard Pauli matrix vector such that σ = (σ x , σ y , σ z ). θ represents a control field strength. Note that θ and m are both constant. A real erroneous operationR(m, θ) with a systematic control field strength error is modeled as follows.where ǫ ≪ 1 is an unknown fixed parameter that represents the error. If we find a series of operationsR(m j , θ j ) (m j and θ j are constant), such thatwe call it a fully compensating composite quantum gate which is robust against a control field strength error. 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 n(t) = n(t)|σ|n(t) (∈ R 3 ). Suppose that the Hamiltonian H(t) generates a cyclic time evolution such that |n(T ) = e iγ |n(0) (γ ∈ R). The AA phase γ g is defined as [19] whereis a dynamic phase. Next, suppose |n + (0) and |n − (0) are two states satisfying (a) n + (0)|n − (0) = 0 (or, n + (0) = −n − (0)) and (b) |n ± (T ) = e iγ± |n ± (0) , where γ ± ∈ R. An arbitrary quantum s...

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