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

Abstract

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...

Geometric aspects of composite pulses

“…More precisely, we reveal that composite pulses robust against certain kinds of systematic errors are nothing but GQGs. This has been observed previously in one-qubit operations [36]. Now we elaborate and generalize this observation to two-qubit operations that are indispensable for a universal set of quantum gates in QIP.…”

“…For example, Zanardi & Rasetti [24] proposed to use the Wilczek-Zee holonomy to implement unitary gates. It is also possible to implement unitary gates by using the Aharonov-Anandan phase [10,25,[36][37][38]. To see this, let {|j a } 1≤a≤n be the eigenvectors of a Hamiltonian H (l(0)) and suppose their dynamical evolution is cyclic, that is, When there is no degeneracy, the spectral decomposition of U (T ) is written as…”

“…To see the geometric nature of type A composite pulses, we follow the argument introduced earlier [36], which has been generalized to a multi-qubit system in [10]. Suppose that the systematic error is proportional to W i ,…”

Geometric Property of Off Resonance Error Robust Composite Pulse

“…, and V i denotes exp(−iθ i n i • σ/2) while V 0 is the identity matrix. In Ref [20], it is shown that if the above error robustness condition for PLE is satisfied, the operation by H 0 (t) in Eq. ( 15) is a geometric quantum gate [22][23][24] based on the concept of holonomy [25][26][27][28].…”

Construction of arbitrary robust one-qubit operations using planar geometry

“…For example, it was shown that any sequence robust against amplitude errors is a geometric quantum gate using AharonovAnandan phase [19,20]. Controlled-NOT and SWAP gates robust against coupling strength error were designed for Ising-type interactions [12].…”