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Implementation of controlled phase shift gates and Collins version of Deutsch-Jozsa algorithm on a quadrupolar spin-7/2 nucleus using non-adiabatic geometric phases

In this work Controlled phase shift gates are implemented on a qaudrupolar system, by using non-adiabatic geometric phases. A general procedure is given, for implementing controlled phase shift gates in an 'N' level system. The utility of such controlled phase shift gates, is demonstrated here by implementing 3-qubit Deutsch-Jozsa algorithm on a 7/2 quadrupolar nucleus oriented in a liquid crystal matrix.


INTRODUCTION
The use of quantum systems for information processing was first introduced by Benioff [1]. In 1985 Deutsch described quantum computers which exploit the superposition of multi particle states, thereby achieving massive parallelism [2]. Researchers have also studied the possibility of solving certain types of problems more efficiently than can be done on conventional computers [3,4,5]. These theoretical possibilities have generated significant interest for experimental realization of quantum computers [6,7]. Several techniques are being exploited for quantum computing and quantum information processing, including nuclear magnetic resonance (NMR) [8,9].
NMR has played a leading role for the practical demonstration of quantum gates and algorithms. Most of the NMR Quantum information processing (QIP) experiments have utilized systems having indirect spin-spin couplings (scalar J couplings) [10,11,12,13].
A quadrupolar nucleus of spin I has (2I+1) energy levels, which are equi-spaced in the high magnetic field and the 2I single quantum transitions are degenerate. In the presence of first order quadrupolar interaction this degeneracy is lifted and gives rise to 2I equispaced transitions [29,30]. For molecules partially oriented in anisotropic media, such as liquid crystals, such a situation is often obtained [15]. If (2I + 1) = 2 n , such a system can be treated as an n-qubit system [15,17]. The advantage of such systems and the dipolar coupled systems, over the J-coupled systems is that the coupling values are one to two orders of magnitude larger, allowing shorter gate times or the use of shorter transition selective pulses. So far quadrupolar systems have been used for, quantum simulation, preparation of pseudo pure states, implementation of quantum gates and search algorithms [18,19,20,21,22,23,24,25,26,28]. Recently Das et al have implemented Cleve version of 2qubit DJ algorithm on a spin 7/2 nucleus [27]. In all these cases the controlled gates are implemented by inverting populations between various levels, by using transition selective π pulses. Recently it has been demonstrated that non-adiabatic geometric phases can be used for implementing quantum gates [33,34]. Here we use non-adiabatic geometric phases to implement controlled phase shift gates and Collins version of DJ algorithm on a 3-qubit system obtained by eight eigen states of a spin-7/2 quadrupolar nucleus.
The Hamiltonian of a quadrupolar nucleus partially oriented in liquid crystal matrix, in the presence of large magnetic field B o , having a first order quadrupolar coupling, is given by, where ω o = −γB o is the resonance frequency, S is the order parameter at the site of the nucleus, and e 2 qQ is the quadrupolar coupling. If the order parameter S is small, the effective quadrupolar coupling ∧ can be of the order of a few kHz in spite of e 2 qQ being of the order of several MHz. Thus, it is possible to observe the satellite transitions due to first order quadrupolar coupling.
A 50-50 mixture of Cesium-pentadecafluoro-octanate and D 2 O forms a lyotropic liquid crystal at room temperature [18]. The schematic energy level diagram of oriented spin-7/2 nucleus is shown in Fig. (1A). The 8 energy levels are labeled as the basis states of a three qubit system. In the high field approximation, effective quadrupolar coupling (∧) can be considered as a small perturbation to Zeeman field. Thus for population purposes the equilibrium density matrix can be considered to be proportional to H z (Fig. 1A). Partially oriented 133 Cesium nucleus (I=7/2) gives rise to a well resolved 7 transitions spectrum at room temperatures ranging from 290 K to 315 K Fig (1B). The effective quadrupolar coupling (∧) changes with temperature, since the order parameter is a function of temperature. All the experiments have been carried out here at temperature 307 K, which gives ∧ = 6856 Hz. The equilibrium spectrum is obtained by applying a (π/2) y pulse, is shown in Fig.(1B).

CONTROLLED PHASE SHIFT GATES
Non-adiabatic geometric phase in NMR was first verified by Suter et.al [32]. Non adiabatic geometric phases in NMR, were used to implement controlled phase shift gates, Deutsch-Jozsa (DJ) and Grover algorithms in weakly J-coupled and strongly dipolar coupled systems [33,34].
A two level subspace (r,s) forms a fictitious spin-1/2 subspace, hence the states |r and |s can be represented on a Bloch sphere (Fig. 2) [34]. The two π pulses applied on a transition (r,s), cyclically rotates each of the states |r and |s [34]. The cyclic path followed by each of the two states, makes an equal solid angle (Ω) at the center of the Bloch sphere [34]. The geometric phase acquired by the states |r and |s is given by e iΩ/2 and e −iΩ/2 respectively [33,34], where the phase difference of e iπ between the states indicate that the two states are traversed in the opposite direction [32,33]. It is possible to obtain various solid angles, hence various phases, by shifting the phases of the two π pulses with respect to each other ( Fig. 2) [33,34]. It is shown that two π pulses with phases θ and (θ + π + φ) respectively, applied on a transition (r,s), subtends a solid angle Ω = 2φ, thus the phase acquired by the states |r and |s is e iφ and e −iφ respectively.
However the internal Hamiltonian evolves during the application of a transition selective π pulse, and gives rise to an additional phase known as dynamical phase. In order to observe only the geometric phase, one has to refocus the evolution of internal Hamiltonian. As shown in Eq. (1), the Hamiltonian consists of Larmor frequency term and quadrupolar coupling term. The Larmor frequency is identical to the frequency of central transition 'd'. Thus the receiver frequency is set at frequency of transition 'd' (Fig. 1A), hence there is no evolution due to the Larmor frequency term. The duration of the selective π pulse is chosen in such a way that the phase acquired by the quadrupolar term makes a complete 2π rotation, in other words e −iH Q tp = 1, where t p is duration of the selective π pulse which is equal to 1.425 ms.
The phase information can be encoded in the coherences, hence in order to observe the phases the initial state should contain the coherences, which is obtained by applying a non-selective (π/2) y pulse on equilibrium state. Figure (3a) shows the implementation of controlled phase shift gate, represented by the diagonal matrix U π (3,4) = diag[1, 1, −1, −1, 1, 1, 1, 1], obtained by applying two selective π pulses on transition (3,4), with phases y and (y + π + π) = y. The two states |010 and |011 acquire π phase shift so the transitions 'b' and 'd' are inverted, confirming the phase gate. since the phases of two π pulses are same, it is possible to combine the two π pulses in to a single 2π pulse. Such pulses are used in section (3.1).
In some quantum computing protocols (for example Grover search algorithm and quantum Fourier transform) one requires a phase gate such as U φ k (Table. 1), which creates the relative phase e iφ at k th state, for example U φ 8 =[1, 1, 1, 1, 1, 1, 1, e iφ ]. Such phase gates can be implemented by sandwiching various phase gates u φ (i,j) , as shown in Table ( indicates the geometric phase shift gate obtained by applying a pair of selective π pulses on a transition (i,j) with phases y and (y + π + φ), thus the diagonal elements corresponding to i and j states, acquire a phase shift e iφ and e −iφ respectively. For example, u φ/8 (1,2) =diag[e iφ/8 , e −iφ/8 , 1, 1, 1, 1, 1, 1], which can be obtained by applying two selective π pulse on transition (1,2) with phases y and (y + π + φ/8).
The method of constructing controlled phase shift gates (Table 1), is explained here.
In Eq. (2), by putting N=8 one can obtain any of the U φ k shown in Table (1). In Table (1), each of the operators, U φ k , requires seven pairs of selective π pulses, where each pair of pulses will act on a single transition. However the π pulses on unconnected transitions can be applied simultaneously. Hence one can simultaneously implement u (1,2) , u (3,4) , u (5,6) , u (7,8) by using a pair of multi-frequency (MF) pulses, similarly u (2,3) , u (4,5) , u (6,7) can be implemented by using another pair of multi-frequency pulses. For example U φ 1 can be implemented by using seven geometric phase shift gates (Table 1), obtained by using two pairs of multi-frequency π pulse, as described below.
The unitary operators of phase shift gates correspond to diagonal matrices which commute with each other, hence U φ 1 of Eq.

COLLINS VERSION OF DJ ALGORITHM
Deutsch-Jozsa (DJ) algorithm determines in a single query, whether a given function is constant or balanced [3,35]. A function is "constant" if it gives the same output for all inputs, and "balanced" if it gives one output for half the number of inputs and another for  the remaining half [6]. Classically for an n bit binary function, at least (2 n−1 + 1) queries are needed to determine whether the function is constant or balanced, whereas the DJ algorithm requires only a single query [6]. The Cleve version of the DJ algorithm requires an extra qubit (ancilla qubit) and uses controlled-not gates [35], whereas Collins version does not require the ancilla qubit but needs controlled phase shift gates [36].
For a three qubit system there are 2 constant and 70 (c N N/2 =c 8 4 ) balanced functions [36,37]. The quantum circuit of the Collins version of the DJ algorithm is shown in Fig. (5). The algorithm starts with a pure state (pseudo pure state in NMR) |000 which is converted to a superposition state of eight basis states |000 , |001 , .....,|111 , by applying a pseudo Hadamard gate on all the three qubits. Thereafter a unitary operator U (oracle) is applied, followed by detection. Theoretically, after U one has to apply the Hadamard gate. In NMR, a Hadamard gate is replaced by a pseudo Hadamard gate, which is implemented by a (π/2) y pulse, and for detection another (π/2) −y pulse is needed. These two pulses cancel each other and the result of the algorithm is available immediately after U.  Fig. (1B). Under the influence of this pulse, the populations of all the states except |000 , start a collective population transfer among them as a linear chain of coupled oscillators [18]. Choosing the correct amplitudes of various harmonics and duration of the pulse, these six transitions can be simultaneously saturated with average population of these seven levels. The relative amplitudes of the six harmonics b, c, ..., and g are respectively given by 0.84, 0.93, 1, 1.03, 1.04 and 1.07 [27]. The duration of the pulse is 2.05 ms. A gradient pulse was applied subsequently to kill (dephase) the coherences created during the process. The final populations were measured using a nonselective small-angle (5 o ) pulse. The small-angle pulse, converts the population differences between adjacent energy levels into single quantum transitions, within linear approximation.
The spectrum of Fig (6b) confirms the preparation of |000 pseudopure state.
(iii) Coherent Superposition: After the creation of |000 PPS, a pseudo Hadamard gate (hard (π/2) y pulse) creates a superposition of 8 basis states. It may be noted that unlike in weakly coupled spins, the state created here is not in equal superposition of eigenstates, since the coefficients of various eigenstates are different. However, as is shown here, the created coherent superposition can be utilized for quantum parallelism, to distinguish different classes of functions. The state of the system after a (π/2) y pulse on |000 PPS, is given by, |ψ = (e −i π 2 Iy ).|000 The corresponding density matrix can be written as, where the elements within the boxes represent single quantum transitions a, b,.., g. Figure   (6c) shows the spectrum of |ψ , where the intensities of transitions are obtained by modulus of the product of single quantum elements of Eq. 8, with corresponding matrix elements of (iv) Implementation of U: For constant function U = U c1 = I (Unit matrix), which requires no pulse, thus the final state ψ c1 is given by, As mentioned in section (2), the controlled phase shift gate U π (i,j) can be achieved by applying two (π) pulses with same phase on transition (i,j), which can be be combined in to a single (2π) pulse. For balance functions, U = U(1, k, l, m) (Eq. 6), can be decomposed in to, (1,2) ....u π (k−1,k) ][u π (l,l+1) .u π (l+1,l+2) ...u π (m−1,m) ] In Eq. (10), u π (1,k) and u π (l,m) are implemented by applying two (2π) y pulses on transitions (1,k) and (l,m) respectively, if (1,k) and (l,m) does not correspond to single quantum transitions, then they can be decomposed in to a series of single quantum transitions, as shown in the second equality of Eq. (10), for example, 5,6) .u π (6,7) .u π where in the last equality of Eq. (11), the pulses acting on unconnected transitions ( Fig.   1) are combined in to a single MF pulse. The required pulses for various U(1, k, l, m) of Eq.
(6) are given in Table (2), the 2π pulses acting on unconnected transitions, are combined in to a single multi-frequency (MF) 2π pulse. The duration of each transition selective pulse, is set such that the evolution due to quadrupolar coupling, makes a complete 2π rotation.
It is to be noted that, in Eq. (11) and Table (2), one can use any phase for (2π) pulses, we have used 'y' phase for all the pulses. The final state |ψ 1,k,l,m is given by |ψ 1,k,l,m = U(1, k, l, m)|ψ .
It can be seen that |ψ 1,k,l,m is same as |ψ , except that the basis states |1 , |k , |l and |m acquire a phase factor e iπ (=-1).
(V) Detection: The single quantum transitions (a, b, ..g) of the final states (Eq. 9,12) are detected. Figure (7) shows the spectrum of |ψ c1 and various |ψ 1,k,l,m . From the shape of the spectrum one can conclude, whether the final state represents a constant (or) a balanced function. For constant function (|ψ c1 ) none of the peaks are inverted, whereas for balanced functions (|ψ 1klm ) atleast one of the peaks is inverted. Furthermore the phases of transitions