Implementation of two-qubit and three-qubit quantum computers using liquid-state nuclear magnetic resonance

Abstract

The field of quantum computing has grown rapidly in the last years. Most realizations of quantum computers were done on liquid-state NMR quantum computers so far. We present in a didactic approach how to build small NMR quantum computers with two and three qubits. The two-qubit quantum computer was implemented on α-D-methylglucose using ¹³C spins, whereas the three-qubit quantum computer was implemented on 4-bromo-1,1,2-trifluoro-1-butene using ¹⁹F spins. We describe in detail how to initialize these systems by generating pseudo-pure states. For initialization, we used the spatial averaging method, which has the advantage that it can be easily understood by product operators. Furthermore, the implementation of quantum logic gate operations will be presented. © 2012 Wiley Periodicals, Inc. Concepts Magn Reson Part A 40: 25–37, 2012.

I. INTRODUCTION

The development of computer science in the last decades led to a revolution in data processing. However, some problems still remain for which no efficient classical algorithms are known, that is, tasks that classical computers cannot solve within finite time (1, 2). Quantum computers are a promising new class of computers that are able to handle these problems. The superiority of quantum computers was, for example, demonstrated by Shor (3, 4) who developed an algorithm that can factorize large numbers efficiently. This is relevant for data encryption because nowadays encryption systems rely on the incapability of classical computers to execute this factorization in an efficient way (5, 6). Besides the potential of quantum computers to overcome common data encryption, they allow for absolute secure data transfer making use of quantum mechanical phenomena such as entanglement (5, 6). A more academic application of quantum computers could be the simulation of quantum systems providing new capabilities as the simulator is a quantum system itself (1, 2).

The first theoretical considerations how to implement quantum algorithms were given by Deutsch in 1985 (7). Since then, it took as long as 10 years to realize the first logic operation on small quantum systems (8). Implementing large-scale quantum computations still faces great practical difficulties (9). The first quantum computers were built using liquid-state NMR that has been widely used in this field over many years (6, 10). Next to this, there are other experimental approaches using quantum particles such as photons, quantum dots, trapped atoms, or ions to build quantum devices (10, 11).

Before the actual computation starts, the quantum computer has to be initialized into a well-defined state. Cory et al. (12, 13) and Gershenfeld and Chuang (14) demonstrated different strategies to prepare proper initial states for NMR quantum computers. Henceforth, research interests of many groups focused on this subject and the first algorithm was implemented on nuclear spin systems shortly after (15–21).

We show how appropriate initial states can be generated on two- and three-qubit systems using liquid-state NMR spectroscopy. Furthermore, the implementation of the quantum logic gate controlled NOT (CNOT) on the two-qubit system will be presented.

II. THEORETICAL APPROACHES

Qubits and Quantum Registers

Quantum bits (qubits) are the basic units of quantum information. A qubit is a two-level quantum system with the eigenstates |0〉 and |1〉. A particle with spin-1/2 has two spin states and is therefore an appropriate two-level quantum system (6, 9, 10). Another example is a photon that has two possible polarization states (6, 22). A classical bit can only take values 0 and 1, whereas a qubit can exist in any superposition of the two states (10, 22, 23). This is expressed by Eq. [1], where ψ is the wave function of the quantum system.

• (1)

The coefficients α and β are complex numbers and satisfy the normalization condition |α|² + |β|² = 1 (10, 24).

The power of quantum computers relies on superpositions of quantum states and the possibility to apply logical operations to them. Therefore, quantum registers containing as many qubits as possible are required instead of a single qubit (6, 23). If a quantum register R consists of N qubits, the state |ψ〉_R of the register is a superposition of all 2^N eigenstates |i〉(1):

• (2)

A two-qubit register, for example, can be found in a superposition of its eigenstates |00〉, |01〉, |10〉, and |11〉 (1, 6, 24).

Reasons for Using Nuclear Spins

Interactions between the individual qubits of a quantum register are essential to implement multiqubit logical operations (23). These interactions are naturally given by nuclear spin–spin coupling in NMR. In liquid samples, only the isotropic part of the J coupling is relevant. Hence the quantum register is confined to individual molecules as these couplings are intramolecular interactions.

Another advantage of using NMR for quantum computing was briefly mentioned above. Spin-1/2 spans a two-dimensional complex vector space (Hilbert space) and therefore exactly realizes a qubit (23, 24). Furthermore, nuclear spins retain the quantum information for a relatively long time due to their rather good isolation from the environment (23). Moreover, NMR is a well-established method providing the necessary technical equipment for quantum computing.

Building an NMR quantum computer consists mainly of choosing suitable molecular systems and converting the quantum algorithms into adequate pulse sequences. After processing the algorithm, the quantum information is read out by, for example, acquiring spectra. Other methods like quantum tomography are also widely applied to get information about the final state of the spin system (6, 9, 25).

Ensembles of Spins

The quantum register must be initialized into a well-defined state before the actual computation starts (4, 12, 22). Usually, this initial state needs to be pure (6, 23). If an ensemble (large collection) of spins is considered, a pure state corresponds to an ensemble with each member being described by the same state vector (22). Unfortunately, nuclear spins in thermal equilibrium are highly mixed, as they cannot be aligned exactly parallel to the magnetic field at temperatures above 1 mK (12, 23). Therefore, it is impossible to prepare a pure state of spins in thermal equilibrium. If quantum algorithms are executed on a mixed ensemble, the averaging over the ensemble might annihilate the results (22).

To describe an ensemble of spins theoretically, it is impractical to treat every spin individually due to the large number of spins in a sample. The density operator approach (26) is a suitable tool to describe an entire ensemble without referring to the individual spin states. For a pure ensemble with state vector |ψ〉, the corresponding density matrix (density operator in its matrix representation) is given by (22, 26)

• (3)

At thermal equilibrium, the density matrix of an N spin system is represented by

• (4)

where is the unity operator and is the so-called deviation density matrix (22). The factor β is defined as

• (5)

with h, γ, B₀, k_B, and T corresponding to the Planck constant, gyromagnetic ratio, magnetic field strength, Boltzmann constant, and temperature, respectively (23).

The identity part of does not evolve under radio frequency (rf) pulses and has no effect on the measured NMR signal (22, 23). Only the deviation density matrix evolves and gives rise to a detectable signal.

Initialization by Preparation of Pseudo-Pure States

A variety of methods were developed to handle the challenge of working with mixed initial states. The most common approach is to create a so-called pseudo-pure state (pps). The preparation of pps aims at converting into an operator that is directly proportional to the density operator of a pure state (22). The pps behaves exactly like a pure state and yields apart from a constant the same signal (22, 27).

In the case of a two qubit system, is described by

• (6)

This is equivalent to the product operator representation I + I, that is, the density matrix can be also expressed as product of the spin angular momentum operators I^k (k = 1,2). To generate a pps, needs to be transformed into a traceless matrix with identical diagonal elements except one having another value (22, 27). The pps density matrix ,

• (7)

fulfils these requirements and corresponds to product operators

• 

This matrix can be fragmented as shown in Eq. [8], where |00〉 〈00| represents the pure density matrix . Inserting Eq. [8] in Eq. [4] gives , shown in Eq. [9].

• (8)

• (9)

As unit operators do not contribute to the NMR signal, it is clear that evolves apart from the factor β/2^{N − 1} in exactly the same way as . The factor β is ∼10⁻⁵ at ambient temperature (22), thus the NMR signal is significantly reduced compared to the actual pure state (23). In addition, 1/2^{N − 1} causes an exponential reduction of the signal with increasing number of qubits (9, 23).

Several techniques were developed to experimentally prepare pps. They can be classified as (i) spatial averaging, (ii) temporal averaging, and (iii) logical labeling (22, 23). In the spatial averaging method, the pps is prepared by suitably chosen sequences of rf pulses with different flip angles, pulsed gradients, and delays (12, 13, 27). This method is straightforward to implement and was used in this project. Temporally averaged pps are prepared by additive averaging of different experiments (28), that is, the thermal state is added to states, in which the populations were permuted by selective π pulses. Logical labeling techniques prepare a subsystem by sacrificing one or more qubits, which are used as labels (14). The state of the subsystem depends on the state of the labeling qubit.

How to Express Product Operators as Density Matrices

To prepare the pps using the spatial average approach, rather complicated pulse sequences are required. Generally, it is easier to deduce the pps using the simple product operator approach. In the following a “recipe” is given, how to express the product operators in their density matrix representation. This is important, for example, to check if the product operators correspond to the correct pps density matrix.

The four angular momentum operators , Î_x, Î_y, Î_z for an isolated 1/2 spin have the matrix representation (26):

• (10)

A two-qubit product operator in its matrix representation is constructed by

• (11)

where Î₁ and Î₂ are the operators of the spins. The symbol ⊗ indicates the Kronecker product. The Kronecker product of two matrices A and B is defined as (12)

• (12)

that is, every matrix element of A is multiplied by matrix B. For example, the product operator 2II is calculated as:

• (13)

Unit operator is involved for product operators like I:

• (14)

The other product operators are calculated analogously. Eq. [11] can be of course expanded for systems with more qubits. For example, for a three for-qubit system the operator 4III is given by

• (15)

Logical Operations

To perform computational steps, algorithms are needed, that is, logic gates. Unlike their classical counterparts, quantum gates must be reversible (9, 29). Reversibility is required as quantum systems naturally evolve by unitary transformations, which are themselves reversible (9, 30). To fulfil the reversibility condition, the reconstruction of the input bits must be possible knowing only the characteristics of the gate and the output bits (6, 9). Consequently, the gate must have the same number of output as input bits. Classical gates such as AND, OR, or XOR (exclusive OR) are therefore not reversible, but even so it is possible to construct reversible counterparts (9).

Gates that act on one single input bit and return a single output bit are nonsurprisingly the simplest type of logic gates. The input bit can be, for example, flipped, which is a logical NOT operation. In NMR quantum computing, the qubit flipping is achieved by a selective π pulse on the target qubit. Using the product operator language, this can be described as: I_z −I_z. The operation CNOT can be considered to be a reversible equivalent to XOR (9, 24). CNOT does not change the first qubit (control qubit) while the second qubit (target bit) is flipped if the control qubit is in state 1. CNOT is a typical two-qubit quantum gate, as it has a two bit input and output. The corresponding truth table is Table 1.

Table 1. Truth Table for the CNOT Gate

Input		Output
0	0	0	0
0	1	0	1
1	0	1	1
1	1	1	0

The Toffoli gate is a three-qubit version of the CNOT gate (13, 31). As shown in Table 2, the first two bits remain unchanged and the third bit is flipped if the first two bits are one. As this gate has two control bits and one target bit, it also referred to as CCNOT (controlled controlled NOT). The realization of the CNOT and the Toffoli gate using appropriate pulse sequences will be discussed in the following sections.

Table 2. Truth Table for the Toffoli Gate

Input			Output
0	0	0	0	0	0
0	0	1	0	0	1
0	1	0	0	1	0
0	1	1	0	1	1
1	0	0	1	0	0
1	1	0	1	1	1
1	1	1	1	1	0

The logical operations mentioned above, NOT, CNOT, and the Toffoli gate, serve just as examples, as a variety of other two- and three-qubit gates exist. However, a comprehensive description can be found in Refs. (6, 9, 17, 18) and shall not be given here.

III. MATERIALS AND METHODS

¹³C-labeled α-D-methylglucose was used as a model for a two-qubit quantum computer. It was synthesized from D-glucose-[C13]₆ (≥98%) received from Martek Biosciences Corperation (Columbis, Maryland). In aqueous solutions, there exist two different isomers of D-glucose each giving rise to a set of signals. The anomeric position was methylated to ensure that only the glucopyranoside isomer is present in solution. Carbon spins 1 and 2 were chosen as qubits. The corresponding labels are shown in Fig. 1(a). The chemical shift of the considered carbon spins and the relevant ¹J_CC coupling of the two spins as well as the longitudinal relaxation time T₁ are given in Table 3. D₂O was used as solvent.

Figure 1. Compounds used for modeling a two qubit (a) and three qubit (b) computer.

Table 3. ¹³C Chemical Shifts and ¹J_CC Coupling Constant for the Two-Qubit System and ¹⁹F Chemical Shifts and Coupling Constants J_FF (Hz) for the Three Qubit System

C atom	δC (ppm)	JCC (Hz)	T1 (s)
The longitudinal relaxation times T1 are given as well.
1	102.0	46.6 (J12)	1.1
2	73.9	-	1.1
F atom	δF (ppm)	JFF (Hz)	T1 (s)
1	−177.0	−114.3 (J12)	4.8
2	−122.4	33.0 (J13)	8.8
3	−102.8	82.6 (J23)	8.3

The ¹⁹F spins of 4-bromo-1,1,2-trifluoro-1-butene (BTF-But) were used as three-qubit system, see Fig. 1(b). BTF-But (≥ 98%) was obtained from ABCR (Karlsruhe, Germany) and dissolved in CDCl₃. The parameters of the spin system are given in Table 3.

Experiments with α-D-methylglucose were carried out on a Bruker DRX-600 spectrometer (600 MHz) using a broadband observe (BBO) probe head, whereas experiments with BTF-But were carried out on a Bruker DRX-400 spectrometer (400 MHz) using a ¹⁹F probe head. Composite pulse decoupling of the protons and pulsed field gradients in z-direction were used in both cases. The Bruker system was controlled by a PC running TopSpin 1.3 software.

IV. REALIZATION OF A HOMONUCLEAR ¹³C TWO-QUBIT COMPUTER

Pulse Sequence for the Preparation of Pseudo-Pure State |00〉

As already outlined in the previous section, the preparation of pps |00〉 aims at converting I + I to I + I + 2II. A pulse sequence containing selective pulses as proposed by Cory et al. (13) could not be used for our system. This is due to small separation of the carbon resonances making large pulse lengths necessary, which are in the range of the coupling evolution 1/(2J₁₂). As this led to unwanted phase modulation of the signals, we designed a pulse sequence without long selective pulses based on the “pulsed field gradient spin echo” (PFGSE) technique (32). The pulse sequence is given in Eq. [16] and visualized in Fig. 2.

Figure 2. Pulse and gradient sequence for the preparation of pps |00〉 on α-D-methylglucose. g1 and g2 represent pulsed field gradients of various strengths.

The strategy of our approach was to excite carbons 1 and 2 as well as all the other carbon spins by a hard [π/4]_x pulse, see first step of Eq. [16]. The subsequent PFGSE period (Eq. [16], second step) was used to achieve a selective excitation of carbons 1 and 2. The first pulsed field gradient [grad]_1z destroys phase coherence of the transversal magnetization of all carbon spins. Afterward a selective spin echo sequence and a second pulsed field gradient of the same strength was applied to refocus carbon spins 1 and 2 and to recover their coherence. The PFGSE does not change the product operators of spin 1 and 2 in total, but it “deactivates” all carbon spins except 1 and 2 by destroying their phase coherence. The coupling constants ²J_CC and ³J_CC are comparatively small and were neglected during the PFGSE.

• (16)

After that, a coupling evolution [1/(4J₁₂)] followed, see third step of Eq. [16]. A selective [π]_x pulse on carbon 3 in the middle of the evolution period was necessary to refocus coupling ¹J₂₃ arising from the presence of I. To obtain the final product operators, a hard [π/4]_−y pulse was applied. The pulsed field gradient [grad]_2z destroyed the transverse magnetization to obtain solely I_z operators.

The other three pps were obtained by selective inversion of spin 1 or 2 yielding states |10〉 or |01〉, respectively. The corresponding product operators are given in Table 4.

Table 4. Coefficients of the Product Operators Representing the Four pps of a Two-Qubit System

pps	I	I	2II
|00	1	1	1
|10	−1	1	−1
|01	1	−1	−1
|11	−1	−1	1

Results for the Preparation of Pseudo-Pure State |00〉

As only operators with transverse pseudo-pure are spectroscopically observable, the I_z operators need to be transferred into I_x or I_y operators. This can be done either by a selective or a hard [π/2] pulse. Reading out pps |00〉 with a hard pulse leads to:

• (17)

The term II represents double antiphase magnetization and cannot be observed, that is, a hard read out is less suitable to corroborate the creation of the pps. A selective pulse, for example, applied to carbon 1 yields in contrast an observable antiphase operator:

• (18)

The line shapes of the in-phase operator I and the antiphase operator II are shown in Fig. 3. The antiphase term partly cancels the in-phase signal and only one line of the doublet remains.

Figure 3. Shape of the observable operators obtained after a selective [π/2] read out of the pps |00〉.

The resulting spectra of the four pps obtained by selective excitation of the respective carbons are shown in Fig. 4. Carbon 1 gives rise to a doublet, which is due to the ¹J_CC coupling to carbon 2. As expected, only one line of the doublet appears when reading out the pps. The coupling pattern of carbon 2 is more complex because of the additional ¹J_CC coupling to carbon 3 and ²J_CC coupling to carbon 4, which evolve during the acquisition time. Therefore, the line is split into a doublet of doublets.

Figure 4. Spectral results for the pps of α-D-methylglucose obtained by selective [π/2]_x read out pulses on carbon 1 (left) and carbon 2 (right) after one scan.

Implementation of the CNOT Gate

The next step following the generation of suitable initial states is the execution of logical operations on the quantum register. In this work, we implemented the CNOT gate on the two-qubit system. This was realized by a pulse sequence consisting of two selective [π/2] pulses applied to the same spin and an intermediate coupling evolution of 1/(2J₁₂) (24).

At first, one has to define the control and target bit. If spin C-2 is selected as control bit, the target bit C-1 is excited by the [π/2] pulse. The latter was chosen as target bit to circumvent the coupling evolution J₂₃, which would occur during 1/(2J₁₂) when exciting C-2. Starting from |10〉, the CNOT operation affects the product operators in the following way:

• (19)

The input state is obviously equal to the output state. The same holds when starting from |00〉. To implement the CNOT gate as done here, the first bit (target bit) is negated if the second bit (control bit) is 1. Accordingly, state |11〉 is transformed into |01〉.

Performing this pulse sequence on the four pps yields the spectra as shown in Fig. 5. A hard read out pulse was used to validate the results of CNOT as solely the different phases of the in-phase operators I and I are of interest.

Figure 5. Spectra of α-D-methylglucose obtained after carrying out the CNOT operation on the four pps. Results were obtained by a hard [π/2]_x read out pulse. Carbon 1 (C-1) is shown on the left and carbon 2 (C-2) on the right side. Note that due to the hard read out, other resonances show up in the spectra, too.

α-D-methylglucose works as two-qubit quantum computer quite well so far. But it is also potentially a six-qubit system when using all ¹³C spins of the pyranoside ring. However, initialization by spatial averaging would not be possible because this method requires cross-coupled spins, that is, the long-range C–C couplings would be too small. Small coupling constants would afford evolution periods that exceed the transversal relaxation time T₂. Another reason is that the separations between the carbon resonances are quite small making selective excitation difficult. Because of these problems, the implementation of a three-qubit quantum computer is demonstrated using a ¹⁹F spin system, as F–F coupling constants are larger and the comparably large separation between the ¹⁹F signals allows for spin excitation by selective pulses.

V. REALIZATION OF A HOMONUCLEAR ¹⁹F THREE QUBIT COMPUTER

Pulse sequence for Pseudo-Pure State |000〉

Preparing the pps |000〉 for a three-qubit system affords to convert the equilibrium deviation matrix

• (20)

to the pps deviation matrix

• (21)

Using product operators, this is equivalent to the transformation of the equilibrium state I + I + I to I + I + I + II + II + II + III. This was achieved by a sequence of selective rf pulses, pulsed field gradients, and coupling evolutions proposed by Cory et al. (13), see Eq. [22]. As this sequence is quite complex, it was sequentially partitioned into four subsequences. The subsequence over the first arrow is referred to as first pulse sequence, the subsequence over the second arrow as second pulse sequence, and so forth. Gradients in z-direction are required to destroy unwanted transverse magnetization at the end of each subsequence.

• (22)

For a better understanding, it is useful to take a closer look at the coupling evolutions. [1/(2J₁₂)] stands for an evolution period containing a spin echo sequence on the third spin to refocus J₂₃. At the start of evolution period [1/(2J)], which is part of the third pulse sequence, only the term I is relevant because all other product operators have z-magnetization. Hence only J₁₃ and J₂₃ couplings evolve. The situation is similar at the start of [1/(4J)]. Only the product operators I and III are affected resulting in an evolution of J₁₃ and J₂₃ couplings.

To obtain the correct product operators, couplings J₁₃ and J₂₃ must evolve [1/(2J₁₃)] and [1/(2J₂₃)], respectively. The fourth pulse sequence is analogous. An evolution period of [1/(4J₁₃)] is necessary for coupling J₁₃ and [1/(4J₂₃)] for coupling J₂₃. To achieve this, appropriate π pulses have to be integrated in the pulse sequence given Eq. [22]. The modified pulse sequence containing all spin echoes is shown in Fig. 6. The assignment of the signals was chosen so that |J₁₃| < |J₂₃|. The strategy was to let coupling J₂₃ fully evolve and refocus the part which evolves during rest of the J₁₃ evolution time. Delays [1/(2J)] and [1/(4J)] were adapted in the following way:

• (23)

The delays d₁ and d₂ were chosen so that

• (24)

where p(π) is the length of the π pulse. Furthermore, π pulses on the third or second spin at the end of pulse sequences 2–4 were necessary to make I or I positive, as they were converted to −I or −I by the π pulse of the spin echo.

Figure 6. Pulse and gradient sequence used for preparing pps |000〉.

Results for the Preparation of Pseudo-Pure State |000〉

Selective excitation of the F–F resonance was possible without any difficulties as the separation of the ¹⁹F signals is 7,000 Hz and 20,000 Hz. Another advantage is that the F–F couplings of BTF-But are quite large. The longest delay needed for coupling evolution is 1/(2J) = 15.2 ms. The problem that the duration of the pulse sequence exceeds the transversal relaxation time T₂, that is, that the spins loose their coherence, surely does not occur here. However, the smallest delay was only about 4 ms. Extremely short selective pulses were accordingly required to avoid the evolution of couplings during pulses. But the shorter the pulse, the longer the band width. Because of the large separation between the signals, Gauss pulses with a length of 250 μs turned out to be still selective. Let us now consider step by step the results obtained after each subsequence.

The first pulse sequence of Eq. [22] transfers the equilibrium state to I + I + I. The ratios of the integrals of the equilibrium state are 1:1:1 because every signal is due to one ¹⁹F atom. According to the product operators, the integral ratios should be 1:0.5:0:25 after the first pulse sequence. The actual observed ratios were 1:0.49:0.27, which is in good agreement with the expected ratios, see also Fig. 7 for details.

Figure 7. BTF-But spectra obtained after the first pulse sequence with a selective 90° read out pulse.

After the first and second pulse sequence, the product operators read I + I + I + II. If a 90° read out pulse about the x axis is applied on spin 1, the in-phase operator I and the antiphase operator II are obtained. The other product operators have z-magnetization and cannot be observed. The in-phase operator gives rise to a double doublet with intensity pattern ++++, whereas the resulting pattern for the antiphase operator is −−++. Consequently, the in-phase term is partly cancelled by the antiphase term, that is, two lines of the double doublet remain, which is shown in Fig. 8. The situation is analogous for a selective read out on spin 2. As no coupling of spin 3 evolved so far, a read out on spin 3 yields the same spectrum as presented in Fig. 7(a).

Figure 8. BTF-But spectrum obtained after the 1. +2. pulse sequence with a selective 90° read out pulse on spin 1.

The product operators after the 1.+2.+3. pulse sequence are

• 

The prediction of the spectra from these product operators is more sophisticated. Consider a selective 90 read out pulse on spin 1: the observable operators I, ċ 2II and ċ 4III are obtained. The term 4III represents magnetization which is doubly antiphase with respect to the coupling to spins 2 and 3. It is important to note that the three operators are multiplied by different coefficients, which has to be taken into account when deriving the spectrum. The in-phase, antiphase and doubly antiphase operators have the intensity patterns ++++, − − ++, and 2(+− −+), respectively. Summing this up, yields the pattern (2+)(2−)(0)(4+), which coincides with the spectral results, see Fig. 9(c). The same consideration can be applied to spin 2. After a selective read out on spin 3, the operators I and ċ 4III with the intensities ++++ and +− −+ can be observed. The superposition of both gives 2(+00+). This corresponds to the spectrum obtained (Fig. 9(a)).

Figure 9. BTF-But spectra obtained after the 1.+2.+3. pulse sequence with a selective 90° read out pulse.

The pps was finally prepared after the fourth pulse sequence. The product operators read:

• 

A selective 90 pulse on, for example, spin 1 results in four observable terms which are shown in Fig. 10. After adding up the intensities, only one line of the double doublet remains. The same holds for spins 2 and 3. The spectra, shown in Fig. 11, match this expectation and confirm that the pps was successfully generated.

Figure 10. Illustration of the form of the spin 1 multiplets expected for the four product operators which lead to an observable signal on spin 1.

Figure 11. BTF-But spectra of the pps |000〉 obtained after the 1.+2.+3.+4. pulse sequence with a selective 90° read out pulse.

There are apparently slight deviations from the ideal spectra, that is, some lines were not perfectly cancelled. This is most likely due to the very large coupling constants requiring such short delays as 4 ms. Therefore, these delays are very sensitive parameters regarding temporal aberrations. It appeared that differences of less than 100 μs have effects on the spectra. This is already in the range of the time that the spectrometer needs for switching between channels. One could circumvent this problem by using a newer instrument with faster electronics.

Since there are 2^N pps for a N qubit system, a three-qubit system has eight pps as given in Table 5. It is clear that the other pps can be generated from |000〉 by NOT operations, that is, inversion. For example, state |001〉 can be obtained from |000〉 by a selective 180° pulse on spin 3. The spectra can be deduced from the product operators as discussed before.

Table 5. Coefficients of the Product Operators Representing the Eight pps of a Three-Qubit System

pps	I	I	I	2II	2II	2II	4III
|000	1	1	1	1	1	1	1
|001	1	1	−1	1	−1	−1	−1
|010	1	−1	1	−1	1	−1	−1
|011	1	−1	−1	−1	−1	1	1
|100	−1	1	1	−1	−1	1	−1
|101	−1	1	−1	−1	1	−1	1
|110	−1	−1	1	1	−1	−1	1
|111	−1	−1	−1	1	1	1	−1

Implementation of the Toffoli Gate

Subsequent to the preparation of the pps, it would be nice to show that the BTF-But quantum computer can perform logical operations. As mentioned before, the Toffoli gate is a three-qubit quantum gate. A pulse sequence to realize the Toffoli gate was proposed by Cory et al. (13). The Toffoli pulse sequence [TOF] is given by Eq. [25],

• (25)

where k = 1,2,3 is the spin excited by selective pulses of the TOF sequence. During the delays all couplings evolve [1/(4J)]. If spin 1 is chosen for the TOF sequence, couplings J₁₂ and J₁₃ can evolve. An evolution period of [1/(4J)] for both couplings is achieved by the method already explained for the preparation of the pps, that is, the larger coupling is refocused by an appropriate spin echo sequence.

The following scheme (13) demonstrates how the TOF sequence on spin 1 affects the operators of the pps |000〉:

• (26)

If one adds up all operators on the right sides of the arrows, the operators representing pps |000〉 are obtained. The TOF sequence applied to |000〉 yields |000〉, which is in agreement with the Toffoli truth table, Table 2. The realization of the Toffoli gate with BTF-But is complicated because the TOF sequence requires three evolution periods. The delays proved to be very sensitive parameters concerning deviations, which caused already problems during the initialization. The situation gets worse when performing the TOF sequence starting from the pps, that is, a proper implementation of the Toffoli gate with the given compound and instrument was not possible.

VI. CONCLUSIONS

We showed how to implement a two-qubit quantum computer on ¹³C labeled α-D-methyl-glucose and a three-qubit computer on BTF-But by applying liquid-state NMR. In the former case, ¹³C and in the latter ¹⁹F spins were used as qubits. It was presented how to initialize these systems by preparing pps within the spatial averaging approach. Furthermore, the two-qubit operation CNOT was implemented on α-D-methylglucose.

α-D-Methylglucose bears the potential as a six-qubit system using all ¹³C spins of the pyranoside ring. In this instance, spatial averaging would not be a suitable method to generate proper initial states as this method requires cross-coupled spins. The long range C–C couplings are partly far too small affording evolution periods that exceed drastically the transversal relaxation time T₂. To circumvent this issue, one could alternatively use another method, that only requires linear coupled systems, for the preparation of pps.

Quantum computing with NMR faces the problem that the interactions between the nuclear spins are generally weak. Moreover, the NMR signal decreases exponentially when increasing the number of qubits. Scaling to a large number of qubits is therefore still challenging. Until now, quantum computers with at most 12 qubits have been realized (6, 33). But quantum registers of at least 20–50 qubits are necessary for a quantum computer to be of any use (6).

Biographical Information

Dorothea Golze was born in Lutherstadt Wittenberg, Germany, in 1986. She received her B.Sc. (2009) and M. Sc. (2011) in chemistry at Leipzig University, Germany. She is currently doing her Ph.D. in theoretical chemistry at the University of Zurich, Switzerland.

Biographical Information

Stefan Berger was intrigued by NMR after having won a bottle of beer during an introductory course in organic NMR led by Prof. H. Suhr at the University Tübingen in 1968. After a Ph.D. thesis with Prof. Anton Rieker, in 1973 he joined Prof. J.D. Roberts at Caltec for postdoctoral work, where he also met Prof. D.M. Grant and Prof. D. Seebach, who were then guest professors in Pasadena. This period was decisive in trying for a habilitation in NMR spectroscopy, which was achieved at the University Marburg. At the University of Leipzig, he tries to combine methodological development of NMR with its application to bioorganic problems.

Biographical Information

Maik Icker born in 1986, received his B.Sc. and M.Sc. in chemistry at the University of Leipzig. During his master thesis, he worked in the field of NMR quantum computing. Currently, he is doing his Ph.D. in Prof. Stefan Berger group. His research interests are now focused on NMR sensitivity enhancement techniques.