Nuclear Spins in a Nanoscale Device for Quantum Information Processing

Coherent oscillations between any two levels from four nuclear spin states of I=3/2 have been demonstrated in a nanometre-scale NMR semiconductor device, where nuclear spins are all-electrically controlled. Using this device, we discuss quantum logic operations on two fictitious qubits of the I=3/2 system, and propose a quantum state tomography scheme based on the measurement of longitudinal magnetization, $M_z$.


INTRODUCTION
It has long been demonstrated that nuclear spins have long relaxation times, thus longer coherence times, making them suitable for quantum information processing devices [1,2] (for a review see [3] and references therein). Most of the works on the manipulation and use of nuclear spins are restricted to those in molecules of liquid solutions. To date, liquid-state NMR is the leader among all quantum computer implementations, both in the number of qubits controlled and the number of gates performed within the coherence lifetime. These successful demonstrations inspired researches to investigate the possibility to use NMR in solid-state systems for the same purpose [4]. Coherent manipulation of nuclear spins in solid-state systems became an exciting field of research which resulted in interesting proposals on nuclear spin qubits, where nuclear spins are individually manipulated [5,6].
An advantage of solid-state NMR over the liquid NMR is that the spins can be highly polarized by dynamic nuclear spin polarization techniques such as polarization transfer from electronic spins. This strong polarization allows a large population available for the preparation of the system close to a pure state. However, only a few quantum information processing experiments with singlecrystal solids have been reported [4,7].
Most of the experiments up to date employ conventional NMR technique, which uses metal coils for excitation and detection of nuclear spin transversemagnetization M xy . In order to obtain a detectable signal, a large number of nuclear spins (10 11 − 10 13 ) should be involved in the process, which limits the sensitivity of these schemes. Moreover, quantum multiple coherence is not directly detected in conventional NMR as it does not produce transverse magnetization.
In a recent study [8], Yusa et al. demonstrated an all-electrical control and detection of nuclear spin magnetization in a nanometre-scale semiconductor device, where longitudinal magnetization M z of nuclear spins in a point-contact channel are detected by the resistivity change of the channel paving the way to the detection of multiple quantum coherence. This device has an improved sensitivity, which requires the involvement of only ∼ 10 8 or less nuclear spins.
In the following, we first describe this nanoscale NMR device, the novel detection technique and the observed multiple quantum coherence. Then, we show how oneand two-qubit operations can be performed on this device, and give the pulse sequences. A method for performing quantum state tomography using M z -detection in this device is also presented, and finally we give a brief discussion and conclusion.

NANOSCALE SEMICONDUCTOR NMR DEVICE
Principle of the device: The nanoscale NMR device is fabricated as a monolithic semiconductor device integrated with a point contact channel and an antenna gate (see Fig. 1). The structure contains a 20-nm GaAs quantum well with AlGaAs barrier layers grown on n-GaAs(100) substrate. This substrate functions as backgate to control the electron density in the point contact region, which is defined by split Schottky gates separated by 600nm. The antenna gate locally irradiates the channel with an alternating rf field for selective and coherent manipulation of the nuclear spins.
By controlling the static magnetic field perpendicular to the grown device surface and the backgate voltage, the point contact region was set to fractional quantum Hall regime at the degenerate Landau-level filling factor ν = 2/3, where there is a strong coupling of nuclear spins to the conduction electrons [9,10,11]. At ν = 2/3, the spin polarized and unpolarized states face each other through a transition region with high resistivity R xx . When sufficient current is driven through the system, the polarized nuclear spins interact with the flowing electrons resulting in an gradual enhancement of R xx . Thus, the current density becomes very high in the constricted region so that nuclear spin polarization occurs only in the point contact region. This enhancement of R xx implies that the polarization of the nuclear spins in the point contact region can be detected by measuring the resistance between the ends of the point contact.
It has already been shown that this resistance values is proportional to the longitudinal magnetization of nuclear spins, ∆M z ∝ ∆R xx [10].
Observation of coherent oscillation with R xx measurement: The point contact channel in this device consists of 69,71 Ga and 75 As isotopes each having total spin I = 3/2. Thus, each nuclide splits into n = 2I + 1 = 4 energy states, |m = |3/2 , |1/2 , | − 1/2 , | − 3/2 , under static magnetic field due to Zeeman effect. This levels are spaced with equal energy separation ofhω 0 . However, the quadrupolar interaction shifts the adjacent states by 2∆ q fromhω 0 allowing three possible transitions at resonanceshω 0 − 2∆ q ,hω 0 andhω 0 + 2∆ q (see Fig. 3). While these transitions are between levels separated by one quantum angular momentum (∆m = 1), energy and angular momentum conservation rules imply that transitions between levels separated by ∆m = 2 and ∆m = 3 are also possible, resulting in a total of six possible coherent oscillations. We named those transitions as one-photon, two-photon and three-photon transitions. Therefore, in this NMR device it is possible to observe six coherent transitions for each of the three nuclei ( 69,71 Ga and 75 As) in the point contact channel.
In order to observe the coherent oscillations, first the nuclear spin polarization is saturated and this is observed by resistance measurement, which is also saturated. Then pulsed rf magnetic field is applied by the antenna gate. If the frequency of the applied pulse is in resonance with the NMR frequency, oscillatory change in M z is observed implying that the superposition coherently rotates between the two energy states. The oscillations in M z is finally detected by the change in the resistance, ∆R xx , before and after the application of the rf pulse. In Fig.2, NMR spectrum for one-and two-photon transitions are given only for 75 As. Coherent oscillations are observed for three-photon transitions, too (not shown here). In the same way, all of the six possible oscillations are observed for 69 Ga and 71 Ga [8].
The decoherence time T 2 of the device is estimated by curve fitting the coherent oscillation between | − 1/2 and | − 3/2 of 75 As. The T 2 was estimated as ∼ 0.6ms without any decoupling process. However, it is enhanced to 1.5ms when the nuclei-electron decoupling is applied [12]. The relaxation time T 1 in this device is longer than 100s.

STATE MANIPULATIONS IN THE NANOSCALE NMR DEVICE
The four-level system in this device forms a quartit, which is equivalent to two logical qubits. This becomes Fig. 2). Thus, the two-qubit state can be written as In the following, we show how different quantum gate operations can be performed on this two fictitious qubit system by applying selective pulses at the resonant frequency between two energy levels, say |m and |n . Here, the free evolution of the coherent system during the finite pulse duration and the time lag between the pulses are ignored, however they can be compensated during experimental realizations.
Rotations of logical qubit A in this quartit can be obtained by applying rf pulses at frequencies ω 0 −∆ q /h and ω 0 + ∆ q /h two induce two-photon transitions:X A (θ) = X 02 (θ)X 13 (θ),Ŷ A (θ) =Ŷ 02 (θ)Ŷ 13 (θ), andẐ A (θ) = Z 02 (θ)Ẑ 13 (θ), whereX nm (θ),Ŷ nm (θ) andẐ nm (θ) corresponds to rotations between levels |m and |n by an angle θ along the corresponding axes. In the same way, rotations on the logical qubit B can be obtained by applying pulses at frequencies ω 0 − 2∆ q /h and ω 0 + 2∆ q /h two induce one-photon transitions: NOT Gate: The transformation of this two-qubit system when a NOT-gate is applied to the first (A) and second (B) logical qubits is described byÛ A NOT |ψ = c 0 |2 + c 1 |3 + c 2 |0 + c 3 |1 andÛ B NOT |ψ = c 0 |1 + c 1 |0 + c 2 |3 + c 3 |2 , which can be implemented by the selective pulse sequencesÛ A NOT = iX 02 (π)X 13 (π) and U B NOT = iX 01 (π)X 23 (π). On the other hand, NOT-gate and coherent oscillations corresponding to single-photon and two-photon transitions between the four spin states observed for 75 As (upper). A three-photon transition was also observed for higher values of the alternating current through the antenna gate (data not shown). Measurements were performed at 100 mK with a magnetic field of 5.5T.
A CNOT gate with the first qubit as the control qubit and the second one as the target qubit is defined by the transformation On the other hand, the CNOT with the first qubit as the target and the second qubit as the control performs the transformationÛ BA CNOT |ψ = c 0 |0 + c 3 |1 + c 2 |2 + c 1 |3 . One can easily see that in this NMR device CNOT-like transformations can be implemented by just one pulse asÛ ′ AB CNOT =Ŷ 23 (π) andÛ ′ AB CNOT =Ŷ 13 (π). These gates differ from the ideal CNOT gate by an extra minus sign in one of the off-diagonal terms of the transformation matrix of the corresponding gate operation, i.e., . One can alternatively perform the same operations by applyinĝ U AB CNOT =X 23 (π) andÛ AB CNOT =X 13 (π). On the other hand, an ideal CNOT gate can be implemented by allowing a more complex sequence of one-photon transition pulses. As an example of this, we give the pulse sequence forÛ AB CNOT , which isÛ AB H is defined as above. One may further simplify this pulse sequence by using combination of one-and two-photon transition pulses.

AB
CNOT . On the other hand, for this device we can obtain SWAPlike gates by a single pulseÛ AB swap =Ŷ 12 (π) orÛ AB swap = X 12 (π). This, too, differs from the ideal operation with a minus sign, which can be compensated by applying more complex pulse sequences.
Effective pure state preparation: For the correct working of the gate operations and consequently the quantum computer, NMR quantum computer should be properly prepared in an effective pure initial state of the nucleus ensemble. Contrary the conventional NMR, in our semiconductor NMR device we can prepare a large population available as the pure state. We owe this to the strongly polarized situation created due to dynamic nuclear spin polarization induced by the current flow. The initial state of the device is far from a pure state as schematically shown in 4. Note that due to dynamic polarization the population becomes larger for the higher spin states. From this initial state, one can ap-plyX 01 (π/2)X 12 (π) to prepare the |11 . Once this state is prepared then we can prepare the pure state |ij by applying the pulseX 2i+j,3 (π) (see Fig. 4). More complicated pulse sequences allows us to prepare effective pure state from any arbitrary distribution. The designed population can be maintained up to T 1 , which is longer than 100s for this device. Experimental results showing effective pure state preparation can be found in Ref. [12].

QUANTUM-STATE TOMOGRAPHY BASED ON
Mz DETECTION Quantum state tomography (QST) is a method for complete reconstruction of a given density matrixρ in a serious of measurements. In general, to reconstruct completely a density matrixρ for a quartit or two qubits, we need to determine 15 real parameters (The 16th element can be found from the normalization condition). Single NMR read-out can only give some of the either diagonal or off-diagonal elements of the given density matrixρ.
In case of the conventional NMR systems, where an M xy measurement is performed, a single measurement gives directly some of the off-diagonal elements of the density matrix. However, in our NMR device, where an M z measurement is employed, a single measurement determines only the population differences ρ nn − ρ mm between the levels |n and |m . This quantity is related to the diagonal elements of the density matrix, where ρ mn ≡ m|ρ|n . In a typical NMR spectrum of our device peaks located at ω 01 , ω 12 , and ω 23 give, respectively, ρ 11 − ρ 00 , ρ 22 − ρ 11 , and ρ 33 − ρ 22 . By imposing the normalization condition, we can easily obtain the diagonal elements ρ ii of the density matrix. The remaining elements of the density matrix can be obtained by rotating the original density matrix by properly chosen rotation operationsR k , which transform the original matrix intô ρ k =R k ρR † k . These rotations move the off-diagonal elements of the original density matrix to the diagonal of the rotated density matrix so that M z measurement provides information on them. By application of a number of such rotations, the off-diagonal elements of the density matrix can be brought into the measurable ones in M z detection. The set ofR k operations is not unique, one can find many different sets of rotations for complete reconstruction of the density matrix using M z detection. We found that the following set of 12 rotations is suitable for the reconstruction of the density matrix using M z detection based state tomography for our devicê R 1 =X 01 (π/2),R 2 =Ŷ 01 (π/2),R 3 =X 12 (π/2), R 4 =Ŷ 12 (π/2),R 5 =X 23 (π/2),R 6 =Ŷ 23 (π/2), R 7 =X 02 (π/2),R 8 =Ŷ 02 (π/2),R 9 =X 13 (π/2), R 10 =Ŷ 13 (π/2),R 11 =X 03 (π/2),R 12 =Ŷ 03 (π/2). (1) One can optimize the rotation set to decrease the number of rotations and/or increase the reconstruction sensitivity. The following is one of such a set composed only of six operations and still exhibiting high scheme sensitivity parameter. The rotations in this set arê R 1 =X 01 (π/2)X 23 (π/2), Initial state Initial state FIG. 4: A schematic illustration of pulse sequences for effective pure state preparation starting from the initial state of the semiconductor NMR device. Note that two-photon and three-photon transitions can be decomposed into one-photon transitions using more complicated pulse sequences, i.e, X02(π/2) = X01(π/2)X12(π). R 2 =Ŷ 01 (π/2)Ŷ 23 (π/2), R 3 =X 01 (π/2)Ŷ 13 (π)X 12 (π/2), R 4 =Ŷ 01 (π/2)Ŷ 13 (π)Ŷ 12 (π/2), R 5 =X 02 (π/2)X 13 (π/2), R 6 =Ŷ 02 (π/2)Ŷ 13 (π/2) (2) CONCLUSIONS All-electrical coherent control of nuclear spins in a nanoscale NMR chip has been demonstrated. Clear coherent oscillations reflect all possible transitions among the four nuclear spin states of each of the nuclide ( 69,71 Ga and 75 As) in the point contact channel. Since arbitrary control of superpositions among the four spin levels can be performed in this device, one and two qubit operations are possible. We have theoretically shown how to utilize this four-level system as a fictitious two-qubit system, and designed the pulse sequences for the realization of one-and two-qubit quantum operations. Moreover, we proposed an NMR state tomography scheme based on the detection of longitudinal magnetization of the nuclear spins.
Although we have not shown in this manuscript, we performed preliminary experiments [12] for the preparation of effective pure states, which are important for the initialization of the NMR quantum computer. Dynamic nuclear spin polarization induced by the current flow allowed us to prepare a large population to use for the preparation of effective pure states.
The results shown in this paper suggest that this nanometre-scale NMR device is a good candidate for quantum-information processing based on solid-state systems.