# Implementing a noise protected logical qubit in methyl groups via microwave irradiation

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## Abstract

We propose a proof-of-principle experiment to encode one logical qubit in noise protected subspace of three identical spins in a methyl group. The symmetry analysis of the wavefunction shows that this fermionic system exhibits a symmetry correlation between the spatial degree of freedom and the spin degree of freedom. We show that one can use this correlation to populate the noiseless subsystem by relying on the interaction between the electric dipole moment of the methyl group with a circularly polarized microwave field. Logical gates are implemented by controlling both the intensity and phase of the applied field.

### Keywords

Logical qubit Symmetry Noiseless subsystems Methyl groups## 1 Introduction

A quantum bit, known as qubit, is a two-level coherent system [1, 2]. A qubit can store quantum information for use in quantum computing, quantum communication, sensing, etc. [1]. Over the past decades, there have been proposals for physical realization of a logical qubit including cold atoms, polarization of a single photon and spins. What is common between all of these physical systems is that there always exist some undesired couplings with their environment, which manifest themselves as a noise process on the system of interest. These undesired couplings shorten both relaxation time and coherence time of a qubit that is equivalent to losing quantum information.

One of the approaches for protecting the quantum information is to encode it in such a way that the logical qubit is not affected by noise in the first place. In fact, the notion of decoherence free subspace or noiseless subsystems have been developed, where the information is encoded in a particular part of the Hilbert space so as to have noise immunity [3, 4, 5]. For example, when the logical states of a qubit are considered as the actual physical states of a spin half particle, one should expect both X-noise (spin flip) and Z-noise (dephasing) which corrupt the information [1]. But, if one considers a group of spins, there are other degrees of freedom rather than the spin magnetization that can be used as a logical basis that is protected against majority of noise processes [6].

Nuclear magnetic resonance (NMR), owing to its long history, has been among the first physical candidates to experimentally demonstrate the quantum information and error correction algorithms [7, 8, 9, 10]. We investigate an NMR implementation of a noiseless logical qubit that is created by using the collective properties of a group of three indistinguishable spins. In particular, we consider a methyl group, which consists of a carbon symmetrically bonded to three protons, and show that the group of protons can serve as one logical qubit that is protected against collective noise.

Since the early age of NMR, the molecular motion and the spin properties of methyl groups have been studied both theoretically and experimentally [11, 12, 13, 14]. Methyl groups have been used for experimental demonstration of classical states with long relaxation times that are known as long-lived state (LLS) [15, 16]. In this report, we show that LLS can be used as an initial step for implementing a coherent superposition of logical basis states. In particular, we analyze the symmetry of the internal rotation and the spin Hamiltonian of methyl groups using a more modern mathematical language which may benefit both the NMR community and the quantum information community. We then use that symmetry analysis to propose a practical way of converting LLS into a logical qubit that can be used for storing quantum information and is immune to collective noise.

There is a challenge for accessing the noise protected logical basis of methyl groups. We show that the noise immunity is due to the spin symmetry of indistinguishable spins and the indistinguishability of spins results in the degeneracy of the subspace of interest. As a result, the logical states are not individually accessible. Nevertheless, we take advantage of the molecular symmetry and the microwave spectroscopy technique [17, 18, 19] to propose an experiment that leads to populating and controlling the noise protected subspace of methyl groups without breaking the degeneracy.

In particular, we show that there is a symmetry correlation between the rotational degree of freedom and the spin degree of freedom due to Pauli exclusion principle. This space-spin correlation suggests that one can populate the noiseless logical states, albeit directly inaccessible due to their degeneracy, by controlling the other degrees of freedom such as rotational motion. We use a symmetry argument based on the group theory to derive the allowed transitions under an electric field irradiation and conclude that an interaction with a *right* or *left* circularly polarized microwave filed results in populating the logical *zero* or *one* states individually without lifting their degeneracy. Moreover, we show a delicate control of the intensity and phase of the applied field can lead to implementing the logical gates. Therefore, we demonstrate a proof-of-principle experiment to use methyl groups for implementing one logical qubit that has a long relaxation and coherence times.

In Sect. 2, we define the logical qubit for methyl groups by looking at the permutation symmetry and the spin Hamiltonian of three indistinguishable spins. Then, in Sect. 3, we study the internal rotational motion of \(CH_{3}\), also known as a *rotor*, followed by Sect. 4 that argues the symmetry of the total wavefunction. Given this symmetry analysis, Sect. 5 seeks for novel means of accessing and controlling the noise protected subspace using the microwave irradiation, leading to preparation and manipulation of one logical qubit.

## 2 Logical qubit in methyl groups

### 2.1 Three indistinguishable spins

*s*indicates the symmetry irreducible representations, \(s \in \{ A, E_{+}, E_{-} \}\) and the

*m*is the eigenvalue of the z component of the total spin angular momentum operator, \(\hbar \hat{\mathbf {S}}_{z}=\frac{\hbar }{2} \sum \nolimits _{i=1}^{3} \sigma ^{(i)}_{z}\). We name \(\{ |s, m\rangle \}\) the cyclic permutation (CP) basis whose explicit expansion in terms of the computational basis is given in Table 1.

The cyclic permutation basis expanded in the computational basis. \(\hbar =1\) is considered

\( | s, m\rangle \) | Expansion in \(\{ |\uparrow \rangle , |\downarrow \rangle \} ^{\otimes 3}\) | \(\hat{ P }_{+}\)’s Eigenvalue |
---|---|---|

\(|A, 3/2\rangle \) | \(|\uparrow \uparrow \uparrow \rangle \) | 1 |

\(|A, 1/2\rangle \) | \(\frac{1}{\sqrt{3}}\left( |\uparrow \uparrow \downarrow \rangle + |\downarrow \uparrow \uparrow \rangle + |\uparrow \downarrow \uparrow \rangle \right) \) | 1 |

\(|A, -1/2\rangle \) | \(\frac{1}{\sqrt{3}}\left( |\downarrow \downarrow \uparrow \rangle + |\uparrow \downarrow \downarrow \rangle + |\downarrow \uparrow \downarrow \rangle \right) \) | 1 |

\(|A, -3/2\rangle \) | \(| \downarrow \downarrow \downarrow \rangle \) | 1 |

\(|E_{+}, 1/2\rangle \) | \(\frac{1}{\sqrt{3}}\left( |\uparrow \uparrow \downarrow \rangle + \varepsilon ^{*}|\downarrow \uparrow \uparrow \rangle +\varepsilon |\uparrow \downarrow \uparrow \rangle \right) \) | \(\varepsilon \) |

\(|E_{+}, -1/2\rangle \) | \(\frac{1}{\sqrt{3}}\left( |\downarrow \downarrow \uparrow \rangle + \varepsilon ^{*} |\uparrow \downarrow \downarrow \rangle +\varepsilon |\downarrow \uparrow \downarrow \rangle \right) \) | \(\varepsilon \) |

\(|E_{-}, 1/2\rangle \) | \(\frac{1}{\sqrt{3}}\left( |\uparrow \uparrow \downarrow \rangle + \varepsilon |\downarrow \uparrow \uparrow \rangle +\varepsilon ^{*} |\uparrow \downarrow \uparrow \rangle \right) \) | \(\varepsilon ^{*}\) |

\(|E_{-}, -1/2\rangle \) | \(\frac{1}{\sqrt{3}}\left( |\downarrow \downarrow \uparrow \rangle + \varepsilon |\uparrow \downarrow \downarrow \rangle + \varepsilon ^{*} |\downarrow \uparrow \downarrow \rangle \right) \) | \(\varepsilon ^{*}\) |

*subsystems*, where the first component refers to the

*s*label and the second component refers to the

*m*label,

*preserved*by the collective spin operators. Therefore, one can choose the

*symmetry label*subsystem of the

*E*subspace ( or \( j=1/2\)) of 3 identical physical qubits to encode one logical qubit because it is

*protected*against collective noise. Explicitly, the logical basis is defined by

### 2.2 Spin hamiltonian of a methyl group

*s*. Therefore, at relatively high field, where the DD couplings and/or the CSA are negligible and for a noise model that the methyl group interacts symmetrically with the environment, neither the environment nor the local spin Hamiltonian in Eq. (7) breaks the cyclic permutation symmetry. Thus, a methyl groups is a candidate for storing one logical qubit that exhibits long relaxation and decoherence times because it is immune to all collective noise.

However, there is a delicate point here that prevents us from doing so. In one hand, the protection against collective noise has its roots in the indistinguishably of spins which allows us to consider the symmetry label \(E_{+}/E_{-}\) as a noiseless subsystem. On the other hand, if neither the local spin Hamiltonian in Eq. (7) nor the coupling to the environment distinguishes spins, the spin eigenstates \(|E_{+},m \rangle \) and \(|E_{-},m \rangle \) are degenerate, and thus, the logical basis \(|\bar{0} \rangle \) and \(|\bar{1} \rangle \) are not accessible individually. When the logical basis states are not individually accessible, implementing any arbitrary logical qubit is not feasible. Nevertheless, in the following we show that it is possible to populate these degenerate states individually without breaking the cyclic permutation symmetry. We propose to use other degrees of freedom of methyl groups, such as internal rotation, to indirectly access and control the degenerate subspace of the spin degree of freedom. Thus, in the next section, we review the symmetry of the rotational degree of freedom of a methyl group.

## 3 Internal rotation of methyl groups

*torsional*degree of freedom of a methyl group which has been extensively studied in the literature [11, 12, 13, 14]. Idealistically, a methyl group is a

*free rotor*that is a rigid body

*freely*rotating around the

*z*axis and its Hamiltonian is simply the

*z*component of the total angular momentum,

*free rotor energy*constant.

*hindering*potential which is imposed by the rest of the molecule as well as other external molecules

^{1}. In ethane, for example, the molecule prefers to be in the staggered configuration rather than the eclipsed configuration [21]. The amount of energy that is required to move from one configuration to another defines the height of the hindering potential or the

*barrier*(Fig. 1). Therefore, in addition to the free rotation term, there is an extra potential that affects the rotational motion [14],

*firm rotor*limit or a rigid rotor with extremely high rotational barrier, the motion along the \(\varphi \) direction is very restricted, because the barrier height is much larger than the free rotor energy, i.e., \( V_{0}\gg F\). Thus, we approximate \(V(\varphi )\) with three quantum wells and treat each well as a harmonic potential,

In non-extreme cases, where the methyl group is neither a free rotor nor a firm rotor (harmonic oscillator), we solve the Schödinger equation numerically and plot the eigenenergies as a function of a parameter \(0< q < 1\), where the \(q=0\) corresponds to a free rotor (\(V_{0}=0\)) and the \(q=1\) corresponds to a firm rotor (\(V_{0} \gg F\)). The first twelve eigenenergies of a methyl group are demonstrated in Fig. 2, from the free rotor limit to the firm rotor limit. The left plot is associated with the nearly free rotor (\(V_{0}\approx 0\)), and as expected, other than the ground state (\(l=0\)), all other energy levels are double degenerate. At an intermediate regime, when \( V_{0}\ge F\), every three energy levels tend to group together and form a *band* that consists of a non-degenerate level and a double degenerate level. Thus, at \( V_{0}\ge F\) limit we label the internal rotation eigenstates with \(\varPhi _{\lambda , n}\) with corresponding energies \(E_{\lambda , n}\) in which *n* refers to the band level (harmonic oscillator level) and \(\lambda \in \{0, \pm 1 \}\). At each band *n*, we denote the *internal* energy difference between the \(\lambda =0\) level and the \(\lambda =\pm 1\) levels with \(\varDelta E_{n}\). At a particular value of *q*, we observe that the sign of the energy *splitting*, \(\varDelta E_{n}\), changes from the *n*th band to the \((n+1)\)th band, and its amplitude gets larger and larger as we go higher in *n*. Because of these two observations, it is reasonable to see at some point the \(E_{\lambda =0,n}\) level and \(E_{\lambda =0, n+1}\) merge together and form a double degenerate level. Indeed, even in the intermediate regime where \(V_{0}\ge F\), the high energy levels behave like that of a free rotor and are doubly degenerate. Because in that energy scale, the methyl group effectively *does not see* the barrier. These numerical observations are better justified in the upcoming section. For larger *q* (or larger barrier height), the internal splitting \(\varDelta E_{n}\) gets smaller and smaller, and eventually, it becomes zero at the firm rotor limit, yielding to a threefold degenerate level. The right plot of Fig. 2 is associated with the nearly firm rotor (\(V_{0}\gg F\)).

Each molecule has a certain geometry and mass and so has a certain free rotation energy, *F*. It also has an associated hindering potential with specific barrier height, \(V_{0}\). Thus, depending on the ratio \(\frac{V_{0}}{F}\), an organic compound may be considered as a free rotor, or an intermediate rotor, or an firm rotor based on the structure of the lowest energy levels.

Examples of chemical compounds with their corresponding barrier height and free rotor energy

### 3.1 The symmetry of the torsional states

A methyl group forms a \(\mathbf C _{3}\) group in the spatial space that has three irreducible representations, \(\{A, E_{+}, E_{-}\}\), with their corresponding character values \(\{1, \epsilon , \epsilon ^{*}\}\) with \(\epsilon =e^{ i\frac{2\pi }{3}}\). In previous section, we solved the Schrödinger equation numerically, yielding the torsional eigenfunctions \(\varPhi _{ \lambda ,n} (\varphi )\) in the intermediate rigid rotor limit (\( V_{0}\ge F\)). Here, we analyze the symmetry of \(\varPhi _{ \lambda ,n} (\varphi )\) and show that the label \(\lambda \) is associated with one of the irreducible representations of \(\mathbf C _{3}\) group.

*A*

*symmetry*, it must be invariant under the \(\frac{2\pi }{3}\) rotation. Therefore, the nonzero terms in the above expansion must be \(l = 3\ k \), where

*k*is an integer. Similarly, if it has \(E_{\pm }\)

*symmetry*, under the \(\frac{2\pi }{3}\) rotation it must pick up a phase \(e^{\pm i\frac{2\pi }{3}}\), i.e., \( \varPhi _{\lambda ,n} (\varphi \pm \frac{2\pi }{3})= e^{\pm i\frac{2\pi }{3}}\ \varPhi _{\lambda ,n} (\varphi )\). Therefore, the nonzero terms in the above expansion must be \(l =3\ k +1\) for \(E_{+}\) symmetry and \( l = 3\ k -1\) for \(E_{-}\) symmetry. This property can be used as a test to see which kind of symmetry each numerical eigenfunction has,

*A*or \(E_{\pm }\) or a combination. We realize that for a methyl group with an intermediate barrier, at the

*n*th band, the \(\lambda =0\) state has only

*A*symmetry and the doubly degenerate states, \(\lambda =\pm 1\), have \(E_{\pm }\) symmetry. Because \(E_{\pm }\) are degenerate, any superposition of them is also a solution to the Schrödinger equation. The first 6 numerical eigenfunctions of the torsional Hamiltonian are plotted in Fig. 3 for the choice of \(V_{0} =32\) meV and \(F= 0.6\) meV.

*V*(

*x*) with

*N*wells (atoms), the lattice (or the molecule) wavefunction, \(\varPhi (x)\), has the following properties:

*a*is the distance between two adjacent wells (lattice constant) and \(\kappa = \lambda \ \frac{ 2\pi }{Na}\) is the pseudo-momentum with \(\lambda = 0, \pm \,1, \pm \,2,\ldots \). Using the Bloch theorem and the linear combination of atomic orbital (LCAO) method with no-hybridization assumption, the lattice wave function of the

*n*th energy level is \(\varPhi _{n}(x) = \sum \nolimits _{j=1}^{N} \ e^{ ika j} \chi ^{(j)}_{n}(x)\) in which \(\chi (x)\) is the solution of each atom.

*n*th eigenfunction of the

*j*th well (harmonic oscillator). The boundary condition, \(\varPhi (\varphi + 2\pi )=\varPhi (\varphi )\), results in \(\lambda =0, \pm 1\). Thus,

*A*symmetry and the \(\lambda =\pm 1\) states have \(E_{\pm }\) symmetry. We should also note that the above LCAO expansions are valid in the limit of no-hybridization assumption where the overlap between \(\chi ^{(j)}_{n}\) and \(\chi ^{(j')}_{n+1}\) is negligible. In that limit, the Hamiltonian is block diagonal and the corresponding Schrödinger equation for the

*n*th block is

*A*symmetry level and the \(E_{\pm }\) symmetry levels. This is an analogy to the energy band gap in the solid-state physics when we deal with a periodic lattice with large

*N*.

The overlap between the wavefunctions of two wells provides a qualitative and yet informative description of the torsional eigenenergies. The symmetry of each well/harmonic oscillator’s wavefunctions changes from the *n*th level to the \((n+1)\)th level. Consequently, the sign of the overlap, \(\beta _{n}\), or the splitting \(\varDelta E_{n}\) changes from the *n*th level to the \((n+1)\)th level, and accordingly, the ordering between the *A* level and the \(E_{\pm }\) levels changes. Moreover, in high energy levels, the wavefunction is less confined within a well, and therefore, its overlap with the neighbor’s wavefunction is larger. This means, as we go higher and higher in energy, the splitting between *A* symmetry and \(E_{\pm }\) symmetry gets larger and larger, and at some point, the labeling \(\{ A, E_{\pm }\}\) is not valid any more, because the no-hybridization assumption breaks. Indeed, if the torsional energy is high enough, the barrier is not effective any more and the methyl group behaves like a free rotor. This symmetry analysis is consistent with the numerical result presented previously.

Now that we have some understanding of the spin symmetry, the torsional symmetry, respectively, we proceed to the argument of the symmetry of the total wavefunction and the correlation between the spatial space and the spin space. One may take advantage of this correlation to create a noise protected qubit.

## 4 The symmetry of the total wavefunction

We are interested in the solid phase of \(X-CH_{3}\) molecule, where the translational and external rotation of this *rigid* body or its center of mass is negligible. Excluding the spin degree of freedom, a methyl group has rotation, vibration and electronic degrees of freedom that are known as Rovibronic for short. At low temperature (solid phase), the external rotational and translational motion are negligible and it is a fair assumption to consider a fixed molecular framework in which the nuclei rotate and/or vibrate near their equilibrium locations relative to a fixed origin (center of mass). Since electrons are much lighter than nuclei, an electron is much faster than a nucleus, and thus, its corresponding energy is very larger than the nuclear energy. This justifies the Born–Oppenheimer approximation [27, 28] that is commonly used in the quantum chemistry to effectively separate the wave function into an electron part and a nuclear part, \(\psi _{e} \otimes \psi _{n}\). In this approximation, one can treat the nuclei at a fixed geometry or in certain configurations in space with a slow motion and solve the Schrödinger equation for the electron wavefunction only, yielding \(\psi _{e}\) which is derived in the nuclear coordinates. In the second step, for each electron state, the nuclear wavefunction is obtained by including an effective potential in the Hamiltonian that serves as a replacement for each electron wavefunction. Indeed, in each electronic level, there is a set of eigenenergies for the nuclear spins.

Further, we need to consider the rotation–vibration part of the wavefunction. In general, the rotation and the vibration are not two independent degrees of freedom, but we might still be able to treat them separately. The vibrational energies are normally in the order of 1000 cm\(^{-1}\), whereas the rotational energies are in the order of 10 cm\(^{-1}\) [29]. Therefore, one can treat the rotational variables as a constant and solve the Schrödinger equation by considering the vibrational part only. In this method, similar to the Born–Oppenheimer approximation, for each vibrational state \(\nu \), an effective potential \(V_{\nu }\) is added to the rotational part of the Schrödinger equation.

*j*th particle with the \((j+1)\)th one, and second the \((j+1)\)th particle with the \((j-1)\)th one, for all \(j\in \{1,2,3\}\). This invariance under the rotation narrows the allowed combinations of the torsional eigenfunctions and the spin eigenstates to those that satisfy

*f*. Thus, the allowed combinations are \(A \times A\), \(E_{+} \times E_{-}\) and \(E_{-} \times E_{+}\). This can be considered as a sort of

*symmetry correlation*between the spatial space and the spin space. One can take advantage of this symmetry induced coupling between the spin space and the torsional space to initialize the spin state of the methyl group in the noise protected subsystem.

Note that an implicit approximation is considered here. When a methyl groups is rotating in the presence of a hindering potential, the three protons are no longer perfectly indistinguishable to the environment, and hence, the spin Hamiltonian in Eq. (7) and the corresponding spin eigenstates are an approximation. But as a first-order approximation, we treat them as identical spins.

## 5 Accessing the noise protected subsystem of methyl groups

The proposed experiment for encoding one logical qubit into the noiseless subsystem of a methyl group consists of two phases. First, we show that at cryogenic condition and at *low field*, where the spin Zeeman splitting is much smaller than the torsional ground state energy splitting, the thermal equilibrium state is highly populated in the \(A \times A\) subspace. Second, we show that an interaction with a microwave field can, in principle, selectively *convert* the population from the \(A \times A\) subspace to the \(E_{-} \times E_{+}\) subspace or to \(E_{+} \times E_{-}\) subspace, depending on the applied being left or right circularly polarized. We elaborate on each of the above phases in the following sections.

### 5.1 Long-lived state by thermal means

*Long-lived state* (LLS) refers to those states that have an imbalance of population between different symmetry subspaces and have been experimentally studied in methyl groups by [13, 15, 16]. LLS exhibits long relaxation times because they are polarized in terms of the symmetry label which is immune to the collective noise. LLS is considered as classical protected states because there is no coherence term between different symmetry subspaces. In the following, we review how an LLS in a methyl group can be implemented just by thermal means and discuss why this state is not capable of storing quantum information. We later use the LLS as an initial state that can be converted to a logical qubit.

*symmetry polarized*states as a set of mixed states that are individually polarized in terms of the symmetry label, but are totally mixed in terms of the total spin magnetization label. They are denoted by \(\rho _{A/E_{\pm }}\) and are given by

*A*subspace and the

*E*subspace and is given by

We show how to populate \(Q_{LLS}\) by thermally cooling a methyl group. Consider an intermediate rigid rotor in the presence of a magnetic field with torsional eigenfunctions \(\varPhi _{\lambda , n}\) and spin eigenstates \(|s, m\rangle \). As discussed in Sect. 4, for a fermionic system, the product of \(\varPhi _{\text {tor}} \otimes \psi _{\text {spin}}\) must be symmetric under the \(\frac{2 \pi }{3}\) rotation, which narrows the allowed combinations of the torsional eigenfunctions and the spin eigenstates to those with symmetry labels \(\lambda \times s \in \{ A \times A, \ E_{\pm } \times E_{\mp }\}\). This *symmetry correlation* between the spatial space and the spin space can be used for implementing a LLS.

Thus, at low field and low temperature (\(\gamma B_{0} \ll k_{b} T \ll \varDelta E_{0}\)) and according to the Boltzmann distribution, there is a significant population imbalance between the \(\varPhi _{A, n=0}\) and the \(\varPhi _{E_{\pm }, n=0}\) in the torsional space. Consequently, due to the spin-space symmetry correlation, there is an imbalance of population imbalance the \(\rho _{A}\) and the \(\rho _{E}\) in the spin Hilbert space. As a result, this thermal process corresponds to initializing the collective spin state of a methyl group in a \(\gamma \)-polarized long-lived state, \(Q_{LLS}\) with \(\gamma = \tanh [ \varDelta E_{0}/k_{b} T]\). This has been experimentally demonstrated in [13, 15, 16].

Note that the \(Q_{LLS}\) that was introduced in Eq. (19) is a classical mixture of \(\rho _{A}\) and \(\rho _{E}\), which are themselves mixed states. When neither the local spin Hamiltonian nor the interaction Hamiltonian distinguishes spin, one is not able to create coherence between the *A* subspace and the *E* subspace and use \(\{A/E\}\) as a logical basis for quantum information processing. In other words, when spins are indistinguishable, the spin Hamiltonian has a block-diagonal form, and hence, the Hilbert space is a direct sum of the two subspaces, \({\mathscr {H}}= {\mathscr {H}}_{A} \oplus {\mathscr {H}}_{E}\). Nevertheless, the subspace \({\mathscr {H}}_{E}\) alone can be used for encoding a logical qubit, which is the subject of discussion in the next section.

### 5.2 Protected state via the electromagnetic field interaction

As mentioned before, the *E* subspace of three identical spins is decomposed into a product of two subsystems, \({\mathscr {H}}_{E}= {\mathbb {C}}^{2} \otimes {\mathbb {C}}^{2}\), where the first subsystem refers to the symmetry, \(s\in \{E_{+}, E_{-}\}\) which can be used as a logical basis, and the second subsystem refers to the total magnetization, \(m \in \{ \pm \frac{1}{2}\}\), which can be corrupted by noise, and hence is not of interest. For storing quantum information, our target is to access the \(E_{\pm }\) subsystem of the spin space individually without removing the degeneracy of these subspaces.

One intuitive solution for accessing the \(E_{+}/E{-}\) noiseless subsystem without breaking their degeneracy is to include a control Hamiltonian that selectively populates either the \( E_{+}\) subsystem (logical \(|\bar{0}\rangle \)) or the \( E_{-}\) subsystem (logical \(|\bar{1}\rangle \)). We propose to first prepare \(Q_{LLS} \) with \(\gamma \approx 1\), as it was explained in Sect. 5.1. This step is done at very low temperature and very low field, which initializes the collective spin state in \(\rho _{A}\) that is polarized in terms of the spin symmetry label. In the next step, we apply a selective “\(\pi \)” pulse that converts the \(\varPhi _{A,0}\) to the \(\varPhi _{ E_{+},0}\) (or \(\varPhi _{ E_{-},0}\)) on the torsional space. This leads to populating \(\rho _{E_{-}}\) (or \(\rho _{E_{+}}\)) on the spin space without removing the degeneracy of them. We have not yet described how to implement these “\(\pi \)” pulses. This section is exploring the possibility of addressing the logical basis (or the \(E_{\pm }\) subsystems) by using an interaction between the rotational degree of freedom of methyl groups and a circularly polarized external electromagnetic field.

We are inspired by microwave spectroscopy [17, 18] which is a well-known technique that uses the microwave irradiation to cause transitions between the rotational states of molecules in the gas phase. In this spectroscopy technique, the emission and the absorption of the electric dipole allowed transitions lead to extracting information about the geometry of rigid bodies such as the bond’s length and angles [17, 30]. We adopt this technique and apply it to our case, where at low temperature the system is in the solid phase rather than the gas phase. The barrier height can be different at solid phase versus the gas phase but not by orders of magnitude. At low field, we ignore the spin space and just focus on the spatial space transitions. Consider a rigid rotor in a symmetric top molecule, \(X-CH_{3}\), whose torsional ground state splitting \(\varDelta E_{0}\) is in the range of GHz. We explore whether a circularly (right or left) polarized microwave filed that is on resonance with \(\varDelta E_{0}\) induces a transition between the \(\varPhi _{A,0}\) and the \(\varPhi _{ E_{\pm },0}\) or not.

*i*th level to the

*j*th level due to an interaction Hamiltonian \(H_{\text {int}}=H_{d}\) is given by

Excluding the spin space and the external rotations, \(\psi _{i}( \mathbf {\mathbf {r}})\) is the internal rotation–vibration wave function. At low temperature, the vibrational wavefunction is symmetric, and therefore, the symmetry of \(\psi _{i}( \mathbf {\mathbf {r}})\) is determined by the symmetry of the torsional wavefunction, \(\varPhi _{\lambda , n}(\varphi )\). Moreover, the electron charge distribution \( \rho _{e}(\mathbf {\mathbf {r}})= |\psi _{e}(\mathbf {\mathbf {r}})|^2\) is also symmetric. Therefore, to find the selection rule (forbidden vs allowed transitions), it is sufficient to analyze the symmetry of \(\varPhi _{\lambda ,n}\ \mathbf {\mathbf {r}}.\mathbf {\mathbf {E}}(t) \ \varPhi _{\lambda ', n'}\). Since we are only interested in the allowed transitions between \(\varPhi _{A,0}\) and \(\varPhi _{E_{\pm },0}\), the only unknown component in the above integral is the symmetry of \(\mathbf {\mathbf {r}}.\mathbf {\mathbf {E}} (t)\).

*symmetry projection*operators to it [32]. In particular, in case of \(\mathbf C _{3}\) group, we apply \(\hat{\textit{P}}_{A}, \hat{\textit{P}}_{E_{+}}\) and \(\hat{\textit{P}}_{E_{-}}\) on \((X + i Y)\) and obtain

*no-rotation*operator. Equation (22) means that the electric dipole interaction with a

*right circularly polarized*electromagnetic field has \(E_{-}\) symmetry. Therefore, for this particular interaction, the transition from \(\varPhi _{A,0}\) to \(\varPhi _{E_{+},0}\) is symmetrically allowed. Similarly, a

*left circularly polarized*field has the \(E_{+}\) symmetry that induces transition from \(\varPhi _{A,0}\) to \(\varPhi _{E_{-},0}\). This observation may enable us to selectively populate the \(E_{+}\) or the \(E_{-}\) subspaces on the spin space as a result of the correlation between the torsional space and the spin space.

*X*” gate which takes \(|\bar{0}\rangle \leftrightarrow |\bar{1}\rangle \) and a “

*Z*” which takes \(|\bar{0}\rangle \rightarrow |\bar{0}\rangle \) and \(|\bar{1}\rangle \rightarrow - |\bar{1}\rangle \). Given the logical gates, “

*X*” and “

*Z*”, one has universal control over a single qubit [1]. In the following, we now show implementing these logical gates is feasible.

In summary, we propose to prepare a logical qubit in a methyl group encoded in spin and rotational degree of freedom. To choose the right molecule, we desire a symmetric top molecule, \(X-CH_{3}\), where the rotational barrier of the methyl group is in the intermediate regime, i.e., it is neither a free rotor nor a firm rotor. We also desire that the energy splitting between the \((n=0, \lambda =A)\) level and the \((n=0, \lambda =E_{\pm })\) of the torsional space to be in the range of GHz. By thermal equilibration at cryogenic temperature (about 0.3 K if \( \varDelta E_{0} \approx 6.5\) GHz), the \(Q_{LLS}\) is well populated. This step is at nearly zero magnetic field but if one desire to confirm that the LLS has been prepared, then one should rapidly warm up the system and transport it into a large magnetic field and measure the NMR spectrum. According to the previous studies [15, 16], one would expect to observe two peaks on the proton channel that have equal intensity and are anti-phase and observe four peaks on the carbon channel that have unequal intensities, but they are two by two anti-phase. To convert this LLS into the logical basis subspace, one can apply a circularly polarized microwave electric field that is tuned to be on resonance with the torsional ground state splitting. Given the dipole moment of the molecule, we control the intensity, the phase and the timing of the microwave field so that we implement an effective \(\pi \) rotation between \(\varPhi _{A,0}\) and \(\varPhi _{ E_{+},0}\) or \(\varPhi _{E_{-}, 0}\) torsional states. To confirm that the logical states have been prepared, one can measure the NMR spectra once again by rapidly transporting to room temperature and high magnetic field. We expect to observe the same anti-phase peaks on the proton channel, but on the carbon channel we should see the signature of an initially prepared logical qubit. Particularly, the two peaks that are associated with \(m=\pm \frac{3}{2}\) on the carbon channel remain the same as that of the LLS, but the other two peaks that are associated with \(m=\pm \frac{1}{2}\) will not be anti-phase anymore. This indicates that the spin population is converted from the \(|A, m=\pm \frac{1}{2}\rangle \) states to the \(|E_{\pm }, m=\pm \frac{1}{2}\rangle \) states. The absolute value of the phase indicates whether we prepared the logical \(|\bar{0}\rangle \) (\(E_{+}\) state) or the logical \(|\bar{1}\rangle \) (\(E_{-}\) state). In principle, any coherent superposition state, \(|\psi \rangle = \cos \alpha \ |\bar{0}\rangle + \sin \alpha \ |\bar{1}\rangle \), can be implemented by controlling the intensity and the phase of the microwave electric field.

For the realization of the above proposal, it is important to find a proper molecule. The proposed experiment demands a molecule whose rotational barrier is not neither extremely large, which has negligible rotational splitting, nor too low, which acts like a free rotor. Moreover, the ground state rotational splitting should be large enough such that a mK or even \(\mu \)K cooling initialize the system in the ground state of the rotational degree of freedom. Moreover, the molecule should have a reasonable dipole moment to interact with the electric field.

## 6 Conclusion

The presented work seeks for novel means of storing quantum information in the noise protected subspace of a group of identical spins. In particular, we investigated the symmetry of the spin degree of freedom and the rotational degree of freedom of methyl groups, respectively, and discussed the symmetry correlation between the two subspaces. That correlation has been used for creating classical states with long relaxation time, and in this report, we moved a step further to propose a scheme that uses this correlation for creating a logical qubit that is robust against collective noise. Our analysis provides a proof-of-principle experiment that relies on symmetry allowed transitions due to the interaction between the dipole moment of the molecule with a circularly polarized microwave field.

A logical qubit with long relaxation and coherence times can find application in magnetometry, sensing and for studying slow dynamics in NMR. If one is interested to use the above-described logical qubit for quantum computing purposes, one should think of implementing a two-qubit logical qubit as well (such as CNOT gate). One possibility is to consider a molecule that has two magnetically in-equivalent methyl groups. An example is ethane where only one of the carbons is labeled, making the two methyl groups individually addressable. One may take advantage of the rotation-rotation interaction between the two methyl groups to implement the two-qubit logical gate.

## Footnotes

## Notes

### Acknowledgements

We acknowledge support from the Canada First Research Excellence Fund (CFREF), Industry Canada, the Natural Sciences and Engineering Research Council of Canada (NSERC RGPIN-418579), Canada Excellence Research Chairs Program (CERC 215284), the Canadian Institute for Advanced Research, the Quantum NanoFab and the Province of Ontario.

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