1. Introduction
Quantum computation has the potential to outperform conventional computation for certain challenging problems [1] and has garnered significant attention within the realms of both quantum physics and computer science. This attention is attributed to its potential to address computational problems that are intractable for classical computing methods [2, 3]. Various physical systems have been harnessed for encoding qubits in quantum computation, including trapped ions [4–6], neutral atoms [7, 8], quantum dots [9], nuclear magnetic resonance [10–12], superconducting circuit [13–18], and optical systems [19–22]. Each of these systems possesses distinct advantages in quantum information processing, thus serving as a potential candidate for materializing universal quantum computers.
Polar molecules exhibit intricate internal energy structures and substantial electric dipole moments [23, 24], which facilitate pronounced interaction with external electric fields [25, 26]. This interaction, in turn, facilitates meticulous manipulation and control of quantum states. As a result, polar molecular systems are regarded as a promising platform for quantum computation [27–29]. In 2002, DeMille initially proposed the application of electric dipole moments of diatomic molecules as qubits. In which these molecules are confined within a 1D optical lattice and are oriented parallel or antiparallel to the electric field [30]. However, the practical coherent control of polar molecules is relatively difficult compared to trapped atoms or ions due to their complex structural characteristics. Fortunately, in recent years rapid advancements have been made in experimental techniques such as laser cooling and trapping for polar molecules [31–40].
To date, substantial theoretical endeavors have been investigated in devising quantum gates and quantum algorithms based on polar diatomic molecular systems [41–47]. For example, Ni et al [42] outlined the production of an iSWAP gate through switchable dipolar exchange interactions between NaCs molecules confined in optical tweezers. Subsequently, Bao et al [45] experimentally demonstrated dipolar spin-exchange and entanglement between single CaF molecules in an optical tweezer array. Holland et al [46] achieved on-demand entanglement of CaF molecules in a reconfigurable optical tweezer array to create Bell pairs deterministically. Moreover, Mur-Petit et al [47] demonstrated the execution of the Deutsch algorithm using the CaF and RbCs molecular qudit. Gregory et al [48] achieved a 0.78(4) seconds Ramsey time in a rotationally-magic trap for RbCs molecules without dipole-dipole interactions.
The utilization of optimal control in quantum information processing has been widely acknowledged [49–57] due to its role in achieving precise manipulation of quantum phenomena, especially for implementing quantum information protocol. Optimal control theory (OCT) presents an avenue for controlling physical systems by optimizing specific observables, and it has been demonstrated in various quantum platforms [58–60]. For instance, a gradient ascent pulse engineering algorithm is used to design the microwave (MW) field for enhancing the robustness of entangling gates based on CaF molecules [54]. The OCT is employed to optimize laser pulses interacting with SrO molecules, leading to the high-fidelity realization of fundamental quantum gates consisting of pendular qubit states [55]. Additionally, Zhang et al [56] examined the generation of high-dimensional entangled states in polar molecular systems via OCT. As we know, a multi-level system, characterized by its large Hilbert space, plays an important role in not only designing a Hamiltonian for quantum simulations [16, 17], but also serving as a flexible platform for the storage and encoding more quantum information [61–63], enabling the execution of more complex quantum algorithms compared to using only qubits. Furthermore, Gedik et al [64] introduced a qutrit-based quantum speedup algorithm designed to solve a black box problem, in which a single qutrit system is capable of implementing the quantum permutation algorithm (QPA), determining the permutation parity with just one call to the quantum oracle, in contrast to the two calls required in the classical setup. Lindon et al [65] experimentally demonstrated arbitrary single-qutrit SU(3) gates in neutral alkali atoms and implemented the Walsh–Hadamard Fourier transform. Motivated by these works, we propose a theoretical scheme for implementing the permutation algorithm within a single SrO molecule qutrit system by designing a series of optimal gate pulses, which was experimentally demonstrated in nuclear magnetic resonance systems [64, 66], superconducting systems [67, 68], and linear optical setups [69–72]. This demonstration provides an elementary illustration of computational speed-up. Our approach exhibits the potential to extend the permutation algorithm into high-dimensional Hilbert spaces, providing a platform for the feasibility of achieving scalable quantum computation with ultracold polar molecules.
This paper is organized as follows. In section 2 , we briefly review the model of polar molecules in an electric field, the QPA, and the multi-target OCT. Section 3 presents the numerical results of the optimal control simulations for the qutrit-based quantum permutation algorithm. We summarize our conclusions in section 4 .
2. Theory
2.1. Hamiltonian
In the context of a single linear polar molecule subject to an external electric field, the translational motion is characterized by a relatively limited range and closely approximates harmonic behavior. For the sake of simplicity, the contributions stemming from translational kinetic energy and potential energy can be neglected. Under this simplifying assumption, the associated Hamiltonian can be expressed as follows [73]:
$\begin{eqnarray}H=B\cdot {{\boldsymbol{J}}}^{2}-\mu \epsilon \cos \theta ,\end{eqnarray}$
where B represents the rotational constant, J signifies the angular momentum operator, and θ denotes the angle between the molecular permanent dipole moment μ and the applied electric field ε. Under a strong electric field, the polar molecules are induced to undergo oscillations along the electric axis within an angular range. Subsequently, the low rotational states are combined to form pendular states [74]. In figure 1(a), the eigenenergies of a SrO molecule are plotted against the parameter με/B for M = 0, where M stands for the projection of angular momentum J along the electric field's direction.
Figure 1. (a) Eigenenergies (in terms of the rotational constant B) of a SrO molecule in an external electric field for M = 0 as a function of μϵ/B. The eigenstates are labeled by ∣0⟩, ∣1⟩, and ∣2⟩, corresponding to ∣s = 0⟩, ∣s = 1⟩, and ∣s = 2⟩, respectively. (b) Molecular orientation cs in three eigenstates ∣s⟩ with s = 0 ∼ 2 as a function of μϵ/B. |
For a molecular system with d-level, the pendular states are ordered by ascending energy levels and encoded as the qudit ∣q⟩, where q = 0, 1, …, d − 1. The pendular qudit state ∣q⟩ can be conceived as a superposition of field-free rotational states, represented as ${\sum }_{l=0}^{\infty }{c}_{{lq}}| {\boldsymbol{J}}=l,M=0\rangle $, where clq symbolizes the coefficients corresponding to the summation of rotational states [30]. Specifically, the three-level qutrit ∣s⟩ with s = 0, 1, 2 can be encoded using the three eigenstates of the SrO molecule, as depicted in figure 1(a). In figure 1(b), the expectation values of $\cos \theta $ in the three eigenstates, which are defined by ${c}_{s}=\langle s| \cos \theta | s\rangle $, are plotted as a function of μϵ/B. The parameter cs generally characterizes the orientations of polar molecules under the electric field.
2.2. Quantum permutation algorithm
The QPA addresses a black box problem characterized by the mapping of d inputs to d outputs following a permutation operation. A qutrit-based algorithm for the black box paradigm has been devised to ascertain the parity of permutations involving three objects via a single oracle invocation. In contrast, the classical algorithm necessitates two separate calls to the oracle [64]. Let us consider the scenario involving three objects, where six possible permutations can be categorized into two groups: even permutations, entailing cyclic rearrangement of elements, and odd permutations, entailing an interchange between two elements. Through the definition of a function f(x), representing the permutation on the set x ∈ {0, 1, 2}, the underlying task is transformed into the determination of the parity of the bijection f: {0, 1, 2} → {0, 1, 2}. The framework of Cauchy's two-line notation is employed to characterize three conceivable even functions fk, as follows:
$\begin{eqnarray}{f}_{1}=\left(\begin{array}{ccc}0 & 1 & 2\\ 0 & 1 & 2\end{array}\right),\ {f}_{2}=\left(\begin{array}{ccc}0 & 1 & 2\\ 1 & 2 & 0\end{array}\right),{f}_{3}=\left(\begin{array}{ccc}0 & 1 & 2\\ 2 & 0 & 1\end{array}\right).\end{eqnarray}$
Conversely, the remaining three odd functions can be expressed as follows: $\begin{eqnarray}{f}_{4}=\left(\begin{array}{ccc}0 & 1 & 2\\ 2 & 1 & 0\end{array}\right),\ {f}_{5}=\left(\begin{array}{ccc}0 & 1 & 2\\ 1 & 0 & 2\end{array}\right),{f}_{6}=\left(\begin{array}{ccc}0 & 1 & 2\\ 0 & 2 & 1\end{array}\right).\end{eqnarray}$
When considering the representation of the elements within the set {0, 1, 2} via three pendular state vectors denoted as ∣0⟩, ∣1⟩ and ∣2⟩, the circuit for the single qutrit permutation algorithm is visually depicted in figure 2. The main purpose is to determine whether a cyclic permutation is even or odd in a black box setup. Upon analysis of equations (2 ) and (3 ), it is evident that in classical scenarios, the permutation operation within the black box must be executed at least twice, employing distinct inputs. By contrast, the QPA, as introduced by Gedik et al [64], accomplishes the resolution of this problem with a single query to the black box. The action of the operation ${U}_{{f}_{k}}$ results in the application of fk to the state ${U}_{{f}_{k}}(| 0\rangle +| 1\rangle +| 2\rangle )\,=(| {f}_{k}(0)\rangle +| {f}_{k}(1)\rangle +| {f}_{k}(2)\rangle )$. Consequently, the quantum algorithm initiates by generating a superposition of eigenstates through the quantum Fourier transformation acting on an initial state (say, the state ∣0⟩). The quantum Fourier transformation UFT is defined as
$\begin{eqnarray}{U}_{\mathrm{FT}}=\displaystyle \frac{1}{\sqrt{3}}\left(\begin{array}{ccc}\exp [{\rm{i}}2\pi /3] & 1 & \exp [-{\rm{i}}2\pi /3]\\ 1 & 1 & 1\\ \exp [-{\rm{i}}2\pi /3] & 1 & \exp [{\rm{i}}2\pi /3]\end{array}\right).\end{eqnarray}$
Figure 2. Schematic view of the quantum circuit for the parity determining algorithm. The input state ∣0⟩ encounters Fourier transformation UFT before entering the black box, an inverse Fourier transformation ${U}_{\mathrm{FT}}^{\dagger }$ is carried out after the black box and the final state is measured. |
Subsequently, an oracle call ensues, characterized as a black box operation in which the unitary operator ${U}_{{f}_{k}}$ (corresponding to the function fk) is enacted upon the superposition state that materializes from the quantum Fourier transformation. The output state then undergoes an inverse Fourier transformation, thereby re-establishing the state as one of the qutrit's eigenstates, albeit altered by a phase factor. The final configuration of the qutrit corresponds to either ∣0⟩ (if fk is characterized as an even function) or ∣2⟩ (if fk is classified as an odd function). Importantly, the ultimate state for the category of even functions is orthogonal to the final state corresponding to the realm of odd functions, thus rendering them distinguishable through the measurement of a single instance. The quantum circuit of the QPA comprises merely three-dimensional gates, as visualized in figure 2.
2.3. Overview of OCT
OCT constitutes a potent computational framework for determining laser pulse profiles that facilitate the guided evolution of a quantum system toward predefined objectives. In the realm of multi-target optimal control theory (MT-OCT), the primary aim centers around deriving an optimal applied field denoted as E(t), which engenders the transition from an initial state $| {{\rm{\Psi }}}_{i}^{k}(0)\rangle =| {{\rm{\Phi }}}_{i}^{k}\rangle $ associated with the kth target system at t = 0 to a specific target state $| {{\rm{\Psi }}}_{f}^{k}(T)\rangle =| {{\rm{\Phi }}}_{f}^{k}\rangle $ of the same kth target system at a fixed time t = T. This optimization is carried out for all target systems encompassed by k = 1 ∼ z, where z denotes the total count of targets characterized by substantial probability. In scenarios involving multiple targets, the formulation of the constrained function within the context of OCT takes on the following expression [49, 50, 55, 56]:5 ) on the right-hand side quantifies the degree of overlap existing between the laser-driven wave functions ${{\rm{\Psi }}}_{i}^{k}(t)$ and the target states ${{\rm{\Phi }}}_{f}^{k}$. The second term embodies the influence of the laser field E(t), responsible for steering the wave functions of the system toward the specified target states. Introducing the penalty factor α0 serves the purpose of constraining the amplitude of the control field, while the laser envelope function $S(t)={\sin }^{2}(\pi t/T)\cos (\omega t)$ is instrumental in ensuring a gradual and experimentally suitable slow turn-on and turn-off in the laser pulse envelope. Notably, in the context of an N-qubit molecular system, the count of transitions requiring optimization is determined as z = 2N + 1. The introduction of an auxiliary constraint serves to rectify the phase of quantum gates and may be formulated in a subsequent manner [49].5 ). By considering variations in $| {{\rm{\Psi }}}_{i}^{k}(t)\rangle $, $| {{\rm{\Psi }}}_{f}^{k}(t)\rangle $, and E(t) for δΓ = 0, a system of the time-dependent coupled Schrödinger equations is given by [49, 55]
$\begin{eqnarray}\begin{array}{l}{\rm{\Gamma }}({{\rm{\Psi }}}_{i}^{k}(t),{{\rm{\Phi }}}_{f}^{k}(t),{\boldsymbol{E}}(t))\\ \quad =\,\displaystyle \sum _{k=1}^{z}\left\{| \langle {{\rm{\Psi }}}_{i}^{k}(T)| {{\rm{\Phi }}}_{f}^{k}\rangle {| }^{2}-{\alpha }_{0}{\displaystyle \int }_{0}^{T}\displaystyle \frac{{\left[{\boldsymbol{E}}(t)\right]}^{2}}{S(t)}{\rm{d}}t\right.\\ \qquad -\,2{\rm{Re}}\left[\langle {{\rm{\Psi }}}_{i}^{k}(T)| {{\rm{\Phi }}}_{f}^{k}\rangle \times {\displaystyle \int }_{0}^{T}\langle {{\rm{\Psi }}}_{f}^{k}(t)| \displaystyle \frac{\partial }{\partial t}\right.\\ \qquad +\left.\left.\,\displaystyle \frac{{\rm{i}}}{{\hslash }}\left[H-{\boldsymbol{\mu }}\cdot {\boldsymbol{E}}(t)\right]| {{\rm{\Psi }}}_{i}^{k}(t)\rangle {\rm{d}}t\right]\right\}.\end{array}\end{eqnarray}$
Here, H corresponds to the time-independent Hamiltonian characterizing the molecular system. The first term within the expression presented in equation ( $\begin{eqnarray}\begin{array}{rcl}| {{\rm{\Phi }}}_{i}^{z={2}^{N}+1}\rangle & = & \displaystyle \frac{1}{\sqrt{{2}^{N}}}\displaystyle \sum _{k=1}^{{2}^{N}}| {\phi }_{i}^{k}\rangle \to | {\phi }_{f}^{z={2}^{N}+1}\rangle \\ & = & \displaystyle \frac{1}{\sqrt{{2}^{N}}}\displaystyle \sum _{k=1}^{{2}^{N}}| {{\rm{\Phi }}}_{f}^{k}\rangle {{\rm{e}}}^{{\rm{i}}\varphi },\end{array}\end{eqnarray}$
where the symbol eiφ represents the global phase. The task of optimizing the field corresponds to identifying the maximum of the expression in equation ( $\begin{eqnarray}\begin{array}{rcl}{\rm{i}}{\hslash }\displaystyle \frac{\partial }{\partial t}{{\rm{\Psi }}}_{i}^{k}(t) & = & \left[H-{\boldsymbol{\mu }}\cdot {\boldsymbol{E}}(t)\right]{{\rm{\Psi }}}_{i}^{k}(t),{{\rm{\Psi }}}_{i}^{k}(0)={{\rm{\Phi }}}_{i}^{k},\\ {\rm{i}}{\hslash }\displaystyle \frac{\partial }{\partial t}{{\rm{\Psi }}}_{f}^{k}(t) & = & \left[H-{\boldsymbol{\mu }}\cdot {\boldsymbol{E}}(t)\right]{{\rm{\Psi }}}_{f}^{k}(t),{{\rm{\Psi }}}_{f}^{k}(T)={{\rm{\Phi }}}_{f}^{k},\\ k & = & 1,\,\ldots ,\,z,\end{array}\end{eqnarray}$
with the control field E(t) takes the following form $\begin{eqnarray}\begin{array}{rcl}E(t) & = & -\displaystyle \frac{z\mu S(t)}{{\hslash }{\alpha }_{0}}\\ & & \times \,\displaystyle \sum _{k=1}^{z}{\rm{Im}}\left\{\left\langle {{\rm{\Psi }}}_{i}^{k}(t)| {{\rm{\Psi }}}_{f}^{k}(t)\rangle \langle {\psi }_{f}^{k}(t)| \displaystyle \sum _{j=1}^{n}\cos {\theta }_{j}| {{\rm{\Psi }}}_{i}^{k}(t)\right\rangle \right\},\end{array}\end{eqnarray}$
where θj denotes the angle between the body-fixed dipole moment μj and the electric field ε.The set of coupled nonlinear equations outlined above typically presents challenges when attempting direct solutions. Nonetheless, to address this issue, the rapidly converging iteration technique is leveraged for the optimization of the control field. This approach can efficiently facilitate the objective function towards the limit of convergence.
3. Quantum permutation algorithm implemented in a SrO molecule
In this section, we focus on the optimization of the control fields interacting with the single molecule system for a single qutrit QPA. In this study, we choose the SrO molecule as a suitable carrier, for which the permanent dipole moment μ = 8.9 debye and the rotational constant B = 0.33 cm−1. The SrO molecule is assumed to be located in the electric field with the intensity being 1.5 kV/cm. Moreover, the fourth-order Runge–Kutta algorithm is adopted to solve the coupled time-dependent Schrödinger equations in equation (7 ). This numerical approach is chosen to guarantee a high level of precision, and a propagation time step of 0.01 picosecond is established.
High-dimensional quantum systems hold significant potential in terms of providing extensive coded space for information storage and manipulation [75, 76]. This attribute plays an important role in the field of scalable quantum computing [65, 77, 78]. Leveraging its intricate internal configuration and prolonged coherence duration, the polar molecule emerges as a natural entity possessing multi-level computational capacity, akin to a quantum computing unit termed a qudit. In the upcoming discussion, a qudit-based algorithm is investigated, which showcases a non-entangled contextual framework surpassing classical methods in terms of problem-solving efficiency [64]. Remarkably, this algorithm exclusively employs a single qudit throughout its execution, eschewing any reliance on either quantum or classical correlations. Consequently, it serves as an exemplary case for investigating the underpinnings of quantum speed-up beyond quantum correlations. Additionally, the strategic deployment of optimal control methodology facilitates the implementation of quantum gates [49, 79]. The practical realization of such gates involves determining the optimal field configuration to steer an initial set of k states $| {{\rm{\Psi }}}_{i}^{k}(0)\rangle $, towards a designated ensemble of final target states $| {{\rm{\Psi }}}_{i}^{k}(f)\rangle $.
Our focus revolves around the encoding of a three-level qutrit using field-dressed states, followed by the execution of QPA on a SrO molecule, without the dipole-dipole interactions. This study presents the introduction of two distinctive criteria devised to assess the effectivity of the optimized gate pulses [50]. The first criterion centers on the phase-insensitive average transition probability, an essential parameter in evaluating the performance of these optimized gate pulses, which is defined as
$\begin{eqnarray}P=\displaystyle \frac{1}{z}\displaystyle \sum _{k=1}^{z}{\left|\langle {{\rm{\Psi }}}_{i}^{k}(T)| {{\rm{\Phi }}}_{f}^{k}\rangle \right|}^{2},\end{eqnarray}$
while the other is the phase-sensitive normalized fidelity, $\begin{eqnarray}F=\displaystyle \frac{1}{{z}^{2}}{\left|\displaystyle \sum _{k=1}^{z}\langle {{\rm{\Psi }}}_{i}^{k}(T)| {{\rm{\Phi }}}_{f}^{k}\rangle \right|}^{2}.\end{eqnarray}$
The initial stage of the qutrit-based quantum permutation process involves the establishment of a Fourier transform operator, denoted as UFT, designed to transform the single-qutrit fiducial state ∣0⟩ into a state characterized by equal amplitudes in a superposed configuration [64]. Figure 3(a) presents graphical depictions of the average transition probability P and fidelity F corresponding to UFT, driven by the optimized control pulse, with the iteration number as an independent variable. Evidently, substantial transition probabilities and high-fidelity outcomes are attained. Convergence is estimated against a threshold δF = 10−5, where δF quantifies the discrepancy in fidelity between consecutive iterations. Remarkably, after 24 iterations, the fidelity F, and transition probability P reach impressive values of 0.9923 and 0.9992, respectively. An interesting observation is the relatively prompt convergence of the transition probability P compared to the fidelity F. This distinction is attributed to the fact that the normalized fidelity evaluates the phase similarity between the output states and the target states, while the average transition probability does not. Moving to figure 3(b), the plot depicts the convergence of the control pulse as a function of the final iteration, with a duration (T) of 100 ns. Subsequently, figure 4 presents the time evolution of the population distribution across states ∣0⟩, ∣1⟩, and ∣2⟩, under the converged pulse corresponding to the Fourier transform UFT. This time evolution reveals the successful realization of transitions $| 0\rangle \to \tfrac{1}{\sqrt{3}}$ $(\exp [{\rm{i}}2\pi /3]| 0\rangle +| 1\rangle \,+\exp [-{\rm{i}}2\pi /3]| 2\rangle )$, $| 1\rangle \to \tfrac{1}{\sqrt{3}}$ (∣0⟩ + ∣1⟩ + ∣2⟩), and $| 2\rangle \,\to \tfrac{1}{\sqrt{3}}(\exp [-{\rm{i}}2\pi /3]| 0\rangle +| 1\rangle +\exp [{\rm{i}}2\pi /3]| 2\rangle )$ in figures 4(a), (b), and (c), respectively.
Figure 3. Fidelity F and average transition probability P versus iteration number for the optimized pulses of UFT [panel (a)]. [Panel (b)] shows the converged laser pulses versus the last iteration for UFT. |
Figure 4. The time evolution of the population of different input states ∣0⟩ (a), ∣1⟩ (b), and ∣2⟩ (c) driven successively by the converged pulse for UFT gate. |
Establishing the parity of a permutation is equivalent to determining the parity of six distinct permutation functions through the utilization of six unitary operators. In accordance with equations (2 ) and (3 ), three even permutations can be generated through the application of their respective unitary matrices as follows
$\begin{eqnarray}{U}_{1}=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right),\ {U}_{2}=\left(\begin{array}{ccc}0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{array}\right),{U}_{3}=\left(\begin{array}{ccc}0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\end{array}\right),\end{eqnarray}$
and three odd permutations can be similarly realized by following the unitary matrices $\begin{eqnarray}{U}_{4}=\left(\begin{array}{ccc}0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0\end{array}\right),\ {U}_{5}=\left(\begin{array}{ccc}0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1\end{array}\right),{U}_{6}=\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{array}\right).\end{eqnarray}$
Subsequently, the control pulses governing the behavior of the six unitary operators ${U}_{{f}_{k}}$ (where k = 1, 2, ..., 6) are designed via OCT. We list the values of fidelity F and average transition probability P for each permutation operation in table 1. It can be observed that both F and P increase monotonously with the iteration number enhancing and tend to the convergence limits asymptotically at the end of the iteration. As depicted in panels (a)–(c) of figure 5, the fidelities of the convergent gate pulses for ${U}_{{f}_{1}}$, ${U}_{{f}_{2}}$, and ${U}_{{f}_{3}}$ can reach 0.9919, 0.9875, and 0.9878, respectively. The respective desired control fields are attained after 63, 40, and 71 iterations. Additionally, the convergence of the control pulses across the final iteration for operations of ${U}_{{f}_{1}}$, ${U}_{{f}_{2}}$, and ${U}_{{f}_{3}}$ are given in figures 5(d), (e), and (f), respectively. One can see three converged control pulses have similar shapes because they are obtained by optimizing the same initial field ${\rm{E}}(t)={\sin }^{2}(\pi t/T)\cos (\omega t)$ with duration time T = 100 ns. Here, ω is set to the transition frequency between ∣0⟩ and ∣2⟩. Moreover, the maximum of the optimal control pulses for ${U}_{{f}_{1}}$, ${U}_{{f}_{2}}$, and ${U}_{{f}_{3}}$ are about 2.3 kV/cm, 2.0 kV/cm, and 3.1 kV/cm, respectively, higher than that of the initial field. It deserves emphasizing that the OCT we adopted in this study is fixed-time, i.e., the pulse length should be determined before optimizing. Note that the free-time OCT [51] can maximize the objective function and exactly estimate optimal temporal duration of the microwave pulses at the same time. Thus, the fidelity F may be further increased by using the free-time OCT method.
Figure 5. Fidelity F and average transition probability P versus iteration number for the optimized pulses of ${U}_{{f}_{1}}$ (a), ${U}_{{f}_{2}}$ (b), and ${U}_{{f}_{3}}$ (c). The converged microwave pulses of ${U}_{{f}_{1}}$ (d), ${U}_{{f}_{2}}$ (e), and ${U}_{{f}_{3}}$ (f) corresponding to the the last iteration. |
Table 1. Fidelity F and average transition probability P of the optimized pulses Uf. Here, f1, f2, f3 and f4, f5, f6 represent even and odd functions, respectively. |
| Uf | Even | Odd | ||||
|---|---|---|---|---|---|---|
| f1 | f2 | f3 | f4 | f5 | f6 | |
| F | 0.9919 | 0.9875 | 0.9878 | 0.9912 | 0.9917 | 0.9896 |
| P | 0.9995 | 0.9995 | 0.9980 | 0.9999 | 0.9995 | 0.9995 |
Figure 6 presents the time evolution of the population distribution across states ∣0⟩, ∣1⟩, and ∣2⟩, each successively driven by the converged pulse corresponding to the ${U}_{{f}_{1}}$ [panels (a1)–(a3)], ${U}_{{f}_{2}}$ [panels (b1)–(b3)], and ${U}_{{f}_{3}}$ [panels (c1)–(c3)] operations, respectively. The operator ${U}_{{f}_{1}}$ acting on the qutrit states ∣0⟩, ∣1⟩, ∣2⟩ results in the same states in the qutrit space. Upon undergoing the ${U}_{{f}_{2}}$ operation, the different input states, namely, ∣0⟩, ∣1⟩, and ∣2⟩, are transformed into ∣2⟩, ∣0⟩, and ∣1⟩, respectively. Similarly, when the ${U}_{{f}_{3}}$ pulse is applied, the transitions ∣0⟩ → ∣1⟩, ∣1⟩ → ∣2⟩, and ∣2⟩ → ∣0⟩ are achieved, accordingly. Therefore, the even permutations of {0, 1, 2} → {0, 1, 2}, {0, 1, 2} → {1, 2, 0}, and {0, 1, 2} → {2, 0, 1} are implemented under the influence of pulses ${U}_{{f}_{1}}$, ${U}_{{f}_{2}}$, and ${U}_{{f}_{3}}$, respectively.
Figure 6. The population evolution of states ∣0⟩ → ∣0⟩ (a1), ∣1⟩ → ∣1⟩ (a2), and ∣2⟩ → ∣2⟩ (a3) for ${U}_{{f}_{1}}$ gate. The population evolution of states ∣0⟩ → ∣2⟩ (b1), ∣1⟩ → ∣0⟩ (b2), and ∣2⟩ → ∣1⟩ (b3) for ${U}_{{f}_{2}}$ gate. The population evolution of states ∣0⟩ → ∣1⟩ (c1), ∣1⟩ → ∣2⟩ (c2), and ∣2⟩ → ∣0⟩ (c3) for ${U}_{{f}_{3}}$ gate. |
In a similar way, the convergent behaviors of the optimized control pulses for the three odd permutation operations are described in panels (a)–(c) of figure 7. The optimized control pulses for ${U}_{{f}_{4}}$, ${U}_{{f}_{5}}$, and ${U}_{{f}_{6}}$ operations can meet the convergence criterion after 9, 19, and 44 iterations, whose fidelities can reach 0.9912, 0.9917 and 0.9896, respectively. Additionally, the convergence of the control pulses across the final iteration for ${U}_{{f}_{4}}$, ${U}_{{f}_{5}}$, and ${U}_{{f}_{6}}$ are graphically illustrated in panels (d)–(f) of figure 7. Moreover, figure 8 displays the population distribution of different input states, which is driven by the obtained convergent pulses for the odd permutations mentioned above. Correspondingly, for the ${U}_{{f}_{4}}$, ${U}_{{f}_{5}}$, and ${U}_{{f}_{6}}$ operations, we exhibit the population dynamics for the transformation of different input states under the converged pulses. Specifically, the ${U}_{{f}_{4}}$ pulse effectively induces the transitions ∣0⟩ → ∣2⟩ (a1), ∣2⟩ → ∣0⟩ (a3), while preserving ∣1⟩ unchanged (a2). As shown in panels (b1)–(b3), after the ${U}_{{f}_{5}}$ operation, the initial set of states, namely, ∣0⟩, ∣1⟩, ∣2⟩ evolve to ∣1⟩, ∣0⟩, ∣2⟩, respectively. Analogously, transition achievements corresponding to ${U}_{{f}_{6}}$ pulse comprise ∣0⟩ → ∣0⟩, ∣1⟩ → ∣2⟩, and ∣2⟩ → ∣1⟩ are shown in panels (c1)–(c3). As a result, the odd permutation of {0, 1, 2} → {2, 1, 0}, {0, 1, 2} → {1, 0, 2}, and {0, 1, 2} → {0, 2, 1} are achieved.
Figure 7. Fidelity F and average transition probability P versus iteration number for the optimized pulses of ${U}_{{f}_{4}}$ (a), ${U}_{{f}_{5}}$ (b), and ${U}_{{f}_{6}}$ (c). The converged laser pulses of ${U}_{{f}_{4}}$ (d), ${U}_{{f}_{5}}$ (e), and ${U}_{{f}_{6}}$ (f) corresponding to the the last iteration. |
Figure 8. The time evolution of the population of states ∣0⟩ (a1), ∣1⟩ (a2), and ∣2⟩ (a3) driven successively by the converged pulses for ${U}_{{f}_{4}}$, respectively. The time evolution of the population of states ∣0⟩ (b1), ∣1⟩ (b2), and ∣2⟩ (b3) driven successively by the converged pulses for ${U}_{{f}_{5}}$, respectively. The time evolution of the population of states ∣0⟩ (c1), ∣1⟩ (c2), and ∣2⟩ (c3) driven successively by the converged pulses ${U}_{{f}_{6}}$, respectively. |
Afterwards, we design the control pulse for the inverse Fourier transform ${U}_{\mathrm{FT}}^{\dagger }$ through OCT. Figure 9(a) demonstrates that the fidelity F and the average transition probability P for ${U}_{\mathrm{FT}}^{\dagger }$ gate can reach 0.9918 and 0.9999 through 21 iterations, respectively. The converged control pulse is graphed as the function of the last iteration in figure 9(b), with a duration of 100 ns. Panels (a)–(c) of figure 10 exhibit the population for the basis states ∣0⟩, ∣1⟩, and ∣2⟩, which are successively driven by the converged pulse for ${U}_{\mathrm{FT}}^{\dagger }$.
Figure 9. Fidelity F and average transition probability P versus iteration number for the optimized pulse of ${U}_{\mathrm{FT}}^{\dagger }$ [panel (a)], the corresponding converged gate pulse versus the last iteration [panel (b)]. |
Figure 10. The population distributions of the evolution of the input states after applying the convergent gate pulse for ${U}_{\mathrm{FT}}^{\dagger }$. Panels (a), (b), and (c) show the distributions of the final population after evolutions when the input states are the basis states ∣0⟩, ∣1⟩, and ∣2⟩, respectively. |
To simulate the QPA, a pulse sequence composed of the operations UFT, ${U}_{{f}_{k}}$, and ${U}_{\mathrm{FT}}^{\dagger }$ is designed with the optimal control method. Specifically, the performance of optimized gate pulses in executing the QPA is illustrated by demonstrating the even permutation scenario ${U}_{{f}_{2}}$ and the odd permutation scenario ${U}_{{f}_{6}}$, which correspond to the output states ∣0⟩ and ∣2⟩, respectively. The quantum permutation dynamics for the desired output state ∣0⟩ or ∣2⟩ is summarized in table 2. In the case of even scenario, UFT, ${U}_{{f}_{2}}$, and ${U}_{\mathrm{FT}}^{\dagger }$ are performed on a SrO molecule. The evolution of the states is as follows: ∣0⟩ → UFT $\tfrac{1}{\sqrt{3}}(\exp [{\rm{i}}2\pi /3]| 0\rangle +| 1\rangle +\exp [-{\rm{i}}2\pi /3]| 2\rangle )$ $\to {U}_{{f}_{2}}$ $\tfrac{1}{\sqrt{3}}(| 0\rangle +\exp [-{\rm{i}}2\pi /3]| 1\rangle +\exp [{\rm{i}}2\pi /3]| 2\rangle )$ $\to \,{U}_{\mathrm{FT}}^{\dagger }$ $\exp [-{\rm{i}}2\pi /3]| 0\rangle $. This sequence of optimized control fields operating on the SrO molecule achieves the desired outcome of the target state ∣0⟩, albeit accompanied by a phase factor. It can be seen from figure 11 that when the UFT pulse is applied, the population of the initial state ∣0⟩ transforms to 0.3169 after a duration of 100 ns. Subsequently, after a single iteration of the QPA for the ${U}_{{f}_{2}}$ pulse (as depicted in figure 11(c)), the population evolves to 0.3211. Furthermore, the application of the pulse sequence on the molecular qutrit leads to the population of the desired state ∣0⟩ reaching 0.9992 upon completion of the evolution, as illustrated in figure 11. To enhance clarity, the visualization of the real part of the density matrix's bar representation for the final state following the application of converged gate pulses for UFT, ${U}_{{f}_{2}}$, and ${U}_{\mathrm{FT}}^{\dagger }$ is presented in the right panels of figure 11, which is in close agreement with our theoretical expectations. Moreover, the odd scenario is also simulated by applying the pulse sequence ${U}_{\mathrm{FT}}{U}_{{f}_{6}}{U}_{\mathrm{FT}}^{\dagger }$ to the molecular qutrit, as shown in figure 12. The population of the desired state ∣2⟩ can reach 0.9976 at the end of the evolution, which is approximate to the theoretical value of 1 for the one-qutrit QPA.
Figure 11. The time evolution of the population (blue solid curves) of state ∣0⟩ driven successively by the converged pulses for UFT (a), ${U}_{{f}_{2}}$ (c), and ${U}_{\mathrm{FT}}^{\dagger }$ (e). The red dashed lines in subplots indicate the expected population values after evolutions. The real part of the density matrix of the final states obtained by applying the gate pluses for UFT (b), ${U}_{{f}_{2}}$ (d), and ${U}_{\mathrm{FT}}^{\dagger }$ (f). |
Figure 12. The time evolution of the population (blue solid curves) of state ∣2⟩ driven successively by the converged pulses for UFT (a), ${U}_{{f}_{6}}$ (c), and ${U}_{\mathrm{FT}}^{\dagger }$ (e). The red dashed lines in subplots indicate the expected population values after evolutions. The real part of the density matrix of the final states obtained by applying the gate pluses for UFT (b), ${U}_{{f}_{6}}$ (d), and ${U}_{\mathrm{FT}}^{\dagger }$ (f). |
Table 2. The quantum permutation dynamics for the desired output state ∣0⟩ or ∣2⟩, respectively. |
| Functions | Output state | Population | ||||
|---|---|---|---|---|---|---|
| Initia value | UFT | Uf | ${U}_{\mathrm{FT}}^{\dagger }$ | |||
| even | f1 | ∣0⟩ | 0.3097 | 0.9981 | ||
| f2 | $\exp [-{\rm{i}}2\pi /3]| 0\rangle $ | 1 | 0.3169 | 0.3211 | 0.9992 | |
| f3 | $\exp [{\rm{i}}2\pi /3]| 0\rangle $ | 0.3736 | 0.9943 | |||
| | ||||||
| odd | f4 | ∣2⟩ | 0.3150 | 0.9990 | ||
| f5 | $\exp [{\rm{i}}2\pi /3]| 2\rangle $ | 0 | 0.3498 | 0.3461 | 0.9979 | |
| f6 | $\exp [-{\rm{i}}2\pi /3]| 2\rangle $ | 0.3396 | 0.9976 | |||
4. Conclusions
In summary, we focused on optimizing the control fields to achieve high-dimensional quantum gate operations within a single molecule system. Multiple pendular states of the SrO molecule were proposed as potential candidates for qutrits. Subsequently, we investigated how to implement a one-qutrit QPA using SrO molecules under an external electric field. Through successive iterations aided by multi-target OCT, we derived a sequence of high-fidelity gate pulses. Our approach involved designing optimal control pulses for the one-qutrit quantum Fourier transform, one-qutrit oracle operation, and one-qutrit inverse quantum Fourier transform, enabling quantum permutation simulation in just three steps. Moreover, we examined two permutation cases for the desired states ∣0⟩ and ∣2⟩ using the optimized pulse sequence, validating the effective achievement in finding the output states. Overall, we theoretically demonstrated the deterministic implementation of the one-qutrit QPA using SrO molecules in an external electric field. This approach decisively determines whether a given permutation of three objects, chosen from a set of six possible functions, is an even or odd permutation with a single query to the black box. Additionally, our findings indicate that employing a single qutrit can provide a two-to-one speedup in determining the parity of cyclic permutations. Our scheme could potentially be extended to multi-level molecular systems. It is worth mentioning that based on an optical tweezer array, experimental demonstrations of iSWAP two-qubit gate operations between individual polar CaF molecules has been achieved [45, 46], with molecular bits encoded in the relevant rotational and hyperfine states in the ground electronic state of CaF. The SrO molecule is also a diatomic molecule, and with the continuous breakthroughs in molecular cooling and manipulation techniques, its application to QPA may have certain experimental feasibility. We believe that our study might pave the way for understanding and implementing high-dimensional quantum algorithms utilizing polar molecules in pendular states.