Implementation of the Quantum Fourier Transform on a molecular qudit with full refocusing and state tomography

TL;DR

This work demonstrates a high-fidelity Quantum Fourier Transform on a molecular spin qudit (173Yb(trensal)) by embedding a full-refocusing protocol to combat inhomogeneous broadening inherent to ensemble experiments. Complementary simulations pinpoint hyperfine-coupling strain as the main source of dephasing and validate refocusing as essential for preserving coherences across a multi-pulse QFT sequence. Tomography of both initial and post-QFT states confirms robust quantum-control capabilities, with refocused sequences delivering fidelities approaching 0.98 for several basis and superposition states. Overall, the results establish the feasibility of executing complex quantum algorithms on molecular spin qudits and highlight refocusing as a practical tool for scaling qudit-based quantum information processing.

Abstract

Molecular spin qudits based on lanthanide complexes offer a promising platform for quantum technologies, combining chemical tunability with multi-level encoding. However, experimental demonstrations of their envisaged capabilities remain scarce, posing the difficulty of achieving precise control over coherences between qudit states in long pulse sequences. Here, we implement in 173Yb(trensal) qudit the Quantum Fourier Transform (QFT), a core component of numerous quantum algorithms, storing quantum information in the phases of coherences. QFT provides an ideal benchmark for coherence manipulation and an unprecedented challenge for molecular spin qudits. We address this challenge by embedding a full-refocusing protocol for spin qudits in our algorithm, mitigating inhomogeneous broadening and enabling a high-fidelity recovery of the state. Complete state tomography demostrates the performance of the algorithm, while simulations provide insight into the physical mechanisms behind inhomogeneous broadening. This work shows the feasibility of quantum logic on molecular spin qudits and highlights their potential.

Figures (9)

• Figure 1: Characterisation of the qudit.a Molecular structure of $^{173}$Yb(trensal). b Energy level diagram of $^{173}$Yb(trensal) in a static magnetic field $\textbf{B}_0$ perpendicular to the molecular C$_3$ axis. The states of the subspace targeted in this work, highlighted in pink, are labelled $\ket{0}$, $\ket{1}$ and $\ket{2}$ from lowest to highest energy, with transition frequencies $f_{01}$ and $f_{12}$. c Phase memory time $T_2$ of both transitions, measured at $B_0=0.2$ T and $T=1.4$ K (scatters) and fit to a Gaussian-shaped decay (solid line). d Coherent Rabi manipulation of both transitions under these same experimental conditions. Error bars on all data are of the order of scatter dimension; they were obtained by applying to the noise the same analysis we applied for the data (see Methods) and taking the root mean square.
• Figure 2: Coherent manipulation and tomography of the qutrit state.a. Density matrix $\rho_{\rm ideal} = \ket{\psi}\bra{\psi}$ for the ideal superposition state $\ket{\psi} = \tfrac{1}{\sqrt{2}} (\ket{0} - i\ket{1})$. b. Density matrix $\rho_{\rm exp}$ for the final state after the experimental manipulation of the initial state of the qutrit, $\ket{0}$, with a $\theta=\pi/2$ pulse with phase $\phi=0$ and carrier frequency $f_{01}$. This pulse ideally generates the superposition state of panel a. The elements of $\rho_{\rm exp}$ were measured by performing a complete tomography experiment of the final state. c. Measured echo traces yielding the tomography of panel b. We estimated the error in $\mathcal{F}$ by applying the same noise analysis of Fig. \ref{['fig1:characterization']} to these traces.
• Figure 3: Pulse sequence for the QFT. a Decomposition of the 3-level unitary transformation $U_d$ performing the qutrit QFT into 9 planar rotations. b Detail of a qutrit refocusing block. After a pulse ($\theta$, $\phi$) of the QFT sequence, a state $\ket{\psi} = a \ket{0} + b \ket{1} + c \ket{2}$ is encoded into the three-level subspace with amplitudes $a$ (green), $b$ (pink), and $c$ (orange). Due to inhomogeneous broadening, the free evolution of each spin in the ensemble is slightly different, and each amplitude of each spin collects a different phase during the time $\tau$ between pulses. This phase depends on the basis state ($\ket{0}$, $\ket{1}$, $\ket{2}$) in which the amplitude is encoded ($\phi_0$, $\phi_1$ and $\phi_2$, respectively). A $\pi$ pulse swaps the amplitudes between the basis states. By applying a sequence of five $\pi$ pulses, equally spaced in time, amplitudes are moved around all the basis states, thus collecting the same global phase $2 \phi_0 + 2 \phi_1 + 2 \phi_2$. This final refocused state includes a swap between two states. c Pulse sequence implementing $U_d$ with embedded refocusing blocks (see panel b) for any pulse with $\theta \neq \pi$. Any such pulse must be sent to a refocused state after a refocusing block. Conversely, $\pi$-pulses can be integrated in them to reduce the number of pulses. Only relative phases between some of the $\pi$-pulses are set by the QFT operation, we can freely chose $\phi_a$, $\phi_b$, $\phi_c$, $\phi_d$, $\phi_e$ and $\phi_f$. All results here were obtained with $\phi_a = \phi_b = \phi_c = \phi_d = \phi_e = 0$ and $\phi_f = -\pi/6$. Detection was performed when the density matrix was refocused, before the last $\pi$ pulse. Both sequences A and B implement the QFT, each one allowing for the detection of different density matrix elements (see Methods).
• Figure 4: Tomography of the QFT acting on the basis states $\ket{0}$ and $\ket{2}$ of the qutrit, and on the superposition state of Fig. \ref{['fig2new']}. We compared the density matrices $\rho_{\rm exp}$ obtained from tomography experiments after implementing the QFT sequences (both refocused and non-refocused) with the density matrix $\rho_{\rm ideal}$ obtained by applying $U_d$ to the density matrix of the initial state ($\rho_0$) extracted from tomography experiments (see SI for the tomography of $\rho_0$). For each $\rho_{\rm exp}$ we calculated the fidelity as $\mathcal{F} = \rm{Tr} \left( \sqrt{ \sqrt{\rho_{\rm ideal}} \rho_{\rm exp} \sqrt{\rho_{\rm ideal}}} \right)$, which measures only the performance of pulse sequence implementing the QFT. An error in the order of 0.01 was obtained for all matrix elements from the standard deviation of repeated measurements. The error on the fidelity was obtained by propagating the single element errors.
• Figure S1: Relaxation of the two addressable qutrit transitions.Left: relaxation of the $\ket{0} \leftrightarrow \ket{1}$ transition, with frequency $f_{01} =$ 333.0 MHz. Right: relaxation of the $\ket{1} \leftrightarrow \ket{2}$ transition, with frequency $f_{12} =$ 359.9 MHz. In both cases, relaxation is best fit as a bi-exponential decay, with characteristic times $T_1^{(1)}$ (fast) and $T_1^{(2)}$ (slow), which is not unusual for a multilevel system with different possible relaxation pathways. The relative weight of both contributions is indicated in $\%$.