Accelerating two-dimensional electronic spectroscopy simulations with a probe qubit protocol

TL;DR

This work addresses the computational bottlenecks in simulating two-dimensional electronic spectroscopy (2DES) on classical hardware. It introduces the probe qubit protocol (PQP), which attaches a single ancilla qubit to the system during the detection phase to extract spectral information at selected frequencies, avoiding full Fourier reconstruction and enabling Heisenberg-limited frequency resolution. Compared with the standard quantum simulation protocol (SQSP), PQP reduces the measurement burden to a single-qubit readout per run and scales more favorably with system size, yielding substantial speedups and memory savings, particularly when probing a few detection frequency lines. Numerical studies on small and medium systems, including the Fenna–Matthews–Olson (FMO) complex, show PQP can reproduce key spectral features with far fewer measurements, while outlining practical considerations for near-term hardware, such as connectivity, noise, and potential extensions like additional probes or matrix completion to reconstruct full 2D spectra.

Abstract

Two-dimensional electronic spectroscopy (2DES) is a powerful tool for exploring quantum effects in energy transport within photosynthetic systems and investigating novel material properties. However, simulating the dynamics of these experiments poses significant challenges for classical computers due to the large system sizes, long timescales and numerous experiment repetitions involved. This paper introduces the probe qubit protocol (PQP)-for quantum simulation of 2DES on quantum devices-addressing these challenges. The PQP offers several enhancements over standard methods, notably reducing computational resources, by requiring only a single-qubit measurement per circuit run and achieving Heisenberg scaling in detection frequency resolution, without the need to apply expensive controlled evolution operators in the quantum circuit. The implementation of the PQP protocol requires only one additional ancilla qubit, the probe qubit, with one-to-all connectivity and two-qubit interactions between each system and probe qubits. We evaluate the computational resources necessary for this protocol in detail, demonstrating its function as a dynamic frequency-filtering method through numerical simulations. We find that simulations of the PQP on classical and quantum computers enable a reduction on the number of measurements, i.e. simulation runtime, and memory savings of several orders of magnitude relatively to standard quantum simulation protocols of 2DES. The paper discusses the applicability of the PQP on near-term quantum devices and highlights potential applications where this spectroscopy simulation protocol could provide significant speedups over standard approaches such as the quantum simulation of 2DES applied to the Fenna-Matthews-Olson (FMO) complex in green sulphur bacteria.

Figures (8)

• Figure 1: Schematic of a phase-cycled 2D spectra $\langle \hat{F}(\omega _1, t_2, \omega _3)\rangle$ for a specific $t_2$ using PQP (a) and the SQSP (b) for a molecular system with energy diagram (c). In the PQP simulation (a), we attach a probe qubit and repeat the experiment with different probe qubit energy gaps, $\omega_3=\omega_{pr}$ and $\omega_3=\omega'_{pr}$, in order to resolve the relevant peak amplitudes located at and close to the chosen $\omega_3$. This protocol functions as a dynamical frequency-filtering method. Orange (purple) circles represent resolved (unresolved) peak amplitudes. The width of blue lines corresponds to the frequency resolution $\Delta \omega_3$ of the peak amplitude estimation limited by the finite interaction time interval $t_3$ (more details in Sec. \ref{['sec: applicability']}). (b) The SQSP resolves only the full 2D spectra with an uncertainty dictated by the number of data points taken over time interval $t_3$ (more details in Sec. \ref{['sec: standard']}), without enabling the extraction of peak amplitudes in single, arbitrary detection frequencies $\omega_3$. The estimation of peak amplitudes over $\omega_1$ of both protocols follows the same procedure (with uncertainty limited by the number of data points taken over $t_1$). (c) Energy diagram of the molecular system considered in this work. Two molecules with excited states $\ket{E_1}$ and $\ket{E_2}$, respectively, are coupled via a electronic dipolar coupling leading to a four-level excitonic system as shown in the middle of the figure, with energy states $\ket{g}, \ket{e_1}, \ket{e_2}$ and $\ket{f}$, ordered respectively by ascending energies.
• Figure 2: Overview of the implementation of the standard quantum simulation protocol of 2DES experiments. Qubits are initialized in the ground-state and evolved with two different types of evolution operators, namely $\hat{U}_{I}(t_p)$ encoding the system-pulse interaction and $\hat{U}_S(t_j)$ encoding the free system evolution, as defined by Eq. \ref{['eq_Ui_trotter']} and Eq. \ref{['eq_Us_trotter']}, respectively. Here, $t_p=D_p \Delta t_p$ and $t_j = D_j \Delta t_j$, with $j \in \{1,2,3\}$, and $D_p$ and $D_j$ denote the number of Trotter layers used for the respective evolution block. The Trotter time-steps $\Delta t_p$ and $\Delta t_j$ are also customizable for each evolution block. Lastly, the qubits are measured in the computational basis.
• Figure 3: Overview of the probe qubit protocol to simulate 2D spectroscopy experiments on a quantum computer. (a) Quantum circuit with the probe qubit interacting with the system during the time interval $t_3$. The operations in the orange box do not need to be applied in this protocol, hence they can be discarded. The Pauli observable $\hat{O}_{pr}$ is measured after $t_3$. (b) The system-probe interaction consists of a weak energy transfer process. Since the probe qubit is initialized in the ground-state $\ket{0}_{pr}$, one can control the back-reaction on the system from the probe to the system so that it remains negligible over the time interval $t_3$ unless $t_3$ is too long (see Appendix A). (c) Trotter layer used to implement the evolution over $t_3$ with system-probe interactions. The probe weakly couples to each molecule via an electronic dipole interaction during $t_3 = D'_3 \Delta t_{3}$, where $D'_3$ is the number of Trotter layers over $t_3$ in this scheme and $\Delta t_{3}$ is the Trotter time-step. $\ket{\Psi}_{S}$ denotes the system's state after the third pulse interaction. The rotation applied to the probe qubit is defined as $\hat{R}_{Z}(\theta)=e^{i\theta\hat{Z}/2}$.
• Figure 4: Absolute 2D spectra obtained from rephasing 2DES experiment simulations using the proposed PQP in this work (a,b) and the SQSP (c,d). The system energy eigenvalues are $E_1 \approx 12141$$cm^{-1}$ and $E_2 \approx 11859$$cm^{-1}$. We normalize the magnitudes to the interval $[0,1]$ independently for the PQP (a,b) and SQSP (c,d). Here we chose $t_1 = 500$$fs$ ($\Delta \omega_1 \approx 67$$cm^{-1}$) and $t_2 = 600$$fs$ for both PQP and SQSP simulations. We chose different $t_3$ for each protocol, namely, $t_3^{(SQSP)} = t_1$ for the SQSP simulation and $t_3^{(PQP)} = 1.45 t_1$ for the PQP simulation to achieve similar detection frequency resolutions ($\Delta \omega_3 \approx \Delta \omega_1$).
• Figure 5: Absolute 2D spectra obtained from rephasing 2DES experiment simulations with shot noise using the proposed PQP in this work (a,b,c) and the SQSP (c,d,e), for $\omega_3 \approx \omega_{pr} = E_1$. For each obtained expectation value in SQSP and PQP simulations shown in Fig. \ref{['fig:results_main_text']}, we added a random number sampled from a Gaussian probability distribution with mean $0$ and standard deviation $\varepsilon = 1\times 10^{-5}, 5 \times 10^{-5}$ and $1 \times 10^{-4}$ to mimic the finite number of shots at estimating the expectation values in a realistic quantum simulation scenario. The simulation parameters are the same as in Figure. \ref{['fig:results_main_text']}.