Extracting the spin excitation spectrum of a fermionic system using a quantum processor

TL;DR

This work develops a unitary quench spectroscopy approach to extract spin-excitation spectra in the 1D Fermi-Hubbard model using a digital quantum computer, connecting dynamical response functions to experimentally accessible observables. By perturbing with a local quench $Q_j$ and monitoring the time evolution of $S^x_i$, it recovers the retarded spin Green's function $G^x_{jk}(t)$ and, via space-time Fourier transform, the dynamical spin structure factor, with a double-quench variant offering exact results under symmetry constraints. To make the method practical on NISQ devices, the authors introduce a Dense Givens Approximate (DGA) initial-state preparation that yields a high-fidelity free-fermion starting point with shallow circuits, and they employ first-order Trotterization to simulate dynamics efficiently; they also optimize qubit ordering and apply post-selection on fixed particle number to mitigate noise. Hardware experiments on IBM devices up to $L=15$ demonstrate the recovery of key features of the two-spinon continuum and show resilience to noise when combined with Pauli twirling, suggesting a viable path toward scalable spectroscopic studies of fermionic systems on near-term quantum hardware, without heavy error mitigation. The work thus provides a practical framework for quantum simulation of dynamical properties in fermionic models, with implications for understanding strongly correlated phases and spin-charge dynamics.

Abstract

Understanding low-energy excitations in fermionic systems is crucial for their characterization. They determine the response of the system to external weak perturbations, its dynamical correlation functions, and provide mechanisms for the emergence of exotic phases of matter. In this work, we study the spin excitation spectra of the 1D Fermi-Hubbard model using a digital quantum processor. Introducing a protocol that is naturally suited for simulation on quantum computers, we recover the retarded spin Green's function from the time evolution of simple observables after a specific quantum quench. We exploit the robustness of the protocol to perturbations of the initial state to minimize the quantum resources required for the initial state preparation, and to allocate the majority of them to a Trotterized time-dynamics simulation. This, combined with the intrinsic resilience to noise of the protocol, allows us to accurately reconstruct the spin excitation spectrum for large instances of the 1D Fermi-Hubbard model without making use of expensive error mitigation techniques, using up to 30 qubits of an IBM Heron r2 device.

Figures (16)

• Figure 1: Density plot of the coefficient of $G^x_{jk}(t)$ in the double quench spectroscopy protocol of Eq. \ref{['eq:GF_double_quench_spin']}, $\sin\theta+\sin\phi$. The black contours denote the region of the $\theta-\phi$ plane in which Eq. \ref{['eq:cos_condition']} is satisfied. The yellow dot, corresponding to $(\theta, \phi) \approx (1.07\pi, \pi/3)$, represent an almost optimal pair of angles for the protocol.
• Figure 2: TDVP simulation of the quench protocol for the 1D Fermi-Hubbbard model with $L=51$ sites. (a),(c) Dynamics of the observable $\braket{\hat{S}^x_i(t)}$ for $U=3$ and filling $n_e=2/3$ for total running times $T=10$ and $T=3$, respectively. (b), (d) Normalised absolute value of the corresponding quench spectral functions in the momentum-frequency domain after applying a two-dimensional Fourier transform to the space-time domain data. The maximum bond dimension for the TDVP simulation is $\chi = 128$.
• Figure 3: Simulation of the quench spectroscopy protocol for a 1D Fermi-Hubbard model initialized in the free-fermion ground state. (a) Energy per site for different initial states in a system with $L=15$ sites, setting the ground state energy of the Fermi-Hubbard Hamiltonian $H_{\rm FH}$ to zero. The green curve represents the energy of the state obtained by applying the quench to the Fermi-Hubbard ground state, $\hat{Q}_{\lfloor L/2 \rfloor}\ket{\Psi}$, the blue curve shows the expectation value of the Fermi-Hubbard Hamiltonian $H_{\rm FH}$ on the free-fermion ground state $\ket{\psi_{\rm FF}}$, and the orange curve corresponds to the energy of the state obtained by applying the quench to the free-fermion ground state, $\hat{Q}_{\lfloor L/2 \rfloor}\ket{\psi_{\rm FF}}$, all as functions of the on-site interaction $U$. Also shown is the energy range (shaded region) of the subspace spanned by $\ket{\Psi}$ and $\hat{Q}_{j}\ket{\Psi}$ with $j = 0,...,L-1$, i.e., the low-energy subspace of $H_{\rm FH}$ containing at most a single spin excitation. (b) Same as for (a), but for a system with $L=51$ sites. (c) Structural similarity index measure (SSIM) between the space-time signals obtained starting from the Fermi-Hubbard and free-fermion ground state with $L=51$ (blue curve) and their Fourier transforms (yellow curve) for a total evolution time of $T=10$. Values of the SSIM close to 1 indicate that two images are similar, with ${\rm SSIM} = 1$ for identical signals.
• Figure 4: Trotterized time evolution of the 1D Fermi-Hubbard model with $L=15$, $U/J=3$ and $n_e=2/3$, using the ground state as the initial state. The latter has been obtained using the density-matrix renormalization group (DMRG) algorithm. The exact simulation has been performed using the TDVP algorithm, while the Trotterized simulations have been obtained with time-evolving block-decimation (TEBD) and expressing the Hamiltonian as a product of the interacting part and the hopping term on even and odd sites. DMRG and the time evolution algoritihms were executed with maximum bond dimension $\chi=64$. Top: (a) exact and Trotterized time dynamics with (b) three and (c) five fixed Trotter steps using the first-order Trotter formula. Bottom: (e), (f), (g) absolute value of the QSF obtained by Fourier transforming and normalizing the time-dependent signals in panels (a), (b), (c), respectively. In the last column, (d) SSIM and (h) root mean square error (RMSE) between the QSFs obtained from exact and trotterized time-evolution simulations are shown for increasing numbers of Trotter steps. Both first- and second- order Trotter are simulated.
• Figure 5: Complete circuit for a fermionic system of size $L=3$. Note the fermion-to-qubit ordering is changed after the initialisation with the Givens rotations. We first encode all the spin up modes in the first $L$ qubits and the spin down modes in the last $L$ qubits. We then move the qubits around with FSWAP gates (represented as swap gates in red) to encode every pair of modes $(i,\uparrow)$ and $(i, \downarrow)$ next to each other. After this reordering, the quench operator $\hat{Q}_j$ is applied to the two central qubits, corresponding to the spin up and down modes of fermion at $i=\lfloor L/2\rfloor$ site. The time-evolution operator is implemented using a first-order Trotter formula, consisting of even and odd hopping, and on-site interaction terms. Note that due to the (up, down)(up, down) JW ordering, implementing each hopping layers requires 3 layers of FSWAP gates per Trotter step. In the figure, only one Trotter step is shown. Finally, the unitary $B_{XY}$ diagonalizing the spin operator $\hat{S}^x_i$ is applied to each pair of qubits, which are then measured in the computational basis.