Superconducting pairing correlations on a trapped-ion quantum computer

TL;DR

This work demonstrates that a trapped-ion quantum computer can prepare and measure states with superconducting pairing correlations in Hubbard-model regimes, bridging theory and experiment for nontrivial off-diagonal observables. By employing an Octagon fermion-to-qubit encoding, leakage-heralding, and stabiliser-based error mitigation, the authors access η-, d-, and s-wave pairing in half-filled, doped, and bilayer Fermi-Hubbard systems, respectively. The study combines perturbative effective-Hamiltonian insights, tailored state-preparation circuits, and specialized measurement schemes to extract pairing correlations, providing a scalable blueprint for exploring superconductivity with quantum hardware. While ambitious, the results also highlight practical limits from entropy and noise for adiabatic-state preparation, outlining pathways—advanced error correction, refined thermometry, and larger, more connected devices—for future digital quantum simulations of correlated electron phenomena.

Abstract

The Fermi-Hubbard model is the starting point for the simulation of many strongly correlated materials, including high-temperature superconductors, whose modelling is a key motivation for the construction of quantum simulation and computing devices. However, the detection of superconducting pairing correlations has so far remained out of reach, both because of their off-diagonal character - which makes them inaccessible to local density measurements - and because of the difficulty of preparing superconducting states. Here, we report measurement of significant pairing correlations in three different regimes of Fermi-Hubbard models simulated on Quantinuum's Helios trapped-ion quantum computer. Specifically, we measure non-equilibrium pairing induced by an electromagnetic field in the half-filled square lattice model, d-wave pairing in an approximate ground state of the checkerboard Hubbard model at $1/6$-doping, and s-wave pairing in a bilayer model relevant to nickelate superconductors. These results show that a quantum computer can reliably create and probe physically relevant states with superconducting pairing correlations, opening a path to the exploration of superconductivity with quantum computers.

Figures (32)

• Figure 1: The Hubbard model at half-filling and light-induced $\eta$-pairing. (a) To benchmark the quantum computer, a region $\mathcal{A}$ of a periodic $N=6 \times 6$ lattice is densely packed with 36 non-interacting fermions. The system is evolved and the imbalance $I_\mathcal{A}=n_\mathcal{A}-n_{\overline{\mathcal{A}}}$ measured. (b) To prepare low-energy states at $U=8$, an approximate ground state of the $6 \times 6$ Heisenberg model is prepared using a classically optimised circuit with $D_\mathrm{Heisenberg}$ layers. (c) The qubit state is then injected into the fermionic Fock space and the fermions delocalised using a Trotterised adiabatic ansatz circuit with $D_\mathrm{Hubbard}$ layers. (d) Energy and temperature corresponding to a thermal state with the same energy. (e) Spin-spin correlations $S(x,y) = 1/N\sum_i \langle S^z_i S^z_{i+(x,y)} \rangle - \langle S^z_i \rangle \langle S^z_{i+(x,y)} \rangle$ for ansatz circuits with ($D_\mathrm{Heisenberg}, D_\mathrm{Hubbard})$ layers. Avg. (max.) standard error on the mean is 0.011 (0.014). (f) The low-energy state is subjected to a light pulse modelled by a time-dependent electric field shown in (g), leading to an increase in $\eta$-pairing correlations \ref{['eq_Peta']}, especially for the staggered average $P_\eta^\mathrm{stag}$. Avg. (max.) standard error on the mean of $P_\eta(x,y)$ is 0.004 (0.009). Note that $S(x,y) = S(-x,-y)$ and $P_\eta(x,y)=P_\eta(-x,-y)$ by definition but we choose to show all sites for visual completeness.
• Figure 2: $D$-wave pairing in the doped checkerboard model. (a) Using a brickwall circuit, an approximate ground state of the ferromagnetic $3 \times 3$ lattice XXZ model is prepared. (b) The state is injected into the fermionic Fock space via a local isometry $V_\mathrm{plaquette}$ which maps the qubit states to the ground states of the $2 \times 2$ Hubbard model in the sector with two $(s)$ and four $(d)$ fermions, respectively. (c) The resulting wave function is a superposition of states with three $s$- and six $d$-plaquettes (one configuration shown). It approximates the ground state of the $6 \times 6$ square lattice with inhomogeneous hopping $t \gg t'$, $U/t=2$ and $1/6$ doping. (d) Pairing correlations \ref{['eq_Pb']} of the approximate ground state which show $d$-wave symmetry. For bonds on the left of a strong plaquette $P_b(x,y)=P_b(-x,-y)$ by definition, but all bonds are shown for visual completeness. Avg. (max.) standard error on the mean is 0.016 (0.017). (e) Sketch of the phase diagram of the checkerboard model at low doping, as conjectured in tsai_optimal_2008, with the prepared state indicated by the red circle (left). We evolve the prepared state by a one-step adiabatic evolution towards $U/t=8,\ t'/t=1/2$ and measure the energy density with respect to the target parameters (right).
• Figure 3: A Bilayer Hubbard model. (a) The approximate ground state of a ferromagnetic XXZ model at $\delta=-2/3$ on a periodic $4 \times 4$ lattice is prepared using a brickwall circuit. (b) The state is injected into the Fock space of a $4 \times 4 \times 2$ bilayer Hubbard model by means of an isometry $V_\mathrm{rung}$ which maps the qubit states to either a hole-pair or a singlet on a given rung. (c) The state thus prepared is an approximate ground state of a bilayer Hubbard model in the limit of strong interlayer exchange coupling $J \gg t$. (d) Rung-rung pairing correlations \ref{['eq_Pr']}. Note that $P_r(x,y)=P_r(-x,-y)$ by definition, although we show all rungs for visual completeness. Avg. (max.) standard error on the mean is 0.0266 (0.0271). (e) The initial state is used to test an adiabatic evolution using $M=1,2$ Trotter steps of size $\tau=0.4$, using a linear ramp towards $t/J=0.7$, while keeping $U=10t$. A lowering of the energy density is observed for $M=1$ before noise heats the state. Pairing correlations between singlets orthogonal to the $A$- and $B$-layers decrease rapidly, but correlations between slanted pairs are likely to increase. Both adiabatic evolutions are complete.
• Figure S1: Comparison of the$\textrm{L}=0$, $\textrm{L}=\pm 1$and$\textrm{L}=$NaN strategies. Expectation values obtained for the imbalance benchmark setup. (a) shows unmitigated data, and (b) mitigated data with the noise mitigation described in section \ref{['sec_supplement_error_mitigation']}. The exact value (without Trotter error) is shown with the black continuous line as indication.
• Figure S2: Octagon Fermion-to-Qubit Encoding for the operators and observables in the single-layer Hubbard Model. Each fermionic site is represented by an up-qubit (blue) and a down-qubit (red). Ancilla qubits (green) are added into all odd faces of the lattice. Nearest-neighbour hopping operators $c^\dagger_{i \sigma} c_{j \sigma}$ are mapped to three-qubit operators (boundary-crossing-term shown) and five-body operators (triangular shape) which act on one ancilla each. On-site repulsion $U n_{i \uparrow} n_{i \downarrow}$ is implemented using to the qubit $n=(1-Z)/2$-operator. Including both up- and down-fermions in the same encoding allows us to simultaneously measure many $\eta$-pairing correlation observables $\Delta^\dagger_i \Delta_j = c^\dagger_{i \uparrow} c^\dagger_{i \downarrow} c_{j \uparrow} c_{j \downarrow}$ (trapezoidal shape). To avoid the presence of magnetic fluxes through even plaquettes, we initialise the stabilisers $S_j = +1$, by preparing a toric code state on the green qubits and a fixed fermionic parity state on the blue and red qubits. The logical operator $B_H$ sets the magnetic flux through one handle of the torus (+1 for anti-periodic, -1 for periodic boundary conditions). Not shown is another logical operator $B_V$, which is obtained by rotating $B_H$ by 90 degrees and swapping $Y \rightarrow X$ on the ancilla qubits.