Fermion-Fermion Scattering in Quantum Field Theory with Superconducting Circuits

We propose an analog-digital quantum simulation of fermion-fermion scattering mediated by a continuum of bosonic modes within a circuit quantum electrodynamics scenario. This quantum technology naturally provides strong coupling of superconducting qubits with a continuum of electromagnetic modes in an open transmission line. In this way, we propose qubits to efficiently simulate fermionic modes via digital techniques, while we consider the continuum complexity of an open transmission line to simulate the continuum complexity of bosonic modes in quantum field theories. Therefore, we believe that the complexity-simulating-complexity concept should become a leading paradigm in any effort towards scalable quantum simulations.

Figure 1

(a) Schematic representation of our proposal for simulating fermion-fermion scattering in quantum field theories. An open transmission line (blue) supporting the continuum of bosonic modes interacts with two superconducting qubits (green) simulating the fermions and one ancilla qubit (grey). The second one-dimensional waveguide (red), forming a resonator due to the capacitors at each edge, supports a single mode of the microwave field and interacts with two superconducting qubits. Each qubit can be individually addressed through on-chip flux lines producing fluxes ${\mathrm{\Phi }}_{\text{ext}}^j$ and ${\overline{\mathrm{\Phi }}}_{\text{ext}}^j$ to tune the coupling strength and its corresponding energies. (b) Sequence of multiple and single qubit gates, inside a Trotter step, acting on superconducting qubits to generate two-qubit interactions coupled to the continuum, where $U_{\mathrm{MS}}(-\pi /2,0)=\exp (i\pi {\sigma }^x\otimes {\sigma }^x/4)$, and $U_{C,A}=\exp [-\varphi {\sigma }_{1,A}^{y,z}\int dk{g}_k(a_k^{†}{e}^{-ikx}-a_k{e}^{ikx})]$.

Figure 2

(a) Scheme for the implementation of a set of $N$ fermionic modes coupled to a continuum of bosonic modes. Each fermionic mode is encoded in a nonlocal spin operator distributed among $N$ superconducting qubits. (b) Feynman diagrams associated with the quantum simulation of two fermionic modes coupled to a continuum of bosonic modes in a superconducting circuit setup, as explained in the text. We point out that our proposal contains all orders of Feynman diagrams for a finite number of fermionic modes.