A relativistic discrete spacetime formulation of 3+1 QED

Abstract

This work provides a relativistic, digital quantum simulation scheme for both $2+1$ and $3+1$ dimensional quantum electrodynamics (QED), based on a discrete spacetime formulation of theory. It takes the form of a quantum circuit, infinitely repeating across space and time, parametrised by the discretization step $Δ_t=Δ_x$. Strict causality at each step is ensured as circuit wires coincide with the lightlike worldlines of QED; simulation time under decoherence is optimized. The construction replays the logic that leads to the QED Lagrangian. Namely, it starts from the Dirac quantum walk, well-known to converge towards free relativistic fermions. It then extends the quantum walk into a multi-particle sector quantum cellular automata in a way which respects the fermionic anti-commutation relations and the discrete gauge invariance symmetry. Both requirements can only be achieved at cost of introducing the gauge field. Lastly the gauge field is given its own electromagnetic dynamics, which can be formulated as a quantum walk at each plaquette.

Figures (7)

• Figure 1: (\ref{['fig:sub-relativisticspacetime']}) In relativistic, digital quantum simulation, the light-like worldlines of the simulated theory coincides with circuit wires, yielding strict causality. (\ref{['fig:sub-timetrotterizedspacetime']}) In non-relativistic, Trotterized analogue quantum simulation, light-like worldlines are approximately recovered through a Lieb-Robinson bound, and are slower. Thus, the simulation is running slower. As typical decoherence times match the depth of the circuit, the QFT is simulated over a shorter period.
• Figure 2: Visualization of locality for $s^\dagger_{x:\nu} a_x$. The coloured dots and lines corresponds to $Z$ operators acting on fermions and gauge fields respectively. Each operator $Z_y$ is applied exactly twice which cancels them out except on site $x$. Here, only one fermionic mode per site is represented, for clarity.
• Figure 3: Transport with qubit $(x,0)$ moving right (and $(x,1)$ moving left) and $T$ updating the gauge fields accordingly (as represented as a single wiggly line for conciseness).
• Figure 4: The 3 steps of the free evolution in the multi-particle sector. The gauge field is omitted for clarity.
• Figure 5: Plaquette states and operator represented in the subspace $\mathbb{Z}^4$ containing four gauge field values.

Theorems & Definitions (1)

• Remark 1: Truncation of the electric field