Efficient Simulation of Loop Quantum Gravity: A Scalable Linear-Optical Approach

The problem of simulating complex quantum processes on classical computers gave rise to the field of quantum simulations. Quantum simulators solve problems, such as boson sampling, where classical counterparts fail. In another field of physics, the unification of general relativity and quantum theory is one of the greatest challenges of our time. One leading approach is loop quantum gravity (LQG). Here, we connect these two fields and design a linear-optical simulator such that the evolution of the optical quantum gates simulates the spin-foam amplitudes of LQG. It has been shown that computing transition amplitudes in simple quantum field theories falls into the bounded-error quantum polynomial time class, which strongly suggests that computing transition amplitudes of LQG are classically intractable. Therefore, these amplitudes are efficiently computable with universal quantum computers, which are, alas, possibly decades away. We propose here an alternative special-purpose linear-optical quantum computer that can be implemented using current technologies. This machine is capable of efficiently computing these quantities. This work opens a new way to relate quantum gravity to quantum information and will expand our understanding of the theory.

Figure 1

From spin-foam amplitudes to quantum gates. (a) A 4-simplex whose boundary is made by 5 tetrahedra. Each tetrahedron is quantized to $|{\psi }_i⟩\in {\mathcal{H}}_{\mathrm{tet}}$ ($i=1,\dots ,5$). ${\mathbf{E}}^{(i)}$ is the oriented area vector of the $i$th tetrahedron face and is quantized as quantum angular momenta. (b) The quantum gate $A$ with 3 input quantum tetrahedra and 2 output. We note that at least 3 qubits are needed to operate this nonunitary gate and even more if a unitary expansion of the gate is used. The spin-foam vertex amplitude is the matrix element of $A$. (c) An example of LQG spin-foam amplitude made by connecting 5 quantum gates $A$. Postselection and feed forward make simulation of this spin-foam amplitude possible with a linear-optical quantum computer [2].

Figure 2

The $12\times 12$ unitary transformation. The input state is 1 for one mode, $1,\dots ,8$ and 0 for all other 11 modes. The output state is the evolved state in modes, $1,\dots ,4$ and conditioned vacuum is measured in modes, $5,\dots ,12$.

Figure 3

Linear-optical circuit representation of $U$ [Eq. (S5)] for $N=12$ spatial modes. For consistency with notation, the top-left input corresponds to the $N$th (12th) mode. The one below that is the ($N-1$)th mode, and so on. The location of the corresponding output modes is found by following the transmission path of the input. For example, the bottom-right output is the $N$th mode. The one above that is the ($N-1$)th mode, and so on. The yellow boxes before the MZIs represent the phases $\varphi$, the blue boxes within the MZIs represent the phases $\omega$, and the red boxes on the far right represent elements of the diagonal phase compensation matrix $D$ (see the Supplemental Material for more details [40]).