The quantum simulation plays an irreplaceable role in various areas that go beyond the scope of classical computers. In a recent study, Keren Li and an interdisciplinary research team at the Center for Quantum Computing, Quantum Science and Technology, and the Institute of Physics and Astronomy in China, the US, Germany, and Canada. Experimentally simulated spin-network states by simulating quantum space-time tetrahedra on a four-qubit NMR quantum simulator. The experimental fidelity was over 95 percent. The research team used the nuclear magnetic resonance quantum tetrahedra to simulate a two-dimensional (2-D) Spinfoam vertex (model) amplitude and to show the local dynamics of quantum space time. Li et al. have measured the geometric properties of the corresponding quantum tetrahedra to simulate their interactions. The experimental work is a first attempt and a basic module to present the Feynman diagram vertex in the Spinfoam formulation and to investigate the quantum loop quantum gravity (LQG) quantum information processing. The results are now available in communications physics.

Classic computers can not study large quantum systems despite successful simulations of various physical systems. The systematic limitations of classical computers occurred when the linear growth of quantum system magnitudes corresponded to the exponential growth of Hilbert space, a mathematical basis of quantum mechanics. Quantum physicists are trying to solve this problem using quantum computers that process information intrinsically or quantum mechanically to exponentially outperform their classical counterparts. The physicist Richard Feynman defined quantum computers in 1982 as quantum systems that can be controlled to mimic or simulate the behavior or properties of relatively inaccessible quantum systems.

In the present work, Li et al. used nuclear magnetic resonance (NMR) with a high controllable power on the quantum system to develop simulation methods. The strategy allowed the representation of quantum geometries of space and space-time based on the analogies between nuclear spin states in NMR samples and spin-network states in quantum gravity. Quantum Gravity aims to combine Einstein gravitation with quantum mechanics to extend our understanding of gravity to the Planck scale (1.22 x 10 ^{ 19 } GeV). On the Planck scale (magnitudes of space, time, and energy), Einstein gravity and the continuum of space-sharing can be replaced by quantum-space time. Research approaches to understanding quantum space times are currently based on spin networks (a graph of lines and nodes representing the quantum state of space at a given time), which represent an important, non-disruptive framework of quantum gravity.

In 1971, the physicist Roger Penrose suggested spinnets that were motivated by twist theory and then applied them to loop quantum gravity (LQG). The spin networks were quantum states that represented fundamentally discrete quantum geometries of space on the Planck scale. In the present study, the research team visualized the spin network using a graph of spin-halftoned links and nodes. For example, each node with edges corresponded to a geometry, and so a graph containing quadrivalent nodes corresponded to quantum tetrahedral geometry.

The research team developed a "network" that contained a series of three-dimensional (3-D) world sheets (2-D surfaces) and their intersections. They showed that every vertex at which the surfaces met resulted in a quantum transition that changed the spin network to represent the local dynamics of quantum geometry. Similar to Feynman diagrams (schematic representations of mathematical expressions describing the behavior of subatomic particles), the quantum space times encoded the transition amplitudes and spinfoam amplitudes between the initial and the final spin network. The quantum space times and spinfoam amplitudes developed in the study provided a consistent and promising approach to quantum gravity. Li et al. characterized the NMR simulation by the ability to control individual qubits with high precision. The quantum tetrahedral and peak amplitudes served as building blocks for LQG (Loop Quantum Gravity) to open a new window for the inclusion of LQG in quantum experiments.

The scientists first derived equations to describe a quantum tetrahedron in a spin network. In a schematic 3 + 1-dimensional dynamic quantum space time model, they demonstrated an atom as a 3-sphere, which includes part of the quantum space around a vertex. The team accurately modeled the boundary of the included quantum space time as a spider web and showed the possibility of simulating large quantum space times with many vertices by quantum bonding of the atoms. The resulting structure was similar to the vertex amplitude of quantum space time, similar to Ooguri's previously developed topological lattice models in four dimensions. The researchers showed that LQG identifies quantum tetrahedral geometries with the quantum momentum pulses. The identification enabled them to simulate quantum geometries with quantum registers (quantum mechanical analogue of a classical processor register). In general, a quantum register can be achieved mathematically using tensor products.

Li et al. Simulated 10 quantum tetrahedra by establishing the corresponding invariant tensor states. They marked these states with 10 colored dots on the Bloch sphere (geometric representation) and performed the experiments with a 700 MHz DRX Bruker spectrometer at room temperature. For all experiments, the research team used the crotonic acid molecule with four ^{ 13 } C cores suitable for the four-qubit system. The scientists developed the experimental system to produce quantum tetrahedra and to simulate their local dynamics in three parts.

- To prepare the state, they first initialized the entire system to a pseudo-pure state. They achieved a fidelity of over 99 percent according to the spatial average method. They then drove the system into 10 invariant tensor states or transforms, which they translated with 10 20 ms shaped pulses.
- The team then presented the measured geometry properties for geometry measurements using a 3D histogram. The experimental uncertainty at this point resulted from the NMR spectrum adaptation process. The combination of experimental and theoretical simulations implied that the invariant tensor states created in the experiments matched the building blocks – quantum tetrahedra.
- During the amplitude simulation, the spin-network states served as boundary data for the 3 + 1-dimensional quantum space time. The vertex amplitude defined in the study determined the spinfoam amplitude and described the local dynamics of quantum gravity in 4-D quantum space time to represent the properties of these boundary data.

To obtain the vertex amplitudes, researchers calculated the internal products between five different quantum tetrahedral states. Ideally, researchers could have used a 20-qubit quantum computer to create two qubit maximum entanglement states between two arbitrary tetrahedra. Since a quantum computer of such dimensions is currently not state-of-the-art, the researchers alternately performed a complete tomography of the state preparation to obtain information about quantum tetrahedral states. When the scientists calculated the accuracy between the experimental quantum tetrahedral states and the theory, the results were well over 95 percent. With the quantum tetrahedra, the research team simulated the peak amplitude. They compared the results between the experiment and the numerical simulation of all five tetrahedra. Accordingly, saddle points of amplitude occurred in the experiments in which the five interacting tetrahedra showed a simple geometric meaning when glued together to form a geometric four-simplex.

In this way, Keren Li et al. A quantum register in the NMR system to generate 10 invariant tensor states representing 10 quantum tetrahedra. They achieved a fidelity of over 95 percent and then measured the surface angles (two flat surfaces) of the model. They considered the spectrum matching errors and geometric identification to understand the success of quantum tetrahedron simulation in the study. The new research was a first step to investigate spin network states and spinfoam amplitudes using a quantum simulator. The accompanying work also showed valid experiments for the study of LGQ.

Quantum computer to clarify the connection between quantum and classical world

**Further information:**

Keren Li et al. Quantum Space Time on a Quantum Simulator,

*Communications Physics*(2019). DOI: 10.1038 / s42005-019-0218-5

Richard P. Feynman. Simulating Physics with Computers, * International Journal of Theoretical Physics * (2007). DOI: 10.1007 / BF02650179

p. Lloyd. Universal Quantum Simulator, * Science * (2006). DOI: 10.1126 / science.273.5278.1073

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