### Quantum Simulation with Light: Frustrations between Photon Pairs

**Philip Walther**

**Author: Philip Walther**

Affiliation:

Affiliation:

**Vienna Center for Quantum Science and Technology (VCQ),**

**Faculty of Physics, University of Vienna, Austria.**

Quantum information science will revolutionize our society if we are able to harness its power. Therefore, worldwide theoretical and experimental efforts are focused on the realization of the Holy Grail of quantum-enhanced applications: the quantum computer. But the difficulties encountered in realizing hundreds of coherent quantum gate operations that act on almost the same number of qubits for the implementation of useful quantum algorithms [1, 2] may seem very discouraging.

On the other side, more than a quarter of a century ago, Richard Feynman [3, 4] envisioned that a well-controlled quantum-mechanical system can be efficiently used for the simulation of other quantum systems and thus is capable of calculating properties that are unfeasible for classical computers. Quantum simulation promises potential returns in understanding detailed quantum phenomenon of inaccessible quantum systems, from molecular structure to the behavior of high-temperature superconductors [5]. Moreover, quantum simulations are conjectured to be less demanding than quantum computations by being less stringent on explicit gate operations or error corrections [6]. The tradeoff, however, is that the level of coherent control of quantum systems necessary for the physical realization of quantum simulators is very demanding.

Photonic quantum systems are not only the natural choice for quantum communication and metrology applications, but have also been proven to be a suitable system for scalable quantum computing [7]. Moreover, the single-particle addressability and tunable measurement-induced interactions make single photons also very promising for the simulation of other quantum systems. In addition to the superior level of quantum control, such photonic quantum simulators allow to utilize quantum interference at beamsplitters which can lead to interesting photon-entanglement that corresponds to ground states of complex correlations in chemical or solid-state systems.

The ground state properties of frustrated quantum systems raised significant interest due to the conjecture that these quantum phenomena may be important for the understanding of high-temperature superconductivity [8]. A quantum system is frustrated if competing requirements cannot be satisfied simultaneously.

A research team from the University of Vienna and the Institute of Quantum Optics and Quantum Information at the Austrian Academy of Sciences headed by Anton Zeilinger realized for the first time an experimental quantum simulation, where the frustration regarding the pairing of quantum correlations was closely investigated [9]. Such pairing of quantum correlations is an important mechanism in chemical or so-called valence bonds, where two electrons from different atoms share an anti-correlated spin state due to the Pauli principle. Obviously this leads to maximally entangled spins, where the spins are always oriented in opposite directions.

The same quantum correlation of valence-bond states can be simulated by a pair of photons that is maximally entangled in polarization, i.e. that the two photons are always orthogonal polarized. Therefore, we were able to simulate four spin-1/2 particles whose ground states exist in two valence-bond states by using two entangled photon pairs. For simulating the interesting entanglement dynamics of these states, however, an effective nonlinear interaction among the photons was implemented by superimposing photons from each pair (Figure 1) at a beamsplitter with a tunable splitting ratio, followed by a measurement of the photons in the output ports.

**Figure 1 : Scheme of the photonic setup for studying frustrated spin-1/2 tetramers. The variable beamsplitters (VBS) of this analog quantum simulator allows for tuning the interaction from localized to delocalized of frustrated valence-bond states.**

Depending on the interaction strength single photons from originally localized valence-bonds are facing the conflict over partnerships between each other. Each photon can establish a single bond to only one partner exclusively, but wants to get correlated with several partners – obviously this leads to frustration. As a result, the quantum system uses “tricks” that allow quantum fluctuations in which different pairings can coexist as superposition. In the scientific community such superpositions of valence-bond states are called spin liquid states and are an active area of research.

**Figure 2: Photonic quantum simulation of the ground state energy of a frustrated Heisenberg spin system.**

The precise quantum control of our single photons enabled us to extract the total energy (Figure 2) and the pair-wise quantum correlations for arbitrary configurations within this spin tetramer (Figure 3). Our simulation also proves that the pairwise quantum correlations and energy distribution are restricted by the role of quantum monogamy: only two photons can share a valence-bond independent of the coupling to other single photons. This can be nicely seen as complementarity relation among the various valence-bond configurations when tuning the coupling by changing the beamsplitter's reflectivity.

**Figure 3: Experimental observation of the complementarity relation for the pair-wise quantum correlation (or Heisenberg energy) among the individual partners when changing the interaction.**

This recent work underlines that quantum simulation is a very good tool for calculating quantum states of matter, and is thus opening the path for the investigation of more complex systems.

**References**

**[1]**P. W. Shor, "Algorithms for quantum computation: discrete logarithms and factoring", Proc. 35th Annual Symposium on "Foundations of Computer Science" (ed. S. Goldwasser) 124-134 (1994). Abstract.

**[2]**L. K. Grover, "Quantum Mechanics Helps in Searching for a Needle in a Haystack", Phys. Rev. Lett. 79, 325 (1997). Abstract.

**[3]**R. Feynman, "Simulating physics with computers," International Journal of Theoretical Physics, 21, 467 (1982). Abstract.

**[4]**R. P. Feynman, "Quantum mechanical computers," Foundations of Physics 16, 507 (1986). Abstract.

**[5]**I. Buluta, F. Nori, "Quantum Simulators," Science 326, 108 (2009). Abstract.

**[6]**A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, M. Head-Gordon, "Simulated Quantum Computation of Molecular Energies," Science 309, 1704 (2005). Abstract.

**[7]**E. Knill, R. Laflamme, G. J. Milburn, "A scheme for efficient quantum computation with linear optics," Nature 409, 46 (2001). Abstract.

**[8]**P. W. Anderson, "The Resonating Valence Bond State in La

_{2}CuO

_{4}and Superconductivity," Science 235, 1196 (1987).

**[9]**Xiao-song Ma, Borivoje Dakic, William Naylor, Anton Zeilinger & Philip Walther, "Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems," Nature Physics, 7, 399 (May, 2011). Abstract.

Labels: Complex System, Quantum Computation and Communication 4, Superconductivity

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