Errors and decoherence
Although quantum simulators are influenced by the environment in the same way as quantum computers, it is generally believed that the decoherence effects are much lower. This is most commonly seen in AQS, which may require less precision (or only a qualitative response). As a result, several defects in the AQS particles may not affect the overall behavior of the set, so AQS may still produce useful results even in the presence of these defects. In addition, decoherence of simulator may be useful, as it can be used as an approximate way to model decoherence of the simulated systems. A simple argument is: If the noise level naturally presented in the simulator is lower than the noise level of the simulated system, one can easily complete the noise in the simulator such that it mimics the noise of the simulated system. This idea has been proven in experiments. In essence, the researcher can determine how decoherence affects a simulation, and then by choosing the appropriate relationship between the system and the simulator, he can use natural symmetries to modify effective decoherence of the simulator. Decoherence can provide a useful tool for extracting information about a critical system.
Unfortunately, there are certain limitations, and the decoherence of simulator in the simulation must be carefully considered. The interaction between the system and the environment can be qualitatively different from the interaction between the simulator and its environment. For example, when simulating spin Hamiltonians with the degenerate states using trapped ions, ions spontaneous irradiation drives the system to the states outside the Hilbert space used in the simulator-system mapping. This indicates that the researcher should be careful when considering decoherence in the simulation. First, the researcher must understand how decoherence will affect the simulation and always find clever mappings to take advantage of the uncontrollable properties of simulator. Second, the researcher should pay attention to how the system and simulator are described. Therefore, the role of errors in AQS should be given more attention. Note that the simulation of open quantum systems does not necessarily need to include decoherence of simulator. Therefore, in ideal conditions, only uncontrollable errors should be minimized as much as possible.
The reliability, complexity, and efficiency of analog quantum simulations are discussed in detail in the literature. Reliability means that the simulation results reliably reflect the simulated system. Validation can be used on a number of different physical systems, and in this way, certain defects of any implementation can be violated as sources of error. On the other hand, the simulation results can be validated against numerical and analytical predictions, but this is only possible for small systems. Complexity/efficiency refers to the fact that the quantum simulator is able to solve problems that cannot be solved on a quantum computer (i.e. the simulator is more efficient than a quantum computer). Note that for quantum simulation of challenging experimental problems, this is not a required necessity. Disorder, noise and other defects may affect the reliability of quantum simulation.
Physical implementation of a quantum simulator requires a controllable quantum system. Any physical system that can be used as a quantum computer will also be a universal machine for DQS. Over the past few decades, possible pathways and experimental developments for the construction of a quantum computer have been widely discussed. However, a quantum system that is not a potential quantum computer can still run AQS. For example, to study cosmic inflation, the propagation of sound waves in a two-component BEC was suggested. A rotating Fermi gas can be used to understand the phenomena of nuclear physics. For many different quantum simulators in condense matter physics, an array of qubits in addition to their controls will make an ideal quantum simulator, since such a system can be viewed as an enlarged, simplified lattice structure of a solid and manipulated in different ways to test different models. Each qubit resides in its own potential energy well and is used to encode a particle with a spin of 1/2. This array can be configured so that its dimensions and geometry can be changed. Such an array can be realized, for example, by atoms in optical lattice, atoms in cavities arrays, ions in micro-trap arrays, or in two-dimensional crystals, electrons in quantum dots arrays, and so on. The desired evolution of the system can be induced through simulator control fields.
Quantum simulators will have many applications in the fields of physics, chemistry, and possibly biology. Quantum simulation will lead to new results that cannot be simulated in any other way, and will also allow the testing of various theoretical models. In general, using a quantum simulator, a researcher can solve problems that are either intractable by classical computers or are experimentally challenging. In addition, the quantumness of the systems itself will give a new perspective to quantum simulators that did not exist in classical simulators (i.e. the decoherence effects and the transition from classical to quantum). While some problems are classically intractable (Hubbard models, spin disorders and frustration, lattice gauge theory and quantum chemistry calculations), others can be solved classically.
Demonstration of quantum simulations with more than a few tens of qubits leads us to conclude that quantum computers (whether digital or analog) will outperform their traditional counterparts, at least for certain applications; a remarkable advance for physics and computer science. But there are still issues to be resolved. From the experimental point of view, in all the quantum simulators proposed, improved controllability and scaling are required. Quantum simulators, except atoms in optical lattices, still cannot manage and control large arrays of qubits. On the other hand, it is difficult to control and read individually atoms in optical lattices, while for other systems it is not. Problems where collective properties are concerned may not require individual control and measurement.
Theoretically, further studies of decoherence and control, especially the estimation of experimental requirements for any quantum simulator can be very useful. It is also useful to consider both the theoretical and practical aspects of when and how much the researcher can use the decoherence of the simulator.
Quantum simulators not only give new insights into new physical phenomena, but also help solve difficult quantum many-body problems. In addition, theoretical and experimental advances in quantum simulation will have a positive impact on the development of other areas such as adiabatic quantum computing, measurement-based quantum computing, topological quantum computing and quantum computing theory. For example, adiabatic quantum computing can be considered as a special case of quantum simulation that the ground state of simulated Hamiltonian encodes the solution of a computational problem. Therefore, the ability to simulate different Hamiltonians can be useful for realizing practical adiabatic quantum calculations. Studying the entanglement in many-body systems and its relation to quantum phase transitions should be considered as another exciting path which is related to quantum simulation.
Quantum simulation will profoundly affect physical researches and will provide new tools for testing physical theories or predicting the behavior of physical systems under different conditions. On the other hand, quantum simulation enables access to new physical realms that are currently beyond experimental access. Even in the absence of a theory to be tested, quantum simulators may open up a new realm for scientists to discover. Curious experiments by quantum simulators may lead to unpredictable discoveries.
 D. G. Angelakis, Quantum Simulations with Photons and Polaritons, Springer, 2017.
 I. L. M. J. P. H. George F. Viamontes, Quantum Circuit Simulation, Springer, 2009.
 R.-Q. Z. H. R. T. Jun Zeng, Quantum Simulations of Materials and Biological Systems, Springer, 2012.
 S. Will, From Atom Optics to Quantum Simulation, Springer, 2013.
 S. A. F. N. I. M. Georgescu, "Quantum Simulation," arXiv:1308.6253v3, 2014.
 B. C. R. O. C. R. V. R. S. Roberto Dovesi, "Ab Initio Quantum Simulation in Solid State Chemistry," Reviews in Computational Chemistry, 2005.
 C. Orzel, "Quantum Simulation," IOP Publishing, 2017.
 J. D. W. P. L. M. N Cody Jones, "Faster quantum chemistry simulation on fault-tolerant quantum computers," New Journal of Physics, 2012.
 R. P. Feynman, "Simulating Physics with Computers," International Journal of Theoretical Physics, 1981.
 C. R. M. T. E. Tobias Schaetz, "Focus on quantum simulation," New Journal of Physics, 2013.
 H. Weimer, Lecture Notes on Quantum Simulation, arXiv:1704.07260v1, 2017.
 G. K. F. Z. E. S. R. Gerritsma, "Quantum simulation of the Dirac equation," NATURE, 2010.
 S. V. R. T. S. Ruwan Senaratne, "Quantum simulation of ultrafast dynamics using trapped ultracold atoms," NATURE COMMUNICATIONS, 2018.
 J. I. C. a. P. Zoller, Goals and opportunities in quantum simulation, NATURE PHYSICS, 2012.
 M. Ljubotina, Quantum Simulation, University of Ljubljana, 2015.