QUIE2T Quantum Information Entanglement-Enabled Technologies

Complexity of simulating many-body systems

In recent years, a strong link between quantum information science and the study of condensed matter systems has been established, in particular to research on strongly correlated quantum systems, so systems that play a key role in the understanding of phenomena such as high-temperature superconductivity. This link is less surprising as it may at first seem: After all, quantum correlations are distributed and shared in an intricate manner in ground states of local quantum many-body systems. The quantitative theory of entanglement can provide new insights into the exact structure of such quantum correlations, in turn opening up new perspectives for the development of new algorithms for the simulation of such quantum many-body problems. Indeed, the significant findings in this field may be seen as a further justification for the importance of the study of entanglement.

Notably, ground states of local systems typically satisfy what is called an "area law", in that the entanglement of a subregion scales only with the surface area of that region. That is to say, they have very little entanglement, an assertion that can be made quantitative. Exploiting this observation, one arrives at the insight that only few effective degrees of freedom are being exploited by natural systems, compared to the exponentially larger Hilbert space. Suitably parameterizing this set by means of what is called tensor networks hence gives rise to new efficient simulation algorithms for the study of strongly correlated systems. Matrix-product states, projected entangled pair states, tree tensor networks or states from entanglement renormalization from a real-space renormalization ansatz are examples of such an approach. These are sets of states, described by polynomially many real parameters, for which one can still efficiently compute local expectation values by means of suitable tensor contractions, and which still grasp the essential physics of the problem.

In such a language, certain elementary obstacles of classical simulations of quantum systems such as in time evolution also become clear, and quantitative links to the theory of criticality and quantum phase transitions can be established. Ideas like Lieb-Robinson bounds, relating to the speed of information propagation in quantum lattice systems, provide key insights into the distribution of correlations in local quantum many body problems with respect to static of dynamical properties. Ideas of quantum information science can hence relate to

• Fundamental issues of the complexity of a classical description of quantum many-body systems in a language of computer science;
• A reassessment of the functioning of existing methods such as the Density Matrix Renormalization Group (DMRG) approach, and
• The development of novel feasible and efficient algorithms specifically for two-dimensional or fermionic systems, opening up new perspectives in the simulation of strongly correlated quantum many-body systems.

This demonstrates that the research into entanglement, its characterization, manipulation and quantification will not only continue to have impact within quantum information but is now reaching the stage where its insights are being applied to other areas of physics, with potentially enormous benefits, both intellectually but perhaps also commercially.

Key references
[1] K. Audenaert, J. Eisert, M. B. Plenio, and R. F. Werner, “Entanglement properties of the harmonic chain”, Phys. Rev. A 66, 042327 (2002)
[2] J. I. Latorre, E. Rico, and G. Vidal, "Ground state entanglement in quantum spin chains", Quant. Inf. Comp. 4, 048 (2004)
[3] M. B. Plenio, J. Eisert, J. Dreissig, and M. Cramer, "Entropy, entanglement, and area: Analytical results for harmonic lattice systems", Phys. Rev. Lett. 94, 060503 (2005)
[4] F. Verstraete and J. I. Cirac, "Renormalization algorithms for quantum many-body systems in two and higher dimensions", cond-mat/0407066
[5] J. Kempe, A. Kitaev, and O. Regev, "The complexity of the local Hamiltonian problem", SIAM Journal of Computing, Vol. 35, 1070 (2006)
[6] G. Vidal, "Entanglement renormalization", Phys. Rev. Lett. 99, 220405 (2007)
[7] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, "Entanglement in many-body systems", Rev. Mod. Phys. 80, 517 (2008)
[8] F. Verstraete, J. I. Cirac, V. Murg, "Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems", Adv. Phys. 57, 143 (2008)
[9] J. Eisert, M. Cramer, M. B. Plenio, "Area laws for the entanglement entropy", Rev. Mod. Phys. 81 (2010)

Connection between QIP and quantum chemistry

Related to the previous field, quantum information theory can help in gaining an understanding the quantum correlations that are present in physical problems from quantum chemistry. Ideas of monogamy and entanglement distribution are related to the quantum representability problem, being of key importance in theoretical quantum chemistry. New ideas inspired by quantum information theory relate to proofs of hardness of certain questions in quantum chemistry, as well as to new simulation methods of such physical systems, contributing to the wider context of gaining a deeper understanding of complex quantum systems.

Key references
[1] A. Klyachko, "Quantum marginal problem and N-representability", J. Phys. A Conf. Ser. 36, 72 (2006)
[2] Y.-K. Liu, M. Christandl, and F. Verstraete, "N-representability is QMA-complete", Phys. Rev. Lett. 98, 110503 (2007)