What’s tensor network, and what is it good for?

A tensor network is a representation of a physical object as a partially or fully contracted product of many tensors. An example is the partition function of a statistical physical model, in which the tensor product is fully contracted to produce one number, i.e., the partition function. Another example is the tensor network representing a wave function of a quantum many-body system, in which indices specifying basis vectors (“physical indices”) are left uncontracted.

For example, the partition function of the square-lattice Ising model is usually expressed as the summation of a product of many factors of the form \(\exp(-KS_iS_j)\) with respect to the Ising spin variables, \(S_i\). By a simple mathematical transformation, we can re-express this as a contraction of the product of tensors each being defined on a unit square.

A wave function in quantum many-body system can also be expressed as a tensor network. In this case, it is not a partition function but the coefficient of the wave function that the contraction of the tensor product expresses. For example, in the case of \(S=1/2\) Heisenberg model on a square lattice, we often take the simultaneous eigenstates of all the z-components of spin operators. That is, a basis wave function is specified by a set of \(N\) eigenvalues, \((S_1, S_2, \dots, S_N)\), where \(N\) is the total number of spins in the system. While the coefficient is generally expressed as \(C(S_1, S_2, \dots, S_N)\), here we let it be expressed as the contraction of \(N\) tensors, \(T(S_1), T(S_2), \dots, T(S_N)\). We call such a tensor network as TNS (tensor network state) or PEPS (Projected Entangled Pair State). In contrast to the classical example of the Ising model, in this case, we still have unfixed degrees of freedom, \(S_1, S_2, \dots, S_N\), which we call physical indices, even after the contraction. (So, this contraction is partial contraction.) We use these degrees of freedom in the estimation of the expectation values of physical quantities, i.e., we consider two identical tensor networks corresponding to the bra and the ket vectors and contract the two by the physical indices.

These are merely ways of re-expressing things, and not solutions to our problems. However, there are many studies for efficient ways of contracting the tensor networks. The results of such studies are not only published in papers, but also found in some software packages, such as iTensor, uni10, mptensor, etc. In addition, the tensor network representation is not only useful as a method of high precision computation but also serves as a framework of developing new concepts, e.g., the real-space renormalization group transformation and characterization of novel quantum state including topological states.