Optimal Renormalization Group Transformation from Information Theory

Abstract

The connections between information theory, statistical physics and quantum field theory have been the focus of renewed attention. In particular, the renormalization group (RG) has been explored from this perspective. Recently, a variational algorithm employing machine learning tools to identify the relevant degrees of freedom of a statistical system by maximizing an information-theoretic quantity, the real-space mutual information (RSMI), was proposed for real-space RG. Here we investigate analytically the RG coarse-graining procedure and the renormalized Hamiltonian, which the RSMI algorithm defines. By a combination of general arguments, exact calculations and toy models we show that the RSMI coarse-graining is optimal in a sense we define. In particular, a perfect RSMI coarse-graining generically does not increase the range of a short-ranged Hamiltonian, in any dimension. For the case of the 1D Ising model we perturbatively derive the dependence of the coefficients of the renormalized Hamiltonian on the real-space mutual information retained by a generic coarse-graining procedure. We also study the dependence of the optimal coarse-graining on the prior constraints on the number and type of coarse-grained variables. We construct toy models illustrating our findings.

Machine learning for quantum matter

“…Motivated by the structural and inner workings of deep neural networks, which have been seen to operate by extracting a hierarchy of increasingly higher-level concepts in its layers, a number of studies have established a series of connections between RG and deep learning [106][107][108][109][110][111][112][113] One of the earliest connections between RG and deep learning appeared in Re.f. [106].…”

Relevance in the Renormalization Group and in Information Theory

“…As mentioned, the RSMI algorithm [36,37] is closely related to the IB. Specifically, it also maximizes the relevant information IðH; EÞ, however contains no tradeoff β I , but instead a fixed cardinality jHj.…”

“…The former being entirely determined by the relevance variable, we need to define E ensuring the IB retains precisely the RG-relevant information, and prove this is indeed the case. An appropriate definition for the real-space RG was postulated in the context of RSMI [36,37]: for a random variable V [36,37]: an optimal encoder extracting information about relevance variable E contained in V is constructed. Right: IB curves depicting relevant information IðH; EÞ retained by solutions to the IB equations (encoders), as a function of the tradeoff β I [see Eq.…”

“…The thickness of the excluded buffer, formally taken to infinity, sets the length scale separating shortrange correlations to be discarded, from information about long-range properties of the system. Despite conceptual appeal (the system itself defines relevance), and partial numerical [36] and theoretical evidence [37],t h e validity and the relation of this approach to field theory were unclear. There are also subtle differences between the IB and RSMI approaches.…”

QuCumber: wavefunction reconstruction with neural networks

“…More recently, RBMs have been adopted by the physics community in the context of representing both classical and quantum many-body states [15,16]. They are currently being investigated for their representational power [17][18][19], their relationship with tensor networks and the renormalization group [20][21][22][23][24], and in other contexts in quantum many-body physics [25][26][27].In this post, we present QuCumber: a quantum calculator used for many-body eigenstate reconstruction.QuCumber is an open-source Python package that implements neural-network quantum state reconstruction of many-body wavefunctions from projective measurement data. Examples of data to which QuCumber could be applied might be magnetic spin projections, orbital occupation number, polarization of photons, or the logical state of qubits.…”