Surveying Structural Complexity in Quantum Many-Body Systems

Abstract

Quantum many-body systems exhibit a rich and diverse range of exotic behaviours, owing to their underlying non-classical structure. These systems present a deep structure beyond those that can be captured by measures of correlation and entanglement alone. Using tools from complexity science, we characterise such structure. We investigate the structural complexities that can be found within the patterns that manifest from the observational data of these systems. In particular, using two prototypical quantum many-body systems as test cases—the one-dimensional quantum Ising and Bose–Hubbard models—we explore how different information-theoretic measures of complexity are able to identify different features of such patterns. This work furthers the understanding of fully-quantum notions of structure and complexity in quantum systems and dynamics.

Technical Appendix

Tensor Network Theory (TNT) [67] is a set of powerful and efficient numerical methods for classically simulating quantum many-body systems. In this Appendix, we briefly review matrix product states (MPS), matrix product operators (MPO), and the density matrix renormalisation group (DMRG) in the context of our work. MPS and MPO provide efficient descriptions of states and operators of quantum many-body systems respectively, while DMRG is an iterative procedure that variationally minimises the energy of Hamiltonians to obtain the ground states of quantum many-body systems.

MPS [18] are widely used as efficient representations of low energy states of one-dimensional quantum systems. In a quantum many-body chain, each lattice site is represented by a tensor, and the tensors are connected to their neighbours. Consider a quantum many-body chain of size N in a quantum state

$$\begin{aligned} \left| \psi \right\rangle&= \sum _{i_1 i_2 \dots i_N}c_{i_1i_2 \dots i_N} \left| i_1 \right\rangle \left| i_2 \right\rangle \cdots \left| i_N \right\rangle \end{aligned}$$

(14)

where $\{ \left| i_j \right\rangle \}$ are the local orthonormal basis states. We can perform repeated Schmidt decompositions [52] at each site, splitting the tensor $c_{i_1i_2 \dots i_N}$ into local tensors $\Gamma ^{[j]}$, and Schmidt coefficients $\lambda ^{[j]}$ that quantify the entanglement across the split, which gives us the canonical form of the MPS representation of the state:

$$\begin{aligned} \left| \psi \right\rangle&=\sum _{\{ i \},\{ \alpha \}}\!\!\left( \Gamma ^{[1]i_1}_{\alpha _1} \lambda ^{[1]}_{\alpha _1} \Gamma ^{[2]i_2}_{\alpha _1 \alpha _2} \lambda ^{[2]}_{\alpha _2}\ldots \lambda ^{[N-1]}_{\alpha _{N-1}}\Gamma ^{[N]i_N}_{\alpha _{N-1}}\right) \left| i_1 \right\rangle \left| i_2 \right\rangle \ldots \left| i_N \right\rangle , \end{aligned}$$

(15)

where $\alpha _j$ takes positive integer values up to the rank of $\Gamma ^{[j]}$. By contracting the Schmidt coefficient tensors $\lambda ^{[j]}$ into the local tensors $\Gamma ^{[j]}$, we obtain a more generic form:

$$\begin{aligned} \left| \psi \right\rangle&= \sum _{i_1 i_2 \dots i_N} A^{x_1}A^{x_2}\cdots A^{x_N} \left| i_1 \right\rangle \left| i_2 \right\rangle \cdots \left| i_N \right\rangle , \end{aligned}$$

(16)

where $A^{x_j}$ is a matrix with the same dimension as the local basis states.

In a similar fashion, a quantum operator can be written in the form of MPO [68]:

$$\begin{aligned} \mathcal {H}&= \sum _{i,k} H^{i_1,k_1}H^{i_2,k_2} \cdots H^{i_N,k_N} |i_1 i_2\cdots i_N\left\rangle \right\langle k_1 k_2 \cdots k_N |, \end{aligned}$$

(17)

where $H^{i_j,k_j}$ is a matrix with the dimension of the local basis state. With quantum states and Hamiltonians represented in MPS and MPO forms respectively, ground states $\left| \psi _g \right\rangle$ may then be obtained by minimising $\left\langle \psi \right| \mathcal {H}\left| \psi \right\rangle$ across all states using the DMRG algorithm.

The DMRG algorithm [69, 70] is an iterative, variational method that truncates the degrees of freedom of the system, retaining only the most significant features required to accurately describe the physics of a target state. The algorithm achieves remarkable precision in describing one-dimensional quantum many-body systems [71].

In the DMRG algorithm, the elementary unit is a site, described by the state $d_i$ where $i=1,\dots , D$ is the label of the states accessible to a given site. A block $B(L,v_L)$ consists of L sites, and has total dimension $v_L$; $H_B$ is the Hamiltonian of the block, containing only terms that involve the sites inside the block. Whenever a block is enlarged, a site is added to the block, forming an enlarged block $B^e$ with a Hilbert space dimension that is the product of the Hilbert space of $B(L,v_L)$ and a site, i.e. $v_L \times D$. An important step in the algorithm is the formation of superblock Hamiltonians, consisting of two enlarged blocks connected to each other. The superblock ground state is calculated using Lanczos [72] or Davidson [73] methods. The ground state is then truncated by discarding the least-probable eigenstates.

The algorithm itself consist of two parts: the warm-up cycle, and finite-system algorithm. The warm-up cycle is designed to create a system block of the desired length of at most dimension $\chi$, before the finite-system algorithm is applied to compute the ground state. Starting from a block B(1, D), each step of the warm-up cycle is carried out as follows [74]:

• 1. Start from a left block $B(L,v_L)$, and enlarge the block by adding a single site.

• 2. Form a superblock by adding a reflected copy of the enlarged block to its right.

• 3. Obtain the ground state of the superblock, and the $v_{l+1}=\min (v_l D,\chi )$ eigenstates of the reduced density matrix of the left enlarged block with largest eigenvalues.

• 4. The truncated left enlarged block is used for the next iteration.

• 5. Renormalise all operators to obtain block $B(L+1,v_{L+1})$.

These steps are repeated until the desired length $L_{\mathrm {max}}$ is reached. Once the infinite-system algorithm reaches the desired length, the system consist of two blocks of $B(L_{\text {max}}/2 - 1,\chi )$ and two free sites. The subsequent step is called the “sweep procedure", the goal of which is to enhance the convergence of the target state. The sweep procedure consists of enlarging the left block with one site and reducing the right block correspondingly to keep the length fixed. While the left block is constructed by the usual enlarging steps, the right block is recalled from memory, as it has been built in the infinite-system algorithm and saved. This procedure is repeated until the left block reaches the length $L_{\text {max}}-4$. At this point the right block B(1, D) with one site is constructed from scratch and the left block $B(L_{\text {max}},\chi )$ is obtained through renormalisation. The sweep procedure is then repeated from right to left, and at each iteration, the renormalised block has to be stored in memory. The procedure is stopped when the system energy converges.

In this manuscript, we use the implementations of these algorithms as described in [67], and the ground states are computed with $\chi =150$ for the one-dimensional quantum Ising chain, and $\chi =80$ for the one-dimensional Bose–Hubbard chain. The resulting ground states are accurate up to $\mathcal {O}(10^{-14})$.