Rigorous Free-Fermion Entanglement Renormalization from Wavelet Theory

We construct entanglement renormalization schemes that provably approximate the ground states of noninteracting-fermion nearest-neighbor hopping Hamiltonians on the one-dimensional discrete line and the two-dimensional square lattice. These schemes give hierarchical quantum circuits that build up the states from unentangled degrees of freedom. The circuits are based on pairs of discrete wavelet transforms, which are approximately related by a “half-shift”: translation by half a unit cell. The presence of the Fermi surface in the two-dimensional model requires a special kind of circuit architecture to properly capture the entanglement in the ground state. We show how the error in the approximation can be controlled without ever performing a variational optimization.

Popular Summary

Recent progress in understanding the physics of quantum information has led to novel methods to simulate quantum physics on existing classical computers and on future quantum computers. Crucial to these developments are operational procedures to prepare interesting quantum states, especially procedures that make efficient use of scarce quantum resources. Addressing the physical properties of electrons is a particularly exciting direction; electronic properties are important both for chemistry and for materials science, but these properties are hard to calculate because electrons are fermions, for which quantum effects are often strong. Towards that end, this work draws on insights from many-body physics, quantum information science, and signal processing to derive novel preparation procedures for several nontrivial fermionic states.

Our results take the form of “quantum circuits,” which are sequences of physical operations that prepare a state of interest from a simple initial state. We consider metallic states, which have proven challenging to address because of their high degree of quantum entanglement. By drawing on the theory of wavelets, we are able to provide preparation procedures for metallic states in one and two dimensions. These results come with mathematically rigorous guarantees of correctness and were obtained without any numerical optimization.

These techniques will plausibly serve as a key element in addressing more complex electronic states that include the effects of electron interactions. The results could also lead to new methods to produce low-energy quantum states of atomic gases and to a better understanding of renormalization in quantum field theory.

Figure 1

MERA quantum circuit preparing an approximate ground state of the one-dimensional hopping Hamiltonian (1), using the notation of Ref. [29]. As discussed in Sec. 3, the tensor network consists of an initial “preprocessing” step followed by repeated layers of “renormalization-group” circuits. Each repeated layer consists of a finite depth circuit followed by projection of half the d.o.f. onto a product state. In more detail, we first apply phase gates (yellow circles) and swapping sublattices in order to let the following gates act either on the even or the odd sublattice. The bulk of the MERA corresponds to two independent wavelet transforms, one for $h_s$ (even sublattice) and one for $g_s$ (odd sublattice). Each step of the two wavelet transforms gives rise to a layer (red boxes) of two parallel quantum circuits of depth $K+L$, consisting of $2\times 2$ unitary gates (solid or hatched blue boxes). At the end of each layer, we apply second quantized Hadamard unitaries (green boxes) that couple both sublattices, and we occupy the negative-energy modes. The figure illustrates the case of $K=L=1$ and $\mathcal{L}=2$ layers. As we increase $K$, $L$, and the number of layers $\mathcal{L}$, we systematically obtain better and better approximations to the ground state (see Sec. 5).

Figure 2

Fourier spectrum of the outputs of the inverse wavelet transform for levels $\ell =1$, 2, 3 and wavelet parameters $K=4$, $L=6$. Both filters have equal magnitude in momentum space (shaded regions). The relative phase difference of their Fourier transforms (solid lines) approximates the exact phase difference of the two components of Eq. (3) or, equivalently, of the negative-energy eigenstates of the single-particle Hamiltonian (black dotted line).

Figure 3

Left panel: Plots of the eigenvalues $2^{\ell }{e}_{\ell }(k)$ of the scaling Hamiltonian (solid lines) and the eigenvalues $2^{\ell }{\epsilon }_{\ell }(k)$ of the wavelet Hamiltonian (dashed lines) for levels $\ell =1,\dots ,5$ and wavelet parameters $K=4$, $L=6$. We observe convergence to a fixed-point Hamiltonian. The original dispersion $e(k)$ is shown by the solid black line. The residual off-diagonal interaction between scaling and wavelet modes explains why the eigenvalues $e_1(k)$, ${\epsilon }_1(k)$ deviate slightly from $e(k/2)$, $e(k/2+\pi )$ (dotted blue lines). Right panel: The single-particle modes obtained from level $\ell =1$ of the wavelet transform, translated back to the original lattice before blocking, should have momentum-space support inside (blue lines) or outside (red lines) the Fermi sea $[-\pi /2,+\pi /2]$. While the wavelets exactly vanish at the Fermi points, small errors originate from the side lobes on the opposite side of the Fermi points. The solid lines are $K=L=4$. For fixed $L=4$ and $K=6$, 8, 10, the support is pushed away from the Fermi surface, but the magnitude of the side lobes does not decrease much (dotted lines). For fixed $K=4$ and $L=6$, 8, 10, the side lobes appear to decrease exponentially fast (dashed lines).

Figure 4

Left panel: The relative phase difference between the Fourier transforms of $h_{ww}$, $g_{ww}$ (smooth surface), which approximates the exact phase difference of the two components of Eq. (8) or, equivalently, of the negative-energy eigenstates of the single-particle Hamiltonian (wireframe mesh). The colored shading of the coordinate plane indicates the momentum-space support of $h_{ww}$ and $g_{ww}$. It is concentrated around $k_x=k_y=\pm \pi$ and vanishes for $k_x=0$ or $k_y=0$. Right panel: The positive-energy branch of the original Hamiltonian (wireframe mesh) and of the renormalized Hamiltonian (smooth surface) after six layers, where it has approximately reached its fixed point. The eigenmodes of both Hamiltonians are exactly characterized by the relative phase difference displayed in the left panel.

Figure 5

Branching MERA quantum circuit preparing an approximate ground state of the two-dimensional hopping Hamiltonian (7). The bottom layer consists of applying one-dimensional wavelet transforms in both the $x$ and $y$ directions of the 45-degree rotated lattice. This gives rise to four outputs, as illustrated in detail in Fig. 6. The wavelet-wavelet output corresponds to approximate eigenstates and hence can be projected out after a subsequent Hadamard transform that couples the even and the odd sublattice, indicated by yellow triangles. The mixed wavelet-scaling outputs are disentangled in one direction and therefore have to be processed further by one-dimensional transformations in the scaling directions, giving rise to two branches that take the form of binary trees as in Fig. 1. The scaling-scaling output contains no disentangled d.o.f. and is thus connected to a copy of the circuit on the renormalized lattice.

Figure 6

One layer of the branching MERA for two-dimensional free fermions. (a) The 2D lattice of fermionic modes. (b) A series of 1D wavelet transformations, as depicted in Fig. 1, applied diagonally. (c) Lattice sites corresponding either to scaling (solid) or wavelet (striped) outputs from the preceding transformations. (d) A series of 1D wavelet transformations applied across the other diagonal. (e) The lattice, which now contains five different types of sites (labeled A–E). (f) Sites A and B, differentiated by the application of Hadamards and truncated, with new sublattices formed separately from sites C–E. (g) The Brillouin zone of the 2D free fermions, where the shaded region denotes the Fermi sea. (h) The sites in (e), approximately corresponding to the distinct regions of the Brillouin zone as shown. Sites A and B contain modes in the occupied $|1⟩$ or unoccupied $|0⟩$ states, respectively, and may be truncated. Sublattices from sites C and D consist of products of 1D chains that are only correlated in the back-sloping diagonal or forward-sloping diagonal directions, respectively. The sublattice from site E is self-similar to the original lattice.

Figure 7

Coarse-grained Hamiltonians for the wavelet-scaling and scaling-wavelet branches. We show the positive-energy branch (surface) and the relative phase difference between the two components of the positive-energy eigenmodes (color coding of the coordinate plane). Because the eigenmodes are independent of one of the two directions and take the form of the one-dimensional problem studied in Sec. 3, the ground state can be disentangled by a tensor product of one-dimensional MERAs as in Fig. 1.

Figure 8

Illustration of the error term $ϵ$ in Theorem 1. The error $ϵ$ decreases exponentially for Selesnick’s construction as the parameters $K$ and $L$ are increased.

Figure 9

Relative error in the energy density for the approximate ground states prepared by our quantum circuits in one dimension (left panel) and in two dimensions (right panel), for various values of the wavelet parameters $K$, $L$ (dashed lines indicate the same $L$). For comparison, we also display the relative error for the analytical construction from Ref. [29] and for the numerically optimized non-Gaussian MERA from Ref. [47].

Figure 10

Two-point function $C(x,y)$ of the approximate ground states in one dimension for wavelet parameters $K=L=1$ (left panel) and $K=L=3$ (right panel).