Criticality and entanglement in nonunitary quantum circuits and tensor networks of noninteracting fermions

Models for nonunitary quantum dynamics, such as quantum circuits that include projective measurements, have recently been shown to exhibit rich quantum critical behavior. There are many complementary perspectives on this behavior. For example, there is a known correspondence between $d$-dimensional local nonunitary quantum circuits and tensor networks on a $[D=(d+1\left)\right]$-dimensional lattice. Here, we show that in the case of systems of noninteracting fermions, there is furthermore a full correspondence between nonunitary circuits in $d$ spatial dimensions and unitary noninteracting fermion problems with static Hermitian Hamiltonians in $D=(d+1)$ spatial dimensions. This provides a powerful perspective for understanding entanglement phases and critical behavior exhibited by noninteracting circuits. Classifying the symmetries of the corresponding noninteracting Hamiltonian, we show that a large class of random circuits, including the most generic circuits with randomness in space and time, are in correspondence with Hamiltonians with static spatial disorder in the 10 Altland-Zirnbauer symmetry classes. We find the criticality that is known to occur in all of these classes to be the origin of the critical entanglement properties of the corresponding random nonunitary circuit. To exemplify this, we numerically study the quantum states at the boundary of Haar-random Gaussian fermionic tensor networks of dimension $D=2$ and 3. We show that the most general such tensor network ensemble corresponds to a unitary problem of noninteracting fermions with static disorder in Altland-Zirnbauer symmetry class DIII, which for both $D=2$ and 3 is known to exhibit a stable critical metallic phase. Tensor networks and corresponding random nonunitary circuits in the other nine Altland-Zirnbauer symmetry classes can be obtained from the DIII case by implementing Clifford algebra extensions for classifying spaces.

Figure 1

A diagram showing the correspondences among nonunitary quantum circuits of noninteracting fermions in $d$ dimensions, Gaussian tensor networks in $D=d+1$ dimensions, and static Hermitian Hamiltonian systems of noninteracting fermions in $D=d+1$ spatial dimensions.

Figure 2

(a) A generic tensor network on the square lattice is depicted. The horizontal and vertical directions of the tensor network are labeled as $u$ and $v$. (b) The graphical representation of the four-leg tensor $T_{ijkl}$ is shown. (c) The contraction between two tensors $T^{\left(1\right)}$ and $T^{\left(2\right)}$ is graphically represented by connecting the contracted legs.

Figure 3

(a) We can rotate the square-lattice tensor network shown in Fig. 2 by ${45}^{\circ }$ and view it as a quantum circuit acting on a one-dimensional qudit chain along the $x$ direction. The $t$ direction is viewed as the physical time direction of the quantum circuit. (b) Each four-leg tensor can be viewed as a (nonunitary) quantum gate acting on two neighboring sites on the one-dimensional qudit chain. By the polar decomposition, this quantum gate can be factored into the product of a unitary operator $U$ and the positive-semidefinite Hermitian operator $K$.

Figure 4

(a) The Gaussian tensors $\mathrm{\Gamma }$ and $\mathrm{{\rm Y}}$ are contracted. The Majorana modes on the contracted legs of the Gaussian tensors $\mathrm{\Gamma }$ and $\mathrm{{\rm Y}}$ are denoted as ${\widehat{\gamma }}_{1,2,\cdots ,d}$ and ${\widehat{\eta }}_{1,2,\cdots ,d}$, respectively. (b) We can view both of the Gaussian tensors $\mathrm{\Gamma }$ and $\mathrm{{\rm Y}}$ as two-leg tensors when implementing the contraction between them.

Figure 5

In the numerical study, we consider the square-lattice random GTN with periodic boundary condition in the $u$ direction. At $v=0$, we start with the initial Gaussian state represented by the Gaussian tensor ${\mathrm{\Gamma }}_0$ which is a product state. The contraction of the tensor ${\mathrm{\Gamma }}_0$ with $v$ (“depth”) rows of the random GTN yields the Gaussian state represented by the tensor ${\mathrm{\Gamma }}_v$.

Figure 6

Correlation function $C\left(r\right)$ [cf. Eq. (13)] obtained from averaging over 20 disorder realizations for Majorana bond number $\chi =1$ (a) and $\chi =6$ (b) and $r$ up to $L/2$. Points indicate raw data, while the black line indicates a fit of the data for the largest system size to Eq. (14).

Figure 7

(a) Scaling of bipartite entropy of half of the system with total system size in the Haar-random ensemble. Majorana bond numbers are $\chi =1$ through $\chi =10$ from bottom to top. Crosses indicate the numerical data, while the dashed lines indicate fits to the form $S_{L/2}={\zeta }_{1}log(L/L_0)$. (b) Dependence of the entropy scaling prefactor ${\zeta }_1$ on the Majorana bond number $\chi$. Inset of (b): convergence of the entropy at the center of the system with depth $v$ for $L=128$ and $\chi =1,2,10$.

Figure 8

Coefficient of the logarithmic term in the entanglement entropy for Renyi entropies of different index $\alpha$; see Eqs. (16) and (17). Crosses represent raw data, and dashed lines fits to the form ${\zeta }_{\alpha }=B(1+1/\alpha )$, where $B$ is the only fit coefficient.

Figure 9

Mutual information between the disjoint segments $[x_1,x_2]$ and $[x_3,x_4]$ versus the cross ratio $\eta$ of Eq. (20). In both panels, $L=100$. We exclude any data points where the intervals $[x_1,x_2]$ or $[x_3,x_4]$ are shorter than four sites, or the ends of the intervals are closer than four sites.

Figure 10

(a) A one-dimensional GTN is shown. (b) A two-leg tensor $\mathrm{\Gamma }$ in the one-dimensional GTN is shown. Its two legs are labeled by $L$ and $R$, respectively. (c) The contraction of the two two-leg tensors $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\prime }$ yields a third two-leg tensor ${\mathrm{\Gamma }}^{″}$. This contraction can be equivalently captured using the transfer matrix formalism.

Figure 11

We can view the square-lattice GTN as a quasi-one-dimensional GTN along the $t$ direction. Each two-leg tensor ${\mathrm{\Gamma }}^{\left(t\right)}$ in the quasi-one-dimensional GTN consists of all four-leg tensors ${\mathrm{\Gamma }}^{(x,t)}$ in the square-lattice GTN that share the same $t$ coordinate.

Figure 12

A generic pure-state Gaussian tensor $\mathrm{\Gamma }$ can be mapped to a unitary scattering problem with its transfer matrix given by $\mathfrak{t}\left[\mathrm{\Gamma }\right]$. Every Majorana mode associated with the Gaussian tensor $\mathrm{\Gamma }$ corresponds to a pair of counterpropagating modes in the scattering problem. The scattering problem respects unitarity, TR symmetry, PH symmetry, and chiral symmetry.

Figure 13

(a) Each four-leg tensor in the square-lattice GTN is mapped to a unitary scattering problem with TR, PH, and chiral symmetries on a four-leg geometry. (b) Applying this mapping to each four-leg tensor, we map the square-lattice pure-state GTN to a network model of unitary scatters on the square lattice. The network model also respects TR, PH, and chiral symmetries.

Figure 14

(a) For the four-leg tensor with Majorana bond number $\chi =1$, the four Majorana modes associated with this tensor are labeled according to this figure. (b) The random GTN can be driven into an area-law entanglement phase upon staggering of the tensors on the two sublattices. The sublattice $A$ is colored green while the sublattice $B$ is colored orange.

Figure 15

(a) Inverse correlation length $1/\xi$ extracted from the squared two-point correlation function in a staggered GTN with Majorana bond number $\chi =1$, as function of the disorder strength dictated by $\sigma$ (see main text). For $\sigma \gtrsim 0.45$ the correlation length diverges, signaling a phase transition into a critical entanglement phase. (b) Scaling of the entanglement entropy of half of the system with system size for different values of the disorder strength $\sigma$.

Figure 16

Illustration of a quasi-three-dimensional tensor network, which can be understood as a stack of $W$ (here $W=3$) layers of the two-dimensional network connected in the $z$ direction. We apply periodic boundary conditions in the $u$ direction.

Figure 17

Entropy scaling in three-dimensional systems. The left panel (a) shows the scaling for fixed bond number $\chi =8$ (where we use the same bond number in all three directions) with the length $L$ of the system for a variety of $W$. Note that the $y$ axis is entropy divided by thickness $W$ for better readability. In all cases, a logarithmic scaling with $L$ is observed. The right panel (b) shows the prefactor of this logarithmic scaling as a function of thickness $W$ and for various bond numbers $\chi$. Together, these results clearly establish $WlogL$ scaling of the entropy. This is using periodic boundary condition in the $L$ and open boundary condition in the $W$ direction, but results for periodic boundary condition in both directions are qualitatively similar.

Figure 18

Probability distribution of the ratio of consecutive entanglement spectrum gaps [see Eq. (D1)] for system size $L=256$. The green line shows the expected distribution for the Gaussian unitary ensemble (GUE), Eq. (D2).

Figure 19

A one-dimensional GTN consists of tensors $\left\{{\mathrm{\Gamma }}^{\left(n\right)}\right\}$ is shown. The tensor ${\mathrm{\Gamma }}^{\left(n\right)}$ of the GTN is located in-between the coordinates $t=n-1$ and $t=n$.