Unitary circuits of finite depth and infinite width from quantum channels

We introduce an approach to compute the spectra of reduced density matrices for local quantum unitary circuits of finite depth and infinite width. Suppose the time-evolved state under the circuit is a matrix-product state with bond dimension $D$; then the reduced density matrix of a half-infinite system has the same spectrum as an appropriate $D\times D$ matrix acting on an ancilla space. We show that reduced density matrices at different spatial cuts are related by quantum channels acting on the ancilla space. This quantum channel approach allows for efficient numerical evaluation of the entanglement spectrum and Rényi entropies and their spatial fluctuations at finite times in an infinite system. We benchmark our numerical method on random unitary circuits, where many analytic results are available, and also show how our approach analytically recovers the behavior of the kicked Ising model at the self-dual point. We study various properties of the spectra of the reduced density matrices and their spatial fluctuations in both the random and translation-invariant cases.

Figure 1

A depth $d=4$ unitary circuit of the type considered in this work.

Figure 2

Left: the reduced density matrix ${\rho }_A$ can be expressed as a matrix product operator involving matrices in left canonical form (squares), and a matrix $R$ that comes from the complement, $\overline{A}$. Rényi entropies are proportional to $tr{\rho }_A^n$; when the MPS is in left canonical form, $tr{\rho }_A^n=tr{R}^n$. Right: application of this idea to a unitary circuit. The matrix-product state $\left|\psi \right(t)\rangle$ is constructed by cutting diagonally through the circuit (solid lines); the leftover gates in the shaded region form the matrix $R$.

Figure 3

Entanglement density of states (i.e., histogram of entanglement energies), at depth $t=8$, computed both exactly and by low-rank approximation of the quantum channel (Sec. 3c). For rank $\gtrsim 20$ the low-energy behavior of the entanglement spectrum is well captured.

Figure 4

(a) Cumulative distribution function of the entanglement eigenvalues under random unitary dynamics, for various circuit depths, extracted from exact evolution of the reduced density matrix under the quantum channel corresponding to the unitary circuit. At the largest depths, an appreciable fraction of the entanglement spectrum consists of eigenvalues below machine precision. (b) A more detailed view of the entanglement spectrum near its “low-energy” edge at various depths $t$ under random unitary dynamics. Recall that the rank of the density matrix increases exponentially with the depth. All points except $t=14$ are averaged over 3000 realizations; $t=14$ is averaged over 300 realizations. Inset: slope of the linear growth of the entanglement density of states with energy.

Figure 5

Rényi-index dependence of the entropy $S_{\alpha }$. The fits are to the form $S_n=S_{\infty }(1-1/n)$, which works for large $n$, with $S_{\infty }\simeq t/6$. Deviations from this form grow with time.

Figure 6

Histograms of $S_2$ and $S_{\infty }$ for $t=12$; lines are fits to a Gaussian.

Figure 7

Top: power spectrum of the Fourier transform of min-entropy across spatial cuts, normalized to one for all circuits. Note the narrowing of the Fourier transform with increasing $t$. Data are for a single system of length 5000. Bottom: estimate of the correlation length $\xi$ extracted from the width of the Fourier peak.

Figure 8

Average purity of a random unitary circuit with $q=2$ computed by applying the quantum channel (red circles) and by the trajectory method (green bars) for $L=1000$ steps. Good agreement is found with the result $\overline{\gamma }={(4/5)}^{t-1}$ from Ref. [10] (dashed blue line).

Figure 9

Entanglement spectra for circuits with a conservation law. Upper panel: behavior of the entanglement spectrum near its edge for three classes of initial states: random product states, random computational-basis bit strings (“number sharp”), and the nonrandom Néel state. Lower panel: fluctuations of the min-entropy for these three classes of states. All data are for depth $t=12$, averaged over 3000 samples. Only the initial Néel state approaches a Gaussian distribution.

Figure 10

Power spectrum of min-entropy fluctuations in the $XXZ$ chain; the initial state is a random product state in the computational basis. As in random unitary circuits, the correlation length of the entanglement fluctuations grows in time.

Figure 11

Evolution of the min-entropy under the quantum channel, after $L$ steps, for depth $t=12$. The data shown are for the Trotterized $XXZ$ chain at two values of the anisotropy, and (for comparison) for a translation-invariant circuit with a randomly chosen gate. The integrable $XXZ$ chain has nonmonotonic entanglement growth, which is generically absent. In all cases the initial state is the Néel state $\uparrow \downarrow \uparrow \downarrow ...$.

Figure 12

Upper panel: entanglement spectra starting from the Néel state, under Trotterized $XXZ$ dynamics, vs anisotropy parameter $\eta$, at depth 12. For large $\eta$ the dynamics freezes, so $S_{\infty }$ remains close to zero out to late times. Lower panel: time evolution of $S_{\infty }$ for a fixed anisotropy $\eta =1.5i$.