Entanglement spectroscopy on a quantum computer

We present a quantum algorithm to compute the entanglement spectrum of arbitrary quantum states. The interesting universal part of the entanglement spectrum is typically contained in the largest eigenvalues of the density matrix which can be obtained from the lower Renyi entropies through the Newton-Girard method. Obtaining the $p$ largest eigenvalues (${\lambda }_1>{\lambda }_2\cdots >{\lambda }_p$) requires a parallel circuit depth of $O\left[p{({\lambda }_1/{\lambda }_p)}^p\right]$ and $O[plog(N\left)\right]$ qubits where up to $p$ copies of the quantum state defined on a Hilbert space of size $N$ are needed as the input. We validate this procedure for the entanglement spectrum of the topologically ordered Laughlin wave function corresponding to the quantum Hall state at filling factor $\nu =1/3$. Our scaling analysis exposes the tradeoffs between time and number of qubits for obtaining the entanglement spectrum in the thermodynamic limit using finite-size digital quantum computers. We also illustrate the utility of the second Renyi entropy in predicting a topological phase transition and in extracting the localization length in a many-body localized system.

Figure 1

Quantum circuit to calculate $R_2$ for a quantum state with a Hilbert space that spans 7 qubits. The qubits labeled ${\alpha }_i$ are in subsystem A and those labeled ${\beta }_i$ are in subsystem B. The operator ${\text{Swap}}_A$ is implemented using three controlled swap gates between qubits ${\alpha }_i$ and ${\alpha }_i^{\prime }$.

Figure 2

Quantum amplitude estimation to calculate $R_2$. Top: Quantum phase estimation on the operator $O={\text{Swap}}_{A}V$ will give $R_2$. $F{T}^{†}$ refers to the inverse Fourier transform operator. Bottom: Quantum circuit showing controlled implementation of the operator $O$. The operator $Q$ has the desired wave function as an eigenstate with the corresponding eigenvalue $q_{\mathrm{\Psi }}$ known a priori. The Fourier transform of this value $\mathcal{F}\left[q_{\mathrm{\Psi }}\right]$ is stored in the ancilla qubits that remain unchanged at the end of the computation. $Z$ refers to the Pauli-$Z$ gate.

Figure 3

Schematic of cylindrical surface showing $N_{\text{orb}}$ one-body Landau gauge wave functions which are Gaussian along the length of the cylinder ($x$ axis) and periodic along the circular direction. The cut in orbital space preserves momentum along the $y$ axis of the cylinder.

Figure 4

The entanglement spectrum for the ground state of a system of $N_e=10$ electrons in $N_{\text{orb}}=28$ orbitals on the surface of a cylinder. The system is cut in the center with an equal number of electrons on each side. Blue crosses correspond to results from exact diagonalization of ${\rho }_A$ while red circles correspond to results from the algorithm.

Figure 5

Spread of the entanglement spectrum of the Laughlin wave function at $K_A=3$ as a function of system size. The solid line is the least-squares fit of the form $a{(1/N_e)}^c+b$, with values $a=10.22,b=0,c=0.6$. The inset shows the accuracy required to get the correct number of nonzero eigenvalues at $K_A=3$ versus the number of qubits required to represent the wave function. The least-squares fit $log\left(\epsilon \right)=a^{\prime }{N}_q^{-c^{\prime }}$ gives $c^{\prime }=0.8$ which is close to the value of $c$ obtained from the fit in the main figure as predicted by the arguments in the text.

Figure 6

The second Renyi entropy $S_2$ diverges at the phase transition between the Laughlin state and a topologically trivial phase for $N_e=8$ electrons in $N_{\text{orb}}=22$ electrons, cut in the center with equal electrons on either side at $K_A=0$.

Figure 7

(a) The second Renyi entropy $S_2$ as a function of the disorder parameter $w$ for different system sizes obeying the Hamiltonian in Eq. (17). (b) The parameter $\gamma$ which is a measure of the many-body localization length as extracted from a fit to the first few levels of the entanglement spectrum, and the same as extracted from $R_2$ using Eq. (16).