Analyzing Many-Body Localization with a Quantum Computer

Many-body localization, the persistence against electron-electron interactions of the localization of states with nonzero excitation energy density, poses a challenge to current methods of theoretical and numerical analyses. Numerical simulations have so far been limited to a small number of sites, making it difficult to obtain reliable statements about the thermodynamic limit. In this paper, we explore the ways in which a relatively small quantum computer could be leveraged to study many-body localization. We show that, in addition to studying time evolution, a quantum computer can, in polynomial time, obtain eigenstates at arbitrary energies to sufficient accuracy that localization can be observed. The limitations of quantum measurement, which preclude the possibility of directly obtaining the entanglement entropy, make it difficult to apply some of the definitions of many-body localization used in the recent literature. We discuss alternative tests of localization that can be implemented on a quantum computer.

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With recent advances in the control of quantum systems, a small quantum computer with 50–100 qubits seems likely within the next few years. A logical question is, can we already use such a small computer to perform calculations that cannot be performed on a classical computer? This question is nontrivial, as many known quantum algorithms become advantageous over their classical counterparts only in an asymptotic regime, i.e., for a very large quantum computer with thousands of qubits. We explain how quantum computers can be used to study the dynamics and eigenstates of disordered, interacting electrons. We obtain numerical estimates for how many gates need to be coherently executed by emulating the quantum simulation on a classical computer.

The phenomenon of many-body localization has recently attracted significant attention. It revolves around the question of whether Anderson localization—the famous suppression of transport in disordered electron systems discovered by P. W. Anderson in 1958—remains stable in the presence of interactions between electrons. Numerical and analytical evidence has shown that Anderson localization does indeed remain stable, and tremendous effort has been expended to understand the details of this many-body localized phase. Significant research has been based on computer simulations; however, the best classical algorithms for the simulation of quantum systems are ill suited for this problem, and the system sizes that can be simulated have thus far been very small. We propose that a quantum computer can be used to look for signatures of many-body localization. Our goal is to measure eigenstates (which we prepare using an iterative quantum phase estimation algorithm), as well as the dynamics following local quenches. We argue that these quantum simulations can yield results that are complementary to findings from classical computers.

Our results indicate that valuable insights into many-body localization can be obtained with only moderate quantum resources.

Figure 1

Overview of the quantum phase estimation algorithm discussed in the main text for $k=4$ ancilla qubits. The lowest line in the figure indicates the $N$ qubits used for the physical system. Here, QFT denotes the quantum Fourier transformation. After the measurements, the readout on the $k$ ancilla qubits contains an estimate for the energy, whereas the $N$ qubits for the physical system contain the final state $|{\mathrm{\Psi }}_f⟩$ of Eq. (12), which is an approximation to an eigenstate and can be processed further to obtain measurements on the physical system.

Figure 2

Dashed lines: Density of states $\rho (E)$ obtained using full diagonalization. Points: $\rho$ obtained using IQPE for $k=16$ bits. All results are for $L=12$.

Figure 3

Energy standard deviation density $\mathrm{\Delta }E/L$. Dotted lines are $W=1$; solid lines are $W=8$. Results have been obtained using exact time evolution to exclude any Trotter errors.

Figure 4

Trotter errors for $L=16$, averaging over 1000 instances. Here, we use a first-order Trotter decomposition.

Figure 5

The entanglement entropy at the center of the system as a function of $\mathrm{\Delta }E/L$ for various system sizes and initial states used in the IQPE algorithm. Dashed lines show data for initial product states in the $S^z$ basis, while solid lines show data for initial states with a mix of $S^z$ and $S^x$ bases, as explained in the text. System sizes are, from top to bottom, $L=12$, 10, and 8.

Figure 6

Entanglement entropy at the center of the system. Dashed lines indicate a fit to $S=S_0+a{T}_{\text{tot}}^{-b}$, where $a$ and $b$ are fit parameters. Inset: $S(T_{\text{tot}})-S(T_{\text{tot}}\to \infty )$, where $S(\infty )$ has been obtained from the dashed fit in the main panel, on a double-logarithmic scale.

Figure 7

The bipartite entanglement entropy for a center cut as a function of system size $L$ in the strong disorder limit, as obtained by IQPE to accuracy (from top to bottom) $k=10,11,\dots ,20$. As the accuracy is increased, and the state begins to approach an energy eigenstate, the entanglement entropy decreases.

Figure 8

Quantum circuit to measure the current operator $i(c_i^{†}{c}_j-c_j^{†}{c}_i)$, as described in the main text. Here, the top two qubits are the qubits corresponding to the physical sites $i$ and $j$, and the bottom qubit is an ancilla qubit.