Exploiting Quantum Parallelism to Simulate Quantum Random Many-Body Systems

We present an algorithm that exploits quantum parallelism to simulate randomness in a quantum system. In our scheme, all possible realizations of the random parameters are encoded quantum mechanically in a superposition state of an auxiliary system. We show how our algorithm allows for the efficient simulation of dynamics of quantum random spin chains with known numerical methods. We propose an experimental realization based on atoms in optical lattices in which disorder could be simulated in parallel and in a controlled way through the interaction with another atomic species.

Figure 1

Comparison of an exact calculation (full line) of a random field $XY$ spin chain with our numerical results with $D=20$ (stars) and $D=30$ (circles). The relative errors are plotted in the insets. Figs. 1a, 1b show the time evolution of the averaged correlations $⟨⟨\sum _{\ell }{S}_{\ell }^x{S}_{\ell +1}^x+S_{\ell }^y{S}_{\ell +1}^y⟩⟩$ (dark curve, blue online) and $⟨⟨\sum _{\ell }{S}_{\ell }^z{S}_{\ell +1}^z⟩⟩$ (light curve, orange online). The evolution Hamiltonian is (3) with (a) $B_0=0$, $J/B=-2$ and $p(\mathbf{b})=1/2^N$, and (b) $B_0=0$, $J/B=-4$ and $|{\psi }_a⟩$ the ground state of $H_0$ with $B_0/J=1.4$. For both figures the chain is prepared initially in the ground state of $H_0$ with $B_0=0$. The time step for the numerical simulations is $\Delta t=0.01J/\hslash$. (c) shows the correlation functions $⟨⟨\sum _{\ell }{S}_{\ell }^x{S}_{\ell +\Delta }^x+S_{\ell }^y{S}_{\ell +\Delta }^y⟩⟩$ (dark curve, blue online) and $⟨⟨\sum _{\ell }{S}_{\ell }^z{S}_{\ell +\Delta }^z⟩⟩$ (light curve, orange online) averaged over all possible ground states of Hamiltonian (3) for $B_0=0$, $J/B=-1$ and $p(\mathbf{b})=1/2^N$ as a function of the distance $\Delta$ between spins. The numerical simulation performs an adiabatic evolution with Hamiltonian (4) with $\beta (t)=Bt/T$ and $T=100J/\hslash$.

Figure 2

Correlation function $⟨⟨c_k^{†}{c}_k⟩⟩$ of a random field $XY$ spin chain with $N=40$, as a function of time and momentum $k$. Here $c_k\propto \sum _{\ell }\sin (k\ell ){\tilde{c}}_{\ell }$, $k=\frac{\pi }{N+1},\dots ,\frac{\pi N}{N+1}$ and ${\tilde{c}}_{\ell }=\prod _{\ell <{\ell }^{\prime }}{S}_{{\ell }^{\prime }}^z(S_{\ell }^x+i{S}_{\ell }^y)$ are the fermionic operators given by the Jordan-Wigner transformation [21]. The evolution Hamiltonian is (3) with $H_0$ being the $XY$ Hamiltonian with $B_0=0$, $B/J=-4$, and $p(\mathbf{b})=1/2^N$. The initial state $|{\psi }_0⟩$ is the ground state of $H_0$. As the system evolves in time the initially sharp Fermi sea disappears.

Figure 3

Correlation function $⟨⟨\sum _{\ell }{S}_{\ell }^z{S}_{\ell +\Delta }^z⟩⟩$ of a random field antiferromagnetic Heisenberg chain with $N=40$, as a function of time and the separation $\Delta$ between the spins. Parameters are as in Fig. 2. As the system evolves in time the antiferromagnetic correlations are smeared out.

Figure 4

Time evolution of the averaged density of a Tonks gas in an optical lattice with $M=40$ sites and $\nu =1/2$ in the random potential generated by another Tonks gas with $\nu =1/5$.