Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions

References

I Supplementary Material for “Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions”

II Implementation of fermion lattice models in trapped ions: Examples

To illustrate our method, we propose several examples with increasing degree of complexity. In our first example, in order to implement the evolution associated with $H_{1}=g(b^{\dagger}_{j+1}b_{j}+b^{\dagger}_{j}b_{j+1})=g(I_{N}\otimes I_{N-1}\otimes...\otimes\sigma^{+}_{j+1}\otimes\sigma^{z}_{j}\sigma^{-}_{j}\otimes I_{j-1}\otimes...\otimes I_{1}+I_{N}\otimes I_{N-1}\otimes...\otimes\sigma^{-}_{j+1}\otimes\sigma^{+}_{j}\sigma^{z}_{j}\otimes I_{j-1}\otimes...\otimes I_{1})$ in terms of Mølmer-Sørensen gates, we first write $H_{1}$ as a sum of tensor products of Pauli matrices. This can always be done for Hermitian fermionic Hamiltonians: using $\sigma^{\pm}=(\sigma^{x}\pm i\sigma^{y})/2$, we get $H_{1}=g(\sigma^{x}_{j+1}+i\sigma^{y}_{j+1})/2\otimes\sigma^{z}_{j}(\sigma^{x}_{j}-i\sigma^{y}_{j})/2+(\sigma^{x}_{j+1}-i\sigma^{y}_{j+1})/2\otimes(\sigma^{x}_{j}+i\sigma^{y}_{j})\sigma^{z}_{j}/2=-(g/2)(\sigma^{x}_{j+1}\otimes\sigma^{x}_{j}+\sigma^{y}_{j+1}\otimes\sigma^{y}_{j})$. Once the fermionic Hamiltonian is in the form of sum of products of Pauli matrices, we use Trotter techniques to decompose the total evolution operator in product of exponentials associated with these products of Pauli matrices. Here, each of these exponentials is $\exp(i(g/2)t\sigma^{x}_{j+1}\otimes\sigma^{x}_{j})$, and $\exp(i(g/2)t\sigma^{y}_{j+1}\otimes\sigma^{y}_{j})$. The first one can be implemented using Eq. (2) in the article, with ${\cal U}_{\rm MS}(-\pi/2,0)$, ${\cal U}_{\rm MS}(\pi/2,0)$ acting on ions $j$ and $j+1$, and ${\cal U}_{\sigma_{y}}(\phi^{\prime}=-\phi=-gt/2)$ acting on ion $j+1$, plus a local rotation upon ion $j+1$ to change from the $z$ to the $x$ basis. Equivalently, for the second exponential, one can proceed similarly, substituting the Mølmer-Sørensen gates ${\cal U}_{\rm MS}(-\pi/2,0)$, ${\cal U}_{\rm MS}(\pi/2,0)$ by ${\cal U}_{\rm MS}(-\pi/2,\pi/2)$, ${\cal U}_{\rm MS}(\pi/2,\pi/2)$, acting on ions $j$ and $j+1$, and ${\cal U}_{\sigma_{x}}(\phi^{\prime}=\phi=gt/2)$ acting on ion $j+1$, plus a local rotation upon ion $j+1$ to change from the $z$ to the $y$ basis.

A second example describes the implementation of the nonlinear dynamics associated with $H_{2}=g(b^{\dagger}_{j}b_{j}b^{\dagger}_{j+1}b_{j+1})$, where $j$, $j+1$ may represent two different modes in the same site, or in different sites. The $j$ index labels the fermionic mode representing any degree of freedom (e.g., spatial, momentum, or spin). In this case, $H_{2}=g\sigma^{+}_{j}\sigma^{-}_{j}\otimes\sigma^{+}_{j+1}\sigma^{-}_{j+1}$, which can be written as a sum of products of Pauli matrices, obtaining $H_{2}=g(\sigma^{z}_{j}+I_{j})\otimes(\sigma^{z}_{j+1}+I_{j+1})/4=g\sigma^{z}_{j}\otimes\sigma^{z}_{j+1}/4+g\sigma^{z}_{j}/4+g\sigma^{z}_{j+1}/4$. Finally, we apply Trotter methods to decompose ${\cal U}=\exp(-iH_{2}t)$ in three exponentials: $\exp(-igt\sigma^{z}_{j}\otimes\sigma^{z}_{j+1}/4)$ will be implemented, according to Eq. (2) in the article, with ${\cal U}_{\rm MS}(-\pi/2,0)$, ${\cal U}_{\rm MS}(\pi/2,0)$ acting on ions $j$ and $j+1$, and ${\cal U}_{\sigma_{y}}(\phi^{\prime}=-\phi=gt/4)$ acting on ion $j$, plus a local rotation upon ion $j+1$ to change from the $x$ to the $z$ basis. The other two exponentials, $\exp(-igt\sigma^{z}_{j}/4)$ and $\exp(-igt\sigma^{z}_{j+1}/4)$ can be implemented easily by local gates.

As a third example, we consider the implementation of the evolution associated with a tunneling Hamiltonian,

in trapped ions (see Fig. 1 in the article). Here, modes 1 and 10 represent nearest neighbours in a 3D lattice. Note that due to the standard mapping of Jordan-Wigner transformation in three dimensions, the effective spin Hamiltonian is highly nonlocal. We first decompose the evolution operator $\exp(-iH_{3}t)$ with a Trotter expansion in terms of the exponentials of each of the two terms in $H_{3}$, i.e., $\exp(-igt\sigma^{x}_{1}\otimes\sigma^{z}_{2}\otimes\sigma^{z}_{3}\otimes...\otimes\sigma^{z}_{9}\otimes\sigma_{10}^{x})$ and $\exp(-igt\sigma^{y}_{1}\otimes\sigma^{z}_{2}\otimes\sigma^{z}_{3}\otimes...\otimes\sigma^{z}_{9}\otimes\sigma_{10}^{y})$. In order to obtain the first of these operators, we apply the Mølmer-Sørensen gates ${\cal U}_{\rm MS}(-\pi/2,0)$, ${\cal U}_{\rm MS}(\pi/2,0)$ to ions 1 to 10, applying in between of them the local ${\cal U}_{\sigma_{y}}(\phi^{\prime}=-\phi=gt)$ gate upon the second ion. Notice that, despite the large number of ions, each of these gates requires just two lasers globally addressed upon all ions. We consider the second ion in this case because, according to Eq. (2) in the article, the resulting nonlocal spin operator acts with a $\sigma^{z}$ operator upon the ion which was acted upon with ${\cal U}_{\sigma_{y}}(\phi^{\prime}=-\phi=gt)$. In our case, we want it to be ion 2, given that in $\exp(-igt\sigma^{x}_{1}\otimes\sigma^{z}_{2}\otimes\sigma^{z}_{3}\otimes...\otimes\sigma^{z}_{9}\otimes\sigma_{10}^{x})$ ion 2, and not ion 1, is acted upon by $\sigma^{z}$ operator. Instead of ion 2, each of the ions 3-9 could also have been chosen for this purpose. In this way, we reduce the amount of local gates needed afterwards. Accordingly, we have so far that ${\cal U}_{\rm MS}(-\pi/2,0){\cal U}_{\sigma_{y}}(\phi^{\prime}=-\phi=gt){\cal U}_{\rm MS}(\pi/2,0)=\exp(-igt\sigma^{x}_{1}\otimes\sigma^{z}_{2}\otimes\sigma^{x}_{3}\otimes...\otimes\sigma^{x}_{9}\otimes\sigma_{10}^{x})$. To obtain the desired exponential, we rotate qubits 3-9 with a global single qubit rotation to change the $x$ basis to the $z$ basis. In order to obtain the second exponential operator, a similar combination of gates should be employed, but in this case the specific Mølmer-Sørensen gates and single qubit gates would be different, as seen in Ref. Mueller11b .

III Analysis of the Trotter error

In this Section we show that the resources needed in our protocol, including number of elementary gates and time scaling, are polynomial on the Trotter error, the total time simulated, and the total size of the system, in terms of the number of fermionic modes.

The Trotter expansion is a useful tool to express the evolution operator of a Hamiltonian that can be written as a sum of efficiently-implementable Hamiltonians, in terms of a certain product of the operators associated to each of these individual Hamiltonians. More specifically, if the Hamiltonian H can be written as a sum of $m$ terms, where $m$ is polynomial in $N$, $H=\sum_{j=1}^{m}H_{j}$, then the standard Trotter expansion reads Lloyd96b ; Nielsen00b ; Berry07b ,

Accordingly, by making $n_{T}$ very large, the error can be made as small as possible.

There are more sophisticated, higher order expansions so called Lie-Trotter-Suzuki methods Suzukiab ; Suzukibb ; Suzukicb , that have a better scaling of the errors. Here, we will focus on time-independent Hamiltonians whose evolution operator is expanded in terms of a $k$-th order Lie-Trotter-Suzuki integrator. We will follow the formalism and error analysis of Ref. Berry07b . In this reference it is shown that decompositions of ${\cal U}=\exp(-iHt)$, where $H=\sum_{j=1}^{m}H_{j}$, can be carried out in the general form

provided $\epsilon\leq 1\leq 2m5^{k-1}||H||t$.

Notice that in all fermionic Hamiltonians we are considering, we have i) a polynomial number of nonlocal spin operators, i.e., $m$ is polynomial in $N$, the total number of fermionic modes. ii) each $H_{j}$ is always of the form of a product of arbitrary number of Pauli matrices times a coupling $h_{j}$, such that its norm $||H_{j}||=h_{j}$, given that the 2-norm of a product of arbitrary number of Pauli matrices is always 1. iii) the total norm of $H$ is bounded by $||H||\leq\sum_{j=1}^{m}||H_{j}||=\sum_{j=1}^{m}h_{j}\leq mh_{j}^{\rm max}$, where $h_{j}^{\rm max}$ is the maximum among all $h_{j}^{\prime}s$. Thus, $||H||$ is polynomial in $m$, and in consecuence, also in $N$.

Accordingly, we have shown that the scaling of the number of elementary gates needed in our expansion, is polynomial (more specifically, a power law) in $\epsilon$, $t$, and $N$, such that our method for implementing arbitrary fermionic Hamiltonians that occur in nature is efficient.

IV Final Remarks

We plot Fig. 3a,b,c in order to analyze the convergence of Trotter methods to the exact diagonalization case when increasing the number of Trotter steps $n_{T}$, and comparing with Fig. 2 in the article (for which $n_{T}=15$). These three figures clearly show the fast convergence for a linear increase in $n_{T}$.

In Fig. 3d we plot the fidelity $|\langle\psi(t_{F})|\psi(t_{F})_{T}\rangle|^{2}$ as a function of $n_{T}$, for $Ut_{F}=2.5$, where $|\psi(t)\rangle$ is the state evolved with exact diagonalization, and $|\psi(t)_{T}\rangle$ is the Trotter-evolved state, for $|\psi(0)\rangle=|\psi_{T}(0)\rangle=b^{\dagger}_{1\uparrow}b^{\dagger}_{1\downarrow}|0\rangle$, and for $|w|/U=4$. We show the numerical results with Trotter (dots) and a fit to the function $1-C/n_{T}^{2}$ (line), where $C$ is a free parameter. This curve has a perfect agreement with the Trotter numerics. Thus, the error goes to zero polynomially in $n_{T}$, as expected.

References