# Simulating the Ising-like model as a quantum circuit

We are interested in simulating the 1d Ising model Hamiltonian using a Quantum Circuit (QC). A similar question was posted before with no answers. Here we will assume, for simplicity, 3 lattice sites and $$J=-1$$.

Generically, the Hamiltonian is given as $$H = -J \sum_{ij} \sigma_i^z \otimes \sigma_j^z.\tag{1}$$ For our case of interest this Hamiltonian becomes: $$H = \sigma^z \otimes \sigma^z \otimes \mathbb{1} + \mathbb{1} \otimes \sigma^z \otimes \sigma^z.\tag{2}$$ Obviously I have not included any periodic boundary conditions. There are only three lattice sites so there are only two interaction terms. In what follows I will replace $$\sigma^z$$ with $$Z$$ implying the corresponding quantum gate.

The evolution operator corresponding to this Hamiltonian is given as $$\tag{3} U(t) = e^{-i (Z \otimes Z\otimes \mathbb{1} + \mathbb{1}\otimes Z \otimes Z)t}.$$ Should these operators not commute we would have to use the Trotter-Suzuki formula. However, they do commute and as a result there is no need to use it.

In each of the two summands there exists a unit operator which can be completely ignored from the circuit. Now, for the operator $$Z \otimes Z$$ the curcuit would read $$\mathrm{CNOT} R_z(2t) \mathrm{CNOT}:$$

My question is whether as the generalization to the 3 lattice sites Hamiltonian is as simple forward as this:

Of course the $$R_Z$$ gate runs for $$2t$$ according the unitary $$U(t)$$. Finally, does this generalize as simply to the $$n$$ lattice site Ising model?

If you know the circuit that corresponds to the unitary $$U_{zz}$$ for $$Z_1Z_2$$, then you also can also be sure that the same circuit will correspond to $$U_{zz}$$ for $$Z_2 Z_3$$, except that the circuit will apply to qubits 2 and 3 instead of to qubits 1 and 2. This generalizes to any two qubits in your $$n$$-site Ising model, for example $$U_{zz}$$ for $$Z_4Z_{13}$$ will have the same circuit as $$U_{zz}$$ for $$Z_1Z_2$$ except that the gates will apply to qubits 4 and 13 instead of 1 and 2.

The last remaining aspect of your question is perhaps themost interesting one. If you can factorize a unitary $$U$$ into $$U=U_AU_B$$, and you know the circuit that corresponds to $$U_A$$ and $$U_B$$, then yes you implement each circuit one after the other, and since in your case $$U_A$$ and $$U_B$$ commute, it doesn't even matter the order in which you implement them, meaning that you could also switched the order so that your sub-circuit for $$U_{zz}$$ on qubits 1 and 2 would be switched with your sub-circuit for $$U_{zz}$$ on qubits 2 and 3. This also generalizes to $$n$$-sites in your model, or $$n$$ different unitary operators and their corresponding $$n$$ sub-circuits!

If you're ever not sure about something like this, you can always double-check yourself in Octave-Online, for example try this code:

Z = [ 1 0 ; 0 -1];                % Define  Sigma Z
Z1 = kron(Z,eye(4));              % Define Z1 = Z \otimes I \otimes I
Z2 = kron(eye(2),kron(Z,eye(2))); % Define Z2 = I \otimes Z \otimes I
Z3 = kron(eye(4),Z);              % Define Z3 = I \otimes I \otimes Z
UA = expm(-1i*Z1*Z2);             % Define UA = exp(-i*Z1*Z2*I)
UB = expm(-1i*Z2*Z3)              % Define UB = exp(-i*I*Z2*Z3)
isequal(UA*UB,UB*UA)              % Check if UA and UB commute


The answer you'll get is 1, whcih means that $$U_A$$ and $$U_B$$ do multiplicatively commute, and therefore it's okay to switch $$U_A$$ and $$U_B$$.

You can also verify that this circuit will correctly implement $$U_AU_B$$ by defining the gates and circuits as follows:

CNOT = [1 0 0 0 ; 0 1 0 0 ; 0 0 0 1; 0 0 1 0 ];
CNOT_12 = kron(CNOT,eye(2));                    % CNOT on q1q2
CNOT_23=kron(eye(2),CNOT);                      % CNOT on q2q3
RZ_piBy2=[exp(-1i*pi/4) 0 ; 0 exp(1i*pi/4)]*2;  % Rz(pi/2) for 2 units of time
RZ_qubit2=kron(eye(2),kron(RZ_piBy2,eye(2)));   % Rz on q2
RZ_qubit3=kron(eye(4),RZ_piBy2);                % Rz on q3
CNOT_12*RZ_qubit2*CNOT_12*CNOT_23*RZ_qubit3*CNOT_23


You can also see some related questions on the topic:

• But is this the optimal circuit? In the # of CNOTS for example or $R$s. Commented Dec 31, 2021 at 13:53
• That's a diffetent question. I'd recommend to post it separately since the question I answered asked nothing about optimality! Commented Dec 31, 2021 at 17:47