To simulate quantum system of 3 spin-1/2 particles system model on a particular state |110> over time t =0 to t = pi i.e., 𝑋𝑋𝑋 Heisenberg spin model, where each qubit represents a quantum spin-1/2 particle in a 1D chain.

Objective 1:

Create XXX Heisenberg model for 3 spin-1/2 particles in a line.

we have a Hamiltonian (𝐻Heis3), we can use it to determine how the quantum system of

3 spin-1/2 particles changes in time.

But Hamiltonian Hheis3 does not change in time.

Hamiltonian Hheis3 equation

Compute the unitary time evolution under the Hamiltonian for a given time t.

By knowing Hamiltonian, we can determine how quantum states of that system evolve in time by solving Schrodinger equation.

Calculating the time evolution operator 𝑈Heis3(𝑡).

we have the operator 𝑈Heis3(𝑡) which describes how any state of the quantum system changes in time by doing simulation on classical computer.

time evolution operator 𝑈Heis3(𝑡)

Simulate using operator 𝑈Heis3(𝑡) on target phase |110> from t =0 to t = pi (100 steps).

we have the operator 𝑈Heis3(𝑡) which describes how any state of the quantum system changes in time.

size of 𝑈Heis3(𝑡) 8X8 2³ = 8 states for N=3

time t = 0 to t = pi

Simulation Plot run on classical computer

Objective 2:

𝑈Heis3(𝑡) must be decompose into product of single and two-qubit gates using Trotterization.

we will use Trotterization for this product decomposition can be approximated with

Trotterization, where n is the number of trotter steps >=4, as n increases, the approximation becomes more accurate. with the help of it we now have approximately decomposed 𝑈Heis3(𝑡) into two-qubit gates XX(t), YY(t), ZZ(t).

N = number of repeated times

Trotterization decomposition output of 𝑈Heis3(𝑡)

Evolution of the state |110> must be done using Trotterization with at least 4 steps n>=4.

Fig above — Generated state tomography circuits to evaluate fidelity of simulation

Calculate the state tomography fidelity on noiseless, noisy, real manila backend simulators and compare.

compete based on the fidelity of the evolved state at time 𝜃=𝜋 using state tomography.

State tomography is a way of determining the exact quantum state — even phase information — at

the end of a quantum circuit by running the circuit multiple times and measuring in different bases.

State tomography will evaluate how well the quantum simulation matches to the expected state at 𝜃=𝜋.

The closer your final quantum state to the expected state, the higher the fidelity.

we will be using 3 out of 7 qubits of IBM manila [2,1,0] in order Ibmq_manila’s remaining qubits will not be evaluated by the state tomography calculation.

expected final state — |110>

noiseless simulator

QasmSimulator()

Day run: state tomography fidelity on qasm_simulator (noiseless) = 0.0003 (mean) ± 0.0002 (standard deviation)

Night run: state tomography fidelity on qasm_simulator = 0.0002 ± 0.0002

n=4 jobs executed on quantum computer using noiseless simulator

sim_noisy_manila — simulator that mimics ibmq_manila

QasmSimulator.from_backend(provider.get_backend(‘ibmq_manila’))

Day run: state tomography fidelity on qasm_simulator(ibmq_manila) = 0.0430 ± 0.0010

Night run: state tomography fidelity on qasm_simulator(ibmq_manila) = 0.0376 ± 0.0010

n = 4 jobs

manila — real manila backend

provider.get_backend(‘ibmq_manila’)

state tomography fidelity on ibmq_manila = 0.1017 ± 0.0035

n = 4 jobs