## Local Quantum State Simulation

**Note**

**Installation Required:** This functionality requires MATLAB Support Package for Quantum Computing.

This topic describes how to simulate a quantum circuit locally and analyze the simulation
results. Building quantum circuits requires iteration on circuit design to refine circuit
gates and confirm behavior. Also, because measurements on quantum hardware are probabilistic,
it is useful to examine the probabilities of final states to determine the most likely outcome
of a measurement. Use the `simulate`

method
of `quantumCircuit`

to
simulate circuits on your local computer. Once you simulate a circuit, use the methods of
`quantum.gate.QuantumState`

to inspect the results.

### Create Circuit

Create a quantum circuit with three qubits and three gates. Apply a Hadamard gate to the first qubit, and apply CNOT gates to the second and third qubits, using the first qubit as the control. Plot the circuit to view its qubits and gates.

gates = [hGate(1); cxGate(1,2); cxGate(1,3)]; C = quantumCircuit(gates); plot(C)

Each horizontal line in the plot represents one of the qubits, and the gates are arranged from left to right in the order that they are applied.

### Simulate Circuit

Simulate the quantum circuit by using the `simulate`

method. All qubits start in the $$|0\rangle $$ state by default, but you can use a second input argument to specify a
different starting state for the qubits. Specify that each qubit has an initial state of $$|1\rangle $$.

`S = simulate(C,"111")`

S = QuantumState with properties: BasisStates: [8×1 string] Amplitudes: [8×1 double] NumQubits: 3

Display the basis states and corresponding amplitudes by inspecting the properties of
the resulting `quantum.gate.QuantumState`

object.

S.BasisStates

ans = 8×1 string array "000" "001" "010" "011" "100" "101" "110" "111"

S.Amplitudes

ans = 0 0 0 0.7071 -0.7071 0 0 0

### Display State Formula

Display a formula representation of the simulated quantum state by using the `formula`

method.
The formula combines the information in the `BasisStates`

and
`Amplitudes`

properties of `S`

. By default, the
formula uses the Z basis.

f = formula(S)

f = "0.70711 * |011> + -0.70711 * |100>"

Specify the `Basis`

name-value argument to display the formula using
the X basis. The formula now shows linear combinations of $$|+\rangle $$ and $$|-\rangle $$ states.

`f2 = formula(S,Basis="X")`

f2 = "-0.5 * |++-> + -0.5 * |+-+> + 0.5 * |-++> + 0.5 * |--->"

### Plot Possible States

Use `histogram`

to
plot the possible states and their probabilities. The probability of each state is equal to
its amplitude squared. This circuit has only two possible states, and each state has a 50%
chance of being measured.

histogram(S)

### Query Possible States

You can query the possible states and their probabilities using the `querystates`

method. This method returns the same information that the histogram shows, but at the
command line.

[states,P] = querystates(S)

states = 2×1 string array "011" "100" P = 0.5000 0.5000

### Query Qubit State Probabilities

You can query the probability that specific qubits will be measured in specific states
by using the `probability`

method. For example, determine the probability that the second qubit will be measured in the $$|1\rangle $$ state.

`p = probability(S,2,"1")`

p = 0.5000

### Simulate Quantum State Measurement

Quantum measurements are probabilistic, so the results can differ between trials. The
`randsample`

method is useful for simulating the aggregated results of many such measurements. You can
specify the number of shots, or trials, and each shot returns a single state.

For example, simulate 50 quantum measurements of the circuit.

M = randsample(S,50)

M = QuantumMeasurement with properties: MeasuredStates: [2×1 string] Counts: [2×1 double] Probabilities: [2×1 double] NumQubits: 3

Display the counts and estimated probabilities of the measured states.

T = table(M.Counts,M.Probabilities,M.MeasuredStates, ... VariableNames=["Counts","Probabilities","States"])

T = 2×3 table Counts Probabilities States ______ _____________ ______ 21 0.42 "011" 29 0.58 "100"

## See Also

`quantumCircuit`

| `quantum.gate.SimpleGate`

| `quantum.gate.CompositeGate`

| `quantum.gate.QuantumState`