Initialising impure density matrices

Question

I wish to initalise the state $\rho=(1-\frac{p}{2})|0\rangle \langle0|+\frac{p}{2}|1\rangle\langle1|$, where p is some measure of decoherence. This is a mixed state. There are some suggestions on here for how to implement this with ancilla qubits and extra gates. However I am now trying to run a quantum circuit with this as my initial state on the actual IBM quantum computers. The problem is that to intialise two qubits in this state requires 6 ancilla qubits using my current approach, meaning I have to use the Melbourne quantum computer which has moderately high gate error rates. It also increases my circuit depth. In order to simplify my circuit I tried something like this

r=random.choices([0,1],weights=(1-p/2,p/2),k=1)
    r.append((r[0]+1)%2)
    circuit2 = QuantumCircuit(3,3);
    circuit2.initialize(r,0)
    circuit2.initialize(r,1)

Although this is statistically correct over many runs it does not give what I want. In each run of the quantum circuit (say 1000 shots), the same intial state is used for all 1000 shots. Is there any way I can make it so that the circuit reevaluates what the initial state should be for each shot?

I do not wish to have to set the number of shots to 1 and evaluate the circuit thousands of times, as the queue time to get my circuit evaluated would be huge.

Answer 1

Why do you need 6 ancilla qubits? Surely you'd just produce the two-qubit state $$ \sqrt{1-p/2}|00\rangle+\sqrt{p/2}|11\rangle $$ by starting with two qubits in $|00\rangle$, performing a single qubit unitary that maps $|0\rangle\rightarrow \sqrt{1-p/2}|0\rangle+\sqrt{p/2}|1\rangle$ (this is a $Y$ rotation of a suitable angle), and then perform a controlled-not controlled from that qubit, targeting the other. The reduced density matrix of either qubit is then what you want (but you should only use one of the two qubits in subsequent measurements etc).