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In the Qiskit tutorial for e.g. Grover's algortihm, there is a 3-qubit implementation on a real quantum computer https://qiskit.org/textbook/ch-algorithms/grover.html#3.1.2-Experiment-with-Real-Devices--

Since we have 3 qubits, there are 2^3=8 states. How does a practical implementation of the search come about to reconstruct all the probabilities for the 8 states when there are only 3 qubits to measure on? What is the cost of doing this, and how is it implemented for a much larger qubit case (say 50, where we have 2^50 states where we can encode information onto)?

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Welcome to QCSE!

Indeed, as you mentioned - A closed qunatum system of $3$ qubits has a state space spanned by $2^3 = 8$ computational basis states. During the computation the quantum statevector of the system can be in many linear combinations of these states. However, upon measurement in the computational basis - The statevector of the system collapses into one of the basis states and we get only one result - which has to be one of the computational basis states.

The graph posted above is showing the distribution of the results being measured. For example, if the program ran for $1000$ "shots" so the results above implies that $000$ has been measured $245$ times, $001$ has been measured $111$ times, $010$ has been measured $63$ times, and so on..

The default "shots" value in qiskit is $1024$, so I guess what we see is the distribution of the $1024$ results being measured. You can set any desired amount of shots using the backend.run(qc, shots = XX) or execute(qc, shots = XX) commands.

I would like to emphasize that quantum programs don't "reconstruct probabilities" - All we can do is maniuplating the statevector of the system throughout the computation and get a single result for each run ("shot") upon measurement.

Having said that, when using classical simulators only, we can keep track on the statevector of the system - which can be seen as a list of probability amplitudes for each computational basis state. The probability to measure each basis state is the magnitude squared of its probability amplitude. For example let $|\psi\rangle = \alpha_{00}|00\rangle + \alpha_{01}|01\rangle + \alpha_{10}|10\rangle + \alpha_{11}|11\rangle $ be the statevector of the system at a given instant. Then the probability to measure $|00\rangle$ is $|\alpha_{00}|^2$, the probability to measure $|01\rangle$ is $|\alpha_{01}|^2$, and so on. That's the closest thing to "extract state probabailites" being asked upon in the title - and again, it's not possible in a real quantum computer because the only information we can extract from a quantum system is a result of a measurement.

Indeed, for a system with $50$ qubits the state space is spanned by $2^{50}$ computational basis states - but it works exactly the same as in the case of $3$ qubits being explained above.

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  • $\begingroup$ So clear, thanks! I forgot that we get either 0 or 1 in all 3 states when we measure each, and the histogram is simply a recording of these ordered events. $\endgroup$
    – MBache
    Commented Aug 26, 2022 at 13:37
  • $\begingroup$ You mean in all 3 qubits, not states. $\endgroup$
    – Ohad
    Commented Aug 26, 2022 at 14:43

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