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By preparing $n$ qubits in $|0 \rangle ^{\otimes n}$ and then passing through the Hadamard gate, one can obtain a system superposition in $\{0,1 \}^n$, each state with equal probability.

I’m wondering if it’s possible to prepare the system in a state where some of the states $|i\rangle$ have a probability of zero.

And $i$ need not be constant in some digit.

Using two qubits as an example: input $|00 \rangle$, expecting $|\psi \rangle = \frac{1}{\sqrt{3}}(|10\rangle +|01\rangle+|00\rangle)$. Do there exists some gate doing this?

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  • $\begingroup$ What do you mean “has probability $0$”? You can choose to apply an arbitrary gate to any qubit that puts it into an unequal superposition. There are a number of questions on this site that address how to do that. $\endgroup$
    – Mark Spinelli
    Jun 5, 2022 at 4:18
  • $\begingroup$ For example, two qubits has zero probability in 00, but then 01, 10 are allowed $\endgroup$
    – yupbank
    Jun 5, 2022 at 4:20
  • $\begingroup$ you can find a unitary sending any initial state into any final state, if that's what you're asking. Or more precisely put, given any pair of states $|\psi\rangle,|\phi\rangle$, there is some unitary $U$ (actually, infinitely many such $U$) such that $U|\psi\rangle=|\phi\rangle$. See eg quantumcomputing.stackexchange.com/q/15807/55 $\endgroup$
    – glS ♦
    Jun 5, 2022 at 10:40
  • $\begingroup$ In the general case, there is no single answer on how to create such states efficiently and it is still an ongoing field of research. A related question to your is this one: quantumcomputing.stackexchange.com/questions/15692/… which may provide some insight into why it is not an obvious solution $\endgroup$
    – sheesymcdeezy
    Jun 5, 2022 at 11:33
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    $\begingroup$ Possible duplicate of quantumcomputing.stackexchange.com/q/17358/10454 $\endgroup$
    – Tristan Nemoz
    Jun 5, 2022 at 17:24

1 Answer 1

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you can check out qiskit textbook,

  1. https://qiskit.org/textbook/ch-gates/phase-kickback.html
  2. https://qiskit.org/textbook/ch-quantum-hardware/density-matrix.html#5.-Mixed-States-in-the-Bloch-Sphere--

This is how you do it:

import numpy as np
from qiskit import QuantumCircuit, QuantumRegister, assemble, Aer

q = QuantumRegister(2, name = 'q')
circuit = QuantumCircuit(q)   

#Define initial state    
initial_state = [1/np.sqrt(3), 1/np.sqrt(3),1/np.sqrt(3),0]   
circuit.initialize(initial_state, [0,1])   

circuit.decompose().decompose().decompose().decompose().decompose().draw("mpl")

enter image description here

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