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quantum-computingq#qubit

Quantum V gate 1/sqrt(5) ( I + 2iZ)

By definition, the gate 1/sqrt(5) (I + 2iZ) should act on a qubit a|0> + b|1> to transform it into 1/sqrt(5) ((1+2i)a|0> + (1-2i)b|1>) but transformations of each RUS step does the following- The ancillas are in |+> state at first

  1. Starting form: 1/sqrt(2) (a,b,a,b,a,b,a,b)
  2. CCNOT(ancillas, input): 1/sqrt(2) (a,b,a,b,a,b,b,a)
  3. S(input): 1/sqrt(2) (a,ib,a,ib,a,ib,b,ia)
  4. CCNOT(ancillas, input): 1/sqrt(2) (a,ib,a,ib,a,ib,ia,b)
  5. Z(input) : 1/sqrt(2) (a,-ib,a,-ib,a,-ib,ia,-b)

Now measuring the ancillas in PauliX basis is equivalent to PauliZ measurement after applying H() to the state. Now I have 2 confusions, should I apply H x H x I or H x H x H to the combined state. Also neither of these transformations turn out to be equivalent to the V-gate defined in the first paragraph when both measurements are Zero. Where did I go wrong?

Reference: https://github.com/microsoft/Quantum/blob/master/samples/diagnostics/unit-testing/RepeatUntilSuccessCircuits.qs (1st sample code)

Solution

  • The transformation is correct, though it takes some time with pen and paper to verify it.

    As a side note, we start with a state |+>|+>(a|0> + b|1>), which is 0.5 (a,b,a,b,a,b,a,b) in vector form (both |+> states contribute a 1/sqrt(2) to the coefficients). It will not affect our calculations of the state after the measurement, since it will have to be renormalized, but it's still worth noting.

    After a sequence of CCNOT, S, CCNOT, Z we get 0.5 (a,-ib,a,-ib,a,-ib,ia,-b). Since we're measuring only the first two qubits in PauliX basis, we need to apply Hadamards only to the first two qubits, or H x H x I to the combined state.

    I'll take the liberty to skip writing out the whole expression after applying Hadamards and fast-forward to the results of measurements, and here is why. We're only interested in the state of the input qubit if both measurements yielded 0, so it's sufficient to gather only the terms of the overall state which have |00> as the state of the first two qubits.

    The state of the third qubit after measuring |00> on the first qubit will be: (3+i)a |0> - (3i+1)b |1>, multiplied by some normalization coefficient c. c = 1/sqrt(|3+i|^2 + |3i+1|^2) = 1/sqrt(10)).

    Now we need to check whether the state we got, |S_actual> = 1/sqrt(10) ((3+i)a |0> - (3i+1)b |1>) is the same state as we'd expect to get from applying the V gate, |S_expected> = 1/sqrt(5) ((1+2i)a |0> + (1-2i)b |1>). They do not look the same, but remember that in quantum computing the states are defined up to a global phase. Thus, if we can find a complex number p with an absolute value 1 for which |S_actual> = p * |S_expected>, the states will be effectively the same.

    This translates into the following equations for p and amplitudes of |0> and |1>: (3+i)/sqrt(2) = p (1+2i) and -(3i+1)/sqrt(2) = p (1-2i). We solve both equations to get p = (1-i)/sqrt(2) which has indeed the absolute value 1.

    Thus, we can conclude that indeed the state we got after all the transformations is indeed equivalent to the state we'd get by applying a V gate.