@cirosantilli/_file/qiskit/qiskit/hello.py Updated 2024-12-15 +Created 1970-01-01
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Our example uses a Bell state circuit to illustrate all the fundamental Qiskit basics.
Sample program output, counts are randomized each time.
First we take the quantum state vector immediately after the input.
input:
state:
Statevector([1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j],
            dims=(2, 2))
probs:
[1. 0. 0. 0.]
We understand that the first element of Statevector is , and has probability of 1.0.
Next we take the state after a Hadamard gate on the first qubit:
h:
state:
Statevector([0.70710678+0.j, 0.70710678+0.j, 0.        +0.j,
             0.        +0.j],
            dims=(2, 2))
probs:
[0.5 0.5 0.  0. ]
We now understand that the second element of the Statevector is , and now we have a 50/50 propabability split for the first bit.
Then we apply the CNOT gate:
cx:
state:
Statevector([0.70710678+0.j, 0.        +0.j, 0.        +0.j,
             0.70710678+0.j],
            dims=(2, 2))
probs:
[0.5 0.  0.  0.5]
which leaves us with the final .
Then we print the circuit a bit:
qc without measure:
     ┌───┐
q_0: ┤ H ├──■──
     └───┘┌─┴─┐
q_1: ─────┤ X ├
          └───┘
c: 2/══════════

qc with measure:
     ┌───┐     ┌─┐
q_0: ┤ H ├──■──┤M├───
     └───┘┌─┴─┐└╥┘┌─┐
q_1: ─────┤ X ├─╫─┤M├
          └───┘ ║ └╥┘
c: 2/═══════════╩══╩═
                0  1
qasm:
OPENQASM 2.0;
include "qelib1.inc";
qreg q[2];
creg c[2];
h q[0];
cx q[0],q[1];
measure q[0] -> c[0];
measure q[1] -> c[1];
And finally we compile the circuit and do some sample measurements:
qct:
     ┌───┐     ┌─┐
q_0: ┤ H ├──■──┤M├───
     └───┘┌─┴─┐└╥┘┌─┐
q_1: ─────┤ X ├─╫─┤M├
          └───┘ ║ └╥┘
c: 2/═══════════╩══╩═
                0  1
counts={'11': 484, '00': 516}
counts={'11': 493, '00': 507}
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@cirosantilli/_file/qiskit/qiskit/qft.py Updated 2024-12-15 +Created 1970-01-01
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This is an example of the qiskit.circuit.library.QFT implementation of the Quantum Fourier transform function which is documented at: docs.quantum.ibm.com/api/qiskit/0.44/qiskit.circuit.library.QFT
Output:
init: [1, 0, 0, 0, 0, 0, 0, 0]
qc
     ┌──────────────────────────────┐┌──────┐
q_0: ┤0                             ├┤0     ├
     │                              ││      │
q_1: ┤1 Initialize(1,0,0,0,0,0,0,0) ├┤1 QFT ├
     │                              ││      │
q_2: ┤2                             ├┤2     ├
     └──────────────────────────────┘└──────┘
transpiled qc
     ┌──────────────────────────────┐                                     ┌───┐   
q_0: ┤0                             ├────────────────────■────────■───────┤ H ├─X─
     │                              │              ┌───┐ │        │P(π/2) └───┘ │ 
q_1: ┤1 Initialize(1,0,0,0,0,0,0,0) ├──────■───────┤ H ├─┼────────■─────────────┼─
     │                              │┌───┐ │P(π/2) └───┘ │P(π/4)                │ 
q_2: ┤2                             ├┤ H ├─■─────────────■──────────────────────X─
     └──────────────────────────────┘└───┘
Statevector([0.35355339+0.j, 0.35355339+0.j, 0.35355339+0.j,
             0.35355339+0.j, 0.35355339+0.j, 0.35355339+0.j,
             0.35355339+0.j, 0.35355339+0.j],
            dims=(2, 2, 2))

init: [0.0, 0.35355339059327373, 0.5, 0.3535533905932738, 6.123233995736766e-17, -0.35355339059327373, -0.5, -0.35355339059327384]
Statevector([ 7.71600526e-17+5.22650714e-17j,
              1.86749130e-16+7.07106781e-01j,
             -6.10667421e-18+6.10667421e-18j,
              1.13711443e-16-1.11022302e-16j,
              2.16489014e-17-8.96726857e-18j,
             -5.68557215e-17-1.11022302e-16j,
             -6.10667421e-18-4.94044770e-17j,
             -3.30200457e-16-7.07106781e-01j],
            dims=(2, 2, 2))
So this also serves as a more interesting example of quantum compilation, mapping the QFT gate to Qiskit Aer primitives.
If we don't transpile in this example, then running blows up with:
qiskit_aer.aererror.AerError: 'unknown instruction: QFT'
The second input is:
(1)
and the output of that approximately:
[0, 1j/sqrt(2), 0, 0, 0, 0, 0, 1j/sqrt(2)]
which can be defined simply as the normalized DFT of the input quantum state vector.
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qiskit.transpile() Updated 2024-12-15 +Created 1970-01-01
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This function does quantum compilation. Shown e.g. at qiskit/qft.py.
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ZX-calculus Updated 2024-12-15 +Created 1970-01-01
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How can we easily prove that that quantum circuit equals the state:
(1)
?
The naive way would be to just do the matrix multiplication as explained at Section "Quantum computing is just matrix multiplication".
However, ZX-calculus provides a simpler way.
And even more importantly, sometimes it is the only way, because in a real circuit, we would not be able to do the matrix multiplication
What we do in ZX-calculus is we first transform the original quantum circuit into a ZX graph.
This is always possible, because we can describe how to do the conversion simply for any of the Clifford plus T gates, which is a set of universal quantum gates.
Then, after we do this transformation, we can start applying further transformations that simplify the circuit.
It has already been proven that there is no efficient algorithm for this (TODO source, someone said P-sharp complete best case)
But it has been proven in 2017 that any possible equivalence between quantum circuits can be reached by modifying ZX-calculus circuits.
There are only 7 transformation rules that we need, and all others can be derived from those, universality.
So, we can apply those rules to do the transformation shown in Wikipedia:
Figure 1.
GHZ circuit as ZX-diagram
. Source.
and one of those rules finally tells us that that last graph means our desired state:
(2)
because it is a Z spider with and .
Video 1.
Working with PyZX by Aleks Kissinger (2019)
Source. This video appears to give amazing motivation on why you should care about ZX-calculus, it mentions
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