Why do I get positive amplitudes when I create a qiskit StateVectorCircuit from only negative amplitudes?

Question

I am using amplitude encoding to encode my features in a quantum circuit. With this I expect to encode e.g. 32 features in five qubits.

For the encoding I use qiskit's StateVectorCircuit. I expect to receive the same state vector when using the StateVectorCircuit with the 'statevector_simulator' as I put in.

However, I experienced, that the sign of the real part (I don't use complex numbers) doesn't matches.

Here is a small example:

state_vector = [-1/2, -1/2, -1/2, -1/2]

state_vector_circuit = StateVectorCircuit(state_vector).construct_circuit()

job = execute(state_vector_circuit, Aer.get_backend('statevector_simulator'), optimization_level=0)
result = job.result()

outputstate = result.get_statevector(state_vector_circuit)

print(outputstate)
print(state_vector)

Running this code, I receive the following output, where the second line is the expected one:

[0.5+0.j 0.5+0.j 0.5+0.j 0.5+0.j]
[-0.5, -0.5, -0.5, -0.5]

Measuring the circuit will have the same results. However, I don't want to measure it immediately instead I have this circuit as an input for my Quantum Neural Network. And, in this case I guess the sign matters.

Where is the wrong sign coming from? Is it the StateVectorCircuit or the 'statevector_simulator'? And more importantly is there a way from preventing this?

On the other hand, I figured I could use only positive amplitudes. However, I feel this would be a limitation.

Edit: I created an example jupyter notebook on my GitHub page: StateVectorCircuitTest.ipynb

Answer 1

Your target state and the state that you get only differ by a total factor of $-1$. In other words, if $|\psi\rangle$ is the state that you get and $|\psi_{t}\rangle$ is the target state, we have:

$$ |\psi\rangle = (-1)\times|\psi_{t}\rangle = -|\psi_{t}\rangle. $$ Such a overall factor is known as a global phase. People think of it as a phase, because we can write $-1 = e^{i\phi}$ for $\phi=\pi$. Note that we're not only considering global phases when $\phi$ is equal to $\pi$, but $\phi$ can be anything between $0$ and $2\pi$.

We don't care about global phases, because whenever we try to retrieve information (i.e. a measurement) from a state the global phase cannot play a role. Therefore, we normally just disregard the global phase, and we set the first entry in our statevector to a real, positive value.

The sign difference between elements in your statevector, however, does play a vital role. We call this the relative phase, as it is a phase of one element of the statevector relative to the other. Also note that if you limit yourself to only real elements (thereby setting any relative phase $\phi_{rel}$ to $0$ or $\pi$), you severely limit the quantum advantage that you can attain.

Answer 2

If your circuit is a component of a larger circuit then global phase may matter. See relevant discussions in https://github.com/Qiskit/qiskit-aer/issues/353 and https://github.com/Qiskit/qiskit-terra/issues/3083.