Measuring post phase shift on digital basis

[juanboschero]

Hi all,

I am quite new to the world of quantum computing with Rydberg atoms and I am trying to use Pulser to pulse elaborate qubit gates. I am trying to incorporate the single qubit Z rotation gate in the digital basis using a virtual post phase shift (as described by the docs) but upon simulating the result in the digital basis, I see that nothing changes. Is there something that I am fundamentally missing or is there another way to incorporate a Z gate for it to appear in the simulation?

Thank you very much!

[HGSilveri]

Hi @Santopelufincho, welcome !
The virtual-Z gate implementation relies on the fact that Z-rotations right before a measurement have no influence on the outcomes, so in fact this last rotation is not even done. Could it be that the last thing you are doing is the Z-rotation?
Feel free to share a snippet of your code so that we can see what you are trying out and give more concrete tips.

[juanboschero]

Oops I do not know why I thought having a 2π-pulse would work... However, let's say I apply a Y-pulse to my [[1],[0]] state, I then get a [[0],[1]]. I do this below and this works perfectly.

reg = Register({"q0": (0, 0)})
device = Chadoq2
seq = Sequence(reg, device)
seq.declare_channel("ch0", "raman_local", initial_target="q0")

def y_pulse(target,channel):
    pulse = Pulse.ConstantDetuning(BlackmanWaveform(250, np.pi), detuning=0, phase= -np.pi / 2)
    seq.target(target,channel)
    seq.add(pulse,channel)

y_pulse("q0","ch0")
seq.draw(draw_phase_shifts=True)
sim = Simulation(seq,sampling_rate=0.1)
results = sim.run(progress_bar=True)
result_basis = results.get_final_state(reduce_to_basis = "digital")
print(result_basis)

Result:

Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket
Qobj data =
[[-8.76161965e-05]
 [ 9.99999996e-01]]

However, now if I want to apply a Z rotation to this state of π/2 to get a final state [[0],[i]], how would I go around doing that? I tried applying the double hadamard but it does not seem to change my final state,

reg = Register({"q0": (0, 0)})
device = Chadoq2
seq = Sequence(reg, device)
seq.declare_channel("ch0", "raman_local", initial_target="q0")

def y_pulse(target,channel):
    pulse = Pulse.ConstantDetuning(BlackmanWaveform(250, np.pi), detuning=0, phase= -np.pi / 2)
    seq.target(target,channel)
    seq.add(pulse,channel)
    
def r_pulse(theta,target,channel): 
    pulse = Pulse.ConstantDetuning(BlackmanWaveform(250, np.pi/2), detuning=0, phase=np.pi / 2)
    seq.target(target,channel)
    seq.add(pulse,channel)
    seq.phase_shift(np.pi, "q0", basis="digital")
    
    pulse = Pulse.ConstantDetuning(BlackmanWaveform(250, np.pi/2), detuning=0, phase=np.pi / 2)
    seq.target(target,channel)
    seq.add(pulse,channel)
    seq.phase_shift(np.pi, "q0", basis="digital")

    seq.phase_shift(theta, "q0", basis="digital")
    
y_pulse("q0","ch0")
r_pulse(np.pi/2,"q0","ch0")

seq.draw(draw_phase_shifts=True)
sim = Simulation(seq,sampling_rate=0.1)
results = sim.run(progress_bar=True)
result_basis = results.get_final_state(reduce_to_basis = "digital")
print(result_basis)

Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket
Qobj data =
[[-8.77531281e-05]
 [ 9.99999996e-01]]

[HGSilveri]

[0, i] is actually the same as [0, 1], they only differ by a global factor of i.

[juanboschero]

To apply a global factor would I need to introduce the global channel? Also, would applying a global i factor work for something like a controlled phase gate? I ask because I am trying to implement the phase estimation algorithm using pulses. Below is how I apply a CPHASE gate. I first pulse both qubits with a Y-pulse to have them in the |11> state, and then I make a controlled phase gate using the identity

------------------0---------------------0
----Rz(theta)----+ -----Rz(-theta)----+

where,
---0--- is a CNOT gate in which 0 is the control and + is the target.
---+---

I implement the CNOT by using placing a Hadamard on the target followed by a controlled z between both qubits and then another Hadamard. Below is the code,

q_dict = {"q0": (0, 0),"q1": (0, 4)}
reg = Register(q_dict)
device = Chadoq2
seq = Sequence(reg, device)
seq.declare_channel("ch0", "raman_local", initial_target="q0")
seq.declare_channel("ch1", "rydberg_local", initial_target="q0")

def ry_pulse(target,channel):
    pulse = define_pulse(250,np.pi,0,-np.pi/2,0)
    seq.target(target,channel)
    seq.add(pulse,channel)

def apply_cz(channel1,channel2,control,target):
    pi_wf = BlackmanWaveform(200,np.pi)
    pi_pulse = Pulse.ConstantDetuning(pi_wf,0,0)
    max_val = Chadoq2.rabi_from_blockade(8)
    two_pi_wf = BlackmanWaveform.from_max_val(max_val=max_val,area=2*np.pi)
    two_pi_pulse = Pulse.ConstantDetuning(amplitude=two_pi_wf,detuning=0,phase=0)
    seq.align(channel1,channel2)
    seq.target(control,channel2)
    seq.add(pi_pulse, channel2)
    seq.target(target,channel2)
    seq.add(two_pi_pulse, channel2)
    seq.target(control,channel2)
    seq.add(pi_pulse, channel2)
    seq.align(channel1,channel2)    

def apply_cnot(channel1,channel2,control,target):
    apply_hadamard(target,channel1)
    apply_cz(channel1,channel2,control,target)
    apply_hadamard(target,channel1)

def apply_zrot(theta,target,channel): 
    pulse = Pulse.ConstantDetuning(BlackmanWaveform(250, np.pi/2), detuning=0, phase=np.pi / 2)
    seq.target(target,channel)
    seq.add(pulse,channel)
    seq.phase_shift(np.pi, "q0", basis="digital")
    
    pulse = Pulse.ConstantDetuning(BlackmanWaveform(250, np.pi/2), detuning=0, phase=np.pi / 2)
    seq.target(target,channel)
    seq.add(pulse,channel)
    seq.phase_shift(np.pi, "q0", basis="digital")

    seq.phase_shift(theta, "q0", basis="digital")

def apply_phase(theta,channel1,channel2,control,target):
    apply_zrot(theta,target,channel1)
    apply_cnot(channel1,channel2,control,target)
    apply_zrot(-theta,target,channel1)
    apply_cnot(channel1,channel2,control,target)

ry_pulse("q0","ch0")
ry_pulse("q1","ch0")
apply_phase(np.pi/4,"ch0","ch1","q0","q1")

seq.draw(draw_phase_shifts=True)

sim = Simulation(seq,sampling_rate=0.5)
results = sim.run(progress_bar=True)
result_basis = results.get_final_state(reduce_to_basis = "digital", tol = 1)
print(result_basis)

In this situation I get [[0],[0],[0],[1]] but if I apply a global phase, I would essentially add the phase to all states regardless if my basis was in the |11> state, right?

Thank you again for your time.

[HGSilveri]

When we speak of two states (|phi_1>, |phi_2>) being equal up to a global phase, it just means there is some phase alpha that makes
|phi_1> = exp(i*alpha) |phi_2>. In QC this global phase factors are always ignored because, when measuring some observable Ô, <phi_1| Ô |phi_1> = <phi_1| exp(-i * alpha) Ô exp(i*alpha) |phi_1> = <phi_2| Ô |phi_2>, so it can always be ignored.

What you are asking for (adding a global phase to a state) is something that is never considered in QC and there is no gate that will do so.

I understand that you are trying to see the effect of a CPHASE gate, but it will never show in an eigenstate. You need to have your state be in a superposition of basis states to see an effect. For example, if you start with

(|00> + |01> + |10> + |11>)/2

and apply CPHASE(theta) you will get

(|00> + |01> + exp(i*theta)|10> + exp(i*theta)|11>)/2

Note how the phase acquired is not global (ie it can't be factored out as a multiplicative factor on all states) and so it can't be ignored anymore.

Nonetheless, if you try this in Pulser with virtual-Z gates you might still not see this phase appear in the state vector for the same reason as before - phase gates before a measurement don't affect the result and are thus not applied. You need to apply another gate afterwards that relies on the fact that the CPHASE gate worked to reach some result (similary to what I suggested you tried out with the two Hadamards).

[juanboschero]

Hi @HGSilveri,
I think the issue I had was not fully understanding the fundamentals of tensor measurements. Moreover, I realized that the way I defined my CPHASE was wrong, by applying this instead, I managed to properly make the complete algorithm. Thanks again for all the help!