Understanding Paddle Quantum's MBQC simulation

Question

Paddle Quantum have a toolkit for simulating measurement-based QC patterns in Python, and I'm having a hard time understanding how this works.

For example, the following code snippet considers the pattern of two qubits joined by an edge. We initialise the two qubits in the state $|+\rangle |+\rangle$, before applying a measurement on the 1st qubit with respect to the $\{ |+\rangle, |-\rangle \}$ basis:

from paddle import to_tensor
from paddle_quantum.mbqc.simulator import MBQC
from paddle_quantum.mbqc.qobject import State
from paddle_quantum.mbqc.utils import basis

# Define MBQC graph as two vertices joined by an edge
G = [['1','2'], [('1','2')]]
# Initialise MBQC model
mbqc = MBQC()
# Set the graph state
mbqc.set_graph(G)

# Define input state as |+>|+>
input_state = to_tensor([[0.5],[0.5],[0.5],[0.5]], dtype='complex128')
input_state = State(input_state, ['1','2'])
# Set the input state
mbqc.set_input_state(input_state)

# Print initial state
print('Input state:')
print(mbqc.get_quantum_output().vector.numpy())
# Measure qubit 1
mbqc.measure('1', basis('XY', to_tensor(0, dtype='float64')))
# Print measurement outcome and resultant state of qubit 2
print('Measurement outcome of 1st qubit:')
print(mbqc.sum_outcomes(['1']))
print('Post-measurement state:')
print(mbqc.get_quantum_output().vector.numpy())

As I understand it, the measurement outcome of the first qubit should always be 0, since the first qubit has state $|+\rangle$ and we are measuring in the $\{ |+\rangle, |-\rangle \}$ basis. However, the code above sometimes gives a measurement outcome of 1 in the console:

Input state:
[[0.5+0.j]
 [0.5+0.j]
 [0.5+0.j]
 [0.5+0.j]]
Measurement outcome of 1st qubit:
1
Post-measurement state:
[[0.+0.j]
 [1.+0.j]] 

Hence, my question is simply: which part of the above simulation am I misunderstanding?

Answer 1

Thanks very much for using our MBQC toolkit, and also thanks for your feedback. In the first part of your code, you set the underlying graph of your MBQC algorithm.

# Define MBQC graph as two vertices joined by an edge
G = [['1','2'], [('1','2')]]
# Initialise MBQC model
mbqc = MBQC()
# Set the graph state
mbqc.set_graph(G)

So you have set two vertices and an edge between them. This edge represents a controlled-Z gate between these qubits.

All the vertices are initially plus states in MBQC by default. But we do allow users to replace it to any given state for flexibility. This is equivalent to specify an input state in the quantum circuit model. In your case, you set these two vertices again to two plus states, which means doing nothing.

# Define input state as |+>|+>
input_state = to_tensor([[0.5],[0.5],[0.5],[0.5]], dtype='complex128')
input_state = State(input_state, ['1','2'])
# Set the input state
mbqc.set_input_state(input_state)

Then you apply X measurement on the first qubit.

# Measure qubit 1
mbqc.measure('1', basis('XY', to_tensor(0, dtype='float64')))

This is equivalent to perform a Z measurement on a bell state 1/sqrt(2) (|00>+|11>) (initialize two plus state, perform a CZ gate and the measure the first qubit in X basis). So you get zero or one with equal probability. If you get outcome 0, the post-measurement state on the second qubit is |0>. If you get outcome 1, the post-measurement state is |1>.

So I guess you miss the CZ gate here. This gate is automatically applied when you measure a qubit.

Also note that once a graph is specified, the computation is completely driven by the measurements. In your code

# Print initial state
print('Input state:')
print(mbqc.get_quantum_output().vector.numpy())

no measurement has been applied yet. So the computation does not get started. The state you print is exactly the input state you set.

Hope everything makes sense now. Let me know if you need more help.

We do hope to make Paddle Quantum better together with the community!