Methods for stabilizer states
Contents
import sys
sys.path.insert(0, '..')
import numpy as np
import qutip as qt
import src
from src import (utils, paulialg, stabilizer, circuit)
2.6. Methods for stabilizer states#
2.6.1. Measurement#
StabilizerState.measure(obs) measure the stabilizer state on a set of commuting observables.
Parameters:
obs- Observables to measure. The following types are supported:PauliList- a list of Pauli operators (user must ensure that operators in the list are commuting, otherwise they can not measured simutaneously).StabilizerState- stabilizers of a stabilizer state is always commuting, which can be treated as commuting observables for measurement.
Returns:
out- measuremnt outcome, can only be \(0\), \(1\) for independent Pauli observables on stabilizer state, where 0 means positive (+1) eigenvalue, and 1 means negative (-1) eigenvalue.log2prob- the log2 of the probability of realizing this particular outcome.
state = stabilizer.ghz_state(3)
state
StabilizerState(
+ZZI
+IZZ
+XXX)
state.measure(paulialg.paulis('ZII','IZI','IIZ'))
(array([1, 1, 1]), -1.0)
In the above example, I created a GHZ state, and did single qubit Z-basis measurements on each of the qubit. We get -1 for all qubits with probility \(2^{-1}=0.5\)
state = stabilizer.ghz_state(3)
state.measure(paulialg.paulis('ZII','IZI','IIZ'))
(array([0, 0, 0]), -1.0)
state.measure(paulialg.paulis('ZII','IZI','IIZ'))
(array([0, 0, 0]), 0.0)
We see the second measurement will return the same measurement result with 100% probability
2.6.2. Expectation#
StabilizerState.expect() provide fast algorithm to evaluate expectation value of:
Pauli operator
Pauli Monomial/Polynomial
Overlap between stabilizer states Warning: currently it only supports pure state overlap with general stabilizer states.
state = stabilizer.ghz_state(5)
state.expect(paulialg.paulis('XXXXX','IZIII','-ZZIII'))
array([ 1, 0, -1])
pauli_poly = 0.5*paulialg.pauli('XXXXX')+0.2j*paulialg.pauli('-ZZIII')
state.expect(pauli_poly)
(0.5-0.2j)
2.6.3. Overlap between two stabilizer states (fidelity)#
state = stabilizer.ghz_state(3)
state2 = stabilizer.random_clifford_state(3)
print("State overlap is:", state.expect(state2))
State overlap is: 0.0
tate = stabilizer.ghz_state(3)
state2 = stabilizer.random_clifford_state(3)
state2.set_r(1)
print("State overlap is:", state.expect(state2))
State overlap is: 0.125
2.6.4. The probability of getting a bit-string readout (Bit-string probability)#
StabilizerState.get_prob(bitstring) will return the probability of measuring given bitstring at current stabilizer state.
Input
bitstring: Integer type 1D array Output
probability
state = stabilizer.ghz_state(625)
state.get_prob(np.ones(625))
0.5
2.6.5. Transform stabilzier state by rotation/Clifford transformation#
As a derived class of PauliList, StabilizerState also support StabilizerState.rotate_by(pauli) and StabilizerState.transform_by(clifford_map)
rho = stabilizer.ghz_state(4)
rho.rotate_by(paulialg.pauli([1,3,1,0]))
StabilizerState(
-YIXI
-XIYI
-XZYZ
+IYIX)
rho.transform_by(stabilizer.random_clifford_map(4))
StabilizerState(
+IIIZ
-YXZZ
+ZIXI
-XIYZ)
2.6.6. Sample stabilizers from the stabilizer group#
Stabilizer.sample(L) will sample \(L\) stabilizers in the stabilizer group
rho = stabilizer.ghz_state(4)
rho.sample(4)
+IIZZ
-YXXY
-YXYX
-XXYY
2.6.7. Get density matrix of a stabilizer state#
@property StabilizerState.density_matrix will return density matrix of stabilizer state as PauliPolynomial
stabilizer.ghz_state(2).density_matrix
0.25000 II +0.25000 XX +0.25000 ZZ -0.25000 YY
2.6.8. Calculate Entropy of a stabilizer state#
StabilizerState.entropy(subsys) will calculate the second Renyi entropy of the stabilizer state on subregion.
Parameter
subsys: List containing the location of region
state = stabilizer.random_clifford_state(20)
state.entropy([1,5,10])
3
2.6.9. Copy a stabilizer state#
StabilizerState.copy() returns a copy of the state, such that the original state will not be touch by modification on the copy state. It is useful to copy the state for measurement (as measurement changes the state).
Example:
state = stabilizer.random_clifford_state(4).copy()