# State Preparation and Measurement Noise¶

import trueq as tq
import matplotlib.pyplot as plt
import numpy as np


The most convenient way to add measurement noise is through the add_readout_error() noise source. There are several ways to specify readout error with this method.

The most basic method is to provide a single number. In the following example, each qubit will get a (symmetric) 1% bitflip error.

sim = tq.Simulator().add_readout_error(0.01)
circuit = tq.Circuit({range(5): tq.Meas()})
sim.run(circuit, n_shots=1000)
circuit.results


Out:

Results({'00000': 949, '00001': 9, '00010': 11, '00100': 11, '00101': 1, '01000': 12, '10000': 6, '11000': 1})


We can use a dictionary to specify the readout error of particular qubits. Qubits that are not explicitly assigned a readout error get the value attached to None which is equal to 0 by default. In this example, all qubits get 1% readout error, except qubit 1 gets a 50% readout error.

sim = tq.Simulator().add_readout_error(0.01, {1: 0.5})
circuit = tq.Circuit({range(5): tq.Meas()})
sim.run(circuit, n_shots=1000)
circuit.results


Out:

Results({'00000': 501, '00001': 10, '00010': 6, '00100': 8, '01000': 456, '01001': 3, '01010': 6, '01100': 2, '10000': 3, '11000': 5})


In both of the above examples, any probability can be replaced with a pair of probabilities [p10, p01], where p10 is the probability of flipping a ‘0’ to a ‘1’ right before measurement, and p01 is the probability of flipping a ‘1’ to a ‘0’ right before measurement. In the following example, qubits get 1% readout error, except qubit 1 gets 5% readout error, and qubit 3 get asymmetric readout error of 1% on $$|0\rangle$$ and 7% on $$|1\rangle$$.

sim = tq.Simulator().add_readout_error(0.01, {1: 0.05, 3: [0.01, 0.07]})
circuit = tq.Circuit({range(5): tq.Meas()})
sim.run(circuit, n_shots=1000)
circuit.results


Out:

Results({'00000': 919, '00001': 8, '00010': 7, '00011': 1, '00100': 15, '01000': 35, '01001': 1, '01010': 1, '10000': 12, '11100': 1})


Measurement noise is applied only when a simulator’s run() method is called. Behind the scenes, measurement noise is impelemented by applying a positive-operator valued measurement (POVM) channel to the quantum state directly before measurement, thereby modifying the probability distribution from which ditstrings are sampled.

As a quick reminder (look elsewhere for a full description, e.g. Wikipedia or the lecture notes of John Watrous), a POVM is a set of positive-semi-definite matrices $$\{P_i\}_{i=1}^N$$ that sum to the identity matrix, $$\sum_{i=1}^N P_i = \mathbb{I}$$. The probability of observing the outcome $$i$$ when measuring a state $$\rho$$ is equal to $$p_i:=\operatorname{Tr}(\rho P_i)$$. One can also take a POVM defined on a single qubit, and tensor it together to get a POVM on a collection of qubits. For example, if the POVM above describes measurement on a single qubit, the probability of measuring the outcomes $$(i_0, i_1, i_2)$$ on three qubits is simply $$p_{i_0}p_{i_1}p_{i_2}$$. This is why we use the Tensor object as a primitive for specifying POVMs; its purpose is to store tensor project structures sparsely without actually taking kronecker products between subsystems unless necessary.

The add_povm() method can be used to directly add a POVM measurement channel. A POVM is specified as a Tensor object, where the input dimension is the number of POVM elements (per subsystem) and the output dimension pair is the dimension of the system.

In this example, we construct a coherent measurement error on qubit 3.

# Define a set of ideal POVM operators for each subsystem
proj0 = np.array([[1, 0], [0, 0]])  # project onto this one to get a "0"
proj1 = np.array([[0, 0], [0, 1]])  # project onto this one to get a "1"
ideal = np.array([proj0, proj1])

# Define a unitary rotation about X by 20 degrees
u = tq.Gate.from_generators("X", 20).mat

# Define noisy POVM operators by rotating the ideal POVM operators by U, which
# coherently changes the basis of the measurement
twisted = [u @ x @ u.conj().T for x in ideal]

# The spawn is the default POVM, and value customizes the POVM of specific qubit labels
povm = tq.math.Tensor(
2,  # number of POVMs per subsystem
(2, 2),  # number of POVMs, (dimensions of qubit subsystems)
spawn=ideal,  # default measurement operation
value={(3,): twisted},  # measurement operation on qubit 3
dtype=np.complex128,  # tensor is real by default
)

# initialize a simulator with the above POVM specifications and test it
circuit = tq.Circuit({range(5): tq.Meas()})
sim.run(circuit, n_shots=1000)
circuit.results


Out:

Results({'00000': 975, '00010': 25})


In this example, we add a correlated readout error on qubits (0,1).

# Define a confusion matrix for a 2-qubit system where the readout of 00, 01, 10 is
# ideal, but 11 is flipped to 01 5% of the time
confusion = [[1, 0, 0, 0], [0, 1, 0, 0.05], [0, 0, 1, 0], [0, 0, 0, 0.95]]

circuit = tq.Circuit([{(0, 1): tq.Gate.x}, {(0, 1): tq.Meas()}])
sim.run(circuit, n_shots=10000)
circuit.results


Out:

Results({'01': 526, '11': 9474})


In this example, we add a third measurement label even though the simulation is on qubits, so that the outcomes ‘0’, ‘1’, and ‘2’ are all possible.

# Here we specify that the state |0> results in the outcome '0' 99% of the time, but
# also results in the outcome '2' 1% of the time. Similarly, the state |1> results in
# the outcome '1' 70% of the time, but also results in '2' 25% of the time and '0' 5%
# of the time.
povm = tq.math.Tensor(
3,
(2, 2),
spawn=[np.diag([0.99, 0.05]), np.diag([0, 0.7]), np.diag([0.01, 0.25])],
dtype=np.complex128,
)
circuit = tq.Circuit([{1: tq.Gate.h}, {(0, 1): tq.Meas()}])
sim.run(circuit, n_shots=10000)
circuit.results


Out:

Results({'00': 5201, '20': 40, '01': 3439, '21': 30, '02': 1275, '22': 15}, dim=3)


State preparation noise can be added using the add_prep() noise source.

Note

The add_prep() noise source only takes place when a Prep() object is encountered.

We can either enter the noise as the probability of a bitflip during preparation of $$|0\rangle$$:

sim = tq.Simulator().add_prep(0.01)
circuit = tq.Circuit([{0: tq.Prep()}])
tq.visualization.plot_mat(sim.state(circuit).mat())


Or we can specify the density matrix we want to prepare with:

sim = tq.Simulator().add_prep([[0.75, 0], [0, 0.25]])
circuit = tq.Circuit([{0: tq.Prep()}])
tq.visualization.plot_mat(sim.state(circuit).mat())


We can place different preparations on different qubits. Here, we have a 1% preparation bitflip error by default, but a 20 degree rotation error on qubit 3.

u = tq.Gate.from_generators("Y", 20).mat
sim = tq.Simulator().add_prep({None: 0.01, 3: u @ np.diag([1, 0]) @ u.conj().T})
circuit = tq.Circuit([{0: tq.Prep(), 3: tq.Prep()}])
tq.visualization.plot_mat(sim.state(circuit).mat())


We can specify preparation states of larger dimension to add leakage levels.

sim = tq.Simulator().add_prep(np.diag([0.97, 0.02, 0.01]))
circuit = tq.Circuit([{0: tq.Prep()}, {0: tq.Gate.x}])
tq.visualization.plot_mat(sim.state(circuit).mat())


Note that if a preparation takes place mid-circuit, then that system is traced out (with no post-selection) and replaced by the specified preparation. This is a non-unitary action and will generally result in a mixed state.

In the following example we prepare the bell state $$|00\rangle+|11\rangle$$ and then prepare the second qubit in $$|1\rangle$$. Thus, we end up with the final state $$\frac{\mathbb{I}}{2}\otimes|1\rangle\langle 1|$$.

sim = tq.Simulator().add_prep(np.diag([0, 1]))
circuit = tq.Circuit([{0: tq.Gate.h}, {(0, 1): tq.Gate.cnot}, {1: tq.Prep()}])
tq.visualization.plot_mat(sim.state(circuit).mat())


When the simulator is asked to find the operator of a circuit containing a preparation object, the preparation object is simulated as a projection step.

plt.figure(figsize=(10, 5))

tq.visualization.plot_mat(op1, ax=plt.subplot(121))
plt.title("Superoperator without Prep() projection")

op2 = sim.operator(tq.Circuit([{0: tq.Prep()}, {0: tq.Gate.id}])).mat()
tq.visualization.plot_mat(op2, ax=plt.subplot(122))
plt.title("Superoperator with Prep() projection")


Out:

Text(0.5, 1.0, 'Superoperator with Prep() projection')


Total running time of the script: ( 0 minutes 0.399 seconds)

Gallery generated by Sphinx-Gallery