# # Copyright 2022 Keysight Technologies Inc. # """ State Preparation and Measurement Noise ======================================= """ #%% import trueq as tq import numpy as np #%% # Adding Measurement Bitflip Noise # -------------------------------- #%% # The most convenient way to add measurement noise is through the # :py:meth:~trueq.Simulator.add_readout_error noise source. There are several ways to # specify readout error with this method. Measurement noise is applied only when a # simulator's :py:meth:~trueq.Simulator.sample or :py:meth:~trueq.Simulator.run # method is called. # # 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 #%% # We can use a dictionary to specify the readout error of particular qubits. Qubits that # are not explicitly assigned a readout error get no noise, or the specified default # value. 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 #%% # 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 # :math:|0\rangle and 7% on :math:|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 #%% # Specifying Confusion Matrices # ----------------------------- #%% # We can also use the full confusion matrix anywhere in the # :py:meth:~trueq.Simulator.add_readout_error noise source. There isn't any value to # this for a single qubit confusion matrix because it is fully specified by a length-2 # vector as described above, but if we are interested in adding misclassification into a # third level, or correlated readout error, then a confusion matrix gives us control # over all the probabilities. In the following example, we specify that 0 and 1 # are misclassified as 2 with a small probability. sim = tq.Simulator().add_readout_error([[0.98, 0.06], [0.01, 0.9], [0.01, 0.04]]) circuit = tq.Circuit({range(5): tq.Meas()}) sim.run(circuit, n_shots=1000) circuit.results #%% # Similarly, in the following example, we specify a correlated readout error on # qubits 2 and 3, where the rows and columns are in the usual 00, 01, 10, 11 order. confusion_mat = [ [0.99, 0.05, 0.00, 0.00], [0.00, 0.90, 0.00, 0.02], [0.00, 0.05, 0.96, 0.00], [0.01, 0.00, 0.04, 0.98], ] sim = tq.Simulator().add_readout_error(0, {(2, 3): confusion_mat}) circuit = tq.Circuit({range(5): tq.Meas()}) sim.run(circuit, n_shots=10000) circuit.results #%% # Adding Measurement POVM Noise # ----------------------------- #%% # Finally, as a more advanced option, measurement error can also be specified as a # positive-operator valued measurement (POVM). # # 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 :math:\{P_i\}_{i=1}^N that sum to the identity matrix, :math:\sum_{i=1}^N # P_i = \mathbb{I}. The probability of observing the outcome :math:i when measuring # a state :math:\rho is equal to :math: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 :math:(i_0, i_1, i_2) on three # qubits is simply :math:p_{i_0}p_{i_1}p_{i_2}\. This is why we use the # :py:class:~trueq.math.tensor.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 :py:meth:~trueq.Simulator.add_readout_error supports POVMs in addition to all # the formats described in the previous section. A POVM is specified a 3D array where # the first index :math:i ranges over POVM elements :math:P_i\. #%% # 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_povm = [u @ x @ u.conj().T for x in ideal] # initialize a simulator with the above POVM applied to qubit 3 sim = tq.Simulator().add_readout_error(0, {3: twisted_povm}) circuit = tq.Circuit({range(5): tq.Meas()}) sim.run(circuit, n_shots=1000) circuit.results #%% # 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. This particular example could be more efficiently implemented as a confusion # matrix, but is presented as a POVM for demonstration. povm = [np.diag([0.99, 0.05]), np.diag([0, 0.7]), np.diag([0.01, 0.25])] sim = tq.Simulator().add_readout_error(povm) circuit = tq.Circuit([{1: tq.Gate.h}, {(0, 1): tq.Meas()}]) sim.run(circuit, n_shots=10000) circuit.results #%% # Adding State Preparation Noise # ------------------------------ #%% # State preparation noise can be added using the # :py:meth:~trueq.Simulator.add_prep noise source. # # .. note:: # # The :py:meth:~trueq.Simulator.add_prep noise source only takes place when a # :py:class:~trueq.Prep() object is encountered. Many circuits do not have # any :py:class:~trueq.Prep() objects unless explicitly added. # # We can either enter the noise as the probability of a bitflip during preparation of # :math:|0\rangle\: sim = tq.Simulator().add_prep(0.01) circuit = tq.Circuit([{0: tq.Prep()}]) tq.plot_mat(sim.state(circuit).mat()) #%% # Or we can specify the pure state or density matrix we want to prepare with. Here a # density matrix is used: sim = tq.Simulator().add_prep([[0.75, 0], [0, 0.25]]) circuit = tq.Circuit([{0: tq.Prep()}]) tq.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(0.01, {3: u @ np.diag([1, 0]) @ u.conj().T}) circuit = tq.Circuit([{0: tq.Prep(), 3: tq.Prep()}]) tq.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.plot_mat(sim.state(circuit).mat())