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Investigation of IBMQ quantum device hardware calibration with Markovian master equation.

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Where use has been made of the work of others, this is duly noted in the text. The background and theoretical concepts are the focus of the following sections, which will be followed by a review of the current literature.

Gate-based Qubit Operations

The quantum version of the XOR circuit is created by the unity-controlled-not, or CNOT (also called CX in gate terminology), gate. One such gate is the phase gate, which applies a rotation about the Z-axis of the Bloch sphere, with an angle of ⇡/2,.

Figure 1: A Bloch sphere representation of the states | 1 i (a), | + i (b), and | i i (c).
Figure 1: A Bloch sphere representation of the states | 1 i (a), | + i (b), and | i i (c).

Superconducting Qubits

Further generalization is required in the full definition of the QHO from the LC circuit, but those details will be omitted here as they are described in other sources [99, 1010]. 2The concept of the Hamiltonian is shared between classical and quantum mechanics, but is however constructed in the quantum operator formalism.

Transmon Qubits

To introduce the desired anharmonicity into the system developed so far, a new circuit element is introduced: the Josephson junction. For example, looking at the two-qubit case, the dimensionality of the Hilbert space must be expanded so that the Pauli becomes Z operators in the Hamiltonian.

Figure 2: Energy level diagram depicting the di↵erence in distances between the quantum states for the linear QHO inductor and the non-linear transmon qubit circuit.
Figure 2: Energy level diagram depicting the di↵erence in distances between the quantum states for the linear QHO inductor and the non-linear transmon qubit circuit.

Qubit Drive and Control

This part depends on the placement of the qubits in the device and as such has no general form. You can get more insight by looking at this quantity in the actual drive frame instead of the system frame.

Density Operator Formalism

In real quantum devices, there are many more degrees of freedom to take into account, especially considering the fragile nature of quantum states. This is essentially an extension of the normalization condition of (33), which means that the quantum states cannot simply vanish from the Hilbert space for this kind of quantum system.

Quantum Channels

Moreover, the density operator allows a closer analysis of the subsystems of the larger composite quantum system. This will be extremely useful in the study of OQS, and as an example of its use with the qubit states described so far, the density operator allows the study of just one or two qubits from a much larger device.

Quantum Master Equations

The composite Hilbert space of the two systems is defined based on the Hilbert space of the subsystems. A simplified view of the result of a dynamic map like this is shown in Figure 33. The operators of the environment have a simpler behavior, and are expressed in the interaction view as. 84).

The intrinsic evolution time of the system, ⌧S, and the relaxation time of the open system, ⌧R. This is the general form of the quantum master equation used to describe the evolution of the density matrix for simple open systems, such as those of quantum computers.

Figure 3: A commutative diagram showing the action of a dynamical map ⇤(t). Adapted from [15 15].
Figure 3: A commutative diagram showing the action of a dynamical map ⇤(t). Adapted from [15 15].

Quantum Noise

The first of the noise errors we will discuss are bit-flip/phase-flip channels. A bit-flip channel is a qubit state error that is released or excited between states |0i and |1i, as shown in the left panel of Figure 44, with probability 1 p, so that the elements of the operation. A quantum channel is again defined in terms of a 1 p depolarization probability expressed as

The depolarization channel is the process of di↵using a system into a fully mixed state, in equilibrium with its environment. The performance elements are also analogous to the amplitude damping channel, since in this case they can be defined according to the probability of photon scattering in the system without energy loss.

Figure 4: A Bloch sphere representation of the noise channels of bit-flip (a) and amplitude damping (b), respectively
Figure 4: A Bloch sphere representation of the noise channels of bit-flip (a) and amplitude damping (b), respectively

Relaxation and Decoherence

Decoherence is the combination of starting in the ground state, moving to the transverse axis, dephasing and going back to the ground state. This is an example of a phase-damping form of noise, so that no energy is lost to the environment, and as such is an elastic process that can be reversed. Transverse relaxation, or decoherence, (Figure 55 right panel) is the combination of the two, as the qubit undergoes dephasing in the transverse state and then relaxes to the |0i state.

This type of relaxation is what is known as a phase-breaking process, because the transition to the ground state, |0i, removes all history of the direction and phase the qubit had in the transverse state. To avoid these noise fluctuations, the T2 experiment can be fitted to the Hahn echo sequence.

Figure 5: A Bloch sphere representation of the noise processes of relaxation (a), pure dephasing (b), and decoherence (c), respectively
Figure 5: A Bloch sphere representation of the noise processes of relaxation (a), pure dephasing (b), and decoherence (c), respectively

Quantum Process Tomography

The 1/f noise of the system generates a coherence function, N(t), for the N ⇡-pulses that are used to refocus the noise. As this process depends on the statistical rigor of the calculations to produce acceptable results, it is important to determine how this success can be measured in tomography. After that, state tomography can theoretically be used to characterize each of the states and construct E (| iih i|), completing the process.

The important result of the practical procedure of process tomography is the scaling in the size of the system. To cover these discussions of topics that will be used in the results of this work, more recent work focused on different aspects of qubit dynamics is required.

Relaxation and Decoherence

Further research on the relaxation and decoherence rates of qubits was carried out by Smirnov, in which the authors considered the expansion of the qubit system to be in a thermal bath of non-zero temperature, giving rise to Rabi oscillations. However, it is noted that there is a need for the description of the qubit dynamics in the presence of a resonant drive. The author goes on to demonstrate that the Rabi oscillations eliminate the divergence in the decoherence of the qubit due to flicker noise.

Finally, the author analyzes the necessary conditions of the Rabi oscillations in the phase qubit with a resonance frequency equal to ⌦R when connected to an LC circuit. Using these results, the authors were able to extract various parameters of the qubit dynamics while fitting various noise functions to the experimental data to quantify their influence on the system and find which noise sources are dominant.

Characterisation and Tomography

This measure of fidelity can be formalized and described quantitatively, in terms of the eigenvalues ​​of the input state. This is used as the starting point for the development of a new method that modifies the maximum likelihood function to include uncertainty from the SPAM gates in the tomography process. The self-consistent method outperforms SQPT in estimation accuracy while using the same number of experiments, with only polynomial incrementing in the classical post-processing sequence.

Through implementations of this method on various quantum circuits, including SQPT gates and a two-qubit version of Grover's algorithm, on IBM's 5-qubit device, the authors found substantial improvements in the device's performance. This programming package enables the precise control and design of the characteristics of microwave pulses used to perform gate checks on the qubits in the quantum devices.

Markovianity and Non-Markovianity

The distinction between Markovian and non-Markovian evolution of quantum devices is a well-researched area in its own right, investigating the mechanisms that give rise to the regimes, as well as how to accurately describe them and model systems based on the systems' memory . . The authors also apply the TTM spectroscopy method to theoretical models, thereby demonstrating an efficient non-Markovian assessment scheme of the quantum process that reconstructs the noise correlation functions for two-qubit circuits where the noise influences are most prevalent. A1,A0}, which are control operations representing all possible manipulations of the system, for any time step, 0 k < K (for K total time steps in the quantum process), to the state⇢k, as.

The authors rigorously develop the mathematical framework of the process tensor, to demonstrate that it is compatible with the broader formalism of non-Markovian processes, and open a way for much broader process characterization of quantum dynamics. This playing field of open access cloud-based experimentation allows for the additional investigation of independent validation of the results demonstrated in this work, as well as the ability to modify the procedures presented in upcoming sections to the researcher's needs.

Methodology

The Hamiltonian is worthy of so much attention and research because it plays a critical role in modeling the qubit dynamics for this study. A solution to this equation can be found for a set of parameters, ~x, which depends on the form of the main equation and the Hamiltonian used. The full solution of the main equation for a set of parameters passed through the Kraus shape of the quantum channel is then a set of values ​​similar to the experimental one.

By using the parameters provided by the periodic device calibration, the master comparison solution can be directly compared to the experimental results to verify the accuracy of the calibration data. In this work, the more advanced and accurate gradient descent method of the Adaptive Moment Estimation (Adam) optimizer [6565] is used.

Experimental Procedure and Results

For example, in the case of experiments T1, the error mitigation changes in Figure 88 are made for the initial value. In applying the optimization algorithm, there was a noticeable discrepancy in its e↵ectiveness depending on the quantum channel to which it was applied. Although the T1 sequences for all qubit sizes and initial states turned out to be very.

Experimental data represented by dots and numerical data for optimized parameters represented by solid lines. stable and reliable, the rest of the experiments yielded more complicated results. This is fortunately not the case for other sequences at the 3-qubit scale, but still limits the focus of the results to the 1- and 2-qubit cases.

Figure 7: Calibration matrices for readout errors. The single-qubit calibration produced a fidelity of 0.912 (a)
Figure 7: Calibration matrices for readout errors. The single-qubit calibration produced a fidelity of 0.912 (a)

Analysis and Discussion

On the contrary, all hardware parameters underwent a significant scaling phenomenon, shifting all of them away from the claimed parameters given by the backend providers. These modifications made it possible to dampen the resonant noise of 2 qubits prone to SPAM errors, and provided a more detailed look at the qubit dynamics. This analysis also goes a step further to the 3-qubit system for an additional point of comparison of the predicted hardware parameters and behaviors.

These additional experiments revealed that the presence of pure dephasing was insignificant in modeling the qubit systems. In the main part of this work, the details of these numerical methods are excluded from the discussion for brevity of the discourse.

Table 1: Claimed and experimental results for the single-qubit experiments to measure the qubit frequency, !, relaxation time, T 1 , and qubit temperature, T
Table 1: Claimed and experimental results for the single-qubit experiments to measure the qubit frequency, !, relaxation time, T 1 , and qubit temperature, T

Adam Optimiser

Numerical Integration

The minimum least-squares error obtained in this example was a modification of the Euler method, which is the simplest method in the Runge-Kutta family of numerical integration, requiring the initial values ​​of f and f0 to be the function point and its first derivative. Grover, "A Fast Quantum Mechanical Algorithm for Database Search," in Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, pp. McRae et al., "Materials Loss Measurements Using Superconducting Microwave Resonators," Review of Scientific InstrumentsReview of Scientific Instruments 9191 (Sept.

Baumet al., "Experimental deep reinforcement learning for designing fault-tolerant gate arrays in a superconducting quantum computer." Alexander et al., "Qiskit Pulse: Programming Quantum Computers through the Cloud with Pulses", Quantum Science and Technology Quantum Science and Technology55 no.

Figure 13: Error curve of the Adam optimiser performing gradient descent to optimise the parameters of a 2-qubit T 1 sequence to minimise the di↵erence between the numerical solution and the experimental data
Figure 13: Error curve of the Adam optimiser performing gradient descent to optimise the parameters of a 2-qubit T 1 sequence to minimise the di↵erence between the numerical solution and the experimental data

Figure

Figure 1: A Bloch sphere representation of the states | 1 i (a), | + i (b), and | i i (c).
Figure 2: Energy level diagram depicting the di↵erence in distances between the quantum states for the linear QHO inductor and the non-linear transmon qubit circuit.
Figure 3: A commutative diagram showing the action of a dynamical map ⇤(t). Adapted from [15 15].
Figure 4: A Bloch sphere representation of the noise channels of bit-flip (a) and amplitude damping (b), respectively
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References

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