# Noise Characterization

Description of the process and algorithms used in the noise characterization.

# General

Noise can be characterized through application of appropriate controls, measurement of experimental system response, and data fusion of measurement outcomes. At this stage controls are restricted to single-qubit driven rotations employed to characterize global noise in a system. Specific control protocols are presented for the two dominant sources of noise described for the single-qubit noise Hamiltonian described in detail; multiplicative amplitude noise on the control field itself (coaxial with the applied driven rotation), and additive dephasing noise ($\propto\sigma_{z}$).

The noise characterization feature is performed in three stages

• Configure: You can pick the type of noise you want to characterize, the amount of detail you need and the best algorithm for your needs.

• Measure: A set of controls will be provided. These should then be applied to your qubits and measure the results. More details in the measurements section

• Reconstruction: The measurements can then be uploaded to the feature where the appropriately matched inversion algorithm is used to reconstruct a noise spectral density.

# Noise Sources

There are two types of noise sources that can be measured as defined below. Both will characterize the target probe qubit that is measured. Hence, it is recommended when characterizing a full quantum computer that each individual qubit is characterized.

## Multiplicative Amplitude Noise

Amplitude noise affecting a single qubit as defined in Noise Hamiltonians , can be reconstructed with the shaped pulses combined with the singletaper or multitaper reconstruction algorithm.

## Ambient Dephasing Noise

Ambient dephasing affecting a single qubit as defined in Noise Hamiltonians , can be reconstructed with the Shaped pulses combined with the suter alvarez reconstruction algorithm .

# Characterization Controls

The following characterization controls are available.

## Pulse Decoupling

The general protocol employs pulsed Carr-Purcell-Meiboom-Gill dynamic decoupling used for ambient dephasing noise, and follows technical procedures whose physical foundations are outlined in previously published research.

Note that the CPMG sequences are not spectrally concentrated, so an inversion algorithm, like Suter Alvarez is required to get an noise spectral density.

We require you specify the following parameters of your device, then we provide an appropriate set of CPMG sequences that will give an accurate reconstruction.

1. Maximum Rabi Rate :Maximum Rabi Rate is the the maximum possible amplitude of the driving field in a real hardware system. We denote this upper bound as $Ω_{max}$.

2. Maximum Achievable Experimental Duration :This is the maximum possible duration of the control experiment. We denote this as $T_{max}$

3. Number of Bands :The number of discrete frequencies measured for ambient dephasing noise characteristics. This number indicates the frequency resolution available to the protocol to estimate dephasing noise. We denote this as $N_{bands}$

The protocol returns $N_{bands}$ number of Carr-Purcell-Meiboom-Gill decoupling sequences. Each of these decoupling sequences operates in one of the $N_{bands}$ frequencies. Carr-Purcell-Meiboom-Gill sequences are characterized by a pre-set of number of quantum bit flips or $\pi$-rotations. These rotations are spaced equally within the duration of $T_{max}$. Therefore, the number of the pulses and the spacing between the pulses define a Carr-Purcell-Meiboom-Gill decoupling sequence and its corresponding operating frequency.

The number of $\pi$-rotations in $i-th$ Carr-Purcell-Meiboom-Gill sequence is

where $i\in {0,1,2,\ldots,N_{bands}-1}$ and

Time instances of each of the $\pi$-rotations are spaced equally with interval $\frac{1}{N_{R,i}$ starting from $\frac{1}{2N_{R,i}}$

The operating frequency of the $i-th$ Carr-Purcell-Meiboom-Gill decoupling sequences is

The protocol collects the experimental measurements corresponding to each of these operating frequencies to estimate the ambient dephasing noise characteristics.

These should be used to perform a set of Measurements . Then they can be reconstructed with the Suter-Alvarez algorithm .

## Shaped Pulses

Here, amplitude modulated control waveforms defined by so-called Slepian functions are used to window the sensitivity of a driven qubit in frequency space to multiplicative noise in the control field. Our implementation is based on previously published research implementing the multitaper method.

To produce an appropriate set of pulses to your device you must provide the following parameters.

1. Maximum Rabi Rate :Maximum Rabi Rate is the the maximum possible amplitude of the driving field in a real hardware system. We denote this upper bound as $Ω_{max}$.

2. Minimum Timing Resolution: This is the minimum discrete time interval allowed in the experiment between samples of control signal. We denote this as $\Delta t$.

3. Maximum Achievable Experimental Duration :This is the maximum possible duration of the control experiment. We denote this as $T_{max}$.

4. Minimum Frequency, Maximum Frequency and Number of Bands: The minimum and maximum frequency values indicate a bound in the frequency domain within which the protocol will be applied. The Number of Bands is the the number of discrete frequencies measured between the given frequency bounds and indicates the frequency resolution available to the protocol to estimate amplitude noise. We denote these as $f_{min}$, $f_{max}$ and $N_{bands}$ respectively.

The pulses are a set of Discrete Prolate Spheroidal Sequences(DPSS) . The DPSS sequences are shown to have energy maximized within a predefined narrow frequency band and minimal out-of-band harmonics. The protocol utilizes this property for an accurate estimation of the noise.

DPSS is a time domain sequence consisting of $N=\frac{T_{max}}{\Delta t}$ elements. The sequence is defined as the real solutions to the eigenvalue problem

where

• $\nu_n^{(k)(N,W)}$ is the $n-th$ element of $k-th$ order DPSS, $n\in {0,1,2,\ldots, N-1}$ and $k\geq 0$
• $W \in (0,\frac{1}{2})$ is the half-bandwidth parameter

The discrete Fourier Transform of $\nu_n^{(k)}(N,W)$ is the real-valued Discrete Prolate Spheroidal Wave Function (DPSWF). We denote this as $U^{(k)}(N,W;l)$ where $l$ is the angular frequency. By definition,

where $l_k = 1(i)$ for even (odd) $k$. This wave function is spectrally concentrated within the frequency band $[-l_B, l_B]=[-2l\frac{W}{\Delta t}, 2l\frac{W}{\Delta t}]$. The DPSS sequence creates a filter function given by

The filter function $FF_{\Omega}(l)$ is spectrally concentrated with the center at $l=0$. In order to sense other frequencies, an amplitude modulation of $\nu_n^{(k)(N,W)}$ is performed as following:

where

• $f_s = f_{min} + (s-1)f_{interval}$, $s\in 0,1,2,\ldots,N_{bands}$, and
• $f_{interval}=\frac{f_{max}-f_{min}}{N_{bands}-1}$.

The co-sinusoidal amplitude modulation shifts the center of the pass band by an amount of $2\pi f_s$. Therefore, by changing $f_s$, the DPSS sequence can be made sensitive at multiple frequencies.

The protocol collects the experimental measurements corresponding to each of these sensitive frequencies to estimate the amplitude noise characteristics. The protocol allows a choice of DPSS order $k=0$ (single taper) or $k\in 0,1,2$ (multitaper)

# Measurements

Th set of controls provided are expected to be applied to the qubit you want to characterize.

Descriptions of the controls provided are in control formats . There is also advice on how to use the file formats provided.

Each control is designed to perform a total identity operation on the qubit if the system is perfect. Any deviation from an identity operation indicates the presence of some noise in the system. Each control is also designed to have a filter function with sensitivity to noises at different frequencies. Hence by determining the deviation from an identity operation for a set of controls a noise spectrum can be reconstructed.

Determining the deviation from identity is as simple as follows:

1. Prepare a qubit in a $| 0 \rangle$ state.

2. Immediately one of the provided controls.

3. Immediately after the control is applied determine the probability that the qubit is in a $| 0 \rangle$. Completing this step may involve a repeated set of preparations and projective measurements. Once the probability has been determined the measurement value is defined as $M_i = 1 - P(|0\rangle)$. If you are also able to produce an estimate of uncertainty in the measurement $\Delta M_i$ this can also be provided and will be appropriately propagated through the reconstruction algorithm.

4. Steps 1. to 3. must be repeated for each control. until you have a full set of measurements $\{ (M_1, \Delta M_1), (M_2, \Delta M_2) \cdots, (M_{N_M}, \Delta M_{N_M}) \}$ where $N_M$ is the number of measurements.

The measurements must then be returned to the noise characterization feature. Details on input and output files is provided in file formats .

# Reconstruction

Given a set of measurements of a particular set of characterization control, the noise characterization can reconstruct a noise spectral density . Some reconstruction algorithms are only compatible with a particular protocol for creating of characterization controls.

## Pulsed Decoupling

The pulse decoupling characterization protocol can be reconstructed using the
reconstruction algorithm originally proposed by Suter and Alvarez .

The algorithm makes a variety of key assumptions to make an accurate reconstruction.

1. The noise spectral density changes slowly between the samples frequencies.

2. The noise spectral density goes to zero quickly after the highest frequency.

When these assumptions are satisfied the reconstructions is normally faithful.

The reconstruction algorithm is implemented according to the original publication. Each of the CPMG sequences have a filter function with a peak at one of the sampled frequencies, and harmonics at appropriate higher frequencies. An integral of the filter function is taken about each sample frequency to determine the sensitivity of pulse sequence to noise about that frequency. Each these weights are formed into a matrix and then inverted. This inversion matrix is then applied to the measurements to reconstruct the noise spectral density.

It is assumed that the uncertainties provided are uncorrelated and approximately follow a gaussian distribution. A co-variance matrix is then produced from the uncertainties provided and similarly transformed by the inversion matrix to get the appropriate uncertainties.

## Single/Multitaper

The shaped pulses characterization protocol can be reconstructed using the multitaper technique described here . The filter functions of the shaped pulses are spectrally concentrated, which simplifies the inversion process dramatically. The inversion process makes one key assumption:

• The noise spectral density changes slowly between the samples frequencies.

The inversion process is very simple when the multitaper option is set to false, which we refer to as the singletaper reconstruction algorithm. Otherwise the multitaper technique is used.

### Singletaper

The single taper technique simply calculates the area of the filter function for each shaped pulse. The measurements are then divided by these areas to reconstruct the noise spectral density. The uncertainties are scaled similarly.

### Multitaper

The multitaper technique uses a weighted sum of the shaped pulses with different orders at each frequency. This weighting is determined using a process described in multitaper technique described here .

The total spectral density is then a weighted sum of the measurements divided by the area of the filter function of the corresponding shaped pulse control.

The uncertainty are assumed to be approximately equivalent to the standard deviation of uncorrelated gaussian processes. The variances are then added with the same weighted sum to get the uncertainty associated with the noise spectral density.