Fiducial and Germ Selection

Fiducial and Germ Selection#

This notebook demonstrates how to generate sets of fiducial and germ sequences which form the building blocks of the operation sequences used by long-sequence GST. As described in GST circuits tutorial, by structuring the GST sequences as

preparation_fiducial + repeated_germ + measurement_fiducial

long-sequence GST is highly sensitive to all possible (within the space of allowed models). Furthermore, by iteratively increase the number of germ repetitions in repeated_germ, pyGSTi’s iterative algorithms are able to avoid (usually!) local optima.

Both germ and fiducial sets are determined for a given “target” model. We currently assume that this model contains unitary gates, such that infinitely many gates may be performed without moving away from a pure state. It is almost always the case that the desired “target” operations are unitary, so this isn’t a burdensome assumption. If this isn’t the case, one should find perform fiducial and germ selection on the nearest unitary model.

Fiducial Selection: the theory#

The purpose of the preparation and measurement fiducial sequences, \(\{F_i\}\) and \(\{H_i\}\), is to prepare a sufficiently diverse set of input states, and a sufficiently diverse set of measurements, to completely probe an operation of interest - defined as the map that lies between the fiducials. This is achieved if (and only if) the input states \(\{\rho_i\}\equiv \{F_i|\rho\rangle\rangle\}\) and the measurement effects \(\{E_j\} \equiv \{\langle\langle E|H_j\}\) are both informationally complete (IC). A set of matrices is IC if and only if it spans the vector space of density matrices. For a Hilbert space of dimension \(d\) this requires at least \(d^2\) linearly independent elements.

In general, any randomly chosen set of \(d^2\) states or effects will be IC. So, for single-qubit GST, we could choose \(d^2=4\) random fiducial sequences. However, while the resulting \(\{\rho_i\}\) and \(\{E_j\}\) will almost certainly be linearly independent, they may be close to linearly dependent.

To evaluate a set of fiducials we form a matrix \(M\), which will allow us to quantify how linearly independent the resulting \(\{\rho_i\}\) or \(\{E_j\}\) will be. If we are evaluating a set of preparation fiducials, then the i^th column of \(M\) is \(F_i|\rho\rangle\rangle\); if measurement fiducials, the columns are \(\langle\langle E|H_i\). This notation assumes a single native preparation or measurement effect; in the case when there are more this simply adds more columns \(M\).

We then form the square matrix \(MM^T\). We either then score the fiducial set as the number of fiducials times the sum of the reciprocals of the eigenvalues of \(MM^T\) (scoreFunc = 'all') or by the number of fiducials times the reciprocal of the smallest eigenvalue of \(MM^T\) (scoreFunc = 'worst'). In both cases, a lower score is better.

In the 'all' case, we are attempting to make all the fiducials as uniformly informationally complete as possible, that is, we are trying to make our fiducial set as sensitive as possible to all directions in Hilbert-Schmidt space. In the 'worst' case, we are instead attempting minimize our insensitivity to the direction in Hilbert-Schmidt space that we are least sensitive to.

Germ Selection: the theory#

The defining property which makes a set of operation sequences a “complete germ set” is the amplification of all possible gate errors. More precisely, the repetition of a complete set of germs, sandwiched between assumed-IC preparation and measurement fiducial sequences, will yield a sensitivity that scales with the number of times each germ is repeated to any direction in the space of Models (defined by the model’s parameterization). This completeness is relative a “target” model, just as in fiducial selection. While the detailed mathematics behind germ selection is beyond the scope of this tutorial, the essence of the algorithm is as follows. The Jacobian \(J\) of a potential set of germs relative to the target Model’s parameters is constructed and the eigenvalues of \(J^\dagger J\) are computed. If the number of large eigenvalues values equals its maximum - the number of non-gauge Model parameters - then the germ set is deemed “amplificationally complete”, and will amplify any gate error. More specifically, a germ set is scored using a combination of 1) the number of eigenvalues values above some threshold and 2) either the 'all' or 'worst' scoring function applied to the eigenvalues of \(J^\dagger J\). Several technical points make this slightly more complicated:

  • only gate errors can be amplified, since only gates can be repeated. Thus, when computing the number of non-gauge parameters of the target model, we really mean the target model without any SPAM operations.

  • typical perfect gates (e.g. \(\pi/2\) rotations) may contain symmetries which decrease the number of non-gauge parameters only at that perfect point in gate-set-space. As such, we add random unitary perturbations to the target Model before performing the Jacobian analysis to mitigate the possibility of mischaracterizing a direction as being amplified when it isn’t for all Models except the perfect one.

  • In the Jacobian analysis, each germ is twirled to simulate the effect of it echoing out all directions except those that commute with it.

If not all that made perfect sense, do not worry. The remainder of this tutorial focuses on how to do fiducial or germ selection using pyGSTi, and does not rely on a rock solid theoretical understanding of the methods.

Fiducial and Germ selection in practice#

The selection of fiducial and germ sequences in pyGSTi is similar in that each uses a numerical optimization which considers different possible “candidate” sets, and tries to find the one which scores the best. The modules pygsti.algorithms.fiducialselection and pygsti.algorithms.germselection contain the algorithms relevant to each type of sequence selection.

import pygsti
import pygsti.algorithms.fiducialselection as fidsel
import pygsti.algorithms.germselection as germsel
from pygsti.modelpacks import smq1Q_XYI
import numpy as np

We’ll begin by constructing a 1-qubit \(X(\pi/2)\), \(Y(\pi/2)\), \(I\) model for which we will find germs and fiducials.

target_model = smq1Q_XYI.target_model('full TP')

Automated “laissez-faire” approach#

We begin by demonstrating the most automated and hands-off approach to computing fiducials and germs – by just providing the target model and accepting the defaults for all remaining optional arguments. Note that one may compute these in either order - fiducial selection is usually much faster, since the required computation is significantly less.

prepFiducials, measFiducials = fidsel.find_fiducials(target_model)

germs = germsel.find_germs(target_model, seed = 1234)

Now that we have germs and fiducials, we can construct the list of experiments we need to perform in order to do GST. The only new things to provide at this point are the sizes for the experiments we want to perform (in this case we want to perform between 0 and 256 gates between fiducial pairs, going up by a factor of 2 at each stage).

maxLengths = [2**n for n in range(8 + 1)]
listOfExperiments = pygsti.circuits.create_lsgst_circuits(
    target_model, prepFiducials, measFiducials, germs, maxLengths)

Less-automated, more control: useful optional arguments#

There are many ways you can assume more control over the experiment design process. We’ll only demonstrate a few here, but all options are discussed in the documentation for the various functions we’ve used.

Different algorithms#

There are a number of different algorithms available for germ selection. You can choose a non-default algorithm by specifying the algorithm keyword argument. Each of the available algorithms has a set of keyword arguments of its own with which you can more precisely specify how you want it to behave. These keyword arguments can be passed as a dictionary to find_germs through the keyword argument algorithm_kwargs.

find_germs and find_fiducials support supports the algorithms: ‘greedy’ (default for find_germs), ‘grasp’ (default for find_fiducials) and ‘slack’.

Each of these algorithms can have different computational performance, and for systems of two-or-more qubits the ‘greedy’ algorithms are typically the most performant computationally (when run with certain mode settings, as discussed in the next section).

graspGerms = germsel.find_germs(target_model, algorithm='grasp', algorithm_kwargs={'iterations': 1}, seed = 1234)

slackGerms = germsel.find_germs(target_model, algorithm='slack', algorithm_kwargs={'slack_frac': 0.25}, seed = 1234)

Fiducial selection can be controlled in much the same way, using the same algorithms.

graspPrepFids, slackMeasFids = fidsel.find_fiducials(target_model, algorithm='slack',
                                                         algorithm_kwargs={'slack_frac': 0.25})

greedyPrepFids, greedyMeasFids = fidsel.find_fiducials(target_model, algorithm='greedy')

Different Modes#

In addition to there being multiple options for the algorithm to use, there are multiple modes that the algorithms can be run in. For find_germs the mode kwarg acts as a flag to indicate the caching scheme used for storing the Jacobians for the candidate germs. Default value of ‘allJac’ caches all of the Jacobians and requires the most memory. ‘singleJac’ doesn’t cache anything and instead generates these Jacobians on the fly. The final option, ‘compactEVD’, is currently only configured to work with the greedy search algorithm. When selected the compact eigenvalue decomposition/compact SVD of each of the Jacobians is constructed and is cached. This uses an intermediate amount of memory between ‘singleJac’ and ‘allJac’. When compactEVD mode is selected we also perform the greedy search iterations using an alternative method based on low-rank update techniques, which means in practice this mode can be orders-of-magnitude faster than the other modes, though typically only for two-or-more qubits. This alternative approach means that this mode also only works with the score function option set to ‘all’. (Note: this mode can also be a bit more finicky than other modes, so be prepared to tinker a bit, you can see hints of this finickiness below).

find_germs also accepts the kwarg float_type. float_type is dynamically inferred based on the target model’s basis (e.g., standard Pauli representations automatically use real-valued arrays). float_type can still be optionally specified for memory tuning (e.g. downcasting to np.single), which can allow for a lower memory footprint. When manually specified float_type is validated for compatibility against the model’s basis.

greedyGerms_compactEVD = germsel.find_germs(target_model, algorithm='greedy', seed = 1234, mode='compactEVD', verbosity=1,
                                            float_type=np.double)

Germ and fiducial lengths#

We can also adjust some algorithm-independent parameters for germ and fiducial selection. For instance, all of the algorithms currently rely on having a pool of circuit from which they construct germs and fiducials. The size of this pool is set by specifying the longest germ or fiducial to include in this pool.

For germ selection, the default maximum germ length is 6.

We can try and set the maximum germ length to 5 and see what we get.

germsMaxLength5 = germsel.find_germs(target_model, candidate_germ_counts={5: 'all upto'}, seed=1234)

If we get too ambitious in shortening the maximum germ length, germ selection won’t be able to find an amplificationally complete germ set. It will send a warning message to stderr if this happens and return None.

germsMaxLength3 = germsel.find_germs(target_model, candidate_germ_counts={3: 'all upto'}, seed=1234)
print(germsMaxLength3)

As was the case with germ selection, if you are too aggressive in limiting fiducial length you may constrain the algorithm to the extent that it cannot even find a set of fiducials to generate an informationally complete set of states and measurements. In that case, it will also send a warning message to stderr and return None for the preparation and measurement fiducial sets.

incompletePrepFids, incompleteMeasFids = fidsel.find_fiducials(target_model, candidate_fid_counts={1:'all upto'})

print(incompleteMeasFids, incompletePrepFids)

Set requirements#

There are several natural things to require of the returned germ and fiducial sets. For germ sets, you will usually want the individual gates to be included as germs. If for some reason you don’t want this, you can set the force keyword argument to None.

nonSingletonGerms = germsel.find_germs(target_model, force=None, candidate_germ_counts={5: 'all upto'},
                                           algorithm='greedy', seed=1234)

In fiducial selection, it is likewise natural to require the empty operation sequence to be in the fiducial set. This requirement may be disabled by setting forceEmpty to False. It is also often desireable for identity gates to be left out of fiducials, since they add no diversity to the set of states and measurements generated. You can allow identity gates in fiducials by setting omit_identity to False.

A more common modification to the fiducial set requirements is to leave out additional gates from fiducials. This might be desireable if you have a multi-qubit system and you expect your 2-qubit gates to be of lower fidelity than your single-qubit gates. In this case you might want to construct fiducials from only single-qubit gates. A list of gates that you would like to omit from your fiducials can be provided as a list of operation labels to the ops_to_omit keyword argument.

Our model doesn’t have multi-qubit gates, but we can demonstrate several pieces of this functionality by setting omit_identity to False and omitting the identity manually using ops_to_omit.

from pygsti.baseobjs import Label
omit_identityPrepFids, omit_identityMeasFids = fidsel.find_fiducials(target_model, omit_identity=False,
                                                                       ops_to_omit=[Label(())])

The ‘Lite’/’Standard’ Germ Set#

So far we have implicitly been constructing examples of what we call the ‘Robust’ germ set. This is a germ set designed to be robust against second-order effects that result in a plateuing of our sensitivity at long circuit depths. Unless your system has very low error rates, it is likely that even with this second order effect you’ll be decoherence limited long before entering the regime where this effect is significant. By setting the kwarg randomize to False you can change the behavior of germ selection such that it produces a significantly smaller, but also somewhat less robust, germ set called the ‘Standard’ or ‘Lite’ germ set. While not the default behavior of find_germs, we’ve found that for most applications the lite germ set is more than sufficient, so we recommend using it unless there is specific reason to prefer the robust experiment design (e.g. if you need high precision estimates for an idle gate known to have a very high fidelity). For more on these different germ sets see this paper.

liteGerms = germsel.find_germs(target_model, randomize=False, algorithm='greedy', verbosity=1)

Verbosity#

The various algorithms can tell you something of what’s going on with them while they’re running. By default, this output is silenced, but it can be turned on using the verbosity keyword argument.

  • A verbosity level of 1 is the default. This prints out what algorithm is being used, the returned set, and the score of that set.

  • A verbosity level of 0 silences all output (other than warnings that things have gone wrong).

  • A verbosity level of \(n+1\) where \(n\geq0\) prints the output of verbosity level 1 in addition to the output that the current algorithm displays when its own verbosity is set to \(n\).