Randomized Benchmarking: Basic Tutorial

This tutorial demonstrates how to perform randomized benchmarking (RB) using pygsti. While RB is a very distinct protocol from Gate Set Tomography (GST), pygsti includes basic support for RB because of its prevalence in the community, its simplicity, and its considerable use of GST-related concepts and data structures. The core protocol is standard Clifford randomized benchmarking defined in "Scalable and Robust Benchmarking of Quantum Processes". Much of the notation is consistent with Wallman and Flammia's "Randomized benchmarking with confidence".

This tutorial will show the following, all in the context of benchmarking a single qubit:

  • How to create a list of random RB sequences (experiments). These are just a list of pyGSTi GateString objects.
  • How to write a template data file from this list.
  • How to compute RB fit parameters from a pyGSTi DataSet filled with RB sequence data.
  • How to compute error bars on the various RB parameters and derived quantities.

We'll begin by importing relevant modules:

In [1]:
from __future__ import print_function #python 2 & 3 compatibility

import pygsti
from pygsti.extras import rb
from pygsti.construction import std1Q_XYI
/Users/enielse/research/pyGSTi/packages/pygsti/tools/matrixtools.py:23: UserWarning: Could not import Cython extension - falling back to slower pure-python routines
  _warnings.warn("Could not import Cython extension - falling back to slower pure-python routines")

Primitive gates, and how they map to Cliffords

First, let's choose a "target" gateset. This is the set of physically-implemented, or "primitive" gates. For this tutorial, we'll just use the standard $I$, $X(\pi/2)$, $Y(\pi/2)$ set. The target gateset should generate the Clifford group (or some other unitary 2-design).

In [2]:
gs_target = std1Q_XYI.gs_target
print("Primitive gates = ", gs_target.gates.keys())
Primitive gates =  [u'Gi', u'Gx', u'Gy']

To generate appropriately random RB sequences, we'll need to know how the set of all the Clifford gates map onto the given primitive set (since RB requires sequences to be random sequences of Cliffords, not of primitive gates). PyGSTi already contains the group of 1-qubit Cliffords. Benchmarking of a different group, or the $n>1$ qubit Clifford group requires the user to define this group.

PyGSTi contains a standard compilation of each 1-qubit Clifford into the gates $\{I,X(\pi/2),Y(\pi/2)\}$, which we will use here.

In [3]:
clifford_to_primitive = std1Q_XYI.clifford_compilation

# get the 1Q Clifford group: the canonical set of superoperator matrices representing the Clifford group, used later.
clifford_group = rb.std1Q.clifford_group

Generating RB sequences

Now let's decide what random Clifford sequences to generate. We use $m$ to denote the length of a Clifford sequence, in Clifford gates and not including the inversion Clifford at the end of each sequence. $K_m$ denotes the number of different random sequences of length $m$ to use. Note: K_m_sched need not be $m$-independent, and can be a dictionary, with $(m,K_m)$ key-value pairs.

In [4]:
m_list = [1,101,201,301,401,501,601,701,801,801,1001]
K_m = 10

Now we generate the list of random RB Clifford sequences to run. The write_empty_rb_files function handles this job, and does a lot. Here's what this one function call does:

  • It creates lists of random RB gate sequences, one list for each $m$, according to the schedule given by $m_{min}$, $m_{max}$, $\delta_m$, and $K_m$. These sequences are expressed as strings of Clifford gate labels and translated using any of the supplied maps (in this case, the string are translated to "primitive" labels also). These lists-of-lists are returned as a dictionary whose keys are "clifford" (always present) and "primitive" (b/c it's a key of the dict passed as alias_maps).
  • The lists for each set of gate labels (the Cliffords and primitives in this case) is aggregated across all $m$ values (so there's just a single list of all the RB sequences) and saved to a file beginning with the given base filename.
  • An empty DataSet is saved in text format using the RB sequences expressed in terms of Clifford gates.
In [5]:
filename_base = 'tutorial_files/rb_template'
rb_sequences = rb.write_empty_rb_files(filename_base, m_list, K_m, clifford_group,
                                       {'primitive': clifford_to_primitive},
                                       seed=0)

There is now an empty template file tutorial_files/rb_template.txt. For actual physical experiments, this file should be filled with experimental data and read in using pygsti.io.load_dataset. In this tutorial, we will generate fake data instead and just use the resulting dataset object.

The files tutorial_files/rb_template_clifford.txt and tutorial_files/rb_template_primitive.txt are text files listing all the RB sequences, expressed in terms of Cliffords and primitives respectively.

Generating fake data

To generate a dataset, we first need to make a gateset. Here we assume a gate set that is perfect except for some small amount of depolarizing noise on each primitive gate.

In [6]:
depol_strength = 1e-3
gs_experimental = std1Q_XYI.gs_target
gs_experimental = gs_experimental.depolarize(gate_noise=depol_strength)

Now we choose the number of clicks per experiment and simulate our data. More information on simulating RB can be found in the following tutorial.

In [7]:
all_rb_sequences = [] #construct an aggregate list of Clifford sequences
for seqs_for_single_cliff_len in rb_sequences:
    all_rb_sequences.extend(seqs_for_single_cliff_len)
    
N=100 # number of samples
rb_data = pygsti.construction.generate_fake_data(
    gs_experimental,all_rb_sequences,N,'binomial',seed=1,
    aliasDict=clifford_to_primitive, collisionAction="keepseparate")

Running the RB analysis

Now that we have data, it's time to perform the RB analysis. The function do_randomized_benchmarking returns an RBResults object which holds all the relevant input and output RB quantities. This object can be used to generate error bars on the computed RB quanties.

Some important arguments are:

  • success_spamlabel : the spam label corresponding to the expected outcome when preparing and immediately measuring.
  • dim : the Hilbert space dimension. This defaults to 2 (the 1-qubit case) and so can usually be left out.
In [8]:
rb_results = rb.do_randomized_benchmarking(rb_data, all_rb_sequences,fit='first order',success_outcomelabel='0', dim=2)

Examining the output

Okay, so we've done RB! Now let's examine how we can use the returned RBResults object to visualize and inspect the results. First let's plot the averaged RB data (i.e., averaged over sequences at each length) and the decay curve that has been fit to the data.

Some useful optional arguments are: xlim, ylim, save_fig_path, loc, which all perform the standard matploblib functions, and also legend (true or false), title (true or false).

In [9]:
#Create a workspace to show plots
w = pygsti.report.Workspace()
w.init_notebook_mode(connected=False, autodisplay=True)