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Example: Running SRB

These examples demonstrate how to use SRB, the most basic QCVV protocol, to estimate the process infidelity of a single-qudit or two-qudit gateset 1 .

Getting started

The simplest thing we can do is run SRB on a single qubit. Qubits are labeled by non-negative integers, and here we target qubit 0. We choose to use sequence lengths [4, 30, 50], and, by default, 30 random circuits are generated per sequence length. Below, we display the first circuit where the 4 random Clifford gates can be seen, plus a fifth gate which is the product of their inverses and a random Pauli matrix, where the additional Pauli matrix is introduced to make the fit more robust. The empty cycles with non-zero markers delimit the random Cliffords, and more generally, in other protocols like make_cb() and make_irb(), contain interleaved cycles of interest.

[2]:
import trueq as tq
import trueq.simulation as tqs

circuits = tq.make_srb(0, [4, 30, 50])
circuits[0]
[2]:
True-Q formatting will not be loaded without trusting this notebook or rerunning the affected cells. Notebooks can be marked as trusted by clicking "File -> Trust Notebook".
Circuit
Key:
  • compiled_pauli: Y
  • measurement_basis: Z
  • n_random_cycles: 4
  • protocol: SRB
  • twirl: Cliffords on [0]
1
Marker 1
Compilation tools may only recompile cycles with equal markers.
(0): Gate.cliff19
Name:
  • Gate.cliff19
Aliases:
  • Gate.cliff19
Generators:
  • 'Y': 69.282
  • 'X': -69.282
  • 'Z': 69.282
Matrix:
  • 0.71 -0.71 0.71j 0.71j
2
Marker 2
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(0): Gate.cliff10
Name:
  • Gate.cliff10
Aliases:
  • Gate.cliff10
Generators:
  • 'Y': 127.279
  • 'X': 127.279
Matrix:
  • -1.00j 1.00
3
Marker 3
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(0): Gate.cliff11
Name:
  • Gate.cliff11
Aliases:
  • Gate.cliff11
Generators:
  • 'Y': -127.279
  • 'X': 127.279
Matrix:
  • 1.00 -1.00j
4
Marker 4
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(0): Gate.s
Name:
  • Gate.s
Aliases:
  • Gate.s
  • Gate.sz
  • Gate.cliff9
Generators:
  • 'Z': 90.0
Matrix:
  • 1.00 1.00j
5
Marker 5
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(0): Gate.sy
Name:
  • Gate.sy
Aliases:
  • Gate.sy
  • Gate.cliff7
Generators:
  • 'Y': 90.0
Matrix:
  • 0.71 -0.71 0.71 0.71
6
Marker 6
Compilation tools may only recompile cycles with equal markers.
(0): Meas()
Name:
  • Meas()

Next, we simulate each of the circuits with the built-in simulator. We can see the bitstring counts of the first circuit below.

[3]:
tq.Simulator().add_overrotation(0.04).add_depolarizing(0.01).run(circuits)
circuits[0].results.plot()
../../_images/guides_error_diagnostics_srb_example_4_0.png

It is important to look at the decay curve to verify that appropriate sequence lengths were chosen. The most informative sequence length to measure occurs around \(m=1/(1-p)\) where the decay is \(y=Ap^m\). If \(A\approx 1\) and \(p\approx 1\), then this corresponds to \(y\approx1/e\approx0.37\). Therefore, in this example, our sequence lengths are a bit too short, and [4, 70] would, in retrospect, have been more appropriate.

[4]:
circuits.plot.raw()
../../_images/guides_error_diagnostics_srb_example_6_0.png

Finally, we can look at the values of the estimate:

[5]:
circuits.fit()
[5]:
True-Q formatting will not be loaded without trusting this notebook or rerunning the affected cells. Notebooks can be marked as trusted by clicking "File -> Trust Notebook".
SRB
Streamlined Randomized Benchmarking
Cliffords
(0,)
Key:
  • labels: (0,)
  • protocol: SRB
  • twirl: Cliffords on [0]
${e}_{F}$
The probability of an error acting on the targeted systems during a random gate.
9.1e-03 (8.1e-04)
0.00906298375159137, 0.0008115815022470435
${p}$
Decay parameter of the exponential decay $Ap^m$.
9.9e-01 (1.1e-03)
0.9879160216645448, 0.0010821086696627246
${A}$
SPAM parameter of the exponential decay $Ap^m$.
9.5e-01 (1.9e-02)
0.945118729291987, 0.01868691153951574

Isolated vs. Simultaneous SRB

The core function make_srb() generates a circuit collection that implements simultaneous SRB, where gates are placed on all specified qubits of a quantum device within each circuit. Different random gates are independently chosen during each cycle. Qubit labels are entered into its first argument and specify which qubits are acted on simultaneously. You can choose which qubits get single-qubit twirls, and which qubits get two-qubit twirls, by nesting the qubit labels appropriately; see the following examples.

[6]:
# simultaneous single-qubit RB on qubits 0 and 1; 60 circuits total
circuits = tq.make_srb([0, 1], [4, 100], 30)

# isolated single-qubit RB on qubits 0 and 1; 120 circuits total
circuits = tq.make_srb([0], [4, 100], 30).append(tq.make_srb([1], [4, 100], 30))

# two-qubit RB on 0 and 1, simultaneous with single-qubit RB on 2; 60 circuits total
circuits = tq.make_srb([[0, 1], 2], [4, 100], 30)

The invocations above all use two sequence lengths (the number of random gate cycles in a circuit), 4 and 100, and produce 30 random circuits per sequence length. Note that specifying qubit labels [0, 1, 2] is a convenient short-hand for [[0], [1], [2]]. This imples, for example, that [[0,1]] has a much different meaning than [0,1].

Isolated SRB

In isolated SRB, we characterize qubits or qubit-pairs one at a time with disjoint experiments. This is done to assess gate quality on a small subregister in the context of an idling environment. In this example, we build a single circuit collection to estimate the average gate infidelity of three pairs on a 6 qubit device.

[7]:
# generate a circuit collection
circuits = tq.CircuitCollection()
for pair in [[0, 1], [2, 3], [4, 5]]:
    circuits += tq.make_srb([pair], [4, 12, 32])

Next, we transpile the circuits into native gates and simulate them.

[8]:
# Initialize a simulator
sim = tq.Simulator().add_overrotation(0.01, 0.02)
sim.add_stochastic_pauli(px=0.01, match=tqs.LabelMatch((0, 1)))
sim.add_stochastic_pauli(pz=0.005, match=tqs.LabelMatch((2, 3, 4, 5)))

# initialize a transpiler into native gates (U3 and CNOT)
cnot_factory = tq.config.GateFactory.from_matrix("cnot", tq.Gate.cnot.mat)
config = tq.Config(factories=[tq.config.u3_factory, cnot_factory])
t = tq.Compiler.from_config(config)

# transpile and simulate
circuits = t.compile(circuits)
sim.run(circuits)

Plot the results.

[9]:
circuits.plot.raw([[0, 1], [2, 3], [4, 5]])
circuits.plot.compare_rb()
../../_images/guides_error_diagnostics_srb_example_16_0.png
../../_images/guides_error_diagnostics_srb_example_16_1.png

Simultaneous SRB

In simultaneous SRB, we characterize qubits or qubit-pairs in a single set of experiments. This captures crosstalk between these subsystems which is usually not present when other qubits are idle. In this example, we add more circuits to the circuits from the previous section.

[10]:
# add simultaneous circuits to our old collection
simul_circuits = t.compile(tq.make_srb([[0, 1], [2, 3], [4, 5]], [4, 12, 32]))
circuits += simul_circuits
sim.run(simul_circuits)

# plot the results
circuits.plot.raw([[0, 1], [2, 3], [4, 5]])
circuits.plot.compare_rb()
../../_images/guides_error_diagnostics_srb_example_18_0.png
../../_images/guides_error_diagnostics_srb_example_18_1.png

Changing the Twirling Group Gateset

The above examples characterize the average gate fidelity of the Clifford group, which is historically by far the most common group to use. However, any unitary 2-design will suffice, and so for SRB, the other notable choice is the set of all unitaries sampled using the Haar measure. We make this alternate choice below.

[11]:
su_circuits = tq.make_srb(tq.Twirl("U", [[0, 1], [2, 3], [4, 5]]), [2, 8, 16])
su_circuits = t.compile(su_circuits)
sim.run(su_circuits)

Our native gates here are still CNOT and U3. It takes on average 1.5 CNOTs to synthesize a two-qubit Clifford gate, but it almost always takes 3 CNOTs to synthesize a Haar-random two-qubit unitary group. Thus we see higher error rates than in the section above.

Simulate and plot the results.

[12]:
su_circuits.plot.raw([[0, 1], [2, 3], [4, 5]])
su_circuits.plot.compare_rb()
../../_images/guides_error_diagnostics_srb_example_23_0.png
../../_images/guides_error_diagnostics_srb_example_23_1.png

SRB with Qudits

The examples above generalize to qudits of higher dimensions and we can use the exact same syntax to generate and simulate the circuits and subsequently analyze their results. Let’s go through an example with a single qutrit:

[13]:
tq.settings.set_dim(3)

circuits = tq.make_srb(0, [4, 32, 64])

# display a sample circuit
circuits[0]
[13]:
True-Q formatting will not be loaded without trusting this notebook or rerunning the affected cells. Notebooks can be marked as trusted by clicking "File -> Trust Notebook".
Circuit
Key:
  • compiled_pauli: W11
  • measurement_basis: W01
  • n_random_cycles: 4
  • protocol: SRB
  • twirl: Cliffords on [0]
1
Marker 1
Compilation tools may only recompile cycles with equal markers.
(0): Gate()
Name:
  • Gate()
Matrix:
  • 0.58 -0.29 -0.50j 0.58 -0.29 -0.50j 0.58 0.58 -0.29 -0.50j -0.29 -0.50j -0.29 0.50j
2
Marker 2
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(0): Gate()
Name:
  • Gate()
Matrix:
  • 0.50 0.29j 0.50 0.29j -0.50 0.29j -0.58j -0.50 0.29j -0.50 0.29j 0.50 0.29j -0.50 0.29j 0.50 0.29j
3
Marker 3
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(0): Gate()
Name:
  • Gate()
Matrix:
  • -0.58j 0.50 0.29j 0.50 0.29j -0.50 0.29j -0.50 0.29j 0.50 0.29j -0.58j -0.50 0.29j -0.58j
4
Marker 4
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(0): Gate()
Name:
  • Gate()
Matrix:
  • 0.58j -0.50 -0.29j 0.58j 0.58j 0.50 -0.29j 0.50 -0.29j 0.50 -0.29j 0.50 -0.29j 0.58j
5
Marker 5
Compilation tools may only recompile cycles with equal markers.
(0): Gate()
Name:
  • Gate()
Matrix:
  • -0.29 -0.50j -0.29 -0.50j -0.29 0.50j 0.58 -0.29 -0.50j 0.58 -0.29 0.50j -0.29 -0.50j -0.29 -0.50j
6
Marker 6
Compilation tools may only recompile cycles with equal markers.
(0): Meas()
Name:
  • Meas()

Next, we simulate each of the circuits with the built-in simulator and display a sample bitstring result:

[14]:
tq.Simulator().add_overrotation(0.04).add_depolarizing(0.02).run(circuits)

circuits[0].results.plot()
../../_images/guides_error_diagnostics_srb_example_27_0.png

The decay curve can be inspected using the plot() method:

[15]:
circuits.plot.raw()
../../_images/guides_error_diagnostics_srb_example_29_0.png

And similarly we can inspect the fit parameters:

[16]:
circuits.fit()
[16]:
True-Q formatting will not be loaded without trusting this notebook or rerunning the affected cells. Notebooks can be marked as trusted by clicking "File -> Trust Notebook".
SRB
Streamlined Randomized Benchmarking
Cliffords
(0,)
Key:
  • labels: (0,)
  • protocol: SRB
  • twirl: Cliffords on [0]
${e}_{F}$
The probability of an error acting on the targeted systems during a random gate.
2.2e-02 (1.6e-03)
0.021827036760267957, 0.0016073262274480847
${p}$
Decay parameter of the exponential decay $Ap^m$.
9.8e-01 (1.8e-03)
0.9754445836446985, 0.0018082420058790954
${A}$
SPAM parameter of the exponential decay $Ap^m$.
1.0e+00 (3.1e-02)
0.9970466318875559, 0.030888982769069474

Footnotes

1

“Streamlined” refers to the method we use to eliminate the constant offset (the \(B\) in \(Ap^m+B\)) that typically appears in standard RB.


Download

Download this file as Jupyter notebook: srb_example.ipynb.