Streamlined Randomized Benchmarking (SRB)

These examples demonstrate how to use SRB, the most basic QCVV protocol, to estimate the process infidelity of a single-qubit or two-qubit gateset 1 . See SRB for more information about this protocol.

Hello World

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 immutable cycles (denoted imm) delimit the random Cliffords, and more generally, in other protocols like make_cb() and make_irb(), contain interleaved cycles of interest.

import trueq as tq

circuits = tq.make_srb(0, [4, 30, 50])
circuits[0]
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: X
  • n_random_cycles: 4
  • protocol: SRB
  • twirl: Cliffords on [0]
 
 
(0): Gate.cliff19
Name:
  • Gate.cliff19
Aliases:
  • Gate.cliff19
Generators:
  • 'X': -69.282
  • 'Y': 69.282
  • 'Z': 69.282
Matrix:
  • 0.50 -0.50j -0.50 0.50j 0.50 0.50j 0.50 0.50j
imm  
 
(0): Gate.id
Name:
  • Gate.id
Aliases:
  • Gate.id
  • Gate.i
  • Gate.cliff0
Likeness:
  • Identity
Generators:
  • 'I': 0
Matrix:
  • 1.00 1.00
imm  
 
(0): Gate.cliff8
Name:
  • Gate.cliff8
Aliases:
  • Gate.cliff8
Generators:
  • 'Z': -90.0
Matrix:
  • 0.71 0.71j 0.71 -0.71j
imm  
 
(0): Gate.s
Name:
  • Gate.s
Aliases:
  • Gate.s
  • Gate.cliff9
Generators:
  • 'Z': 90.0
Matrix:
  • 0.71 -0.71j 0.71 0.71j
imm  
 
(0): Gate.cliff17
Name:
  • Gate.cliff17
Aliases:
  • Gate.cliff17
Generators:
  • 'X': -69.282
  • 'Y': -69.282
  • 'Z': 69.282
Matrix:
  • -0.50 -0.50j 0.50 -0.50j 0.50 0.50j 0.50 -0.50j
imm
(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.

tq.Simulator().add_overrotation(0.04).add_depolarizing(0.01).run(circuits)
circuits[0].results.plot()
srb

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.

circuits.plot.raw()
SRB on [0]

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

circuits.fit()
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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.
9.4e-03 (6.0e-04)
0.00944211595011718, 0.0006001201950034952
${p}$
Decay parameter of the exponential decay $Ap^m$.
9.9e-01 (8.0e-04)
0.9874105120665104, 0.0008001602600046603
${A}$
SPAM parameter of the exponential decay $Ap^m$.
9.9e-01 (8.6e-03)
0.987223971075226, 0.008613720043631125


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.

# 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.

# generate a circuit collection
circuits = tq.CircuitCollection()
for pair in [[0, 1], [2, 3], [4, 5]]:
    circuits += tq.make_srb([pair], [4, 10, 20])

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

# Initialize a simulator
sim = tq.Simulator().add_overrotation(0.01, 0.02)
sim.add_stochastic_pauli(px=0.01, noisy_labels=(0, 1))
sim.add_stochastic_pauli(pz=0.005, noisy_labels=(2, 3, 4, 5))

# initialize a transpiler into native gates (U3 and CNOT)
cnot_factory = tq.config.GateFactory("cnot", matrix=tq.Gate.cnot.mat)
config = tq.Config.from_params("C", factories=[tq.config.u3_factory, cnot_factory])
t = tq.compilation.get_transpiler(config)

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

Plot the results.

circuits.plot.raw([[0, 1], [2, 3], [4, 5]])
circuits.plot.compare_rb()
  • SRB on [0, 1], SRB on [2, 3], SRB on [4, 5]
  • Comparison for Cliffords on [(0, 1), (2, 3), (4, 5)]

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.

# add simultaneous circuits to our old collection
simul_circuits = t.compile(tq.make_srb([[0, 1], [2, 3], [4, 5]], [4, 10, 20]))
circuits += simul_circuits
sim.run(simul_circuits)

# plot the results
circuits.plot.raw([[0, 1], [2, 3], [4, 5]])
circuits.plot.compare_rb()
  • SRB on [0, 1] Cliffords on [(0, 1), (2, 3), (4, 5)], SRB on [0, 1] Cliffords on [(0, 1)], SRB on [2, 3] Cliffords on [(0, 1), (2, 3), (4, 5)], SRB on [2, 3] Cliffords on [(2, 3)], SRB on [4, 5] Cliffords on [(0, 1), (2, 3), (4, 5)], SRB on [4, 5] Cliffords on [(4, 5)]
  • Comparison for Cliffords on [(0, 1), (2, 3), (4, 5)]

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.

su_circuits = tq.make_srb([[0, 1], [2, 3], [4, 5]], [4, 6, 12], twirl="SU")
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.

su_circuits.plot.raw([[0, 1], [2, 3], [4, 5]])
su_circuits.plot.compare_rb()
  • SRB on [0, 1], SRB on [2, 3], SRB on [4, 5]
  • Comparison for SU on [(0, 1), (2, 3), (4, 5)]

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.

Total running time of the script: ( 0 minutes 33.740 seconds)

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