# 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

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 Circuit Key: compiled_pauli: X n_random_cycles: 4 protocol: SRB twirl: Cliffords on  (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.results.plot() 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() 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  ${e}_{F}$ 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(, [4, 100], 30).append(tq.make_srb(, [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 [, , ]. 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

# 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()

• • ## 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()

• • ## 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()

• • 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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