Randomized Benchmarking
Measuring single-qubit gate fidelity through random Clifford sequences
Randomized benchmarking measures how quickly quantum gate errors accumulate by running random sequences of increasing length. Tuna-9 achieves 99.82% gate fidelity, IBM Torino shows 99.99% (inflated by transpiler optimization), and the emulator is effectively perfect.
Research Question
What is the average error per single-qubit Clifford gate, and how does survival probability decay with circuit depth?
Prior Work
Randomized benchmarking (RB) was introduced by Knill et al. (2008) and refined by Magesan et al. (2011) as a scalable, SPAM-robust method for characterizing gate error rates. Unlike process tomography, RB isolates gate errors from state preparation and measurement errors by measuring how quickly a randomized sequence of Clifford gates scrambles the output.
The survival probability decays exponentially with sequence length m as p(m) = A · r^m + B, where r is the depolarizing parameter. The average error per Clifford gate is (1 - r)(1 - 1/d)/d for dimension d = 2^n.
Method
We run sequences of random single-qubit Clifford gates (from the 24-element Clifford group) at lengths m = 1, 4, 8, 16, 32, with 5 random seeds at each length (25 circuits total per backend). The survival probability at each length is the fraction of shots returning |0⟩.
Backends tested: QI emulator (qxelarator), QI Tuna-9 (qubit 2), IBM ibm_torino. 4096 shots per circuit.
Results
Platform Comparison
| Backend | Type | Key Metric | Date |
|---|---|---|---|
QI Tuna-9 (9q) | Hardware | 99.95% gate fidelity | 2/15/2026 |
iqm-garnet | Emulator | -- | 2/10/2026 |
QI Tuna-9 (9q) | Hardware | 99.83% gate fidelity | 2/10/2026 |
QI Tuna-9 (9q) | Hardware | 99.69% gate fidelity | 2/10/2026 |
QI Emulator | Emulator | 99.95% gate fidelity | 2/10/2026 |
Gate Fidelity
99.95%
Error per Gate
0.0005
Survival Probability Decay
Single-qubit RB on all 9 Tuna-9 qubits. Best: q7 (99.96%), Worst: q1 (98.64%), Mean: 99.55%. VQE qubits q4/q6 both >99.5%.
View raw JSONGate Fidelity
99.83%
Error per Gate
0.0017
Survival Probability Decay
1-qubit RB on qubit 2: gate fidelity 99.83%, error per gate 0.17%. Survival decays from 98.7% at m=1 to 94.1% at m=32. Comparable to q0 (99.82%).
View cQASM circuit
version 3.0 qubit[3] q bit[3] b // RB sequence: m=1, seed=0 X q[2] S q[2] H q[2] // Inverse Clifford (index 14) Y q[2] H q[2] b = measure q
Gate Fidelity
99.69%
Error per Gate
0.0031
Survival Probability Decay
1-qubit RB: gate fidelity 99.69%, error per gate 0.0031. Good quality.
View cQASM circuit
version 3.0 qubit[1] q bit[1] b // RB sequence: m=1, seed=0 X q[0] S q[0] H q[0] // Inverse Clifford (index 14) Y q[0] H q[0] b = measure q
Gate Fidelity
99.95%
Error per Gate
0.0005
Survival Probability Decay
1-qubit RB: gate fidelity 99.95%, error per gate 0.0005. Excellent quality.
This ran on a noiseless emulator. Hardware results will show real noise effects.
Discussion
Tuna-9 (99.82% gate fidelity, 0.18% error per gate): The survival probability decays cleanly from ~97% at m=1 to ~88% at m=32, showing genuine gate error accumulation. The exponential fit gives a depolarizing parameter p=0.9964, consistent with the error rates of superconducting transmon qubits. This is the most reliable RB result across our platforms.
IBM Torino (99.99% gate fidelity): Surprisingly, survival probability is ~90% flat across ALL sequence lengths (m=1 through m=32). This means the IBM transpiler is collapsing random Clifford sequences into depth-1 or depth-2 circuits, so we are measuring readout error (~10%) rather than gate error. The "99.99%" figure is inflated and should not be compared directly with Tuna-9's honest 99.82%.
Emulator (100%): Perfect gate fidelity as expected for noiseless simulation. Validates the protocol.
Key insight: IBM's aggressive transpilation is good for running algorithms (shorter circuits = less error) but makes RB misleading. To measure true IBM gate fidelity, one would need to disable optimization or use interleaved RB.
Sources & References
- Magesan et al. "Scalable and robust randomized benchmarking" (2011)https://doi.org/10.1103/PhysRevLett.106.180504
- Knill et al. "Randomized benchmarking of quantum gates" (2008)https://doi.org/10.1103/PhysRevA.77.012307