Case Study 1: Reading a Histogram Like a Diagnostician
"The hardware is noisy" is a shrug. "That tilt is $T_1$" is a diagnosis, and it takes ten seconds to reach.
Executive Summary
Four hardware runs come back looking wrong in four different ways. A shrug attributes all four to "noise" and stops. This case study reads each one — and gets four different answers, each pointing at a different physical mechanism and a different remedy.
The method is a two-axis classification you can apply in your head. And the fourth case is the important one: a histogram that looks perfect while the state behind it has been destroyed.
Skills applied: noise channels and their signatures (§11.7); the two independent axes (§11.7); basis rotations (Ch. 5 §5.6); the entanglement witness (Ch. 4 Case Study 2).
Reproducibility. All figures from AerSimulator with the stated noise models, 8,192 shots,
seed_simulator=1234.
The Two Axes
Every measurement below is a Bell state, whose ideal output is 50% 00, 50% 11, nothing else. Two
numbers summarize any deviation:
$$\text{error fraction} = \frac{n_{01} + n_{10}}{N}, \qquad \text{imbalance} = \frac{n_{00} - n_{11}}{n_{00} + n_{11}}$$
These are independent. A channel can move one, the other, both, or neither — and which it moves identifies it.
Case A: Impossible Outcomes, Balanced Peaks
{'00': 3991, '01': 108, '10': 98, '11': 3995}
error fraction 0.0251
imbalance -0.0005
Reading: impossible outcomes present, peaks balanced.
Diagnosis: depolarizing or readout error. Something is scrambling outcomes with no preferred direction.
Distinguishing the two requires one more experiment, and it is the cheapest in this book: run a circuit with no gates at all. Prepare $|00\rangle$, measure immediately. Whatever error appears is pure readout — Chapter 2's Case Study 2 built exactly this.
qc = QuantumCircuit(2, 2)
qc.measure([0, 1], [0, 1]) # no gates. Any error here is readout.
If the gate-free circuit shows most of the 2.5%, it is readout. If it shows almost none, the error is in the gates.
Remedy: readout mitigation (Chapter 13 §13.2) if readout; circuit depth reduction (Chapter 28) or better qubits (Chapter 29) if gates.
Case B: Clean Outcomes, Tilted Peaks
{'00': 4687, '11': 3505}
error fraction 0.0000
imbalance +0.1443
Reading: no impossible outcomes at all — and a 14% tilt toward 00.
Diagnosis: amplitude damping. $T_1$.
This one is unambiguous, and the reasoning is worth spelling out. Depolarizing and readout noise both
create forbidden outcomes; here there are none. What has happened instead is that population has
drained from $|1\rangle$ toward $|0\rangle$ — energy relaxation — which shifts weight from the 11
peak to the 00 peak without ever producing a 01 or 10.
Remedy: shorten the circuit, or use qubits with better $T_1$. Readout mitigation would do nothing here, and reaching for it would waste a day.
⚠️ Common Pitfall — Do not diagnose from the error fraction alone.
Case B has an error fraction of exactly zero. A pipeline that only tracks "fraction of shots in impossible states" — which is the natural metric, and the one this book used through Chapters 2 and 4 — reports this state as perfect.
It is not perfect. It is 14% skewed, and for a VQE that skew is a systematic bias in every energy you compute.
Track both axes. The second one costs a subtraction.
Case C: Both Signatures
T1=200 μs, T2=120 μs, gate duration 5000 ns:
error fraction 0.0225
imbalance +0.0240
Reading: impossible outcomes and a tilt, both modest.
Diagnosis: thermal relaxation over a long circuit. Real decoherence combines $T_1$ (which tilts) and $T_2$ (which does other things, see Case D), and over a long enough exposure both show.
The diagnostic detail that pins it down: scale with duration. Thermal relaxation grows with $t/T_2$, so the same circuit run with longer gates — or the same gates on a deeper circuit — should show proportionally more of both signatures:
gate duration error imbalance
533 ns 0.0021 +0.0028
2000 ns 0.0089 +0.0097
5000 ns 0.0225 +0.0240
Both columns scale together, roughly linearly in duration. That co-scaling is the fingerprint — depolarizing error would not tilt, and readout error would not grow with circuit duration at all.
Remedy: reduce depth (Chapter 28), use dynamical decoupling on idle qubits (Chapter 13 §13.6, Chapter 31 §31.4), or accept it and mitigate.
Case D: A Perfect-Looking Histogram
{'00': 4099, '11': 4093}
error fraction 0.0000
imbalance +0.0007
Reading: flawless. No impossible outcomes, perfectly balanced peaks. By both axes, this is a clean Bell state.
It is not. This is phase damping at $\gamma = 0.30$ — severe $T_2$ decoherence.
And here is the part that should be alarming. Run it at three different damage levels:
phase damping γ=0.1: {'00': 4099, '11': 4093} error 0.0000 imbalance +0.0007
phase damping γ=0.3: {'00': 4099, '11': 4093} error 0.0000 imbalance +0.0007
phase damping γ=0.6: {'00': 4099, '11': 4093} error 0.0000 imbalance +0.0007
Identical. Bit for bit. A sixfold difference in decoherence strength, and the computational-basis histogram does not move at all.
The diagnostic is blind here for the reason this book has returned to five times: a
computational-basis measurement cannot see phase. Phase damping destroys the coherence between
$|00\rangle$ and $|11\rangle$ without moving any population. What remains is a classical mixture —
half the time 00, half the time 11 — which is precisely Chapter 4's Case Study 2 impostor, and it
produces identical counts.
The remedy is a second basis, and you already have it:
qc.h(0); qc.h(1) # measure in X instead
The same three runs, in the $X$ basis:
γ = 0.0: <XX> = +1.0000
γ = 0.1: <XX> = +0.9482
γ = 0.3: <XX> = +0.8396
γ = 0.6: <XX> = +0.6379
Cleanly separated, monotonically, by one extra circuit. The correlator that was invisible in one basis is a precise damage gauge in another.
That is Chapter 4's entanglement witness doing exactly the job it was built for — and note that at $\gamma = 0.6$, $\langle ZZ\rangle + \langle XX\rangle = 1 + 0.64 = 1.64$, still above the separable bound of 1, so the state is degraded but demonstrably still entangled. Push the damping further and that witness crosses its threshold, at which point you can no longer prove entanglement at all — which is the point Chapter 4 §4.9 anticipated about evidence degrading before the physics does.
🔬 Honest Assessment — Why this case matters most.
$T_2$ decoherence is arguably the most important noise mechanism in quantum computing. It is what destroys superposition, it is the reason circuits have a depth limit, and it is the thing error correction primarily exists to fight.
And it is invisible to the default measurement.
That has a consequence worth stating plainly: a great deal of quantum "verification" cannot see the failure mode that matters most. A GHZ-state fidelity reported as "fraction of shots that were all-zeros or all-ones" — which Chapter 4 §4.9 used, and flagged at the time — would score a fully decohered classical mixture at 100%.
This is not a hypothetical concern. It is why Chapter 30's benchmarks use entanglement witnesses and randomized benchmarking rather than success fractions, and why Chapter 4 Case Study 2 insisted on a witness with a proven threshold rather than a comparison against one impostor.
The rule, one more time: your measurement can only see what it is sensitive to.
The Decision Procedure
measure the Bell state in the COMPUTATIONAL basis
|
+-- impossible outcomes? ---- no ----+
| | |
| yes |
| | |
| peaks balanced? peaks balanced?
| / \ / \
| yes no no yes
| | | | |
| depolarizing thermal T1 LOOKS CLEAN
| or readout relaxation (amplitude -- but you have
| | (check it damping) not tested phase
| | scales with |
| gate-free duration) v
| circuit MEASURE IN THE X BASIS
| separates <XX> near +1 -> genuinely clean
| them <XX> near 0 -> T2 decoherence
Four measurements total — the original, a gate-free calibration circuit, a duration sweep, and an X-basis run — and every mechanism in this chapter is distinguished.
Lessons
- Two independent axes: impossible outcomes, and peak imbalance. Which moves identifies the channel.
- Impossible outcomes + balanced peaks = depolarizing or readout. A gate-free circuit separates them.
- No impossible outcomes + tilted peaks = $T_1$. Readout mitigation would not help.
- Both, scaling with duration = thermal relaxation. The co-scaling is the fingerprint.
- Neither does not mean clean. Phase damping is invisible in the computational basis.
- An error fraction of zero is not evidence of a good state. Track imbalance too, and then measure in another basis.
- $T_2$ is the most important mechanism and the default measurement cannot see it. Use a witness.
- Four cheap measurements distinguish every mechanism here: the circuit, a gate-free version, a duration sweep, and a second basis.
Questions
-
Reproduce all four cases with the noise models given, and confirm the two-axis numbers.
-
Build the gate-free calibration circuit and use it to separate depolarizing from readout error in Case A. What fraction of the 2.5% was readout?
-
Implement
classify(counts)returning a diagnosis string. Test it on all four cases plus a fifth of your own construction. Where does it fail, and is that a classifier bug or a data limitation? -
Case D's remedy is an $X$-basis measurement. Run it on both an intact Bell state and one under phase damping at $\gamma = 0.3$, and report $\langle XX\rangle$ for each. How much damping can you detect at 8,192 shots? (Chapter 5 §5.8's power analysis.)
-
Case C's fingerprint is co-scaling with duration. Design an experiment on real hardware that varies circuit duration while holding gate count fixed. (Hint:
delay.) -
A colleague reports a GHZ fidelity of 98%, measured as the fraction of all-zeros and all-ones outcomes. Write the two-sentence question you would ask.
-
Hardest. Extend the two-axis diagnostic to a three-qubit GHZ state. How many independent summary statistics are there now, and which noise channels do they separate? Is there a channel pair that becomes distinguishable at three qubits but not at two?