Chapter 11 — Key Takeaways (Simulation and Noise Models)

The simulator page. The noise signature table is the diagnostic skill.

The four methods

Method Cost Practical limit Use for
statevector $2^n$ ~30 qubits the default; exact
density_matrix $4^n$ ~15 qubits noise, purity, fidelity
stabilizer $O(n^2)$ thousands Clifford only — QEC, benchmarking
matrix_product_state $O(n\chi^2)$ hundreds, if entanglement bounded shallow, 1-D circuits
extended_stabilizer exp. in $T$ count near-Clifford approximate

Measured on a GHZ chain, 128 shots:

                       n=5     n=10     n=15     n=20     n=25
statevector          101 ms   64 ms    66 ms    71 ms   351 ms
density_matrix        37 ms   49 ms  9,148 ms  FAILED       —
stabilizer           269 ms  332 ms   300 ms   294 ms   297 ms
matrix_product_state  38 ms   43 ms    45 ms    50 ms    52 ms

Density matrix costs the SQUARE — every noise simulation needing one is half as wide.

Density matrix gives what counts cannot

qc.save_density_matrix()
sim = AerSimulator(method="density_matrix", noise_model=noise)
rho = DensityMatrix(sim.run(transpile(qc, sim)).result().data()["density_matrix"])
depolarizing $p$ purity fidelity
0.00 1.0000 1.0000
0.05 0.9269 0.9625
0.10 0.8575 0.9250
0.20 0.7300 0.8500

Getting the equivalent from counts requires tomography and exponentially many settings (Ch. 30).

★ Stabilizer: the wall has a hole

Gottesman–Knill: Clifford circuits ($H$, $S$, CNOT, Paulis) are simulable in polynomial time, at any size — track $n$ Pauli generators ($O(n^2)$ bits), never write down the state.

Follows Does not follow
Clifford circuits give no quantum advantage that entanglement is not the resource
T-count is the currency of resource estimation (Ch. 23) — a 1000-qubit GHZ state is maximally entangled and easy
QEC is simulable (Ch. 25)

⚠️ The testing trap. Appending $T$ gates before a measurement did not break the stabilizer method — RemoveDiagonalGatesBeforeMeasure deleted them first (a diagonal gate before a $Z$ measurement changes nothing observable). Only H T H, where $T$ genuinely matters, made it fail.

Check that your test case survives to the thing you are testing. Print the transpiled circuit.

  method                  P(0)     error    (exact = 0.8536)
  statevector           0.8563    0.0028
  stabilizer            FAILED             <- correct behavior
  extended_stabilizer   0.8318    0.0218   <- APPROXIMATE
  matrix_product_state  0.8563    0.0028

MPS: entanglement, not qubit count, is what costs

n=10 n=16 n=20 n=24
low entanglement, sv / MPS 65/46 67/46 77/49 214/49 ms
high entanglement, sv / MPS 67/45 81/54 125/66 1563/85 ms (18×)

⚠️ Caveat: the "high entanglement" circuit is H + CNOT = Clifford — entangled but structured. Genuinely hard needs high entanglement AND non-Clifford gates.

The inference is one-way. MPS fast → your circuit is easy (a genuine finding). MPS slow → only that this method struggled. Proving easy needs one algorithm; proving hard needs ruling out all of them.

Hardness needs both:

              |  low entanglement  |  high entanglement
  Clifford    |  easy (stabilizer) |  easy (stabilizer)
  non-Clifford|  easy (MPS)        |  HARD

Noise models from a backend

sim = AerSimulator.from_backend(FakeSherbrooke())     # THE most useful line here

Basis gates, coupling map, and calibrated errors — locally, no queue. Reproduced Chapter 2's Bell error fraction (0.0439) at 0.0447 from a completely different code path.

Limits: static snapshot · independent Markovian errors · no crosstalk correlations · no drift. Order of magnitude right, third decimal wrong.

★★ Noise signatures — two INDEPENDENT axes

$$\text{error fraction} = \frac{n_{01}+n_{10}}{N}, \qquad \text{imbalance} = \frac{n_{00}-n_{11}}{n_{00}+n_{11}}$$

Channel Error fraction Imbalance Reading
Depolarizing $p{=}0.05$ 0.0251 ($\approx p/2$) −0.0005 impossible outcomes, balanced
Readout $p{=}0.05$ 0.0897 ($\approx 2p$) −0.0042 same shape, much larger per $p$
Amplitude damping $\gamma{=}0.15$ 0.0000 +0.144 no impossible outcomes, tilted
Thermal, 533 ns 0.0021 +0.003 both, small
Thermal, 5000 ns 0.0225 +0.024 both, scaling with duration
Phase damping, any $\gamma$ 0.0000 +0.0007 INVISIBLE

Reading rule: impossible outcomes → depolarizing or readout (a gate-free circuit separates them). Peak imbalance → $T_1$. Both, co-scaling with duration → thermal relaxation.

★ The channel this cannot see

  phase damping γ=0.1 / 0.3 / 0.6:  {'00': 4099, '11': 4093}   -- IDENTICAL, bit for bit

A sixfold difference in decoherence, and the histogram does not move. The computational basis is blind to phase (Ch. 3 §3.5) — a decohered Bell state is Chapter 4's classical impostor and gives identical counts.

Measure in the $X$ basis:

  γ = 0.0 → <XX> = +1.0000      γ = 0.3 → <XX> = +0.8396
  γ = 0.1 → <XX> = +0.9482      γ = 0.6 → <XX> = +0.6379

Cleanly separated, monotonically, by one extra circuit — Chapter 4's entanglement witness doing its job.

$T_2$ is arguably the most important noise mechanism in quantum computing, and the default measurement cannot see it. A GHZ "fidelity" reported as the fraction of all-zeros/all-ones outcomes scores a fully decohered classical mixture at 100%.

Fifth appearance of this book's most persistent theme: your measurement can only see what it is sensitive to.

Testing in three tiers

Tier What it asserts Cost
1. exact, noiseless, seeded the algorithm is correct milliseconds
2. sampled, seeded, tolerance the measurement layer is correct ~1 s
3. noisy, device model the circuit is runnable ~1 s

Tier 3 catches a design regression — an ansatz change that tripled routing cost (Ch. 10 CS1) — in a second, with no hardware.

Common pitfalls

  • Diagnosing from the error fraction alone (misses $T_1$ entirely).
  • Reading a clean histogram as a clean state (misses $T_2$ entirely).
  • Concluding hardness from MPS being slow.
  • Calling a Clifford demonstration a computation.
  • Using density_matrix above ~15 qubits.
  • A test whose hard case gets optimized away before it is tested.

Project piece added this chapter

vqelab/tests/test_sim.py — noiseless_energy() (the reference every hardware result is measured against), noisy_energy(), noise_signature() (the two-axis classifier), and five tests across the three tiers, all passing.

Without the reference, "the energy was −1.125" is uninterpretable. With it, "12 mHa above the noiseless value" is a statement you can act on.