27 Hardware characterization in qg – Quantum Computing, Information and Physics in qang Space

27.1 From the base: an idle qubit is a readout

A real qubit is never completely cold: at rest it has a residual excited population \(p\), and its equilibrium readout is \(qg_{eq}=1-2p\), in m-qg \(m_{eq}=1000(1-2p)\). For a qubit of frequency \(f\) in thermal equilibrium,

\[ qg_{eq}=\tanh\!\left(\frac{hf}{2k_BT_{\text{eff}}}\right), \]

so the equilibrium readout is a thermometer. The instrument’s readout is also affine in qg: if it confuses 0→1 with probability \(e_{01}\) and 1→0 with \(e_{10}\), the measured readout is \(a+b\,qg\), with \(a=e_{10}-e_{01}\) and \(b=1-e_{01}-e_{10}\) (notes §31). The problem is that in the Z basis only \(a+b\,qg_{eq}\) is seen: temperature and readout error are mixed.

27.2 T1, T2, temperature and readout in one sweep

The standard calibration measures “\(|0\rangle\)” and “\(|1\rangle\)”, which are really the thermal state and X applied to it, and counts the thermal population as readout error. The qg protocol adds a herald: measure, apply I, X or \(R_y(\pi/2)\), wait and measure again. The pairs at zero delay give, in closed form, the mean \(m\), the readout covariance \(V=b^2(1-qg_{eq}^2)\) and \(C_X=a^2-b^2\), from which \(a\), \(b\) and \(qg_{eq}\) follow. The delays add T1 (after X) and T2 (the decay of \(qg_X\)). The script repeats the notes’ experiment with 40 repetitions of 16 circuits of 2000 shots (T1 = 100 µs, T2 = 70 µs, \(e_{01}=0.015\), \(e_{10}=0.04\), 5 GHz):

Table 27.1: Standard calibration against the heralded qg sweep (notes §31, recomputed).

true thermal population (temperature)	\(e_{01}\) standard	\(e_{01}\) qg	temperature qg
0 (0 mK)	0.0145	0.0149	\(11\pm15\) mK (bound)
0.01 (52 mK)	0.0240	0.0149	\(52.5\pm1.0\)
0.03 (69 mK)	0.0418	0.0149	\(68.9\pm0.9\)
0.08 (98 mK)	0.0915	0.0153	\(98.3\pm0.8\)

The standard calibration reads temperature as readout error: it overestimates \(e_{01}\) by exactly \(p\cdot b\) and sees the qubit perfectly cold at any temperature.
The qg sweep is unbiased in \(e_{01}\), \(e_{10}\), \(p\) and the temperature, and resolves it to 1 mK from 52 mK up (at \(p=0\) it only gives a bound).
T1 and T2 come out unbiased in both, but the joint qg fit is more precise (T1 spread 1.5–2.1 µs against 2.8–3.1; T2, 2.2–2.5 against 3.1–4.8), partly because of the maximum-likelihood fit itself.

As IBM circuits (§86). With 32 one-qubit circuits with mid-circuit measurement, on three IBM device noise models, all three committed predictions held: T1 and T2 within 25% and readout errors within 0.01. The qg T2 sits closer to the reported value than the standard one, which comes out 10% high on two of them. What a simulated model cannot test is precisely the thermal population (Aer has none). The real run is pending and is the first one worth doing on hardware.

What a real chip already says (§168, §170). On IQM Garnet the readout was corrected with the two poles, on the same qubit: \(e_0=0.0055\) and \(e_1=0.029\) in §168, and \(0.0030\) and \(0.0265\) in §170. That is the standard calibration of the table above, so any thermal population the qubit had is counted inside \(e_0\). The two numbers do not separate it, and the change of \(e_0\) between runs does not say whether the readout or the temperature changed. The two runs also found something the affine readout model does not have: an error that depends on the angle. With the poles corrected, \(R_y\) read 16–55 m-qg low at the intermediate angles of §168 (over-rotations of \(1.3^\circ\) to \(3.9^\circ\)), and the §170 eigenstate 26–71 m-qg low (Section G.6). Calibrating the readout is not enough: the angle of the gates has to be characterized too, and in m-qg that error shows without any model.

Figure 27.1: Left: estimated readout error \(e_{01}\) against the true thermal population; the standard calibration inflates it by \(p\cdot b\), the qg sweep stays at the true value. Right: estimated effective temperature against the true one; the standard one is always 0.

27.3 Which noise is it? The mean readout against XEB and HOP

With \(N\) shots of a 4-qubit quantum-volume circuit, can one decide whether the noise is dissipative (T1) or unital (dephasing or depolarizing)? The notes (§18) set it up with 24 circuits, three channels and 10 strengths, and each measure is turned into a classifier with a single threshold fitted on 16 circuits and scored on 8 it has not seen. XEB and HOP need the ideal distribution (a classical simulation); the mean readout and \(qg_S\) only need counts. The script recomputes it:

Table 27.2: Accuracy on held-out circuits (0.67 = always answer ‘unital’).

noise per gate	shots	mean readout	\(qg_S\)	XEB	HOP
strong, 2–50%	100	0.88	0.72	0.70	0.71
strong, 2–50%	1000	0.91	0.72	0.74	0.73
strong, 2–50%	10,000	0.91	0.73	0.76	0.75
weak, 0.5–5%	100,000	0.67	0.67	0.67	0.67

With strong noise, the mean readout identifies T1 much better than the standard measures, from only 100 shots and without the ideal distribution: 0.88 with 100 shots and 0.91 with 1000 or more. Its accuracy grows with the strength (0.75 below 10% per gate, 0.98 above 25%). XEB and HOP measure how much noise there is, not which kind.
At realistic strengths everything is at chance, even with 100,000 shots. The limit is not shot noise: the ideal mean readout already varies from circuit to circuit (spread 0.139, from −0.325 to +0.334), more than the T1 shift.

The practical consequence: the mean readout is a T1 detector on circuits whose ideal value is known, such as an idle wait before measurement, not a passive detector on any circuit.

27.4 Coherent errors and the always-on ZZ

A conditional-phase error (an extra ZZ) conserves the weight, so one might expect it to pass the weight filter unnoticed. The notes tested this twice with committed predictions, and both times the prediction failed in an informative way:

A coherent RZZ after every native gate (§145): the kept fraction fell by 0.009–0.020 at \(\varepsilon=0.10\) rad, and the filtered error grew only 4% on the ECR devices but 62% on FakeTorino (CZ). The error acts inside the decomposition of the logical gate, where the qubits are not in the computational basis, and part of it leaves the sector.
Always-on ZZ between neighbours (§150, 5 and 15 kHz): the filter removed 32–60% of the added error; the filtered error grew 14–38% at 15 kHz and stayed 4–5 times below raw. Short circuits suffer least.

On these devices coherent errors are filtered in proportion to the time the qubits spend outside the computational basis.

27.5 Control quantization: an even grid in the angle or in qg

The electronics sets each angle from a grid of \(2^b\) values. The usual choice is a grid even in \(\theta\); a grid even in \(qg=\cos\theta\) is even in the Born probability and, on the Bloch sphere, is made of equal-area bands (notes §17). The script recomputes it:

One qubit, random targets: the qg grid’s worst probability error is exactly \(\pi/2\) times smaller (1.568, 1.570 and 1.570 with 4, 8 and 10 bits; it saves \(\log_2(\pi/2)\approx0.65\) bits). But its worst infidelity is worse, by a factor that grows like \(2^b\) (from 2.7·10⁻³ to 1.7·10⁻² with 4 bits), because it is coarse near the poles.
Loading a distribution with a tree of \(R_y\) rotations (the state-preparation step of quantum Monte Carlo): in qg each rotation is set directly by a conditional probability, \(qg=2p-1\). In the 45 cases (4, 6 and 8 qubits; 4, 6 and 8 bits; five distributions), the qg grid lowers the total variation distance in 43 of 45 (median 1.56 times, up to 2.9).
For smooth distributions the infidelity is also lower with the qg grid (25 of 27, median 1.98 times); for sparse or sharply peaked ones the \(\theta\) grid wins (15 of 18, median 3.3 times, up to 56).

Design rule: a grid even in qg when the task is to reproduce probabilities (sampling, Monte Carlo, smooth distributions); a grid even in \(\theta\) when state fidelity matters and the targets sit near the poles.

Figure 27.2: Left: accuracy in telling T1 from unital noise with four measures, at 100 and 10,000 shots and strong noise. Right: ratio between the total variation distance with a \(\theta\) grid and with a qg grid in the 45 loading cases; above 1 the qg grid wins.

27.6 What remains open

The heralded sweep (§86, 32 one-qubit circuits) is ready for a real IBM device and is the first thing worth running: the thermal population, its central point, does not exist in the simulators. T1 detection at realistic strengths needs circuits with a known ideal value.

27.7 Without qang and with qang

Table 27.3: The chapter’s actions without the qang scale and with it.

action	without qang	with qang	difference
calibrate the readout	two circuits, “\(\|0\rangle\)” and “\(\|1\rangle\)”: \(e_{01}\) inflated by \(p\,b\)	heralded sweep: unbiased	in favour; separating by repeated measurement is known practice, qg adds the closed form
measure the temperature	the standard calibration always sees 0	\(T_{\text{eff}}\) from \(qg_{eq}\), to 1 mK	in favour
measure T1 and T2	separate fits	joint qg fit: 1.5 times more precise	in favour, partly because of the joint fit
tell T1 from unital noise	XEB or HOP, with ideal simulation: 0.70–0.76	mean readout, counts only: 0.88–0.91	in favour with strong noise; none with realistic noise
anticipate coherent ZZ errors	“the filter does not see them”	the filter removes 32–60% of the always-on ZZ	in favour, although the committed prediction failed
quantize the control	grid even in \(\theta\)	grid even in qg: probability \(\pi/2\) times better, 43 of 45 loads better	in favour for probabilities; against for fidelity near the poles

Balance. This is the chapter where the qang scale contributes most on its own, without depending on the weight filter: the equilibrium readout is a thermometer, the instrument’s readout is affine in qg and separates from the temperature in closed form, the mean readout tells dissipation from unital noise, and a grid even in qg is the right grid for probabilities. The limits are written down: with realistic noise the mean readout does not classify arbitrary circuits, and the qg grid loses in fidelity near the poles.

27.8 In code

import numpy as np

f = 5e9                                  # qubit frequency (Hz)
h, kB = 6.62607015e-34, 1.380649e-23
p = 0.03                                 # residual thermal population
qg_eq = 1 - 2 * p                        # equilibrium readout: 940 m-qg
T_eff = h * f / (2 * kB * np.arctanh(qg_eq))
T_eff * 1e3                              # ~69 mK

27.9 Summary

The equilibrium readout \(qg_{eq}=\tanh(hf/2k_BT_{\text{eff}})\) is a thermometer, and the instrument’s readout is affine in qg.
The standard calibration counts temperature as readout error; the heralded sweep separates the two without bias and measures the temperature to 1 mK.
The mean readout tells T1 from unital noise with 100 shots when the noise is strong (0.88 against 0.70–0.72); with realistic noise nothing manages it on arbitrary circuits.
Coherent ZZ errors are not invisible to the filter: they are filtered according to the time spent outside the computational basis.
A grid even in qg improves 43 of 45 distribution loads; one even in \(\theta\) is better for fidelity near the poles.

Bibliographic notes

For T1, T2, read-out and ZZ coupling in superconducting qubits, see Krantz et al. (2019). XEB is Arute et al. (2019); HOP and quantum volume, Cross et al. (2019); randomized benchmarking, Magesan et al. (2011). The numbered research notes (§) cited in this chapter and the scripts that check it are in the repository (Monteverde 2026a); the definition of the qang unit is in Monteverde (2026b).