Appendix H — Catalogue of qang sub-measures
The qang unit does not reformulate any physical law. Each sub-measure in this catalogue is a reading of one probability, or of a few, and equals a quantity that physics or statistics already has. What a sub-measure can bring is a decision: fewer shots, a shorter pulse, a compilation chosen before running, a reading that noise does not bias. For each one the catalogue records its definition, what it equals, which decision it improves and with what evidence, and classifies it with one of the statuses of Table H.1.
The tables are generated from the registry qang.measures with tools/make_apendice_h.py, the same source for both languages of the book. A new sub-measure enters the catalogue as unverified and changes status only with a test, preferably pre-registered (Chapter 28).
| status | meaning | sub-measures |
|---|---|---|
| exact | exact identity: it names or re-scales something; no gain is claimed | 18 |
| measured gain | tested in simulation or analysis, with a measured gain over the usual practice | 14 |
| tested on hardware | tested on a real device | 3 |
| mixed | a gain in some regimes, none or a loss in others | 7 |
| no gain | tested: equivalent to the usual practice | 4 |
| against | tested: worse than the usual practice | 2 |
| unverified | untested proposal: a candidate for a pre-registered experiment | 3 |
| not adopted | proposal rejected as wrong or unsupported | 1 |
H.1 Base readings
| sub-measure | definition | equals | decision it improves | status | evidence | where |
|---|---|---|---|---|---|---|
| \(qg_Z\) | \(2p_0-1=\langle\sigma_z\rangle=\cos\theta\) | expectation value of \(\sigma_z\) | working coordinate measured without inversion | exact | identity; holds for mixed states | ch. 1, §7 |
| m-qg | \(m=1000\,qg_Z\) | \(\langle\sigma_z\rangle\) times a thousand, linear in \(p_0\) | budget gate error and shot noise on one scale | tested on hardware | IQM Garnet: 1.64 m-qg per \(X\), over-rotations of 16–55 m-qg hidden from the readout model; M2 failed | ch. 1, §168 |
| \(\theta\) | \(\arccos qg_Z\) | arcsine transformation of a proportion | error bar \(1/\sqrt N\) equal over the whole circle | exact | exact binomial sums; asymptotic (fails at \(Np\approx2.5\)) | ch. 1, §125 |
| \(qg_S\) | \(H(p_0)=H\big((1+qg_Z)/2\big)\) | Shannon entropy of the outcome | certify randomness together with the mean reading | mixed | alone it is fooled under T1; paired with the mean reading it is not | chs. 1 and 26, §7, §51 |
| qga | \(1\ \text{turn}=1000\ \text{qga}\) | an angle in another unit | write gate angles without factors of \(\pi\) | exact | convenience; changes neither gradients nor plateaus | ch. 1 |
| \((qg_Z,s)\) | \(s=\operatorname{sign}(qg_X)\) | the sign of the angle’s sine | cross a pole during an optimization | measured gain | H\(_2\): with \(qg_Z\) alone it stops at Hartree–Fock; with the sign it reaches the exact energy | ch. 1, §65 |
H.2 Derived quantities
| sub-measure | definition | equals | decision it improves | status | evidence | where |
|---|---|---|---|---|---|---|
| \(qg_\kappa\) | \(1/[\ln2\,(1-qg_Z^2)]\) | Fisher information about \(qg_Z\), in bits | damp an optimizer’s step near the poles | no gain | removes the divergence of the gradient in \(q\), but it is descent on \(\theta\): 103 steps both (ch. 30); the JAX bench repeats it | chs. 1, 23 and 30 |
| \(qg_{SNR}\) | \(\lvert qg_Z\rvert/\sqrt{1-qg_Z^2}\) | signal-to-noise of one shot, \(\lvert\cot\theta\rvert\) | shots to read a sign: \((z/qg_{SNR})^2\) | exact | 27 shots at 60° for 3σ; exact binomial: error < 1 % | chs. 1 and 3 |
| \(qg_L\) | \(\lvert\theta-\theta'\rvert/2\) | Bhattacharyya angle; Fubini–Study on a common meridian | distance between states from two \(Z\) histograms | exact | with different azimuths it is only a lower bound | chs. 1 and 3 |
| \(qg_\tau\) | \(\sqrt{1-r^2}=2\sqrt{\det\rho}\) | lack of purity | see a pulse’s area where the m-qg reading does not shift | exact | 0.277 for \(R_X(324^\circ)\) against 0.031 for \(R_X(-36^\circ)\); needs three bases | chs. 1–3 |
| \(\kappa_{qg}\) | \(1000\,\lvert\cos\theta\rvert\) | curvature of the reading with respect to the angle | predict the shift from amplitude error, \(\tfrac12\theta^2\sigma^2\kappa_{qg}\) | exact | QuTiP: agrees to second order | chs. 1 and 2, §160 |
| qFi | \(4\pi^2\cdot10^{-6}\ \text{qga}^{-2}\) | Fisher information of the qang angle | fixed angular resolution: \(159.15/\sqrt N\) qga | exact | not a flat manifold: the sphere has curvature 4 | ch. 1 |
| qHz | \(\text{qga}/\text{s}\) | angular frequency, \(2\pi/1000\) rad/s | none so far | exact | unit with no use in the book | ch. 1 |
H.3 States and registers
| sub-measure | definition | equals | decision it improves | status | evidence | where |
|---|---|---|---|---|---|---|
| \(1-r^2\) | \(4p_0(1-p_0)\) at definite weight | local purity | measure purity with one basis instead of three | measured gain | holds only with a definite Hamming weight | ch. 5 |
| \(\arccos(qg_Z/r)\) | \(qg_Z/r\) | Bloch polar angle of a mixed state | direction of a noisy state | mixed | in favour with depolarizing noise; against with pure dephasing | ch. 5, §45 |
| \(qg_{XX},qg_{YY},qg_{ZZ}\) | \(\langle\sigma_i\otimes\sigma_i\rangle\) | Pauli correlators | diagnose a Bell pair and predict teleportation without teleporting | measured gain | 3 settings instead of 9; 4–5 two-qubit gates instead of 10–11, maximum error 0.029 | chs. 12 and 13 |
| \(m_E\) | \(\sqrt{1000^2-m_\perp^2}\) | Eve’s information (Helstrom with purification) | how many test rounds certifying bits needs | exact | the computation is standard; the reading makes it geometric | ch. 18 |
| \((1000-m_r)/2000\) | \(m_r=1000\,r\) | energy of the passive state | know how much energy nobody can extract | exact | modest: the split is read at a glance | ch. 19 |
| syndrome \(qg_Z,qg_X\) | ancilla readings | the \(T_2/T_1\) rule | choose and recalibrate an error-correcting code without test shots | measured gain | 6.4 % over the oracle against 64–220 % for a fixed code; drift 13.1 % against 20.4 % | ch. 26 |
| \(\gamma\) | \(1+2\sqrt{1-qg^2}\) | overhead of cutting a gate | choose which gates to cut | mixed | up to 2.9 times against angle-blind rules; none against qiskit-addon-cutting | ch. 26 |
| \(qg_Z\) on graph states | \(qg_Z=0,\ qg_S=1\) | the local \(Z\) reading | detect entanglement with local \(Z\) readings | no gain | cannot tell a graph state from a product: the limit belongs to the \(Z\) basis | ch. 30 |
| parity \(qg\) | \((dqg)^2/(1-qg^2)=N^2\) | quantum Fisher information of a GHZ state | reach the Heisenberg bound with a parity reading | exact | the one-shot formula of Chapter 1 reaches the bound | ch. 34 |
H.4 Gates and compilation
| sub-measure | definition | equals | decision it improves | status | evidence | where |
|---|---|---|---|---|---|---|
| minimum area | \(\theta\bmod2\pi\); \(R_{ZZ}\) with period \(\pi\) | shortest representative of the same gate | compile with less pulse area: less error | measured gain | 4-qubit QAOA: area 4.4× and error 4.1× lower; the 1.5× prediction for \(R_{ZZ}\) failed (1.14×) | ch. 2, §160 |
| exposure | population outside the sector after each two-qubit gate | noiseless computation on the circuit | choose the compilation before spending shots | measured gain | rank correlation 0.80 with T1 leakage; zero exposure removes ≥ 98 % of T1 | ch. 2, §68 |
| \(N\)-conserving gates | \([U,\sum_i Z_i]=0\) | compilation within the sector | compile so that the filter removes T1 | mixed | the filter removes 99.5 % of T1 against 48.5 % with CNOT; none on ions | ch. 24, §61 |
| qg grid | equal steps in \(qg_Z\) | control quantization | quantize the control angles | mixed | probabilities \(\pi/2\) times better, 43 of 45 loads; against for fidelity near the poles | ch. 27 |
| best_chain | qubit chain of highest predicted \(K\) | layout choice from calibration data | choose the physical qubits before running | measured gain | 6 to 17 points more shots kept than the default layout (simulated) | ch. 32 |
H.5 Noise and readout
| sub-measure | definition | equals | decision it improves | status | evidence | where |
|---|---|---|---|---|---|---|
| weight filter, \(K\) | \(K=(1-\gamma)^{kd}\) | post-selection on the Hamming sector | read, train and differentiate under T1 without bias; shot cost known before running | tested on hardware | IQM Resonance: error 0.0615 → 0.0386 (1.59×) with 88 % of the shots; simulated: exact to \(10^{-12}\) m-qg; against with \(K\le0.14\) (2 of 12) and on deep LiH (76.6 → 85.0 mHa) | chs. 20, 21, 23, 30 and 31, §112 |
| echo, \(M^{1/2}\) | echo transfer matrix | correction of errors within the sector | remove what the filter cannot see | measured gain | −27 to −62 % of the filtered error, with \(\binom nk\) more echo circuits | ch. 22, §106 |
| \(qg_{eq}\), \(T_\text{eff}\) | heralded affine readout | readout calibration with repeated measurements | calibrate readout without mistaking temperature for error | measured gain | effective temperature to 1 mK; avoids an unsafe certifier and recovers 10–50 % of bits | chs. 18 and 27, §31 |
| step \(\times\lvert\sin\theta\rvert\) | \(\theta-\eta\max(\lvert\sin\theta\rvert,\epsilon)\,g\) | trust region near the poles | train with aggressive rates without tuning | measured gain | 2.3 to 13 times more successes from rate 2; little or nothing with safe rates | ch. 23 |
| local cost | \((1-\overline{qg_Z})/2\) | local cost of Cerezo et al. | avoid the barren plateau of a global cost | mixed | 470 times more gradient variance with 10 qubits and short circuits; none with deep circuits | ch. 23 |
| ZNE of the filtered reading | zero-noise extrapolation on the filtered \(qg_Z\) | post-selection and ZNE separately (mitiq) | mitigate T1 and dephasing together | measured gain | 55.9 → 1.1 m-qg with T1 and dephasing; with few shots ZNE does not pay | chs. 21 and 35 |
| pair \((qg_S,\overline{qg_Z})\) | entropy and mean reading of the register | XEB or HOP with an ideal simulation | tell T1 from unital noise from counts alone | mixed | 0.88–0.91 against 0.70–0.76 with strong noise; chance with realistic noise | chs. 27, 29 and 30 |
H.6 Readings of known physical laws
| sub-measure | definition | equals | decision it improves | status | evidence | where |
|---|---|---|---|---|---|---|
| \(\langle\beta\rangle\) | \(mc^2/E\) | expectation value of the Dirac matrix \(\beta\) | read \(E=mc^2\) and charge conjugation as a \(qg_Z\) | tested on hardware | exact and known; charge conjugation flips the reading; run §170 on IQM Garnet | app. G, §170 |
| \(\widehat{qg^2}\) | unbiased squares | standard bias correction | compare two clocks with few shots | measured gain | bias from +11.8 % to +0.9 % with 100 shots | ch. 8 |
| unbiased \(\sqrt{S_1^2+S_2^2+S_3^2}\) | unbiased Stokes squares | degree of polarization | estimate polarization with few photons | measured gain | the bias drops about three times, without vanishing | ch. 6 |
| \(\arccos(m/l)\) | \(\langle J_z\rangle/l\) | direction of \(\langle\mathbf J\rangle\) | read angular momentum as a \(qg_Z\) | exact | the vector model \(\arccos(m/\sqrt{l(l+1)})\) draws something else | ch. 11 |
| flavour \(qg_Z\) | \(1-2P_{\alpha\to\beta}\) | oscillation probability | read the mixing angle with a fixed error bar | exact | identity on the two-flavour formula | ch. 9 |
| \(m_Z\), \(m_X\) | readings of the chronology qubit | comparing density matrices | tell P-CTC from D-CTC with one \(Z\) reading | measured gain | P-CTC moves \(m_Z\), D-CTC does not | ch. 14 |
| \((1000+m_P)(1000+m_F)\) | \(=10^6\) | Yoshida–Kitaev law | judge a wormhole decoder | exact | the law is known; the scale shows that \(m_P=-500\) proves nothing | ch. 15 |
| \(N/2+\sum_n qg_n/2\) | \(G/G_0=\sum_n T_n\) | Landauer formula | read conductance and partition noise on one scale | exact | conductance = mean \(qg\), noise = mean \(qg^2\); the bar of \(\hat\theta\) is constant (0.985–1.009) | ch. 33 |
| line in artanh | \(\operatorname{artanh}qg\) | Fermi fit | measure electron temperature | against | bias of +2.9 % (Fermi fit: −0.14 %) | ch. 33 |
| \(\theta/2=\operatorname{gd}(r)\) | Gudermannian of the squeezing \(r\) | \(\tanh^2r\), \(\bar n\) and the entropy | read squeezing as a qang angle | exact | a new identity, exact | ch. 34 |
| \(\theta=2\pi\Phi/\Phi_0\) | interference fringes | Aharonov–Bohm phase | estimate the flux with an even error bar | exact | deviation \(1/2\pi\sqrt N\) constant; the sign needs another reading | ch. 34 |
H.7 Open and rejected proposals
| sub-measure | definition | equals | decision it improves | status | evidence | where |
|---|---|---|---|---|---|---|
| \(\kappa_{qg}\) compilation | minimum arrival curvature | — | mitigate barren plateaus | unverified | untested | chs. 1 and 2 |
| \(E_\text{total}\) | \(\langle H\rangle+\lambda_\theta L(\theta)+\lambda_\varphi L(\varphi)\) | — | regularize a VQE | unverified | \(L\) and \(\lambda\) undefined | ch. 1 |
| \(R(\theta,\varphi)\) | \(N[p_1+\alpha\cos\varphi]\) | — | model emission from the excited population | unverified | \(N\) and \(\alpha\) not fixed; not compared with an emission model | ch. 1 |
| 500 qga CPT | \(E_-\leftrightarrow\) a 500 qga shift | — | place antimatter on the scale | not adopted | no support; the correct reading is charge conjugation on \(\langle\beta\rangle\) | ch. 1, app. G |
| signs of \(qg_{Z,i}\) | \(\operatorname{sign}qg_{Z,i}\) | bit-string histogram | read the marked item in a noisy search | against | 66 % against 91 % for the histogram mode (5 qubits, 100 shots) | ch. 3, §63 |
| stopping \(qg_{SNR}\), \(qg_\kappa\) | threshold on \(\lvert qg_k\rvert\) | the formula for \(k^\ast\) | stop Grover iterations | no gain | they cannot tell whether the flag passed the pole; they cost one run per \(k\) | ch. 3 |
| qg cycles | angle-to-reading translation in Wigner, Kitaev and OTOC | — | protect states or measure chaos | no gain | they neither protect nor measure chaos | ch. 34 |
H.8 Balance
Of 52 sub-measures, 18 are exact identities, 14 have a measured gain in simulation or analysis, 3 were tested on a real device (with a mixed result for the milli-qang), 7 are mixed, 4 turned out equivalent to the usual practice, 2 lost to it, 3 are unverified and 1 was not adopted. The gains do not come from the unit but from the decisions the reading allows before spending resources: choosing the minimum-area representative, the compilation of least exposure, the code from two readings, or reading only the weight sector that T1 does not contaminate. The unverified sub-measures are the list of pending experiments; the catalogue keeps them in sight so that they are not cited as results.
H.9 In code
from qang import measures
measures.summary() # sub-measures per status
measures.by_status("sin_verificar") # the pending experiments
m = measures.find("min_area")
m.decision[1], m.evidence[1] # the decision it improves and the evidence, in English
measures.resolve(m) # qang.pulse_area.minimal_area_circuit