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Coherence and Fidelity Explained

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TECHNOLOGY EXPLAINER

Coherence and Fidelity
— How to read the yardsticks of a qubit: T1 and T2, error rates, randomized benchmarking

Announcements of quantum hardware are full of numbers such as "T1 of so many microseconds" or "two-qubit gate fidelity of 99.9%". Those numbers cannot be read correctly unless you know what they measure and what they leave out. This article assumes no physics background. With figures and small calculations it explains decoherence, the meaning of T1 and T2, the relation between fidelity and error rate, and how randomized benchmarking (RB) works. The second half covers why numbers from different technologies must not be ranked against each other and what materials cause qubits to lose their lifetime. It is the article to read after "Quantum Computing Fundamentals" and "Quantum Gates and Quantum Circuits".

Sources: a review (Krantz et al.), peer-reviewed papers (Nature, Science, Physical Review and others), arXiv papers, official ISO/IEC and IEEE pages / numerical examples are our own calculations / checked on 16 October 2026

A scene of measuring and recording a qubit's lifetime (a decay curve). It does not depict any real device or measurement result.
Fig. 1. Conceptual illustration generated with ImageGen. A scene of measuring and recording a qubit's lifetime (a decay curve). It does not depict any real device or measurement result. Not an actual product, facility, dimensional specification or measurement. Distinguish enlarged models from full-scale scenes.
In this article
  1. Three-line summary / what to know first / where this sits in the series
  2. The big picture: four yardsticks, what each measures
  3. What is decoherence?
  4. T1: the energy relaxation time
  5. T2: how long the phase survives (T2*, T2E and CPMG)
  6. Time and error: the longer a gate takes, the more finite lifetime adds error
  7. Fidelity and error rate: how to read "99.9%"
  8. Reported values, and cautions for reading them
  9. Randomized benchmarking (RB) and its limits
  10. A materials engineer's view: what shortens a qubit's life
  11. How to read evaluations and standards
  12. Glossary
  13. References (primary sources)
  14. Claim-to-source audit
Notation used in this article

Sourced = stated in a published source (link given)
Our calculation = a value this article computed or checked from stated assumptions
Not yet confirmed = standards under development, company plans and other items that are not settled
Our own reading for materials engineers is marked as commentary; inference is shown in bold. Every analogy comes with a note on where the analogy breaks down.

How to read the numbers in this article

The curves, tables and simulated data in Chapters 3 to 7 and 9 are our own illustrative calculations (with fixed random seeds), not measurements from a real device. The reported values in Chapters 5 and 8 are values each paper reported under its own conditions; they are not meant for ranking devices of different technologies. For standards (ISO/IEC, IEEE) we confirmed only the title, scope and status, not the text. This field moves fast, so standards carry a check date (16 October 2026).

1. Three-line summary / what to know first / where this sits in the series

  • The lifetime yardsticks (T1, T2): T1 is the time for a qubit to lose energy and fall from "1" to "0"; T2 is how long the heading (phase) of the arrow stays in step. T2 cannot exceed 2×T1. A T2 value changes with how it is measured (T2*, T2E, CPMG), so check the kind Sourced [Ref. 1]
  • The accuracy yardsticks (fidelity, error rate): how close one operation is to the ideal. "99.9%" is a rough "one error in 1000". But you cannot read it without knowing which operation, how it was measured, and whether it is an average or the best Sourced [Refs. 2, 3]
  • The conditions behind each yardstick differ between technologies: you cannot rank "microseconds" for superconducting qubits against "minutes" for ions or "seconds" for silicon. A floor on gate error is set by the ratio of gate time to lifetime Our calculation
What to know before reading

We use the arrow picture (north pole = 0, south pole = 1, tilted = superposition) and interference (adding waves) from "Quantum Computing Fundamentals", and the gate (an operation that rotates the arrow) from "Quantum Gates and Quantum Circuits". If you have not read them, the necessary parts are briefly explained again. Formulas are kept to a minimum, and harder material is tucked into folded "More detail" sections.

Where this sits in the series

This is the article to read after "Quantum Gates and Quantum Circuits". Next, "Quantum Algorithms" shows why "how many gates can be run in a row" matters so much for algorithms. Technology-specific articles, quantum error correction, and cryogenics and control are covered in their own articles in the series.

2. The big picture: four yardsticks, what each measures

There is no single yardstick for the "quality" of a quantum computer. It helps to sort them into four.

Four yardsticks for the "quality" of qubits 1. Lifetime T1, T2 (coherence times) How long the arrow stays steady Measured by varying a waiting time Does not show: errors during gates 2. Gate accuracy Fidelity, error rate How close one operation is to ideal Measured by RB and others Does not show: every kind of error 3. Preparation and readout SPAM and readout errors Whether 0/1 is read correctly Measured with known states Separate from gate errors 4. Whole circuit Quantum volume and others Whether long circuits run Measured by running random circuits Does not show: ranking for each use The link: the floor on gate error is set roughly by gate time divided by coherence time So read the numbers in 2 together with 1 (Chapter 6). 3 is measured separately from gates, and 4 is the combined result. Knowing what each yardstick measures, and what it does not, is the first step in reading a number. Never rank devices of different technologies by their numbers (Chapters 8 and 9).
Fig. 2 Conceptual diagram (vector drawing). A sorting of the four yardsticks for qubit quality (conceptual). It shows what each measures and what it does not. The grouping is our commentary, not a standard or a classification from one source.

This article mainly covers 1 (lifetime) and 2 (gate accuracy). Items 3 and 4 are touched on in later chapters and in "How to read the yardsticks and standards". What matters is knowing what each yardstick leaves out. For example, even with a long lifetime (1), inaccurate gate control gives low gate accuracy (2). Conversely, even if the numbers in 2 are good, large preparation and readout errors (3) will spoil the result.

A stumbling block: why can't one number describe "performance"?

As with cars, "top speed" alone does not tell you whether a car is good (you also need fuel economy, brakes, comfort). A quantum computer is similar: the "number of qubits" or "T1" alone does not tell you whether it is usable. Where the analogy breaks down: a car's properties can be measured nearly independently, but a qubit's lifetime, gate errors and readout errors are related to one another (Chapter 6) (our commentary).

3. What is decoherence?

As we saw in the Fundamentals article, interference needs the qubit's arrow to be correct even in its heading (phase). Real qubits, however, are slightly affected by their surroundings (heat, fluctuating electromagnetic fields, defects in materials, nearby atomic nuclei and so on). The resulting degradation of the arrow is called decoherence.

There are two main ways it happens.

Decoherence = the environment degrading the arrow (T1 and T2) T1: energy is lost and the arrow falls north |0⟩ |1⟩ x y t = 2·T1 dashed grey: t = 0 (south = 1) dashed brown: t = T1 → north component 0.26 Analogy: a spinning top loses its tilt to friction and falls over. It ends up north (the low-energy direction) Limit of the analogy: in reality probabilities decay exponentially T2: the heading (phase) falls out of step |0⟩ |1⟩ x y average arrow (short) grey: arrow at t = 0. Thin green dashes: arrows that drifted apart Analogy: many clocks start together but run at slightly different rates, and the hands spread out The average hand (blue) shrinks = phase information is lost Limit: a single qubit is disturbed similarly by fluctuations A perfectly isolated qubit cannot be made, because it must be connected to the outside for control and readout.
Fig. 3 Conceptual diagram (vector drawing). Conceptual diagram of the two faces of decoherence (T1: energy relaxation, T2: loss of phase coherence). The T1 arrows follow 1 − 2e^(−t/T1) as drawn by us; they are not data from any device.
  • T1 (energy relaxation): the qubit loses energy and falls from "1" (the high-energy state) to "0" (the low-energy state). The arrow leans from south to north
  • T2 (phase relaxation): even if the tilt stays the same, the heading around the north pole (the phase) falls out of step. Arrows on the equator point in scattered directions and the average arrow shrinks

These two are measured with separate times (T1 and T2). The review describes relaxation by a longitudinal rate (Γ1 = 1/T1) and a transverse rate (Γ2 = 1/T2), the latter having an extra contribution from "pure dephasing" Sourced [Ref. 1].

Common misconception: "decoherence happens because an observer looked" is wrong

It is often said that "a quantum state breaks when it is looked at", but decoherence happens even when no one is looking, through exchange with the environment. Measurement (Chapter 4 of the Fundamentals article) is an operation by a person or instrument that deliberately reads out the state; decoherence is information about the state leaking into the environment unintentionally. Both leave the arrow unable to return, but they are different things (our commentary).

A stumbling block: if we isolate it completely, does decoherence vanish?

In principle, a qubit completely isolated from its environment does not degrade. In practice you must connect it to the outside to operate it with gates and read out the result. Wherever you open a connection, noise can come in. Balancing "isolation" against "control and readout" is the hard part of design (our commentary; see our article on cryogenics and control).

4. T1: the energy relaxation time

T1 is a time that expresses how fast a qubit falls from "1" to "0". Measuring it is simple. Put the qubit in "1", wait a time τ, and measure. Repeat for various τ, each many times, and count the fraction that is "still 1". That fraction falls off exponentially.

Measuring T1: set to south (1), wait, then measure Procedure (repeat for different waits τ) π to 1 wait τ measure For each τ repeat about 1000 times and count how often it is still "1" π = a 180° rotation (like the X gate) The "still 1" fraction decays exponentially 0% 50% 100% 0 1 2 3 4 5 Waiting time τ (in units of T1) τ = 1·T1: 37% τ = 2·T1: 14% τ = 3·T1: 5% T1 = the time for the "still 1" fraction to fall to 1/e ≈ 37%. Example: in a measurement shown in a review, one transmon qubit had T1 = 85 µs [Ref. 1].
Fig. 4 Chart including our calculation (vector drawing). The T1 procedure and its exponential decay (conceptual). The curve is our drawing of e^(−t/T1), not data from a specific device. 37%, 14% and 5% are our calculation (e⁻¹, e⁻², e⁻³).

The time at which the "still 1" fraction reaches 1/e (about 37%) is T1. At twice T1 it is about 14%, and at three times about 5% Our calculation (e−1 = 0.368, e−2 = 0.135, e−3 = 0.050). The review shows a measured example of a transmon qubit with T1 = 85 µs Sourced [Ref. 1].

Waiting time τRatio to T1Fraction still "1"
0.1·T10.190.5%
0.5·T10.560.7%
1·T1136.8%
2·T1213.5%
5·T150.7%

Our calculation. We evaluated the exponential e−τ/T1.

A stumbling block: T1 of 100 µs does not mean the qubit surely breaks after 100 µs

T1 is not an "average lifetime" but the time for the fraction to fall to 1/e. Even a qubit with T1 = 100 µs has already lost about 10% by 10 µs, yet about 37% remains at 100 µs. Individual qubits also decay at unpredictable moments (probabilistically), so "lifetime" is an analogy (our commentary).

5. T2: how long the phase survives (T2*, T2E and CPMG)

T2 is how long the arrow's heading (phase) stays in step. A single measurement cannot show the phase, so we measure it with a little cleverness. There are three typical ways, each sensitive to a different kind of "fluctuation", so the values differ too.

T2: how long the phase survives. The value depends on how it is measured Ramsey (T2*) Includes slow, static fluctuations Hahn echo (T2E) A π pulse mid-way cancels slow fluctuations CPMG / dynamical decoupling Many π pulses make it longer still π/2 π/2 τ π/2 π π/2 τ/2 τ/2 π/2 π π π π/2 repeated π pulses Measures the "drift" Echo rewinds the drift Depends on the protocol A measurement example on one transmon qubit (values from a figure in a review; just one example) T1 (energy relaxation) 85 µs T2* (Ramsey) 95 µs T2E (Hahn echo) 120 µs 2×T1 (upper bound on T2) 170 µs Relation: 1/T2 = 1/(2·T1) + 1/Tφ (Tφ is the time for pure phase disturbance). Hence T2 can never exceed 2·T1 [Ref. 1]. A value labelled "T2" can be T2*, T2E or CPMG and changes accordingly. Without the kind it cannot be compared.
Fig. 5 Conceptual diagram (vector drawing). The pulse sequences of the three ways of measuring T2 (Ramsey, Hahn echo, CPMG) and a measurement example for one transmon qubit. The numbers are the example in Fig. 5 of the review [Ref. 1]; the sequences are conceptual.
NameProcedureWhat it measuresTypical trend
T2* (Ramsey)π/2 pulse, wait, π/2 pulse, measureThe phase drift, including slow (static) fluctuationsTends to be the shortest
T2E (Hahn echo)π/2, wait τ/2, π (flip), wait τ/2, π/2, measureSlow fluctuations are cancelled by the mid-way flip; fast ones remainLonger than T2*
CPMG / dynamical decouplingInsert many π pulsesCancels still more slow fluctuations (the range cancelled depends on the protocol)Longer still

The procedures follow the explanation in the review [Ref. 1] Sourced. The trend (T2* < T2E < CPMG) is the general tendency from cancelling slow fluctuations; in an individual device the order need not hold (our commentary).

In the review's measured example (one transmon qubit), T1 = 85 µs, T2* from a Ramsey measurement = 95 µs, and T2E from a Hahn echo = 120 µs. These do not reach the limit set by T1 (2×T1 = 170 µs) Sourced [Ref. 1].

The relation between T1 and T2: T2 cannot exceed 2×T1

The phase relaxation always includes a contribution from energy relaxation (T1). So there is a relation 1/T2 = 1/(2·T1) + 1/Tφ (Tφ is the time for the phase alone to be disturbed). Even if Tφ is infinite, T2 only reaches 2·T1 Sourced [Ref. 1].

T1Tφ (phase-only disturbance)T2 (calculated)How to read it
100 µsinfinite (no disturbance)200 µsThe maximum T2 (2×T1)
100 µs200 µs100 µsShortened by the phase disturbance
100 µs100 µs67 µsA large phase disturbance makes T2 shorter than T1

Our calculation. We evaluated 1/T2 = 1/(2·T1) + 1/Tφ with illustrative values (T1 = 100 µs), not values of any real device. In real devices the phase disturbance may decay in a form other than an exponential, and this formula assumes an exponential decay [Ref. 1].

Common misconception: "a long T2 means a good qubit" is wrong

T2 changes a lot with how it is measured (T2*, T2E, CPMG), so a T2 without its kind cannot be compared. A value obtained with dynamical decoupling (repeated π pulses) is a value for "protecting an idle qubit", not a value during a gate. To know the quality of a qubit used for gates, you must also check the gate error rates (Chapters 7 and 9) (our commentary).

6. Time and error: the longer a gate takes, the more finite lifetime adds error

What a long lifetime (T1, T2) means becomes clear only against the gate time. During a gate the qubit also decays, following its lifetimes. For a gate that takes time t, the error set by lifetime alone (a floor) is roughly given by the formula below.

The formula and its meaning

ε ≈ t/(3·T2) + t/(6·T1) (when the gate time t is well below T1 and T2). Meaning: the error from lifetime is "the fraction of the lifetime that the gate takes", times factors of 1/3 and 1/6. With T1 = T2 = 100 µs and a 25 ns gate (1/4000 of the lifetime), ε ≈ 1.3×10−4 Our calculation.

The longer a gate takes, the more finite lifetime adds error (our calculation) 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 1 10 ns 100 ns 1 µs 10 µs 100 µs Gate time t Error from finite lifetime ε (floor) error 1% error 0.1% T1 = T2 = 100 µs T1 = 100 µs, T2 = 200 µs T1 = T2 = 1 ms Example: with T1 = T2 = 100 µs, a 25 ns gate gives ε ≈ 1.3×10⁻⁴, 1 µs gives 5.0×10⁻³, and 10 µs gives 4.8×10⁻². Formula: ε = 1 − (3 + 2e^(−t/T2) + e^(−t/T1)) / 6 ≈ t/(3·T2) + t/(6·T1), for one idle period; control errors are not included. Real gate errors also include control imperfection, leakage and crosstalk, so they are larger than this floor.
Fig. 6 Chart including our calculation (vector drawing). A floor on the error set by gate time and lifetimes (T1, T2) alone (our calculation). The formula is the average gate fidelity of a qubit undergoing amplitude and phase damping (using Nielsen's relation [Ref. 2]), checked numerically by us. It is not a value for any specific device.
Lifetimes (illustrative)25 ns gate100 ns gate1 µs gate10 µs gate
T1 = T2 = 100 µs1.3×10−45.0×10−45.0×10−34.8×10−2
T1 = 100 µs, T2 = 200 µs (the maximum T2)8.3×10−53.3×10−43.3×10−33.2×10−2
T1 = T2 = 1 ms1.3×10−55.0×10−55.0×10−45.0×10−3

Our calculation. We evaluated ε = 1 − (3 + 2e−t/T2 + e−t/T1)/6. This is the average gate fidelity of a qubit that waits for time t while undergoing amplitude damping (T1) and dephasing (T2). We used the relation between average gate fidelity and process fidelity for one qubit (Nielsen, 2002) [Ref. 2] and checked it by numerical simulation. Control errors, leakage and crosstalk are not included; real errors are larger.

A stumbling block: "long lifetime" and "fast gates" are separate efforts

To reduce errors you can either extend the lifetime (better materials and environment) or make the gate faster (better control). As the table shows, a tenfold longer lifetime cuts the error of the same gate tenfold. On the other hand, making gates too fast raises control errors and leakage into nearby states. In ion experiments, one report varied the two-qubit gate time from 3.8 µs to 520 µs to study the relation between speed and fidelity Sourced [Ref. 4] (our commentary).

"Close to the T1 limit": lifetime becomes the ceiling on gate accuracy

Sung and colleagues state that the fidelities of their CZ and iSWAP gates on superconducting qubits (99.76 ± 0.07% and 99.87 ± 0.23%) are close to the T1 limit Sourced [Ref. 5]. In terms of the formula above, "ε is almost entirely set by t/(3·T2) + t/(6·T1)". At that point however far control is polished, the rest is set by lifetime (and therefore by materials).

7. Fidelity and error rate: how to read "99.9%"

Fidelity (F) is a number from 0 to 1 (often written as a percentage) showing how close what was actually produced is to the ideal. Closer to 1 is better, and the error rate is ε = 1 − F. A fidelity of 99.9% means an error rate of 0.1%, roughly one error in 1000.

Fidelity F and error rate ε (= 1 − F): "one error in how many?" Fidelity F Error ε Roughly... 99% 1×10⁻² 1 in 100 99.9% 1×10⁻³ 1 in 1,000 99.99% 1×10⁻⁴ 1 in 10,000 99.999% 1×10⁻⁵ 1 in 100,000 Bar length = digits of "one in how many" (guide) There are three kinds of "fidelity" State fidelity How close a prepared state is to the target Process fidelity How close the whole operation is to the ideal Average gate fidelity Averaged over all input states (as in RB) F_avg = (d·F_process + 1) / (d + 1) One qubit (d = 2): F_process 0.9985 → F_avg 0.9990 When you hear "99.9% fidelity", check these four things 1. Which operation (single-qubit gate / two-qubit gate / readout / idling) 2. For how many qubits 3. Average or best (or worst) 4. How measured (randomized benchmarking, tomography, a state measurement ...)
Fig. 7 Chart including our calculation (vector drawing). The relation between fidelity and error rate (our calculation) and the three kinds of fidelity. The relation between average gate fidelity and process fidelity follows Nielsen (2002) [Ref. 2]. The numerical example on the right is our calculation.

There are at least three kinds of "fidelity". State fidelity (how close a prepared state is to the target), process fidelity (how close the whole operation is to the ideal) and average gate fidelity (fidelity averaged over all input states). Nielsen gave a simple formula (Favg = (d·Fprocess + 1)/(d + 1)) for the average gate fidelity and process fidelity of a d-dimensional system Sourced [Ref. 2]. For one qubit (d = 2) with Fprocess = 0.9985, Favg = (2×0.9985 + 1)/3 = 0.9990 Our calculation.

Common misconception: "two devices with 99.9% fidelity have the same quality" is wrong

The same "99.9%" means different things depending on which operation it is for (single-qubit gates usually give better values than two-qubit gates), for how many qubits (it tends to worsen as qubits are added), whether average, best or worst, and how it was measured (RB, tomography, a state measurement). As Chapter 9 shows, an average gate fidelity cannot be compared directly with fault-tolerance thresholds Sourced [Refs. 6, 7].

What this calculation teaches: the "order of magnitude" of the error rate sets the usable circuit length

As the gates article showed, with error rate ε per gate the success probability of the whole circuit falls to about 37% after about 1/ε gates. With an error rate of 10−2 that is about 100 gates, with 10−3 about 1000, with 10−4 about 10,000 Our calculation. One order of magnitude in error is one order of magnitude in usable circuit length.

8. Reported values, and cautions for reading them

Here are examples of values reported in papers, shown with their measurement conditions. Each is an individual report with its own technology, purpose and conditions, and none is meant to rank technologies.

Examples of long times reported so far (log scale; conditions differ, so do not compare) 1 µs 10 µs 100 µs 1 ms 10 ms 100 ms 1 s 10 s 100 s 1000 s 10⁴ s Superconducting: Google 105-qubit processor, mean mean over an error-correction machine T1 68 µs, T2 (CPMG) 89 µs Superconducting: Ta film (Place 2021) lifetime and with dynamical decoupling above 0.3 ms Superconducting: Ta on Si (Bland 2025) best single qubit T1 up to 1.68 ms Diamond NV, room temperature (2009) crystal with 13C cut to 0.3% T2 1.8 ms Diamond NV: single spin (2018) decoupling tailored to nearby nuclear spins above 1 s Donor electron spins in 28Si (2012) many spins (ensemble) T2 up to 2 s 31P nuclear spin in 28Si (2014) single qubit, CPMG above 30 s Ion 171Yb+ (2017) single qubit, dynamical decoupling above 10 min Ionised-donor nuclear spins in 28Si (2013) many spins (ensemble) 39 min (room temp.) Blue = T1 or lifetime, grey = T2 (CPMG), amber = diamond NV, green = spins in silicon, purple = ion. Conditions differ (single or ensemble, dynamical decoupling or not, temperature, gate qubit or memory). This figure does not rank technologies. Sources: Refs. 8, 9, 10, 11, 12, 13, 14, 15, 16
Fig. 8 Conceptual diagram (vector drawing). Examples of long coherence times reported in papers (log scale). The measurement conditions differ greatly, and some are not values for qubits usable for gates. The especially long values are ensemble averages over many spins or memory use with dynamical decoupling. The figure is not a ranking.
System (report)Reported valueWhat the value is, and conditions
Superconducting transmon
Google Quantum AI, 105-qubit processor (2024) [8]		Mean T1 68 µs, mean T2,CPMG 89 µsMean over the qubits of the machine used in the error-correction experiment
Superconducting transmon, tantalum film
Place et al. (2021) [9]		Lifetime and coherence time (with dynamical decoupling) above 0.3 msTwo-dimensional transmons with niobium replaced by tantalum. Lifetimes improved in 17 devices
Superconducting transmon, tantalum on high-resistivity silicon
Bland et al. (2025) [10]		T1 up to 1.68 ms. T2E exceeds T1The best single qubit. Mean quality factor over 45 qubits 9.7×106
NV electron spin in diamond
Balasubramanian et al. (2009) [11]		T2 = 1.8 msRoom temperature. CVD diamond with 13C reduced to 0.3% by isotope control
NV electron spin in diamond
Abobeih et al. (2018) [12]		Above 1 sA single electron spin. Decoupling tailored to the surrounding nuclear-spin environment
Donor electron spins in 28Si
Tyryshkin et al. (2012) [13]		T2 up to 2 s (10 s extrapolated at 1.8 K with a magnetic-field gradient)High-purity silicon with residual 29Si below 50 ppm. Many spins (donor density 1014 to 1015 cm−3)
31P nuclear spin on a 28Si substrate
Muhonen et al. (2014) [14]		Above 30 s with a CPMG sequence. Control fidelity above 99.99%A top-gated nanostructure on an isotopically controlled 28Si substrate. A single qubit
Ion 171Yb+
Wang et al. (2017) [15]		Above 10 minA single qubit, cooled with 138Ba+, with a few thousand dynamical-decoupling pulses (memory use)
Nuclear spins of ionized donors in 28Si
Saeedi et al. (2013) [16]		39 min at room temperature (3 hours at cryogenic temperature)A value for many spins, operated optically, with dynamical decoupling

Sourced (statements in the abstracts and texts of the papers). The arrangement is our commentary.

System (report)Reported accuracy of operationsWhat the value is, and conditions
Ion 43Ca+
Ballance et al. (2016) [4]		Single-qubit 99.9934(3)%, two-qubit 99.9(1)%Hyperfine qubits in a room-temperature trap. Two-qubit gate time studied from 3.8 to 520 µs
Ion 9Be+
Gaebler et al. (2016) [17]		Single-qubit error 3.8(1)×10−5 per gate, two-qubit gate error 8(4)×10−4Qubits with a memory coherence time above 1 s. The two-qubit value is deduced from Bell-state creation
Ion 40Ca+
Clark et al. (2021) [18]		Infidelity of Bell-state preparation 6(3)×10−4Optical-frequency qubits. 35 µs. Measured directly without subtracting experimental errors
Superconducting
Sung et al. (2021) [5]		CZ 99.76 ± 0.07%, iSWAP 99.87 ± 0.23% (two-qubit interaction fidelities)Tunable coupler. Reported to be close to the T1 limit
Superconducting
Bland et al. (2025) [10]		Single-qubit gate 99.994%Tantalum-on-high-resistivity-silicon transmon
Neutral atom
Evered et al. (2023) [20]		Two-qubit 99.5%Up to 60 atoms in parallel. Evaluated by several methods based on repeated gates
Silicon spin
Xue et al. (2022) [19]		Single- and two-qubit both above 99.5%Extracted by gate set tomography. The average single-qubit fidelity including crosstalk and idling errors on the neighbouring qubit is above 99%

Sourced (statements in the abstracts of the papers). The arrangement is our commentary. How each value was obtained (RB, tomography, deduction from Bell states and so on) differs between papers, so they cannot be compared directly.

Caution: do not rank numbers from different technologies against each other

Listing "minutes for ions, seconds for silicon, milliseconds for superconducting qubits" makes it look like a ranking. But they cannot be compared, for these reasons. (1) The purposes differ: the long values may be for idle storage (memory) or use dynamical decoupling, not gates. (2) The counts differ: an average over many spins (an ensemble) versus a single qubit. (3) The measurements differ: T1, T2*, T2E, CPMG, RB, tomography. (4) The gate speeds differ: a lifetime means something only together with the gate time (Chapter 6). (5) Best and average differ: the best single qubit versus the mean over all qubits. Compare only within the same technology, with the same measurement and conditions (our commentary).

9. Randomized benchmarking (RB) and its limits

(1) Why RB is needed

To learn a gate's error rate one could use "tomography", examining the state produced by the gate in detail. But this method mixes in preparation and measurement errors, and its effort grows steeply with the number of qubits. Knill and colleagues proposed randomized benchmarking as a way to estimate the computationally relevant errors without relying on accurate preparation and measurement, and reported, for trapped ions, an error probability of 0.00482(17) per randomized π/2 pulse Sourced [Ref. 21]. Emerson and colleagues described a scalable method of noise estimation using random unitary operations Sourced [Ref. 22].

(2) How it works: random operations, then the operation that undoes them

Randomized benchmarking (RB): repeat random operations and see how fast it degrades 1. Prepare Start from north (0) 2. m random gates m randomly chosen operations 3. One undoing gate Makes the whole sequence the identity 4. Measure "Success" if 0 is read Repeat for many values of m and count the "success" fraction 50% 75% 100% 1 10 100 1000 Number of random gates m Success fraction p = 0.998 ⇒ r = 0.1% p = 0.990 ⇒ r = 0.5% Dots = simulated data (20,000 shots per m, seed 0). Lines = fits of A·pᵐ + 0.5 Three points to read 1. The decay rate p matters Success falls with m; its speed gives the error per operation 2. Robust to prep/readout errors They change the height A, not the rate p 3. The error rate r r = (1 − p)(d − 1)/d; d = 2 for one qubit, d = 4 for two
Fig. 9 Chart including our calculation (vector drawing). The RB procedure and how to read the decay (our calculation). Dots are our simulated data (seed 0), not measurements from a real device. The formula follows Magesan et al. (2011) [Ref. 3].
  1. Start from north (0) and apply m randomly chosen gates
  2. At the end apply one gate that undoes everything (ideally the whole sequence does "nothing" and the arrow returns north)
  3. Measure; "success" if 0 comes out. Vary m, repeat each many times and count the success fraction
  4. As m grows, the success fraction falls in the form A·pm + B. The decay rate p gives the average error rate r per operation

The error rate is found from r = (1 − p)(d − 1)/d, where d is 2(number of qubits): d = 2 for one qubit and d = 4 for two Sourced [Ref. 3]. For example, for one qubit with p = 0.998, r = (1 − 0.998)/2 = 0.1%. In our simulated data, for true error rates of 0.1% and 0.5%, the error rates found by fitting were 0.1003% and 0.5046% Our calculation (20,000 shots per m, seed 0).

Preparation and measurement errors only enter the heights of the curve (A and B) and do not affect the decay rate p. This is why RB is said to be robust to preparation and measurement errors Sourced [Ref. 23].

(3) The RB family

MethodWhat it measuresSource
Standard RBThe average error rate of the whole set of gates, accounting for preparation and measurement errorsEmerson et al. (2005) [22], Knill et al. (2008) [21], Magesan et al. (2011, 2012) [3, 23]
Interleaved RBThe error rate of one particular gate (the gate under test is inserted between the random gates)Magesan et al. (2012) [24]
Simultaneous RBThe effect (crosstalk) when nearby qubits are driven at the same timeGambetta et al. (2012) [25]
RB with mirror circuitsA scalable RB for many qubitsProctor et al. (2022) [26]
Gate set tomography (GST)Characterises the whole set of gates in detail (assesses the fault-tolerance threshold in the diamond norm)Blume-Kohout et al. (2017) [7]

Sourced (statements in the abstracts of the papers). The arrangement is our commentary.

(4) The limits of RB: not directly comparable with fault-tolerance thresholds

  • Blume-Kohout and colleagues state that the "error rate" reported by RB is not sensitive to all errors and cannot be compared directly with the diamond norm that expresses fault-tolerance thresholds Sourced [Ref. 7]. The same paper used gate set tomography to show, with more than 95% confidence, that operations on a trapped-ion qubit satisfy a threshold of diamond norm 6.7×10−4 or less
  • Sanders and colleagues show how to obtain, from a reported average gate fidelity, an upper bound on the error rate relevant to fault tolerance, and show that a realistic gap can exist between the reported average fidelity and the true error rate Sourced [Ref. 6]
  • Proctor and colleagues, measuring twelve processors, report that on current hardware structured programs fail up to an order of magnitude earlier than random ones, so that standard error metrics inferred from the behaviour of random programs do not accurately predict the performance of useful programs Sourced [Ref. 27]
A stumbling block: the "99.9%" from RB is not a number that counts every error

RB measures "the average error seen when random operations are averaged". Coherent errors that tilt steadily in one direction, or errors that show up only in particular operations, can be buried in the average and look small. Gate set tomography examines them too, but is laborious. When you read "99.9% by RB", read it as "an average-error metric", and note that comparing it with the level needed for error correction calls for a different metric (our commentary).

10. A materials engineer's view: what shortens a qubit's life

A qubit's lifetimes (T1 and T2) are set by defects and impurities in materials, contamination at interfaces, and the surroundings. Let us reread the numbers so far from the materials side.

A scene of checking the quality of qubit materials and interfaces. Lifetime depends on material defects and interface contamination (materials engineer's views 1 to 4). It does not depict any real facility or product.
Fig. 10. Conceptual illustration generated with ImageGen. A scene of checking the quality of qubit materials and interfaces. Lifetime depends on material defects and interface contamination (materials engineer's views 1 to 4). It does not depict any real facility or product. Not an actual product, facility, dimensional specification or measurement. Distinguish enlarged models from full-scale scenes.
A materials engineer's view 1: two-level systems (TLS) and dielectric loss: defects in oxides and at interfaces absorb the energy

Martinis and colleagues showed that dielectric loss from two-level defects (atomic-scale defects with two energy states) is a dominant decoherence source in superconducting qubits, and that better dielectrics and junctions of small area (about 10 µm2 or less) cut the loss sharply. In a phase qubit they report an improvement of the energy relaxation rate by a factor of 20 Sourced [Ref. 28]. The review by Müller and colleagues says that superconducting circuits made of aluminium or niobium are fundamentally limited by loss from two-level defects in the amorphous oxide layers that encase them Sourced [Ref. 29].

Along this line Place and colleagues replaced niobium with tantalum and took lifetime and coherence time above 0.3 ms Sourced [Ref. 9], and Bland and colleagues changed the substrate to high-resistivity silicon and reached a best T1 of 1.68 ms Sourced [Ref. 10]. For a materials engineer, a qubit is also a very sensitive "detector" of material loss (the review by Müller and colleagues notes that superconducting circuits can probe individual defects [Ref. 29]). That Bland and colleagues express lifetime through a quality factor Q shows that the picture of loss translates into the same language as dielectric loss in materials (our commentary).

A materials engineer's view 2: quasiparticles and radiation: environmental radioactivity and cosmic rays also shorten life

The lifetime of a superconducting qubit is affected by electrons whose pairs have been broken (quasiparticles) in the superconductor. Vepsäläinen and colleagues presented evidence that ionizing radiation from environmental radioactive materials and cosmic rays contributes to the observed excess density of quasiparticles, and predicted that this would ultimately limit the coherence times of superconducting qubits of the type they measured to milliseconds. They also showed that radiation shielding reduces the effect (they describe it as a small effect for today's qubits that will be critical for fault-tolerant superconducting machines) Sourced [Ref. 30]. For a materials engineer, even "the radiation environment where the machine sits" and "radioactive elements in surrounding materials" bear on lifetime (our commentary).

A materials engineer's view 3: isotopes and purity: making the nuclear spins and impurities in the crystal "quiet"

The lifetime of a spin qubit is set by the nuclear-spin isotopes and impurities in the crystal. Balasubramanian and colleagues reported a room-temperature electron-spin T2 of 1.8 ms in CVD diamond in which 13C was reduced to 0.3% and other paramagnetic defects were below 1013 cm−3 Sourced [Ref. 11]. Tyryshkin and colleagues reported a donor electron-spin T2 of up to 2 s in 28Si with residual 29Si below 50 ppm Sourced [Ref. 13]. A review of isotope engineering is by Itoh and Watanabe [Ref. 31]. For a materials engineer, controlling the isotope ratio and impurity concentration during crystal growth translates directly into longer qubit lifetimes (our commentary).

A materials engineer's view 4: ion-trap electrode surfaces: electric-field noise from surfaces, and "invisible" causes

In ion traps the ion sits close to the electrode surface, so electric-field noise from the electrode surface causes effects such as heating of the ion. The review by Brownnutt and colleagues summarises measurements of electric-field noise inferred from ion heating rates, and says that the presence of several noise sources and different scalings with distance complicate interpretation, and that the mechanisms behind much of this noise are poorly understood Sourced [Ref. 32]. For a materials engineer this means that electrode material, surface cleanliness and processing bear on gate errors, but which factor dominates has not yet been sorted out (our commentary; a review of materials challenges overall is by de Leon and colleagues [Ref. 33]).

11. How to read evaluations and standards

The quality of a qubit is read with three yardsticks: its lifetime (T1 and T2), the accuracy of operations (fidelity, error rate) and the performance of whole circuits. The table sets out how each is measured (RB, tomography, quantum volume and so on) and the state of standards for terms and performance metrics, as far as we could confirm (checked 16 October 2026).

What is measuredRepresentative benchmark or evaluation method (name, source)Standards and guidance (ISO/IEC, IEEE and others)How to read it
Lifetime (T1, T2)T1 decay measurement, Ramsey (T2*), Hahn echo (T2E), CPMG. Procedures as explained in the review [1] SourcedCould not be confirmed (ISO/IEC 4879:2024 is a vocabulary standard, but whether it defines T1 and T2 is unconfirmed because we did not read the text) [34] SourcedT2 changes with its kind (T2*, T2E, CPMG). Check single or ensemble, memory use, and conditions
Accuracy of operations (error rate, fidelity)Standard RB (Emerson et al., Knill et al., Magesan et al.) [22, 21, 3], interleaved RB [24], simultaneous RB [25], gate set tomography [7] SourcedCould not be confirmed (IEEE P7131 has a working group on performance metrics and benchmarking, but we found no published standard) [35] Not yet confirmedCheck which operation, how many qubits, average or best, and how it was measured. An RB error rate cannot be compared directly with fault-tolerance thresholds [7, 6]
Definition of fidelityThe relation between average gate fidelity and process fidelity (Nielsen, 2002) [2] SourcedCould not be confirmedDistinguish state, process and average gate fidelity
Whole-circuit performanceQuantum volume (Cross et al., 2019) [36], RB with mirror circuits [26], layer fidelity measuring error per layered gate (EPLG; McKay et al., 2023) [37], the speed metric CLOPS (Wack et al., 2021) [38], application-oriented benchmarks [39], measuring capabilities (Proctor et al.) [27] SourcedCould not be confirmed (as above)McKay et al. state that quantum volume focuses on the best subset of a device and cannot show average performance over many qubits, and propose layer fidelity. Each metric measures something different
Overview of benchmarkingThe survey by Lorenz et al. (2025) organises benchmarks at the levels of qubits and gates, processors, software and applications, and argues for standardisation [40] SourcedCould not be confirmedA survey's recommendations are not themselves a standard
Terms— (terminology, not evaluation)ISO/IEC 4879:2024 "Information technology — Quantum computing — Vocabulary" (edition 1, May 2024, IEC/ISO JTC 3) [34] Sourced / IEEE P7130 "Standard for Quantum Technologies Definitions" (on IEEE's page an active PAR, that is, a project under development; PAR approved 23 September 2021) [41] Not yet confirmedA vocabulary, not a standard that decides pass or fail for performance. We have not read the texts and state none of their definitions

The arrangement of this table is our own organisation (Our calculation). The ISO number, title and date were checked on ISO's committee page and the IEEE status on IEEE SA's pages (checked 16 October 2026). Standard texts are paid or unread, so we limit ourselves to title and scope.

Reading notes (for materials engineers)

  • Read T1 and T2 numbers together with their kind and conditions: T1 or T2, T2* or T2E or CPMG, single or ensemble, for gates or memory, at what temperature. Do not rank numbers measured under different conditions
  • Check what the "fidelity" is a fidelity of: single- or two-qubit, average or best, RB or tomography. An RB number cannot be compared directly with a fault-tolerance threshold [Refs. 7, 6]
  • Read lifetime and gate time together: a long lifetime does not reduce errors if the gate is slow (Chapter 6)
  • Check whether a standard is a recommendation or a requirement: ISO/IEC 4879 is a vocabulary, and IEEE P7130 and P7131 are projects under development. For any claim of "compliance", ask which standard and which clause

12. Glossary

Coherence
The wave-like order of a quantum state (the phases being in step).
Decoherence
The environment disturbing a quantum state so that it degrades (the arrow blurs).
T1 (energy relaxation time)
The time for the "1" fraction to fall to 1/e (about 37%).
T2 (phase relaxation time)
How long the phase stays in step. Its value depends on how it is measured. T2 ≤ 2×T1.
Tφ (pure dephasing time)
The time for the phase alone to be disturbed, apart from energy relaxation. 1/T2 = 1/(2T1) + 1/Tφ.
T2* (Ramsey)
T2 measured by Ramsey interference. Includes slow fluctuations and tends to be short.
T2E (Hahn echo)
T2 measured with a mid-way flip (π pulse). Cancels slow fluctuations.
CPMG, dynamical decoupling
Inserting many π pulses to cancel fluctuations and keep the phase longer.
Fidelity
A number showing how close the actual result is to the ideal. Closer to 1 is better.
Error rate
1 − fidelity. A guide to the probability of error in one operation.
Average gate fidelity
A gate's fidelity averaged over all input states. What RB reports.
Process fidelity
A fidelity showing how close the whole operation is to the ideal operation.
Diamond norm
A measure of the size of an operation's error that includes the worst input. Fault-tolerance thresholds are stated in it.
SPAM error
Error in state preparation and measurement.
Randomized benchmarking (RB)
A method that finds the average error rate per operation using random sequences of operations.
Interleaved RB
An RB that finds the error rate of one particular gate.
Gate set tomography (GST)
A measurement that characterises a whole set of gates in detail. Allows comparison with thresholds.
Two-level system (TLS)
An atomic-scale defect in a material with two energy states. It absorbs energy and shortens lifetime.
Quasiparticle
An electron left over when a pair in a superconductor breaks. It shortens a qubit's lifetime.
Transmon
The typical form of a superconducting qubit.
Quantum volume
A metric of whole-device performance based on how well random circuits run.
Layer fidelity (EPLG)
A metric of the fidelity of a layer of many qubits, measured by simultaneous RB.

13. References (primary sources)

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  16. Saeedi K. et al. "Room-Temperature Quantum Bit Storage Exceeding 39 Minutes Using Ionized Donors in Silicon-28", Science 342, 830–833 (2013), DOI 10.1126/science.1239584 — doi.org/10.1126/science.1239584
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14. Claim-to-source audit

Claim in the textBasisCategory
The two faces of decoherence (T1 relaxation, T2 dephasing); how T1 and T2 are measured (T1 decay, Ramsey, Hahn echo, CPMG); 1/T2 = 1/(2T1) + 1/Tφ and T2 ≤ 2T1; the measurement example T1 = 85 µs, T2* = 95 µs, T2E = 120 µsText and Fig. 5 of the review by Krantz et al. (2019)
Reference 1 https://doi.org/10.1063/1.5089550		Sourced
The relation between average gate fidelity and process fidelity, F_avg = (d·F_process + 1)/(d + 1)Abstract of Nielsen (2002)
Reference 2 https://doi.org/10.1016/S0375-9601(02)01272-0		Sourced
The proposals and properties of randomized benchmarking (accounting for preparation and measurement errors, the average error rate, the link to the diamond norm); interleaved RB, simultaneous RB, mirror-circuit RBAbstracts of Emerson et al., Knill et al., Magesan et al., Gambetta et al., Proctor et al.
Reference 22 https://doi.org/10.1088/1464-4266/7/10/021
Reference 21 https://doi.org/10.1103/PhysRevA.77.012307
Reference 3 https://doi.org/10.1103/PhysRevLett.106.180504
Reference 23 https://doi.org/10.1103/PhysRevA.85.042311
Reference 24 https://doi.org/10.1103/PhysRevLett.109.080505
Reference 25 https://doi.org/10.1103/PhysRevLett.109.240504
Reference 26 https://doi.org/10.1103/PhysRevLett.129.150502		Sourced
That the RB error rate cannot be compared directly with fault-tolerance thresholds (diamond norm); that GST showed a trapped-ion qubit meets a 6.7×10⁻⁴ threshold; a method to bound the error rate from average fidelity; that structured programs fail up to an order of magnitude earlierAbstracts of Blume-Kohout et al. (2017), Sanders et al. and Proctor et al. (2022)
Reference 7 https://doi.org/10.1038/ncomms14485
Reference 6 https://doi.org/10.1088/1367-2630/18/1/012002
Reference 27 https://doi.org/10.1038/s41567-021-01409-7		Sourced
The reported values in the Chapter 8 tables (superconducting T1 68 µs, T2,CPMG 89 µs; tantalum above 0.3 ms; Ta/Si T1 up to 1.68 ms; NV 1.8 ms and above 1 s; 28Si donor T2 up to 2 s; 31P nuclear spin above 30 s; ion above 10 min; ionised donors 39 min)Texts and abstracts of each paper
Reference 8 https://doi.org/10.1038/s41586-024-08449-y
Reference 9 https://doi.org/10.1038/s41467-021-22030-5
Reference 10 https://doi.org/10.1038/s41586-025-09687-4
Reference 11 https://doi.org/10.1038/nmat2420
Reference 12 https://doi.org/10.1038/s41467-018-04916-z
Reference 13 https://doi.org/10.1038/nmat3182
Reference 14 https://doi.org/10.1038/nnano.2014.211
Reference 15 https://doi.org/10.1038/s41566-017-0007-1
Reference 16 https://doi.org/10.1126/science.1239584		Sourced
The gate accuracies in the Chapter 8 table (ion 99.9934%, 99.9%, 3.8×10⁻⁵, 8×10⁻⁴, 6×10⁻⁴; superconducting CZ 99.76%, iSWAP 99.87%, single-qubit 99.994%; neutral atom 99.5%; silicon spin above 99.5%)Abstracts of each paper
Reference 4 https://doi.org/10.1103/PhysRevLett.117.060504
Reference 17 https://doi.org/10.1103/PhysRevLett.117.060505
Reference 18 https://doi.org/10.1103/PhysRevLett.127.130505
Reference 5 https://doi.org/10.1103/PhysRevX.11.021058
Reference 10 https://doi.org/10.1038/s41586-025-09687-4
Reference 20 https://doi.org/10.1038/s41586-023-06481-y
Reference 19 https://doi.org/10.1038/s41586-021-04273-w		Sourced
That dielectric loss from two-level systems is a dominant decoherence source in superconducting qubits (relaxation rate improved 20-fold); the fundamental limit from two-level defects in oxide layersAbstracts of Martinis et al. (2005) and Müller et al. (2019)
Reference 28 https://doi.org/10.1103/PhysRevLett.95.210503
Reference 29 https://doi.org/10.1088/1361-6633/ab3a7e		Sourced
That ionizing radiation raises the quasiparticle density and could limit coherence to milliseconds; that shielding lengthens the relaxation timeAbstract of Vepsäläinen et al. (2020)
Reference 30 https://doi.org/10.1038/s41586-020-2619-8		Sourced
Isotopes and purity: T2 = 1.8 ms in diamond with 13C at 0.3%; T2 up to 2 s in 28Si (29Si below 50 ppm); a review of isotope engineering; a review of materials challengesBalasubramanian et al., Tyryshkin et al., Itoh and Watanabe, de Leon et al.
Reference 11 https://doi.org/10.1038/nmat2420
Reference 13 https://doi.org/10.1038/nmat3182
Reference 31 https://doi.org/10.1557/mrc.2014.32
Reference 33 https://doi.org/10.1126/science.abb2823		Sourced
A review of electric-field-noise measurements near ion-trap electrode surfaces (multiple noise sources; many mechanisms poorly understood)Abstract of Brownnutt et al. (2015)
Reference 32 https://doi.org/10.1103/RevModPhys.87.1419		Sourced
Error from finite lifetime over a gate time (the Sung et al. CZ and iSWAP gates being close to the T1 limit); the speed/fidelity study over two-qubit gate times of 3.8 to 520 µs in ionsAbstracts of Sung et al. (2021) and Ballance et al. (2016)
Reference 5 https://doi.org/10.1103/PhysRevX.11.021058
Reference 4 https://doi.org/10.1103/PhysRevLett.117.060504		Sourced
The proposals of quantum volume, layer fidelity (EPLG), CLOPS and application-oriented benchmarks; a survey of benchmarkingCross et al., McKay et al., Wack et al., Lubinski et al., Lorenz et al.
Reference 36 https://doi.org/10.1103/PhysRevA.100.032328
Reference 37 https://arxiv.org/abs/2311.05933
Reference 38 https://arxiv.org/abs/2110.14108
Reference 39 https://doi.org/10.1109/TQE.2023.3253761
Reference 40 https://arxiv.org/abs/2503.04905		Sourced
The number, title and date of ISO/IEC 4879:2024 "Information technology — Quantum computing — Vocabulary" (edition 1, May 2024, IEC/ISO JTC 3); whether it defines T1 and T2 is unconfirmed (text not read)ISO's committee page
Reference 34 https://www.iso.org/standard/80432.html		Sourced
That IEEE P7130 is under development (active PAR, approved 23 September 2021); that the IEEE P7131 working group is active and we found no published standardIEEE SA pages
Reference 41 https://standards.ieee.org/ieee/7130/10680/
Reference 35 https://sagroups.ieee.org/7131		Not yet confirmed
The T1 decay table (e^−1 and so on), the T2 and Tφ table, the lifetime-error formula and table (numerical check of ε = 1 − (3 + 2e^(−t/T2) + e^(−t/T1))/6), the F_avg example, the simulated RB data and fits (0.1003% and 0.5046%)Our own calculation (Python, NumPy; seed 0). Not the results of any real deviceOur calculation
Rankings of numbers across technologies, the text of ISO/IEC 4879, the publication state of IEEE P7131, and a full comparison of mean and best T1/T2 of every deviceCould not be confirmed from the primary sources available in this survey, so not statedcommentary
Readings such as "decoherence happening because someone looked is wrong", "a long T2 means a good qubit is wrong", "long lifetime and fast gates are separate efforts", "RB 99.9% is an average metric", "do not rank technologies under different conditions", and the analogies with their limitsOur own organisation and commentary based on the published material; not views expressed by the organisations citedcommentary
That Figs. 1 to 10 are explanatory drawings and not observations of real data, and that Figs. 1 and 10 are ImageGen-generated conceptual illustrationsNote in this articlecommentary

Last updated: 16 October 2026. Sources are limited to primary material (a review, peer-reviewed papers, arXiv papers, and official ISO and IEEE pages). Values from each paper are shown as values under its own measurement conditions, and we do not rank devices of different technologies. The numerical examples (decay, lifetime-limited errors, simulated RB data) are our own illustrative calculations, not results from real devices. We could not confirm the text of the standards, the publication state of IEEE P7131, or an exhaustive comparison of device values, so we do not state them. All figures are explanatory conceptual diagrams or drawings that include our own calculations; Figs. 1 and 10 are ImageGen-generated conceptual illustrations.

Illustration update: 17 October 2026. References to vector drawings in the article and source tables describe the retained original SVGs. See each current caption for the foreground image format and reading guide. ImageGen illustrations are not measurements or photographs of actual products; original SVGs, values and sources remain available in expandable references.
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