1. The short version, what to know first, and where this fits in the series

• Why cool: a superconducting qubit's energy corresponds to only about 0.24 K as a temperature (5 GHz). At room temperature (300 K), thermal energy is about 1250 times the qubit's, so the arrow is constantly shaken by heat. So qubits are cooled to about 10 to 20 mK. At 20 mK, the chance that heat excites a 5 GHz qubit is about 6 in a million Our calculation
• How cool: a dilution refrigerator stacks five plates at different temperatures; the bottom one (the mixing chamber) reaches about 6 to 20 mK. But the heat it can remove there is only about 19 microwatts (at 20 mK) [Ref. 1], and even the heat leaking down single wires hits that limit as the wires multiply
• Wiring and scale: more qubits need low-heat wiring materials (NbTi, stainless steel), control circuits that are cooled and placed inside (cryo-CMOS and others), and designs that connect several refrigerators. These are also materials and manufacturing problems

2. Why cool — heat "shakes" a qubit

(1) An everyday example: telling a small ripple on a still lake

Suppose you want to see the small ripple of a pebble dropped in a lake. If a strong wind is churning the surface, you cannot. You have to stop the wind. For a qubit, heat is the wind. A qubit's state is set by a very small energy difference, so if the heat around it jostles strongly, the arrow wanders by itself.

(2) In numbers: a qubit's energy is about 0.24 K in temperature

A superconducting qubit's frequency is typically 5 GHz, and the readout resonator is around 7 GHz [Ref. 1]. The energy gap between a qubit's two states is hν (h is Planck's constant). As a temperature, hν/k_B ≈ 0.24 K (5 GHz) Our calculation. Thermal energy at room temperature (about 300 K), k_BT, is about 1250 times that (300 ÷ 0.24). So at room temperature heat is far larger than the qubit's energy gap, and the two states mix about half and half.

Let us see the chance that heat excites a qubit with a simple formula. When two states are in thermal equilibrium, the probability of being in the upper one is 1 / (1 + exp(hν/k_BT)) (our calculation, for 5 GHz).

Our calculation. Assumes the qubit is in thermal equilibrium at the refrigerator's temperature. Real qubits tend to have a higher "effective temperature" than the refrigerator.

In practice a qubit may not cool all the way to the refrigerator's temperature. Jin and colleagues measured the excited-state population of a 3D transmon while varying the refrigerator temperature. From 35 to 150 mK it followed a Maxwell-Boltzmann distribution in equilibrium with the refrigerator, and below 35 mK it saturated at about 0.1%. They concluded the qubit's effective temperature is 35 mK Sourced [Ref. 3]. Even if the mixing chamber is colder, heat still reaches the qubit, and the likely cause is thermal noise through the wiring and similar paths (our commentary).

(3) Heat also arrives through wiring — thermal photons

Heat comes not only through the refrigerator's plates but also as radio waves travelling down the wires (thermal photons). A hot radio wave in a room-temperature line that reaches the qubit shakes the arrow. The mean number n of thermal photons at 6 GHz is n = 1 / (exp(hν/k_BT) − 1): about 1040 at room temperature, about 9 at 4 K, and about 5.6×10⁻⁷ at 20 mK (Our calculation; formula and frequency choice follow Krinner et al. [Ref. 1]). So attenuators are placed in the line to cut thermal photons. To bring the photon number at the qubit down to about 10⁻³ from a room-temperature line, a total of about 60 dB (a millionth) of attenuation is needed, Krinner et al. explain Sourced [Ref. 1].

(4) Different kinds of quantum computer are cooled differently

The arrangement of the table is our commentary. Do not line up temperatures across platforms to rank them.

3. How a dilution refrigerator works — five plates, cooling step by step

(1) An everyday example: five nested rooms

Suppose you nest five rooms to protect an inner room from hot air outside. The outer room is next to the heater, and heat reaches less and less as you go inward. If every room has air conditioning that removes the heat coming in from outside, the innermost room stays very cold. A dilution refrigerator works the same way. Five plates (stages) at different temperatures are stacked on thin rods, and each stage disposes of heat.

The figure above shows temperatures of an empty refrigerator, and cooling powers at the stated temperatures, for the Bluefors XLD400 reported by Krinner et al. Sourced [Ref. 1]. The 50 K and 4 K stages are made by pulse-tube coolers (machines that cool through pressure changes of compressed helium gas). The next three stages, the still, the cold plate and the mixing chamber, are the "dilution" part using a mixture of helium-3 and helium-4, and about 6 mK (when empty) at the mixing chamber is the lowest temperature.

(2) The principle of dilution — absorbing heat as helium dissolves

Helium has two isotopes with different masses (helium-3 and helium-4). A liquid mixture of both separates, below about 872 mK, into a helium-3-rich phase and a helium-4-rich phase (the 3He concentration at the point where separation begins is 67.3%) [Ref. 2] Sourced. The lighter 3He-rich phase floats on top, with a boundary between it and the lower phase. In the cold mixing chamber, 3He dissolves from the rich phase into the lower, dilute phase. This absorbs heat from the surroundings. That is "dilution cooling". Dilution refrigerators have been reported to cool continuously down to about 2 mK [Ref. 2].

The 3He that has dissolved into the dilute phase is carried to a slightly warmer chamber called the still and pumped out as gas. It is then compressed, pre-cooled and returned to the rich phase. The 3He circulates in a closed loop and is not used up each time. But 3He is a scarce and costly isotope [Ref. 2]. Bigger refrigerators need more of it, so it could become an important constraint on scaling (our commentary).

(3) The cooling power is very small

The mixing chamber's cooling power (at 20 mK) is about 19 microwatts [Ref. 1]. That is about one 52,000th of a 1 W light bulb (1 / 0.000019 ≈ 52,600; Our calculation). The lower the temperature, the smaller the cooling power. It is about 200 microwatts on the cold plate (at 140 mK), about 40 milliwatts at the still (at 1.2 K), about 1.5 W at the 4 K stage, and about 30 W at the 50 K stage [Ref. 1]. So the heat generated inside the refrigerator (heat conducted down wires, attenuator dissipation, amplifiers, control circuits) must fit within each stage's cooling power.

4. Wiring and the heat problem — each line is a path for heat

(1) An everyday example: small holes in the wall of a winter house

Suppose you drill a small hole in the wall between a warm room and the cold outside. One hole lets in only a little draught. But with hundreds of holes, the heating cannot keep up. The same goes for a cryogenic refrigerator. Every line from room-temperature equipment is a path for heat. Each one carries little, but as their number grows they eat into the limited cooling power.

(2) Types of lines, and the signal path

Experiments with superconducting qubits mainly use three kinds of lines [Ref. 1] Sourced.

• Drive lines: carry the microwaves that operate the qubits
• Flux lines: carry the current that tunes the frequency of a frequency-tunable qubit through magnetic flux
• Output (readout) lines: bring out the qubit's state as a signal from the readout resonator

The figure is the configuration adopted by Krinner et al. [Ref. 1]. In the control (drive) line, 20 dB attenuators sit at the 4 K stage, the cold plate and the mixing chamber, for 60 dB in total, to reduce room-temperature thermal photons. In the readout line, a TWPA (travelling-wave Josephson parametric amplifier, a low-noise amplifier near the quantum limit [Ref. 14]) sits at the mixing chamber and a HEMT amplifier (gain about 40 dB) at 4 K. To keep noise from returning to the sample, a circulator and isolators (parts that pass signals in one direction only) give more than 60 dB of isolation in total [Ref. 1] Sourced. To avoid "quasiparticles" (broken Cooper pairs in the superconductor) created by infrared light, an optional infrared filter made of an absorber (Eccosorb CR-124) is provided (6 dB attenuation at 6 GHz, in front of the mixing-chamber attenuator) [Ref. 1].

(3) Passive heat: heat conducted down the lines

A line is made of metal, so it also conducts heat. The heat carried by one line between stages of the refrigerator is "thermal conductivity × cross-section ÷ length" summed over the temperature range (Eq. (1) of Krinner et al.) [Ref. 1]. So materials with low thermal conductivity (stainless steel, superconducting NbTi) are used, thin and long. There is a flip side, though: materials that carry electricity poorly tend to turn signal into heat [Ref. 1]. The heat flowing in per line, as measured by Krinner et al., is as follows Sourced [Ref. 1].

Measured values (uncertainty in parentheses). The refrigerator is a Bluefors XLD400, and line lengths between stages are 140 to 290 mm (Tables 1 and 2 of Krinner et al.) [Ref. 1]. The NbTi of the output line is the value for the part that becomes superconducting below 4 K.

From these values and each stage's cooling power, let us compute "how many lines can be run". Dividing a stage's cooling power by the heat flowing in per line gives a rough number of drive lines (Our calculation; passive heat only, not including attenuator dissipation or other lines).

The cold plate (200 μW ÷ 0.4 μW ≈ 500 lines) and the 50 K stage (30 W ÷ 45 mW ≈ 670 lines) hit their limit first. Since several lines are needed per qubit, this combination of refrigerator and line supports a few hundred qubits. Krinner et al. also write that, ignoring space constraints, at least 150 qubits could be operated [Ref. 1].

(4) Active heat: heat from attenuators

An attenuator turns most of the incoming signal into heat (99% for 20 dB) [Ref. 1]. The signal reaching a qubit is tiny (a peak of −66 dBm for a 30 ns π pulse), and on average about −78 dBm is needed at the mixing chamber [Ref. 1]. Even so, a vastly larger signal is sent in at room temperature, so the position and amount of attenuation set how the heat is shared. In an example by Krinner et al., with attenuation split {0, 10, 0, 20, 30} dB (50 K, 4 K, still, cold plate, mixing chamber), the dissipation of 25 drive lines took 35% of the cold plate's cooling power. Removing 10 dB at the cold plate and adding 10 dB at 4 K, to {0, 20, 0, 10, 30} dB, cuts the cold-plate heat to one tenth. In return, the thermal photon number at the qubit side rises from about 0.0012 to 0.002, a bit under double [Ref. 1] Sourced. So where attenuators go is a trade-off between heat and noise.

5. Control electronics — from room temperature into the cold

(1) The present: control equipment at room temperature

The signals that run qubits are made by electronic equipment such as waveform generators, mixers, amplifiers and digitisers. Today the mainstream is to put them in racks at room temperature and send signals to the refrigerator on coaxial lines. An IBM researcher writes that conventional rack-based room-temperature control electronics "impose severe constraints on wiring density, cost and system footprint" and are a barrier to large-scale fault-tolerant computing [Ref. 11] Sourced (a 2026 preprint).

(2) Four ideas for fewer wires

Numbers are as stated in the papers' abstracts. The target qubits (superconducting or silicon) also differ by row. This is not a comparison to rank them.

(3) Why is "heat dissipation" the biggest problem?

A CMOS circuit always gives off heat when it uses electricity. The lower the temperature in the refrigerator, the less heat can be removed, so which stage it goes on is decided by that. Pauka and colleagues noted that CMOS is often discounted because its power dissipation could heat fragile qubits, and showed a chip running at 100 mK with dissipation reduced to 18 nW per cell [Ref. 7] Sourced. In our calculation, dividing the cold plate's 200 μW (at 140 mK; Section 3) by 18 nW gives about 10,000 cells (an upper limit that ignores static power and wiring heat). Xue et al.'s 3 K chip corresponds to being placed at a stage near 4 K (the 4 K stage of Krinner et al.'s refrigerator has about 1.5 W of cooling power [Ref. 1]; the refrigerator is a different one) [Ref. 5].

6. The scaling wall — beyond thousands of qubits

Let us put the above into a sense of scale. The table lists refrigerator examples we could confirm. Makers' stated values and company plans are not achieved facts.

KIDE's cooling power, according to the maker's page, is measured at the KIDE insert flange. Each company's numbers are under different conditions, so do not line them up to rank them.

Even in a large machine like KIDE, line heat matters. Suppose 4000 lines of the same stainless coax Krinner et al. measured (0.4 μW each at the cold plate) were installed: the passive heat on the cold plate would be 1.6 mW (Our calculation). KIDE's page gives over 3 mW of cooling power at 100 mK, so passive heat alone would use about half. High-density wiring lines have different materials and structures, however, so this calculation is only an example to see the order of magnitude.

Paths to larger scale

• Bigger, and several, refrigerators: IBM shows a modular design with box-shaped "cells" side by side. Putting cells together lets quantum cables pass into the neighbouring cell and joins thermal shielding into a "cold tunnel" that links processors. IBM also says the cells can be used to test "l-couplers" (cables operating inside a dilution refrigerator, at the one-metre scale) [Ref. 16] Not yet confirmed (company plan)
• Fewer lines: cryo-CMOS, SFQ, optical fibre (Section 5)
• Lines that carry less heat: NbTi, thin wires, heat anchoring (Section 7)
• Countering radiation and cosmic rays: radiation creates quasiparticles and limits qubit lifetime. Shielding has been reported to correlate with longer coherence [Ref. 18] Sourced. Materials outside the refrigerator and the installation site are also challenges when scaling up
• Securing 3He: 3He is scarce and costly [Ref. 2]. Our view is that supply limits could appear if large refrigerators multiply (Inference; we did not check supply statistics)

7. A materials engineer's view — materials for refrigerators, wiring and insulation

Cryogenic equipment is full of opportunities for materials engineers. Here we use public property data (NIST fits) to compute how much the heat flow changes with the choice of material.

(1) Heat conduction differs by five orders of magnitude between materials

How readily a material passes heat is its thermal conductivity k (in W/(m·K)). The figure shows thermal conductivity from 4 to 300 K drawn from NIST's fits [Ref. 6] (Our calculation).

What the figure shows (Our calculation):

• Copper (oxygen-free, RRR 100) has a very large thermal conductivity from 4 to 50 K, peaking at about 1540 W/(m·K) at 10 K. Krinner et al. also write that copper has an "anomalously large" thermal conductivity at 4 to 50 K [Ref. 1]. It suits parts that carry heat away (heat anchors, plates) and cannot be used where you want to stop heat
• Stainless steel 304 is about 0.27 W/(m·K) at 4 K and about 15 at 300 K, a thousandth to a few tens of a copper's. It is a poor heat conductor for a metal and is used for the outer conductor of lines and support rods
• G-10, PTFE and polyimide are another one to two orders of magnitude lower (about 0.01 to 0.07 W/(m·K) at 4 K) and suit supports, insulation and thermal spacers

Compare rods of the same shape. The heat flowing through a rod of 1 mm² cross-section and 100 mm length connected from room temperature (300 K) down to 4 K is as follows (NIST's fits integrated over temperature; Our calculation).

Our calculation. In practice heat is intercepted at stages such as 50 K on the way, so less reaches 4 K. This is an example for comparison. The G-10 values use the normal-direction fit, which applies from 10 K.

The 4 K stage's cooling power was about 1.5 W [Ref. 1]. A copper rod of the same shape connected straight from room temperature to 4 K would by itself exceed that cooling power. The choice of material decides whether a refrigerator can work at all.

(2) Copper's "RRR" — purity and defects set low-temperature conduction

Copper's thermal conductivity depends strongly on RRR (residual resistivity ratio). RRR is the electrical resistance at room temperature divided by that at very low temperature, and is larger the fewer the impurities and crystal defects (the general definition; our commentary). By NIST's fits, the thermal conductivity at 4 K is about 640 W/(m·K) for RRR 100 and about 3180 for RRR 500 (Our calculation) [Ref. 6]. So even for "copper", purity, working and heat treatment can make a fivefold difference in low-temperature performance. When choosing a material for cooling plates and heat anchors, grade and history matter.

(3) Which material for which part

Rows marked "Our calculation" are our calculations from NIST's fits, and rows marked "Sourced" are as stated in each source. The grouping by part is our commentary.

8. What could not be confirmed, and a summary

What could not be confirmed

• Operating temperature of photonic quantum computers: we could not confirm a number in primary material, so it is not in the table of Section 2.
• Latest specifications of current refrigerators: temperatures and cooling powers are values from the 2018 report (XLD400). Newer models need checking in the makers' official material. KIDE is the maker's stated value.
• Thermal conductivity figures for NbTi: they are not in NIST's tables, so heat flowing down lines uses the measurements of Krinner et al. and no material curve is drawn.
• Contact thermal resistance of heat anchors: important, but we found no figures in the primary material we could check.
• Statistics on 3He supply and price: we confirmed the statement that it is "scarce and costly", but not supply figures.
• Third-party verification of IBM's cryo-CMOS (2026) and modular refrigerator: the former is a preprint (not peer reviewed) and the latter a company blog; we could not confirm independent verification.
• A standard that defines the performance of dilution refrigerators: could not be confirmed (temperature-scale standards are in Section 9).

9. How to read evaluations and standards

The yardsticks of cryogenics and control are the base temperature, the cooling power at a stated temperature, the heat load per line, a qubit's effective temperature (excited-state population) and the number of thermal photons. None has meaning unless you state the temperature and where it was measured (which stage). We could not confirm a standard that defines refrigerator performance. There is, however, an international standard for the temperature scale itself at cryogenic temperatures. Check date: 16 October 2026.

The arrangement of the table is Our calculation (organising). Numbers, titles and adoption dates of standards were checked on each body's page (check date 16 October 2026). Standards texts are paid, so only titles and scope are given.

Cautions in reading (for materials engineers)

• "Lowest temperature of the refrigerator" and "temperature of the qubit" are different: even with a 6 mK mixing chamber, a qubit's effective temperature may be 35 mK. Read cooling power together with the temperature
• Makers' stated values differ in where and how they are measured: KIDE's value is stated as measured "at the insert flange". Do not line it up against other makers or models to rank them
• Company announcements and third-party verification are different: IBM's cryo-CMOS is a non-peer-reviewed preprint and the modular refrigerator is a company blog. They are treated as "company claims"
• Standards are at the stage of temperature scales and terms: PLTS-2000 is an international scale for how to measure temperature; it does not decide pass or fail on a refrigerator's performance or safety. ISO/IEC 4879 is a terminology standard with no legal force (in general a standard becomes binding when cited in law or a contract; our commentary)

10. Glossary

11. References

1. Krinner S. et al. "Engineering cryogenic setups for 100-qubit scale superconducting circuit systems", EPJ Quantum Technology 6, 2 (2019), DOI 10.1140/epjqt/s40507-019-0072-0 (arXiv:1806.07862) — doi.org/10.1140/epjqt/s40507-019-0072-0
2. Kim P. H. et al. "A continuously-cooled 3He/4He phase separation refrigerator", arXiv:2503.10754 (2025) (preprint) — arxiv.org/abs/2503.10754
3. Jin X. Y. et al. "Thermal and residual excited-state population in a 3D transmon qubit", Phys. Rev. Lett. 114, 240501 (2015), DOI 10.1103/PhysRevLett.114.240501 — doi.org/10.1103/PhysRevLett.114.240501
4. Barends R. et al. "Minimizing quasiparticle generation from stray infrared light in superconducting quantum circuits", Appl. Phys. Lett. 99, 113507 (2011), DOI 10.1063/1.3638063 — doi.org/10.1063/1.3638063
5. Xue X. et al. "CMOS-based cryogenic control of silicon quantum circuits", Nature 593, 205–210 (2021), DOI 10.1038/s41586-021-03469-4 — doi.org/10.1038/s41586-021-03469-4
6. NIST "Cryogenic Material Properties" (fits for thermal conductivity and other properties; checked 16 October 2026) — trc.nist.gov/cryogenics/materials/materialproperties.htm
7. Pauka S. J. et al. "A cryogenic CMOS chip for generating control signals for multiple qubits", Nature Electronics 4, 64–70 (2021), DOI 10.1038/s41928-020-00528-y — doi.org/10.1038/s41928-020-00528-y
8. Lecocq F. et al. "Control and readout of a superconducting qubit using a photonic link", Nature 591, 575–579 (2021), DOI 10.1038/s41586-021-03268-x — doi.org/10.1038/s41586-021-03268-x
9. McDermott R. et al. "Quantum–classical interface based on single flux quantum digital logic", Quantum Sci. Technol. 3, 024004 (2018), DOI 10.1088/2058-9565/aaa3a0 — doi.org/10.1088/2058-9565/aaa3a0
10. Liu C.-H. et al. "Single flux quantum-based digital control of superconducting qubits in a multichip module", PRX Quantum 4, 030310 (2023), DOI 10.1103/PRXQuantum.4.030310 — doi.org/10.1103/PRXQuantum.4.030310
11. Underwood D. et al. (IBM Quantum) "Transistors to transmons: large-scale cryogenic CMOS control of superconducting qubits", Research Square preprint (3 July 2026) (not peer reviewed) — doi.org/10.21203/rs.3.rs-10095152/v1
12. Petit L. et al. "Universal quantum logic in hot silicon qubits", Nature 580, 355–359 (2020), DOI 10.1038/s41586-020-2170-7 — doi.org/10.1038/s41586-020-2170-7
13. Yang C. H. et al. "Operation of a silicon quantum processor unit cell above one kelvin", Nature 580, 350–354 (2020), DOI 10.1038/s41586-020-2171-6 — doi.org/10.1038/s41586-020-2171-6
14. Macklin C. et al. "A near–quantum-limited Josephson traveling-wave parametric amplifier", Science 350, 307–310 (2015), DOI 10.1126/science.aaa8525 — doi.org/10.1126/science.aaa8525
15. Bluefors "KIDE Cryogenic Platform" (maker's product page; checked 16 October 2026) — bluefors.com/products/kide-cryogenic-platform/
16. IBM "IBM's new modular architecture for cryogenic systems", IBM Quantum blog (19 August 2026) (company description) — ibm.com/quantum/blog/modular-cryogenics
17. BIPM "Supplementary Information for the Realization of the PLTS-2000" (CCT realisation guide) — bipm.org/…/Guide-PLTS-2000.pdf
18. Vepsäläinen A. P. et al. "Impact of ionizing radiation on superconducting qubit coherence", Nature 584, 551–556 (2020), DOI 10.1038/s41586-020-2619-8 — doi.org/10.1038/s41586-020-2619-8
19. IonQ "IonQ roadmap" (official page; checked 16 October 2026) — ionq.com/roadmap
20. Quantinuum "Technical perspective: By the end of the decade, we will deliver universal, fully fault-tolerant quantum computing", official blog (September 2024) — quantinuum.com/blog/technical-perspective…
21. ISO/IEC 4879:2024 "Information technology — Quantum computing — Vocabulary" (published May 2024). Number, title, date and "Published" status were checked on the listing page of the Institute for Standardization of Serbia (ISS) — iss.rs/en/project/show/iso:proj:80432 (the official ISO page is https://www.iso.org/standard/80432.html; it could not be fetched automatically from our environment, and was confirmed through search-result listings)
22. IEEE P7131 working group "Quantum Benchmarking Working Group P7131" (site of the working group on the draft standard for quantum computing performance metrics and benchmarking) — sagroups.ieee.org/7131

12. Claim-to-source audit

Last updated 16 October 2026. Sources are limited to primary material (peer-reviewed papers, arXiv papers, official pages of NIST and BIPM, official material from refrigerator makers, IBM, IonQ and Quantinuum, and ISO standard listing pages). Makers' stated values, company descriptions and plans are kept separate. Most temperatures and cooling powers are values for one refrigerator reported in 2018, not for current models. Numerical estimates are our own calculations, not results from real machines. The operating temperature of photonic quantum computers, the latest refrigerator specifications, thermal conductivity figures of NbTi, contact thermal resistance of heat anchors, statistics on 3He supply, third-party verification of cryo-CMOS and modular refrigerators, a standard defining refrigerator performance, the text of ISO/IEC 4879 and the publication status of IEEE P7131 could not be confirmed and are not stated here. All figures are explanatory concept diagrams or drawings that include our calculations; Fig. 1 and Fig. 6 are ImageGen-generated conceptual illustrations.