The pigeonhole problem, in plain numbers
A fixed-frequency transmon is born with one number it must hit: its transition frequency, usually somewhere between about 4 and 6 gigahertz. You cannot use the whole radio spectrum — the control electronics, the wiring, and the chip's own physics confine every qubit to a narrow usable band only a couple of gigahertz wide. So the design question is brutally simple: you have a short shelf, and you must place many qubits on it without two of them landing on the same spot, or even too close.
Why can't they be close? Because neighbouring qubits have to talk to each other to do two-qubit gates, and the rules for those gates — and the microwave pulses that drive them — only work cleanly when the two frequencies are far enough apart. Bring them within a forbidden distance and the gate misfires, the pulse meant for one qubit nudges the other, or readout confuses the two. Each kind of clash is a collision, and collisions are what frequency crowding is really about.
Where the collisions come from: junction spread
If you could place every qubit exactly on its assigned frequency, crowding would be a tidy seating chart and collisions would never happen. You cannot. A transmon's frequency is set by its Josephson junction, and the junction's strength depends on an oxide barrier only a few atoms thick. Two junctions drawn identically come out slightly different — this is parameter variability, and for junctions it is stubbornly large. Even good processes leave a spread of roughly one to a few percent in frequency, which at 5 gigahertz is tens to over a hundred megahertz of scatter.
Now do the arithmetic the field actually lives by. Each qubit lands somewhere random near its target. A collision happens when a qubit drifts within a forbidden window of one of its neighbours. The chance that any one pair is safe is high; the trouble is that a big chip has many pairs, and the qubit is fine only if it dodges every collision rule with every neighbour at once. Multiply many almost-certain successes together and the product sags. That is why yield falls off a cliff as chips grow.
Why yield collapses as a chip grows
(toy model, numbers for intuition only):
Let p_ok = chance ONE qubit clears ALL its
collision rules with ALL neighbours.
Whole-chip yield ~ p_ok ^ N (N = qubits)
p_ok = 0.99 p_ok = 0.95 p_ok = 0.90
-------------- -------------- --------------
N= 10 : 90% N= 10 : 60% N= 10 : 35%
N= 50 : 61% N= 50 : 8% N= 50 : 0.5%
N=100 : 37% N=100 : 0.6% N=100 : 0.003%
Message: even a 99%-good single qubit gives a
bad whole-chip yield once N is large. Shrinking
the junction spread RAISES p_ok, and because it
is an exponent, a small gain there is huge here.Read the table and one lesson dominates. The exponent N is set by ambition; the base p_ok is set by fabrication. You cannot make N small — the whole point of scaling is more qubits — so the only lever is to push p_ok closer to one by shrinking the junction spread. Because it sits in an exponent, a modest tightening of the spread can turn a hopeless chip-level yield into a workable one. This single fact is why so much of quantum chip design is, underneath, a fight over a few atoms of oxide.
Two escape routes: trim the spread, or tune it away
Faced with this, engineers fight on two fronts. The first is to make the junctions more uniform in the first place — better lithography, steadier oxidation, tighter process control — so the as-fabricated spread is smaller. The second, and increasingly the workhorse, is to fix the frequency after fabrication: measure where each junction actually landed, then nudge the bad ones toward target. This second move goes by the name junction frequency targeting, and the most common version is laser annealing.
Trimming pulls the spread toward target
(schematic frequency axis, GHz):
AS FABRICATED (wide spread, a collision):
4.8 5.0 5.2
| * * | * * * | * scattered
| ^^ two too close = collision
AFTER LASER-ANNEAL TRIM (onto a grid):
4.8 5.0 5.2
| * * * | * on target
| (collision cleared)
Anneal RAISES a junction's frequency by gently
aging its oxide with heat; you trim UP toward
target, never down. Aim a little low, then trim.- Design the chip with target frequencies deliberately spaced to avoid every known collision rule, leaving a little headroom.
- Fabricate, then measure each qubit's actual frequency — at room temperature you can read the junction resistance, which tracks it.
- For junctions that landed low or collide, apply a brief, local laser pulse that gently ages the oxide and nudges the frequency up toward target.
- Re-measure, repeat the trim if needed, and only then cool the chip down to confirm the collisions are gone.
A defining bottleneck — said plainly
It is worth being blunt: frequency crowding is not a minor nuisance to be engineered away next quarter. It is one of the handful of problems that genuinely caps how large a fixed-frequency superconducting processor can be today. Trimming and aging have pushed chip-level yield from hopeless to usable at the scale of tens of qubits, and that is real progress — but the exponent in the yield equation never goes away. Every doubling of qubit count demands a tighter spread just to stand still.
There are escape hatches, and each carries its own bill. Tunable qubits dodge crowding by moving their frequency on demand — but, as the tunable-junction trick always does, they trade that freedom for sensitivity to magnetic noise, which shortens coherence. Sparser connectivity means fewer neighbours and fewer collision rules to satisfy — but fewer connections can make the qubits harder to use for computation. There is no free lunch here; every cure for crowding spends something somewhere else.