The Coherence Budget

Where the energy leaks

A superconducting qubit is a tiny resonator holding a single packet of microwave energy. Its coherence time is just an honest measure of how long that packet survives before the state is lost. So the central design question is plain: where does the energy go? It does not vanish — it leaks into something, and on a real chip there are only a handful of places for it to leak.

The single biggest culprit at the conditions where qubits actually run — a few photons of energy, and milli-kelvin cold — is a defect called a two-level system, or TLS. A TLS is an atom or small group of atoms, usually sitting in a thin oxide or at a surface, that can flip between two configurations. Unluckily, it can flip at almost exactly the qubit's frequency. When that happens it acts like a tiny resonant antenna, absorbs the qubit's energy, and dumps it as heat.

Participation times loss tangent: the one picture to keep

TLS do not sit out in empty space; they live inside materials — the oxide on a metal surface, the substrate underneath, the interface between the two. Whether a given patch of bad material hurts you depends on two separate things, and keeping them separate is the whole trick. First: how lossy is the material itself? Second: how much of the qubit's electric field actually sits in that material? A terrible material the field barely touches does little harm; a fairly good material soaked in field can dominate.

Loss from one region (plain symbols):

  loss(region) = p * tan(delta)

  p          = participation ratio
               (fraction of the qubit's electric
                energy stored in that region; 0 to 1)
  tan(delta) = loss tangent of the material
               (how lossy the material is per cycle)

Total loss = sum of p * tan(delta) over all regions.
Q_i (internal quality factor) = 1 / (total loss).

So a region only matters when BOTH p and tan(delta)
are appreciable. Either one near zero -> little harm.

That product, p times tan(delta), is the mental model the whole field runs on. The quantity engineers actually measure is its inverse, the internal quality factor Q_i: a high Q_i means low loss, a long-lived resonance, and a longer-lived qubit. You measure Q_i on a bare test resonator first, at single-photon power and milli-kelvin — because that is the regime that matters — and use it as the meter that tells you whether last month's process change actually helped.

A loss budget, region by region

Once you accept that loss is a sum over regions, you can write it down like an accountant's ledger. A planar qubit's electric field is shared among a few named places: the bulk substrate beneath the metal, the metal-air surface, the metal-substrate interface, the substrate-air interface, and the vacuum above. Each gets a participation share p and carries a material loss tangent. The product is that region's line item in the budget.

Illustrative loss budget for a planar qubit
(numbers are rough, for intuition only):

  REGION              p (field    tan(delta)   p*tan
                       share)      (lossiness)
  ------------------  ----------  -----------  --------
  bulk substrate       ~0.90       1e-7         ~1e-7
  metal-air surface    ~0.001      2e-3         ~2e-6
  metal-substrate if.  ~0.002      1e-3         ~2e-6
  substrate-air if.    ~0.003      1e-3         ~3e-6
  vacuum (above)       ~0.09       0            0
  ------------------  ----------  -----------  --------
  TOTAL loss ~ sum of last column ~ 7e-6
  Q_i ~ 1 / total ~ 1.4e5

Note: the substrate holds ~90% of the field but is
very clean, so the THIN, BAD interfaces (tiny p,
large tan delta) end up dominating the total.

Read the ledger and a counter-intuitive lesson jumps out. The substrate holds about ninety percent of the field, yet because a good substrate like high-resistivity silicon or sapphire is extremely clean, it barely costs you anything. The damage comes from the interfaces — layers only a few atoms thick, holding a thousandth of the field, but so lossy that they dominate the total. This is why so much fab effort goes into surfaces almost too thin to see: that is where the budget is actually spent.

1. Fabricate bare test resonators on the candidate substrate and process — no qubits yet, just the loss meter.
2. Cool to milli-kelvin and measure Q_i at single-photon power, where the TLS are at their most damaging.
3. Change one thing — a gentler oxide etch, a wider gap, a cleaner deposition — and measure again to see which line item moved.
4. Only once the bare resonators are clean do you commit the process to real qubits, where many other things can also go wrong.

What the budget honestly buys

The loss budget is a powerful lens, but it is worth saying plainly what it does and does not cover. It captures dielectric loss — energy soaked up by lossy insulators and the TLS inside them — which is the dominant channel for a well-made resonator. It does not, on its own, account for every other way a qubit dies: stray quasiparticles, radiation into wiring, or noise jittering the frequency. The budget is the largest single piece, not the whole story.

There is a humbling subtlety, too: individual TLS are discrete and a little random. One chip might have a single nasty defect parked right on a qubit's frequency, while its neighbour from the same wafer does not. So Q_i and coherence fluctuate from device to device, and even drift over hours as defects wander. A loss budget predicts the average behaviour of a process; it cannot promise that any one chip avoids a bad roll of the dice. Honest design plans for the spread, not just the mean.