Superconducting Qubits and Error Correction

Bigger Is Better

In 2024 Google Quantum AI ran a set of experiments on its Willow chip: grids of 3×3, 5×5 and 7×7 data qubits, each encoding one logical qubit, checked for errors every 1.1 µs.

The logical error per round fell from 0.65 % to 0.30 % to 0.14 %. Each step of 2 in distance divided it by about 2.14. In 2023 the same team had distance 5 beat distance 3 by only about 4 %; this time each step more than halved the error, and the distance-7 logical qubit outlived the chip’s best single qubit (291 µs against 119 µs).

Distance 3 → 5 → 7 · error per round 0.65 % → 0.30 % → 0.14 % · Λ = 2.14

An Artificial Atom

A capacitor and an inductor make an LC circuit, which oscillates. Cold enough to be quantum, its energy comes in steps, all equally spaced: a microwave that lifts it from level 0 to 1 lifts it from 1 to 2 just as well, so its lowest two levels can’t be used on their own.

Replace the inductor by a Josephson junction, two aluminium films with a layer of aluminium oxide one or two nanometres thick between them. Its inductance changes with the current, and each step is smaller than the one below. On this chip 0 to 1 is 5.01 GHz and 1 to 2 is 4.76 GHz. The lowest two levels alone are |0⟩ and |1⟩.

5 GHz corresponds to 0.24 K. So that heat can’t kick it, the chip hangs in a dilution refrigerator at about 10 to 20 mK.

0→1 5.01 GHz · 1→2 4.76 GHz · anharmonicity −246 MHz (this page’s circuit)

Cut Open: One Chip

The chip is a square of silicon 10 mm across and half a millimetre thick, coated with aluminium film about 100 nm thick. Narrow etched gaps draw the circuit: each qubit is a cross-shaped island of aluminium with 130 µm arms; one arm ends in two Josephson junctions to ground, each about 0.2 × 0.3 µm.

The chip sits in a pocket of a gold-plated copper box, framed by a circuit board. Over a hundred aluminium wires 25 µm thick join the pads at the chip’s edge to the board, whose traces run to microwave connectors in the walls. With the lid on, it is a small shielded box that hangs below the coldest plate of the fridge.

Silicon 10 × 10 × 0.5 mm · aluminium about 100 nm · junction about 0.2 × 0.3 µm · oxide 1–2 nm

Turned by Microwaves

A qubit’s state can be drawn as a point on a sphere: |0⟩ at the north pole, |1⟩ at the south. A microwave pulse at the qubit’s 5 GHz turns the point around the sphere, by an angle set by the pulse’s strength times its length.

Half a turn takes |0⟩ to |1⟩. A quarter turn stops on the equator: measured there, it gives 0 or 1 with equal chance. Where on the equator it sits (its phase) changes what later operations do. On Willow, a single-qubit operation takes 25 ns.

Angle ∝ strength × length · half a turn = 0↔1 · single-qubit gate 25 ns

Reading Out

Beside each qubit lies a folded readout resonator, near 7 GHz; the resonators on one feedline each have their own frequency, so they can be read together. Whether the qubit is in |0⟩ or |1⟩ pushes the resonator’s frequency a little one way or the other, here about 0.55 MHz.

A microwave tone near that frequency sent along the feedline comes out with a phase that depends on the qubit’s state. A cold amplifier boosts it, and within a few hundred nanoseconds it reads as 0 or 1. After the measurement the qubit is in the state that was found.

Resonator 7 GHz · shift ±χ ≈ 0.55 MHz · measurement about 500 ns

Where Errors Come From

A qubit in |1⟩ leaks its energy to its surroundings and falls to |0⟩, with a time constant called T1; the phase on the equator slowly wanders, with a time constant T2. On Willow’s error-correction chip, T1 averages 68 µs and T2 89 µs.

Each operation is imperfect too: a single-qubit gate errs about 0.035 % of the time, a two-qubit gate about 0.33 %, a measurement about 0.77 %. Useful quantum algorithms need hundreds of millions of operations in a row. Single qubits are nowhere near; only error correction can close the gap.

T1 68 µs · T2 89 µs · two-qubit gate error 0.33 % (Willow averages)

Redundancy Without Copies

The simplest classical error correction copies: store a bit three times, and if one copy flips, the majority puts it right. A quantum state can’t be copied (the no-cloning theorem), and measuring it directly destroys it.

The way round is to ask only about neighbours’ parity: are these two the same? An extra measure qubit asks, and learns only “same” or “different”, not what either one holds, so the stored information is untouched. Where an error happens, the checks on either side turn to “different”.

No cloning · measure parities, never values

The Surface Code

The surface code lays its data qubits in a d×d square, with measure qubits between them: half check the X parity of the data qubits around them, half the Z parity (four in the middle, two at the edge). One error lights the checks on either side of it; a chain of errors lights only its two ends.

A computer works out the likeliest errors from where the lights are, and corrects them; this is decoding. d is the fewest errors that can fool the code, and a code of distance d corrects any (d−1)/2. Distances 3, 5 and 7 use 17, 49 and 97 qubits.

Distance d uses 2d² − 1 qubits · 3 → 17 · 5 → 49 · 7 → 97

Below Threshold

A bigger code can correct more errors, but it also has more places to go wrong. Only when each qubit’s error rate is below a “threshold” does a bigger distance pay; the further below, the more each step of 2 gains.

This page ran the numbers for a model in which every operation errs with the same chance: the threshold comes out near 1.2 %. At a physical error rate of 0.5 % each step of 2 halves the logical error, close to Willow’s 2.14; at 0.1 % it divides it by ten. Real chips don’t have uniform noise, so there is no single threshold.

This page’s simulation: threshold about 1.2 % · Λ ≈ 2.0 at p = 0.5 % · Λ ≈ 10 at p = 0.1 %

How Far to Go

Willow’s logical qubit only held a state; it did no operations between logical qubits. About once an hour, the experiment saw a burst of correlated errors whose cause is not yet understood. Extrapolating, the paper puts an error per round of one in a million at distance 27: 1,457 physical qubits per logical qubit.

One estimate puts factoring a 2048-bit RSA number at under a million physical qubits for under a week, assuming each operation errs 0.1 % of the time, about three times better than Willow’s two-qubit gate. Every qubit needs lines from room temperature down, and decoding has to keep up with a round every microsecond.

Target 10⁻⁶: distance 27, 1,457 qubits per logical qubit · RSA-2048: < 1 million qubits (estimate)

Taken Apart

The lid, the board, the chip with its wires, the connectors and the copper base come apart, layer by layer.

Whichever part the pointer rests on is lifted out and named; left alone, the parts take turns.