A qubit is a clock, and a clock is only as good as the field it sits in. Every charge and every little magnet nearby pushes on that field, and none of them hold still. This page shows how those pushes turn a sharp pointer into a blur, why fast noise is gentler than slow noise, and what you can do about each.
Around any real qubit there are charges and there are little magnets. Each one makes a field at the qubit, and each one moves.
The field at the qubit is the sum of everything nearby, and nearby things never hold still.
Charges make electric fields that fall off as 1/r². Magnetic dipoles, which is what a nuclear spin or an unpaired electron looks like from a distance, make magnetic fields that fall off as 1/r³. A spin qubit mostly feels the magnetic field. A qubit built on an electric dipole mostly feels the electric one. Both fields wander as the neighbors jiggle, hop, and flip.
A qubit's pointer turns at a rate that follows the field. A wobbling field is a clock with a wobbling tick.
Same qubit, slightly different field, and the two pointers drift apart. That drift is a phase error, and it never stops growing.
The grey pointer is a perfect reference clock. The colored pointer is a qubit whose field is off by a little, or wobbling. Watch the gap between them. With a steady offset the gap grows in a straight line. With a wobbling field it wanders like a drunk walker, and the wandering is the whole story of this page.
Start discrete. A single charge hops between two sites next to the qubit, or a single spin flips. The clock rate jumps between two values.
Run the experiment many times and the pointers fan into two groups. Their average shrinks. If the neighbor is slow, the average even comes back.
Each thin pointer is one run of the experiment with its own copy of the neighbor. The thick arrow is their average, and its length is what any real measurement sees. A slow neighbor splits the runs into two clean branches that meet again half a turn later. A fast neighbor never lets either branch form, and the average just fades, slowly.
Now let the number of neighbors grow. The two-branch picture melts into a smooth blur, and only two numbers about the noise survive.
With many neighbors the field wobble is bell-shaped. All that matters is how big it is and how fast it changes.
Add neighbors. The histogram of the field goes from two spikes to a bell. The pointers stop forming branches and simply spread. The average arrow now shrinks smoothly, and its shape depends on one comparison: does the field change faster or slower than the pointer can respond?
Two sliders, one surprise. At the same size, noise that changes faster does less damage.
Slow noise adds up. Fast noise cancels. The damage rate is the size squared times the memory time.
Hold the size fixed and shorten the memory time. The arrow lasts longer. This is why a molecule in a warm liquid, surrounded by solvent dipoles that reorient in picoseconds, can keep its phase far longer than the size of those fields would suggest. The solvent jiggles so fast that the qubit only feels the average.
Everything so far changed how fast the pointer turns. Noise can also knock it off the equator. That is a different loss with a different name.
Noise along the axis blurs the phase. Noise across the axis, at the right frequency, flips the qubit. T2 counts the first, T1 counts the second, and T2 can never exceed 2T1.
Below is the whole sphere, with a cloud of possible states on it. Pure phase noise squashes the cloud into a line along the axis. The poles survive, the equator is lost. Flipping noise pulls the whole cloud toward the ground state instead, poles included. Real qubits get some of each.
If the field holds still long enough, you can run the clock error backwards. One flip in the middle, and the fan folds back into a pointer.
Flip every pointer across the dial at half time. Fast runners become slow runners, and they all meet again at the finish.
Each run has its own steady field error, so its pointer runs ahead or behind by its own amount. A flip at half time mirrors every pointer. The ones that were ahead are now behind by the same amount, and the same field error walks them back to the start line. It only works if the field error is the same in the second half as in the first. Turn the memory time down and watch it fail.
The clock rate depends on the field through a curve. Bias the qubit at the top of that curve and small field wobbles stop mattering to first order.
A wobble δB shifts the rate by slope × δB. At a stationary point the slope is zero, and the wobble is demoted to second order.
Drag the operating point along the curve. The vertical band is the field wobble, which does not change. The horizontal band is the rate wobble the qubit actually suffers, and it collapses as you approach the top. Engineers know this trick from bridge circuits and from biasing an amplifier at its most linear point. Physicists call it a sweet spot or a clock transition.
Different hardware, different neighbors, same two numbers. Sort each platform by whether its noise is fast or slow and the right fix follows.
Ask two questions of any qubit. What is the nearest neighbor, and does it move faster or slower than the pointer?
Four lines, in the order the page earned them.
The companion pages cover what came before this one. How Many Bits Fit in a Qubit? builds the dial and the shrinking arrow. Bits, Qubits, and Phase shows what the phase is for, and why T2 is the number a sensor lives or dies by.
