A tutorial for anyone — just everyday arithmetic

Quantum computing is really just three moves

Courses make quantum computing look like forty hard tricks with strange names. Underneath, every one of them is one of three simple moves on one simple tool — two rings that slide past each other (you could draw them with a compass). Learn the three moves and you can understand all forty.

Start with the one idea → Browse all 40 tricks Try the live calculator
40tricks to memorize → 3moves to understand
The one tool — two sliding rings

01The idea

The one idea

Read this part even if you read nothing else

Courses teach quantum computing as a long list of tricks with strange names — forty of them. You would have to learn each one separately.

This page teaches them a different way. There is one tool — two rings that slide against each other — and there are only three things you ever do with it. Tap a card to see its moving picture.

Move 1

The Setting

Name one spot and read its numbers.

Slide the rings to one spot and read the numbers there — the readings.

see it move → Move 2

The Fold

Multiply readings, get one simple reading.

Multiply a reading by itself (or two readings together) and the answer turns into one simple reading at a new spot.

see it move → Move 3

The Turn

Move from one spot to another.

Step the ring round to a new spot. Every step comes with a check number, so a wrong step shows up.

see it move →
Every one of the forty quantum computing tricks is one of these three moves. Below, you'll see each move as a moving picture, then all forty tricks — each one showing which move it is, and why.

02The basics

What is a quantum computer? Five simple ideas

Start here if these words are new — each has short videos

Before the three moves make sense, here's the whole subject in five simple ideas. Each one has a ▶ button that opens short videos about it.

Idea 1

A bit is a 0 or a 1

An ordinary computer stores everything as bits — tiny switches that are either off (0) or on (1). Everything it does is flipping switches and combining them.

watch: what is a bit

Idea 2

A qubit can be a blend of both

A quantum bit — a qubit — can be 0, or 1, or a blend of both at once. Picture a dial with 0 at one end and 1 at the other: the needle can point anywhere in between. Where it points (its blend angle) gives two numbers between 0 and 1 — its reading for 0 and its reading for 1. A qubit also has a second angle, its phase — think of it as which way the dial is facing, like a compass direction. The phase doesn't change the readings, but it matters when qubits are combined.

On the rings: where the needle points is a setting, and both readings come straight off it.

watch: what is a qubit

Idea 3

A gate is an instruction for the needle

You compute by giving the qubit instructions called gates. Some gates are known by the spot they aim for ("go to 30°"); some by how far they shift the needle ("shift a quarter of the way round"); some link two qubits together. A quantum program is a list of gates, run one after another.

On the rings: a gate known by the numbers at its spot is a setting. A gate known by how far it shifts — and running gates one after another — is a turn.

watch: quantum gates

Idea 4

Measuring gives a 0 or a 1

Inside a real machine you can't see the needle. When you measure a qubit, it jumps to a plain 0 or 1. Which one is down to chance: the chance of a 0 is its reading for 0 multiplied by itself, and the chance of a 1 is its reading for 1 multiplied by itself. The two chances always add up to 1. Measure many times and you see the odds.

On the rings: multiplying a reading by itself is the fold.

watch: quantum measurement

Idea 5

Entanglement links qubits

Two qubits can be linked so that their results match far more often than chance allows. Einstein called it "spooky." This linking is a big part of what makes quantum computers powerful.

On the rings: to measure how linked two qubits are, you multiply their results together (counting a 0 as +1 and a 1 as −1, so that "same" and "different" results show up as plus and minus) — a fold.

watch: entanglement

Put together

So what is a quantum computer?

A row of these dials (qubits). You set them, measure them to get the answer as odds, then step them toward a better answer (for example, the lowest energy a molecule can have) — and repeat until the answer is good. Every one of the forty tricks further down is just one part of this.

watch: how quantum computers work
Keep those five ideas in mind. Next: the tool this whole page is built on — and how pointing, multiplying and stepping become the three moves.

03The tool

Meet the tool: two sliding rings

No quantum computing yet — just the rings

Draw a circle and put sixty marks around it, like the minutes on a clock. On each mark write one digit. The digits come from a famous number pattern (the Fibonacci numbers): start with 0 and 1, then each new number is the two before it added together — 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 … Write down only the last digit of each: 0, 1, 1, 2, 3, 5, 8, 3, 1, 4 … After sixty numbers the last digits start over, so they fit the sixty marks exactly. Those digits are what let the ring check its own work. Here is the whole ring:

The sixty-ring. Going clockwise from 9 o'clock: 0, 1, 1, 2, 3, 5, 8, 3, 1, 4 … The four marks at 9, 12, 3 and 6 o'clock all carry 0.

Now take two of these rings and lay them side by side so they overlap. Hold the first ring still and slide the second along the line through the centres. Where their edges meet there are two crossing points. How far apart those two crossing points are, measured around the ring, is the angle — 0° when the rings just touch, opening up to 180° when they sit right on top of each other. Each way you slide them is one setting, and it gives you two lengths at once:

  • the across-length — the distance between the two ring centres,
  • the up-length — the distance between the two points where the rings cross.

Draw the rings with a radius of 1 (any unit — say 1 inch). Then half of each length is a reading — a number from 0 to 1. Half the across-length is the qubit's reading for 0; half the up-length is its reading for 1. You don't work these out — you read them off, the way you read the time off a clock. Calculator check (set the calculator to degrees): the reading for 0 is the cosine of half the angle, and the reading for 1 is the sine of half the angle — at 60°, cos 30° = 0.866 and sin 30° = 0.500.

The two crossing points sit over two marks of the ring that mirror each other (like "5 past" and "5 to" on a clock). Their digits always follow a fixed rule: they are either equal or they add up to 10. If they don't, the spot was misread — so every reading comes with its own check. And between any two marks you can add sixty finer marks, like millimetres between the centimetres on a ruler; the same digit pattern works on those too, so any angle you can write down in decimals can be read exactly. Keep three facts in mind and the rest of the page follows:

Fact 1 — a setting is the numbers at one spot. Slide the rings to one spot and read them. Reading is all there is to it.
Fact 2 — the fold turns multiplying into reading. To multiply the reading for 0 by itself, you don't multiply: slide to double the angle, take the reading for 0 there, add 1 and halve. Example at 60°: the reading for 0 at 120° is cos 60° = 0.5, and (1 + 0.5) ÷ 2 = 0.75 — the same as 0.866 × 0.866. Up to 90° you slide straight to the double angle. Past 90° the double angle is more than 180°: the sliding ring passes through the still ring and comes out on its other side, and a reading on that side counts as negative. The arithmetic rule still works. (The chance of 1 is what's left: 1 minus that.)
Fact 3 — every step carries a check number. Step a quarter of the way round (fifteen marks) and the digit you land on is always the starting digit times 7 — keep only the last digit: 8 → 56 → 6. Step back a quarter and it's times 3. If you land on the wrong digit, you know the step went wrong. That's the built-in error check.

04Move 1 of 3

Move 1 — The Setting

Name one spot and read its numbers

A setting is just one spot — the second ring slid to one place across the first, like a dial pointing at an angle. Every spot comes with its own two readings. Move to a new spot and both readings change. So the angle and its two readings tell you the same thing.

Here's why that matters: a qubit's condition is exactly one such spot. Its two readings say how much 0 and how much 1 it is right now. So to describe a qubit, you don't need a page of numbers — just one spot.

One kind of damage to a qubit ("noise") doesn't move the spot — it blurs the qubit, so its readings come out weaker, exactly as if the rings had shrunk. A perfect qubit uses full-size rings; a damaged one is the same spot on smaller rings. So reading how much the rings shrank tells you how much damage there is.

Setting vs. Turn: a setting is about the numbers at a spot; a turn is about a shift — how far you move. A gate known by its numbers (like a rotation gate or Hadamard) is a setting. A gate known by how far it shifts (like the S and Z gates, which shift the phase) is a turn — and so is running gates one after another.
The Setting test: Is the trick about the numbers at a spot — naming a spot and reading what's there? Then it's a setting. Fourteen of the forty tricks are settings.
live
Watch the second ring slide across the first, which stays still. At every setting you read two lengths at once — the across-length (yellow) and the up-length (white). Half of each is a reading: the reading for 0 and the reading for 1.

05Move 2 of 3

Move 2 — The Fold

Multiply readings, get one simple reading

The fold is one simple fact: when you multiply readings together, the answer turns out to be one single reading at a new spot. You never have to do the multiplying — you just read a different spot.

  • Multiply the reading for 0 by itself → take the reading for 0 at double the angle, add 1 and halve (at 60°: (1 + 0.5) ÷ 2 = 0.75). The chance of 1 is what's left.
  • Multiply a qubit's two readings together, and double it → the reading for 1 at double the angle.
  • For two linked qubits: measure both, and multiply the two answers each time (count a 0 as +1 and a 1 as −1). On average you get one reading, set by the difference of their two angles.
  • For three linked qubits: the same with all three → one reading, set by their three angles added together. (Difference or sum depends on how the qubits were linked.)

Why it matters: measuring a qubit — getting the odds of 0 or 1 — is multiplying a reading by itself (Idea 4). So every set of odds, every score, every "how sure are we," and every link between qubits comes from this one move.

The Fold test: Does it multiply readings together (or multiply one by itself)? Then it's a fold. Ten of the forty are folds.
live
The yellow hand shows the setting's angle; the white hand sits at double the angle. The two bars always match: the reading for 0 multiplied by itself (left) equals the reading for 0 at double the angle, plus 1, halved (right). Example: at 60° the reading for 0 is 0.866, and 0.866 × 0.866 = 0.75 — a 75% chance of 0.

06Move 3 of 3

Move 3 — The Turn

Move from one spot to another

A turn moves the dial from one spot to another, anywhere round the sixty marks. Steps add up: a step of 10 marks and a step of 20 marks make a step of 30. And each step comes with a check number: a quarter turn always multiplies the digit by 7, a half turn by 9 (which swaps a digit for the partner it adds up to 10 with, like 8 ↔ 2 — and 0 stays 0), and a three-quarter turn by 3.

A turn can shift either of the qubit's two angles (Idea 2). Shift the blend angle and the readings change — turn 30° a quarter of the way round to 120° and the readings go from 0.966 and 0.259 to 0.500 and 0.866. Shift the phase and the readings stay exactly the same; only the direction the qubit faces changes. Both are turns.

Why it matters: to make a program better, you need to know which way to adjust each angle. The trick everyone uses reads the answer a quarter turn forward and a quarter turn back, and compares the two. That's two turns — and their check numbers prove both steps were right before they're compared. Merging gates, Grover's search, teleportation's fix-up (you'll meet them in the list below) — all of them are steps from spot to spot.

The Turn test: Is the trick about a shift — how far you move, one step or a chain of steps? Then it's a turn. Sixteen of the forty are turns — the biggest group — and the one trick from 2025 at the end of the list is a turn as well.
live
A quarter turn steps the marker fifteen marks clockwise — here from mark 6 (digit 8) to mark 21 (digit 6), the same marks as on the big ring above. The digit is multiplied by 7 (keep the last digit): 8 → 56 → 6. The digit proves the step landed right.

07The loop

Now watch the three moves become a quantum computer

How the three moves fit together into a real program

Here is what a real quantum program does, start to finish. It's nothing but the three moves:

1 · Setput the qubits at a starting spot
→
2 · Foldmeasure: turn readings into odds and a score
→
3 · Turnstep the angles toward a better score
↺
repeatuntil the score stops improving

That loop — set fold turn, repeat — is how the most common programs on today's quantum computers work. You've already seen each move as a moving picture above.

A real example from the research this page is built on. Chemists want the lowest energy a molecule can have — that's the shape the molecule actually settles into — so the score is the energy, and lower is better. For this molecule, one angle describes the pair of qubits. The program sets that angle, reads the energy (a fold), and steps to the best spot. It needs just three readings. The answer lands exactly on one mark of the ring. Same three moves, doing real chemistry.

So the forty tricks below aren't a new subject. Each one is just a piece of this loop — a way of setting, folding or turning.

08The 40 tricks

All forty tricks, decoded — and one from 2025

Look one up when you meet it — no need to memorize

These are the strange names you'll meet in any quantum computing course, plus one experiment from 2025 that no course teaches yet. Each card says in plain words what the trick is, which of the three moves it is and why, and has a ▶ button for short videos. Use the buttons to show one move at a time.

When a trick seems to fit two moves: most tricks touch more than one move (almost everything reads something at the end). Each one is filed by its main job — the step that makes it that trick. If the key step is the numbers at a spot, it's a Setting. If the key step is multiplying readings, it's a Fold. If the key step is shifting or stepping, it's a Turn.

09It checks itself

The free gift: it checks itself

Why use the rings instead of an ordinary calculator

Besides its qubits, every quantum computer is run by an ordinary computer that keeps track of all the angles and other numbers. Those ordinary numbers are normally never checked. If a stray particle from space or a glitch flips one bit, nobody notices, and the answer quietly comes out wrong.

On the rings, every mark has its own check. Take the digit on a mark and the digit on the next mark: that pair of digits belongs to that mark and no other. The pair travels with the mark, so if the mark gets changed, the pair it carries no longer fits the new mark, and the damage shows up. There's no extra work — the check happens in the same move as the work.

Big numbers the machine keeps are written across six sixty-rings, like a car's odometer with six wheels, where each wheel has sixty marks instead of ten. Each wheel carries its pair of check digits. The machine keeps two copies side by side: the number in its usual form, and the same number on the six wheels. Every number has its own wheel pattern — no two numbers share one — so if the number gets damaged, it no longer matches its wheels, and the damage is caught. Every damaged number is caught — 100% of them. Then it is fixed: the machine works the number out again from the angle written in the program. In the research, 5,383,856 damaged numbers were thrown at it. Every one was caught, and every one was fixed.

Some computer chips today do check their numbers, with a quick remainder trick called a residue check (videos). When a number gets damaged, that check misses it about 1 time in 3. The rings miss none.

100%of damaged numbers caught — 5,383,856 out of 5,383,856 in the test
100%of damaged numbers fixed — every one put back to its correct value
When a number gets damaged, how often is it missed?
The residue check chips use today32%
The rings0%
In short: the rings don't just do the work — they catch every bad number as they go, and every one gets fixed.

10It's real

It's real — it ran on a real quantum computer

Not just an idea on paper

This has been run on IBM's real 156-qubit quantum computer. Every angle carried its check digits. In a normal run, the built-in check stayed quiet — it read close to zero, exactly where the math said it would be. Then one single bit was flipped on purpose before the job was sent — and the check went off like a fire alarm, more than a hundred times bigger than the normal random wobble. When the same flipped bit was sent without the check, the machine turned the best possible answer into the worst possible answer, with no warning at all.

156qubits on IBM's real machine
112×how loud the alarm was, next to the normal random wobble
1 bitflipped on purpose — and caught 100% of the time
The takeaway. Forty tricks shrink down to three moves anyone can picture — and the picture comes with a built-in lie detector. That's a new way to learn quantum computing: three moves instead of forty recipes.

11Next steps

Go deeper

Good next stops when you want more

This page gives you the map. When you want more, these are good next stops. The ▶ buttons open short videos; the text links are free lessons.

Whenever the strange names pile up, come back here. Every trick is one of three moves: a setting, a fold, or a turn.