What is a qubit, and what does a quantum computer actually do?

An ordinary bit is a switch: 0 or 1. Everything your phone does comes down to billions of these.

A qubit is a quantum bit. Chapter 2 showed that a quantum state works like a direction. For a qubit, picture an arrow inside a ball. Arrow straight up means 0. Straight down means 1. But the arrow can point anywhere: up, down, sideways, or any angle in between.

Measuring a qubit asks one question, "0 or 1?", and always gets one of those two answers. If the arrow points straight up, you get 0 for certain. If it points sideways, you get 0 or 1 with even odds. At any other angle, the angle sets the odds, exactly as in chapter 2.

Physicists call this the Bloch sphere, after Felix Bloch. Every possible state of one qubit is a point on its surface.

Two reasons. First, you can't read the arrow's direction. Measuring only ever gives 0 or 1. Second, the arrow has a direction around the ball as well as up and down. You can't see that part in a single measurement, but it controls how qubits interfere. That's where the power lives.

An ordinary computer runs on logic gates, such as NOT, which flips 0 to 1. A quantum computer runs on quantum gates, and each one turns the arrow in a precise way.

• The X gate flips the arrow upside down: 0 becomes 1. It's the quantum NOT.
• The H gate (Hadamard) tips the arrow from the pole to the equator: from certain to 50/50.
• The Z gate spins the arrow halfway round the vertical axis. The odds don't change, but the hidden direction does.

A quantum program is a list of gates applied in order, then a measurement. Physicists draw it as a circuit: one line per qubit, with a box for each gate.

Two bits can be 00, 01, 10 or 11, one at a time. Two qubits have an amplitude for each of the four, all at once, and the amplitudes can add and cancel like the paths in chapter 4.

A two-qubit gate called CNOT (controlled NOT) flips the second qubit only if the first is 1. Put the first qubit through H, then apply CNOT, and you get an entangled pair from chapter 5: measure them and you get 00 or 11, each half the time, never 01 or 10.

Each extra qubit doubles the list of amplitudes. There's one amplitude for every possible readout, and each extra qubit splits every readout in two: one ending in 0, one ending in 1. You can't cut corners by describing each qubit on its own, because once qubits are entangled, like the pair above, only the whole list describes them.

2 qubits: 4 numbers. 10 qubits: 1,024. 50 qubits: about a thousand million million. 300 qubits: more numbers than there are atoms in the observable universe.

That's why ordinary computers struggle to imitate quantum ones. Simulating 30 qubits exactly takes about 17 gigabytes of memory, which a laptop can just about manage. 50 qubits takes about 18 petabytes, a warehouse of disks.

Here's the catch from the hook. All those amplitudes exist, but measurement gives you just one result, n bits long, picked with the odds. Read naively, a quantum computer gives you a random answer. The trick is to make that random answer almost certainly the right one.

From interference. A quantum algorithm is a sequence of gates designed so that, by the end, the amplitudes for wrong answers point in opposite directions and cancel, while the amplitudes for the right answer line up and add (chapter 4's puzzle). Then one measurement gives the right answer with high odds.

Designing those cancellations is hard, and it only works for problems with the right hidden structure. That's why there are only a handful of famous quantum algorithms, not a quantum version of everything.

In 1985 David Deutsch at Oxford described how a universal quantum computer would work, and found the first problem where it beats an ordinary computer. Here's the idea as a game.

• Shor's algorithm (1994). Peter Shor found a way to factor large numbers exponentially faster than any known ordinary method. Most internet encryption relies on factoring being hard, so a big enough quantum computer would break it. Chapter 13 covers the threat and the fix. Optional page 7a shows it factoring 15.
• Grover's algorithm (1996). Lov Grover found a way to search an unsorted list of a million items in about a thousand steps instead of a million. It's a square-root speed-up: useful, but modest. Optional page 7b.
• Quantum simulation. In 1981 Richard Feynman proposed using quantum machines to simulate nature, which is quantum to begin with. Molecules and materials are the strongest long-term case: new drugs, batteries, catalysts and fertilisers. Optional pages 7b and 7c.
• Near-term methods. Hybrid methods share the work between a small quantum computer and an ordinary one, for chemistry and optimisation. They're widely tried. None has yet shown a clear advantage on a useful problem (chapter 14).

No. Quantum computers are specialists. For email, video, spreadsheets, games and nearly all of today's computing, an ordinary computer is faster, cheaper and far more reliable. Quantum computers also struggle with big data: loading a large dataset into qubits can take so long that it cancels out any speed-up.

Think of a quantum computer as an accelerator, like a graphics card, that an ordinary computer calls on for particular jobs, usually over the cloud. IBM put a 5-qubit machine on the cloud for anyone to try in 2016, and several companies now offer access.

No. Computer scientists believe quantum computers can't efficiently solve the hardest class of puzzles in general (problems like the travelling salesman, which computer scientists call NP-complete). They help where a problem hides a pattern that interference can find, such as the repeating pattern Shor's algorithm uses.