Qubits & Quantum Computing

A classical bit is any physical system with two distinguishable states labelled 0 and 1, such as a transistor switched on or off. Your laptop runs on billions of them, each definitely 0 or definitely 1 at every instant. A quantum bit, or qubit, also has two distinguishable states, written |0⟩ and |1⟩ in the Dirac notation of Chapter 14, but it obeys quantum mechanics. You have met several: an electron's spin, a photon's polarisation, an atom's two lowest energy levels.

A qubit can therefore occupy a superposition α|0⟩ + β|1⟩, where α and β are complex numbers called probability amplitudes. Measure it and you get exactly one bit: 0 with probability |α|², or 1 with probability |β|²; hence |α|² + |β|² = 1, the normalization condition. As Chapter 15 insisted, this is not ignorance of a hidden value. Amplitudes, unlike probabilities, can be negative or complex, so contributions can cancel. That cancellation, interference, together with the entanglement of Part 4, powers every quantum algorithm.

In 1980 Paul Benioff described a quantum-mechanical model of a computer. In a 1981 lecture at MIT, published in 1982, Richard Feynman argued that ordinary computers seem to need exponentially growing resources to simulate quantum systems, so nature is best simulated by quantum hardware. In 1985 David Deutsch at Oxford defined a universal quantum computer, the quantum heir of Alan Turing's universal machine.

Two complex amplitudes contain four real numbers, but only two matter. Normalization removes one; another vanishes because multiplying both amplitudes by the same complex factor of size one, a global phase, changes no measurement probability. What survives is the balance between the amplitudes, set by an angle θ, and the relative phase φ between them: the amplitude on |0⟩ is the cosine of θ/2, while the amplitude on |1⟩ has a size equal to the sine of θ/2 and carries the phase φ.

Read θ as the angle down from the north pole and φ as the longitude, and every qubit state becomes a point on a sphere of radius one, the Bloch sphere, named for Felix Bloch. |0⟩ sits at the north pole, |1⟩ at the south pole, and the equator holds the equal superpositions, with |+⟩ = (1/√2)(|0⟩ + |1⟩) on the positive x axis and |−⟩ = (1/√2)(|0⟩ − |1⟩) opposite it. The chance of reading 0 is the cosine squared of θ/2, the spin rule of Chapter 17: a spin-one-half particle is a qubit, and the Bloch arrow is its spin direction.

Two cautions. Opposite points on the sphere are orthogonal, perfectly distinguishable states, although they sit 180° apart rather than 90°; the half-angles take care of that. And the sphere pictures one qubit alone. An entangled qubit has no point on the surface: its individual description shrinks inside the ball, and the shared information lives in correlations that no single globe can draw.

Quantum mechanics restricts what a gate can be. Between measurements an isolated system evolves by the Schrödinger equation of Chapter 10, which is linear and preserves total probability. Such operations are called unitary, and every unitary can be undone. For one qubit the result is beautifully visual: every gate rotates the Bloch sphere.

The X gate is the quantum NOT, a half-turn about the x axis that swaps the poles: X|0⟩ = |1⟩ and X|1⟩ = |0⟩. The Z gate, a half-turn about the z axis, has no classical counterpart: it leaves |0⟩ alone and multiplies |1⟩ by −1, a phase flip. Z changes no 0-or-1 probability, yet it turns |+⟩ into |−⟩, a hidden sign that steers any later interference. The Hadamard gate H is a half-turn about the axis midway between x and z, carrying |0⟩ to |+⟩ and |1⟩ to |−⟩, and back again.

Apply H twice and the qubit returns home; seeing why is this chapter's most important calculation. The first H makes (1/√2)(|0⟩ + |1⟩). The second acts on each piece: the |0⟩ piece adds 1/2 to |0⟩ and 1/2 to |1⟩, while the |1⟩ piece adds 1/2 to |0⟩ and −1/2 to |1⟩. The routes to |0⟩ add up to 1; the routes to |1⟩ cancel to zero, the same destructive interference that darkens double-slit fringes.

Circuit diagrams draw each qubit as a wire, time running left to right, gates as boxes. A circuit is a product of unitary matrices, so order matters: H, then Z, then H equals X, while Z followed by two Hadamards is just Z. The trio H, CNOT and T, an eighth-turn about the z axis, is universal: strung together, these gates approximate any unitary as closely as desired.

Rotating qubits one at a time never entangles them; they must interact. The workhorse is the controlled-NOT, or CNOT: if its control qubit is 0 nothing happens, and if the control is 1 the target is flipped. On basis states it acts like a reversible classical gate, sending |10⟩ to |11⟩ and |11⟩ to |10⟩ and leaving |00⟩ and |01⟩ alone. The surprise comes when the control is in superposition.

Start in |00⟩ and apply H to the first qubit, giving (1/√2)(|00⟩ + |10⟩): the first qubit undecided, the second definitely 0. Now apply CNOT with the first qubit as control. The |00⟩ branch is untouched, and in the |10⟩ branch the target flips to |11⟩. The result is (1/√2)(|00⟩ + |11⟩), the Bell state |Φ+⟩ of Chapter 20: measure both qubits and you get 00 or 11, each half the time, never 01 or 10. Two gates have built the correlations Einstein derided as spooky action at a distance, which, as Chapter 20 showed, can never carry a message.

Entanglement also makes quantum states costly to write down. Two qubits need four amplitudes, for 00, 01, 10 and 11, and the Bell state shows they cannot in general be split qubit by qubit; n qubits need 2ⁿ. Fifty qubits require about 1.1×10¹⁵ complex amplitudes, roughly 18 petabytes of memory at 16 bytes apiece, and 266 qubits need more amplitudes than the observable universe has atoms, about 10⁸⁰. No wonder Feynman expected classical computers to choke on quantum physics.

Now the most repeated myth in popular science: a quantum computer puts its input into a superposition of every value, evaluates the function once, and so tries every answer in parallel. The first half is true. Compute f on an equal superposition of all 2ⁿ inputs and you do hold every input paired with its output. But measurement returns one pair at random, exactly what an ordinary computer gets by trying a random input. Superposition alone buys nothing.

Quantum algorithms instead choreograph interference. Every computational path carries an amplitude with a sign or phase, and a good circuit makes the paths to wrong answers arrive out of step and cancel while the paths to the right answer reinforce. Large speed-ups need exploitable structure, such as a hidden period or a global property of a function; without it, the most interference can buy is the square-root gain of Grover's search in Part 6.

The cleanest demonstration is the Deutsch–Jozsa algorithm, published by David Deutsch and Richard Jozsa in 1992. An oracle, a black box, computes a function f from n-bit inputs to 0 or 1, promised to be either constant or balanced, giving 0 for exactly half the inputs. A classical computer that must be certain may need to query one more than half of the 2ⁿ inputs; the quantum algorithm needs one query. Hadamards spread the input across all 2ⁿ values; the oracle, by a trick called phase kickback, multiplies each amplitude by −1 wherever f equals 1; Hadamards fold the paths back together. The final all-zeros amplitude is the average of those signs: ±1 if f is constant, exactly zero if it is balanced.

Since sampling a few random inputs answers classically with high confidence, the Deutsch–Jozsa algorithm is a proof of principle, not a practical tool. Its lesson is the mechanism, phases written by an oracle and read by interference, on which useful algorithms build.

In 1996 Lov Grover of Bell Labs found a quantum algorithm for unstructured search: finding one marked item among N using only a yes-or-no check. Classically you need about N/2 checks on average; Grover needs only about π/4 × √N. Each round flips the sign of the marked item's amplitude, then reflects every amplitude about the average, so the marked one grows while the rest shrink. In effect the state rotates by a fixed small angle toward the answer, so timing matters: overdo it and you rotate past.

The speed-up is quadratic, not exponential: a million items need about 785 rounds instead of about 500,000 checks. In 1997 Charles Bennett, Ethan Bernstein, Gilles Brassard and Umesh Vazirani proved that no quantum algorithm can search an unstructured list with fewer than about √N queries, so Grover is essentially optimal. It roughly halves the effective key length of symmetric ciphers, one reason 256-bit AES keys are recommended for long-term secrets.

Shor's algorithm is the exponential headline. In 1994 Peter Shor, also of Bell Labs, showed that a quantum computer could factor integers and compute discrete logarithms in polynomial time. Factoring N reduces to finding the period r of the remainders left when successive powers of a number a are divided by N; a greatest-common-divisor calculation then yields the factors. A quantum Fourier transform makes the amplitudes interfere so that a measurement is likely to reveal r. The best classical method known, the general number field sieve, slows faster than any polynomial in the number of digits.

On a large enough error-corrected machine, Shor's algorithm would break both RSA, invented by Ron Rivest, Adi Shamir and Leonard Adleman in 1977, and elliptic-curve cryptography, which together protect much of the internet. Estimates of the machine required keep falling: about 20 million noisy qubits for eight hours (Gidney and Ekerå, 2019), fewer than a million for under a week (Gidney, 2025), and a March 2026 preprint argues that 10,000 reconfigurable atoms could suffice, given far longer run times. Actual runs of Shor's algorithm, on simplified circuits, have factored only 15 (2001) and 21 (2012). But data harvested now could be decrypted later, so NIST finalized post-quantum cryptography standards in August 2024.

A good qubit must be isolated enough to keep its superposition yet controllable enough to steer and measure. Decoherence, the environmental eavesdropping of Chapter 22, is the enemy, summarised by two times: T₁, how long |1⟩ survives before relaxing to |0⟩, and T₂, how long the relative phase survives. What counts is how many gates fit inside them.

Superconducting circuits, pursued by Google, IBM and others, are tiny circuits of aluminium or niobium interrupted by Josephson junctions, the tunnelling barriers of Chapter 12, and chilled to about 10 millikelvin. The junction makes each circuit an anharmonic oscillator, unlike the even ladder of Chapter 13, so microwave pulses can address just its lowest two levels. John Clarke, Michel Devoret and John Martinis showed in 1984 and 1985 that such a circuit acts as one quantum object, earning the 2025 Nobel Prize in Physics. Gates take tens of nanoseconds; coherence lasts tens to hundreds of microseconds.

Trapped ions, used by Quantinuum, IonQ and others, are charged atoms held in vacuum by electric fields and driven by lasers or microwaves. Identical by nature, they stay coherent for seconds or longer and post some of the best fidelities, but two-qubit gates take microseconds to hundreds of microseconds. Neutral atoms, pursued by QuEra, Pasqal and others, sit in grids of laser beams called optical tweezers and are entangled through Rydberg states, highly excited states whose enormous electron clouds make neighbouring atoms interact strongly; a Caltech group has trapped more than 6,100 atoms in one array.

Photonic qubits barely decohere and travel at room temperature, though the best detectors need cooling; photons hardly interact, so two-qubit operations rely on measurement tricks, and lost photons are the main danger. Spin qubits, proposed by Daniel Loss and David DiVincenzo in 1998, use the spin of a single electron, or of a hole (a missing electron), trapped in silicon or germanium, and could be made in chip foundries. As of 2026 the best two-qubit gates fail roughly once in a thousand to ten thousand operations, yet Shor's algorithm at cryptographic scale needs billions of operations. Raw qubits will not do.

The simplest classical error correction copies each bit several times and takes a majority vote. Quantum mechanics seems to forbid this twice over: an unknown state cannot be copied, the no-cloning theorem of next session, and checking a qubit destroys its superposition. Peter Shor found a way through in 1995 with a nine-qubit code, followed by Andrew Steane's seven-qubit code in 1996. Spread one qubit across several entangled qubits and measure only collective parity checks, such as whether two qubits agree. The answers, called the error syndrome, reveal the error without revealing the encoded amplitudes.

The protected qubit is a logical qubit. The threshold theorem, proved by several groups in 1996 and 1997, guarantees that if physical operations fail less often than some threshold, enlarging the code suppresses logical errors exponentially. The leading design, the surface code, grew from Alexei Kitaev's 1997 toric code: a checkerboard of data and measurement qubits with a threshold near 1 percent. A patch of code distance d uses d² data qubits and d² − 1 measurement qubits and corrects up to (d − 1)/2 errors.

The decisive test came in December 2024. On its 105-qubit Willow chip, Google Quantum AI ran surface codes of distance 3, 5 and 7, and each step cut the logical error rate by a factor of about 2.14; the 101-qubit distance-7 logical qubit failed in 0.143 percent of correction cycles and outlived the chip's best physical qubit 2.4 times over. Other teams run dozens of logical qubits: in 2023 a neutral-atom processor at Harvard, MIT and QuEra ran 48 logical qubits with error detection, not full correction, and in 2025 Quantinuum reported 48 error-corrected ones on its 98-ion Helios.

The state of play as of 2026: processors hold from dozens to over a thousand physical qubits, no quantum computer has factored a number of cryptographic size or run a long, fully fault-tolerant algorithm, and claimed quantum advantages concern specially designed sampling and simulation tasks, several since narrowed by better classical algorithms. Roadmaps target hundreds of logical qubits around 2029 to 2030; treat them as goals, not results. Nor will quantum computers replace laptops: they are expected to be specialised co-processors reached over the cloud, most promising for simulating molecules and materials.