Intuition

You cannot look at a qubit without disturbing it. But you can ask a group of qubits a yes/no question about how they relate, like "do these two agree?", without learning what each one is. If the answer changes, an error happened.

Core idea

A stabilizer is a product of Pauli operators (X, Y, Z) that leaves every valid code state unchanged: S|ψ⟩ = |ψ⟩. All stabilizers must commute. With n physical qubits and n − k independent stabilizers, the code stores k logical qubits. Measuring the stabilizers gives the syndrome: +1 means "fine", −1 means "something here flipped". A code is written [[n, k, d]], where the distance d is the fewest single-qubit errors that can change the logical state without being noticed. It can correct up to ⌊(d−1)/2⌋ errors.

Example: the 3-qubit bit-flip code has stabilizers Z₁Z₂ and Z₂Z₃. CSS codes use separate X-type and Z-type stabilizers; the surface, colour and most qLDPC codes are all CSS codes.

Research level

The stabilizer formalism (Gottesman, 1997) turned quantum error correction into linear algebra over bits: a code is a pair of binary check matrices H_X, H_Z with H_XH_Z^T = 0. Code design is a set of competing trade-offs, not a checklist every code meets: higher rate k/n usually needs longer-range or heavier checks; larger distance costs more qubits; low-weight, local checks are easier to build but limit rate (the BPT bound for 2D local codes); and some codes with excellent parameters are hard to decode fast. The threshold theorem says that if physical error rates are below a constant threshold, arbitrarily long computations are possible with polylogarithmic overhead, under assumptions about the noise (for example local, weakly correlated errors and fast classical processing). Real devices only approximately satisfy these assumptions.

Intuition

Lay qubits on a chessboard. Each square checks its corner qubits. One error lights up the checks next to it, like footprints. A decoder follows the footprints back to the error.

Core idea

Below is a real distance-3 rotated surface code: 9 data qubits and 8 checks, storing 1 logical qubit. Click a qubit to give it an X, Z or Y error (click again to change it). The checks that light up are the syndrome. "Decode and fix" uses the smallest correction that matches the syndrome. One error is always fixed. Two errors can fool it, because d = 3.

Three different numbers: the code distance d is a property of the code (here 3). The physical error rate p is how often each hardware qubit or gate fails. The logical error rate p_L is how often the encoded qubit fails per round. Error correction helps only when p is below the threshold, and then p_L falls as d grows.

Use the error-chance slider and "Run" to compare one bare qubit with the encoded qubit. At low error rates the code wins. At high error rates it loses. The crossover is the idea behind the threshold.

Research level

This model ignores measurement errors. Real experiments repeat the checks many times and decode in space and time. With circuit-level noise the surface code threshold is about 1%. A distance-d patch uses 2d² − 1 qubits. Logical error per cycle falls roughly as (p/p_th)^(d+1)/2. Google's Willow chip (2024) showed a distance-7 memory with a suppression factor Λ ≈ 2.14 each time d grew by 2, and a logical qubit that outlived its best physical qubit. Being below threshold is necessary but not enough for practical fault tolerance: you also need many logical qubits, logical gates and magic states, fast decoding, and a qubit overhead you can afford (roughly hundreds to a thousand-plus physical qubits per logical qubit for useful error rates), and the threshold number itself depends on the noise model assumed. Open problems: correlated errors (cosmic rays, leakage), and reducing the large qubit overhead.

Intuition

Some quantum circuits look powerful but are secretly easy. If you only use certain gates, a normal computer can follow along cheaply, so there can be no quantum speed-up.

Core idea

Clifford gates (H, S, CNOT and the Paulis) only move stabilizer states to other stabilizer states. For one qubit these are the 6 corners of the octahedron below. Press H, S and X: you only jump between corners. Press T: the state leaves the corners and becomes a "magic" state.

Gottesman–Knill theorem: circuits made only of stabilizer-state inputs, Clifford gates and Pauli measurements can be simulated efficiently on a classical computer. Adding a non-Clifford gate such as T, which can be used as many times as needed, makes the set {H, S, CNOT, T} approximately universal: it can approximate any quantum circuit to any accuracy (efficiently, by the Solovay–Kitaev theorem). This works for this standard set; it does not mean that any single extra gate in any setup gives universality.

Research level

Classical simulation tracks n stabilizer generators in an n × 2n binary table (Aaronson–Gottesman, O(n²) per measurement). Tools such as Stim simulate error-correction circuits with millions of operations per second, which is how most QEC research is done. For stabilizer-rank (Clifford+T) simulators such as Bravyi–Gosset, the cost grows polynomially in qubits and Clifford gates but exponentially in the number of T gates (about 2^0.23t for t T gates in the best known decompositions). Other simulators scale differently: tensor networks depend on entanglement and circuit structure, not T-count. So T-count is a key cost measure for Clifford+T simulation and for fault-tolerant resource estimates, but it is not the only factor that sets classical simulation cost.

Intuition

Error-corrected machines can do Clifford gates safely, but not the T gate. So they make a special "ingredient" state, clean it up, and then use it up to perform one T gate. A quantum computer becomes a factory of these ingredients.

Core idea

The magic state |T⟩ = (|0⟩ + e^iπ/4|1⟩)/√2 can be consumed by a short Clifford circuit with one measurement to apply T to a data qubit (gate teleportation). Injected magic states are noisy. 15-to-1 distillation takes 15 noisy copies with error p and outputs one with error about 35p³, using only Clifford operations.

Research level

Distillation factories can take a large share of a fault-tolerant computer. Newer ideas cut the cost: smaller codes for factories (Litinski, 2019) and magic state cultivation (Gidney, Shutty & Jones, 2024), which grows a T state inside a surface code at a cost close to a CNOT. In 2025, QuEra, Harvard and MIT showed distillation on logical qubits encoded in colour codes on neutral atoms, with output fidelity better than the inputs. Open problem: lowering the total space-time cost of non-Clifford gates.

Intuition

Storing a logical qubit is not enough; you must compute with it. Two ways: do the same gate to every physical qubit at once (transversal), or stitch two code patches together and cut them apart again (lattice surgery).

Core idea

Transversal gate: apply a gate qubit-by-qubit across code blocks. One faulty physical gate can only damage one qubit per block, so errors do not spread. The surface code has a transversal CNOT between two patches; colour codes have all single-qubit Clifford gates transversally.

Lattice surgery: measure new stabilizers along the shared edge of two patches (merge) for a few rounds, which measures the joint operator X_L1X_L2 or Z_L1Z_L2. Then stop measuring them (split). Two such measurements plus corrections give a logical CNOT, using only nearest-neighbour gates.

Research level

The Eastin–Knill theorem says no code can have a universal set of transversal gates, which is why magic states are needed. Lattice surgery costs about d rounds per operation; "A Game of Surface Codes" lays out full architectures around it. Atom-array machines can move whole blocks and do transversal CNOTs in parallel, which can make error correction faster per logical operation. Google teleported logical states between colour codes with lattice surgery (fidelity 86.5% to 90.7%, 2024).

Intuition

Tile a triangle with hexagons coloured red, green and blue so no two neighbours share a colour. Put qubits on the corners. Each tile checks its corners. The extra symmetry gives easier logic.

Core idea

Every tile carries both an X-type and a Z-type stabilizer on the same qubits. The smallest colour code is the 7-qubit Steane code [[7,1,3]]. Colour codes allow every single-qubit Clifford gate (H, S) to be done transversally, and a distance-d code needs fewer data qubits than a surface code of the same distance.

Research level

The price: weight-6 checks and a lower threshold, and each error lights up three checks, so simple matching decoders do not work directly. In 2024 Google ran distance-3 and distance-5 colour codes on Willow: errors dropped by Λ = 1.56 from d=3 to d=5, transversal Clifford gates added only 0.27% error, and injected magic states reached over 99% fidelity with post-selection. Colour codes also underpin magic-state cultivation.

Intuition

The surface code only lets neighbours talk, so each logical qubit needs a big patch of its own. If some checks may reach across the chip, many logical qubits can share one block and protect each other.

Core idea

A quantum low-density parity-check (qLDPC) code keeps each check small (it touches only a few qubits) and each qubit in only a few checks, but allows long-range connections. This lets the rate k/n stay high as codes grow. Compare 12 logical qubits at distance 12:

Compare the qubits needed to store many logical qubits:

Research level

IBM's bivariate bicycle codes have a circuit-level threshold of about 0.8% (similar to the surface code) and need degree-6 connectivity, including long-range couplers. In 2025 the Kunlun superconducting chip ran a distance-4 bivariate bicycle code (4 logical qubits in 18 data qubits) using long-range couplers. A 2026 trapped-ion preprint ran nine codes on one device, including a qLDPC code with 4 logical qubits in 18 physical qubits, with some instances reaching lifetimes comparable to the physical qubits (breakeven). These are small memory demonstrations with a few logical qubits, not yet large-scale fault-tolerant computation. Open problems: fast decoders (BP-OSD is slow), logical gates inside qLDPC blocks, and building long-range links at scale.

Intuition

The fridge produces a new page of syndrome data every microsecond. A classical computer must read each page and decide what went wrong before the next pages pile up. If it falls behind, the machine stalls.

Core idea

A decoder turns syndromes into a correction. You do not have to physically fix errors right away; you can track them in software (the "Pauli frame"). But before any non-Clifford gate, the decoder must have caught up, because the next step depends on its answer. If decoding is slower than syndrome production, the backlog grows exponentially. Try it: each bar is the real time needed to reach the next T gate (outline = no backlog).

Research level

Main decoders: minimum-weight perfect matching (PyMatching / Sparse Blossom), Union-Find (near-linear time), BP-OSD for qLDPC, and neural networks (AlphaQubit). Google's Willow ran a real-time decoder at distance 5 with an average latency of 63 µs over up to a million cycles of 1.1 µs each. Recent work moves decoders onto FPGAs, ASICs and GPUs, including real-time decoding of IBM's gross code, to bring qLDPC decoding toward the microsecond range. Open problem: decoders that are both fast and nearly optimal at large distance.

Intuition

Instead of wires on a chip, the qubits are single atoms held by focused laser beams. The beams can pick atoms up and move them, so any qubit can be brought next to any other.

Core idea

Atoms are trapped in optical tweezers in a vacuum. Two-qubit gates use Rydberg states: lasers puff an atom's outer electron into a huge orbit so nearby atoms strongly affect each other. Moving whole rows of atoms in parallel allows transversal gates between code blocks. Lost atoms can be detected and treated as known "erasures", which are easier to correct than unknown errors.

Research level

Bluvstein et al. (2024) ran up to 48 logical qubits on 280 atoms with transversal gates. In 2025 the Harvard/MIT/QuEra team demonstrated key components of a universal fault-tolerant architecture on arrays of up to 448 atoms (for example 2.14× below-threshold surface-code performance over four rounds, transversal gates and lattice surgery), and separately logical magic-state distillation. These are small-scale building-block demonstrations, not yet a full scalable fault-tolerant computer. Weak points: slow cycles (around milliseconds, versus about a microsecond on superconducting chips), atom loss and reloading, and readout speed. Google announced a neutral-atom effort in March 2026, alongside its superconducting work.

Intuition

A microwave box (a resonator) can hold many photons, so it has many levels instead of two. Spread one qubit cleverly across those levels and some errors can be corrected inside a single piece of hardware.

Core idea

Cat qubits store 0 and 1 as two opposite "clouds" of light (coherent states |α⟩ and |−α⟩). Bit flips need the cloud to jump across, which gets exponentially rarer as the clouds get bigger; only phase flips remain, and a simple repetition code can handle those. Slide the photon number to see the trade-off.

GKP qubits store the qubit in a grid pattern of the oscillator, so small shifts in position or momentum can be detected and undone.

Research level

Yale (2023) ran real-time GKP error correction that beat the best physical qubit in the same system (beyond breakeven). AWS's Ocelot chip (2025) combined cat qubits with a repetition code, showing that logical errors can be reduced with far fewer components. Alice & Bob is building cat-qubit machines. Open problems: noise-bias-preserving gates and scaling the control hardware.

Intuition

Any state of two parts can be written as a short list of paired terms, "this on the left goes with that on the right". If the list has one term, the parts are independent. If it has more, they are entangled.

Core idea

Schmidt decomposition: any pure state of A and B can be written |ψ⟩ = Σ λ_i |a_i⟩|b_i⟩ with λ_i ≥ 0 and Σ λ_i² = 1. The number of terms is the Schmidt rank. Both halves have reduced states with the same eigenvalues λ_i². The entanglement entropy is −Σ λ_i² log₂ λ_i².

Purification: every mixed state ρ of A can be viewed as half of a pure state of A plus an extra system R: |ψ⟩ = Σ √p_i |i⟩_A|i⟩_R. Mixedness is "missing information about a partner".

Research level

The Schmidt coefficients are the singular values of the 2-part amplitude matrix, which is the basis of tensor-network simulation (matrix product states): low entanglement means small matrices and cheap classical simulation. All purifications of ρ are related by a unitary on R (Uhlmann's theorem), which underlies fidelity, channel theory (Stinespring) and the no-cloning and no-hiding results.

Intuition

A normal measurement on a qubit has two answers. A cleverer measurement can have three answers, for example "it was A", "it was B", or "I'm not sure", and never be wrong.

Core idea

A POVM is a set of positive operators E_i that add up to the identity. The chance of outcome i is ⟨ψ|E_i|ψ⟩. Unlike projective measurements, the E_i need not be orthogonal, and there can be more outcomes than dimensions. Example: |0⟩ and |+⟩ overlap, so no measurement can always tell them apart. The best error-free measurement gives a definite answer only 1 − 1/√2 ≈ 29% of the time.

Research level

Naimark's theorem: every POVM is a normal projective measurement on the system plus an extra qubit. POVMs give optimal state discrimination (Helstrom bound for minimum-error), unambiguous discrimination (Ivanovic–Dieks–Peres bound), informationally complete tomography (SIC-POVMs), and security proofs in quantum cryptography.

Intuition

Molecules are quantum objects, so a quantum computer is a natural tool for them. Many scheduling and network problems can be written as "split the dots into two groups so as many lines as possible cross".

Core idea

Chemistry (VQE): map electrons in orbitals to qubits, start from the simple Hartree–Fock guess, add a tunable circuit such as UCCSD, measure the energy, and let a classical optimiser lower it. H₂ in the smallest basis needs 4 qubits (2 after symmetries). The target is "chemical accuracy", 1.6 millihartree (about 1 kcal/mol).

MaxCut (QAOA): try it yourself. Click nodes to move them between side A and side B and count the cut edges. QAOA builds a circuit that makes good cuts more likely to be measured.

Research level

Small molecules are easy for classical methods, so they are tests, not wins. Useful targets, such as the FeMoco enzyme cluster or battery materials, are estimated to need about a thousand or more logical qubits running quantum phase estimation. For MaxCut, the classical Goemans–Williamson algorithm guarantees 87.8% of the best cut, and no quantum advantage for QAOA has been proven. Variational methods also face barren plateaus and noisy measurements. Bottom line: chemistry and optimisation are active research directions; so far there is no proven quantum advantage over the best classical methods on useful, real-world workloads.

Intuition

You do not send a circuit straight to the chip. You send a small job to a service that compiles it for that exact chip, runs it many times, cleans up noise and sends back numbers.

Core idea

IBM's pattern has four steps: map the problem to circuits, optimise them for the target hardware (transpile into "ISA" circuits that use only native gates and real qubit links), execute with a primitive, and post-process. There are two primitives: the Sampler returns measured bitstrings; the Estimator returns expectation values such as energies, with built-in error mitigation. Jobs can run alone, in a batch, or in a session for fast iterative loops.

pip install qiskit qiskit-ibm-runtime

from qiskit import QuantumCircuit
from qiskit.transpiler.preset_passmanagers import generate_preset_pass_manager
from qiskit_ibm_runtime import QiskitRuntimeService, SamplerV2 as Sampler

service = QiskitRuntimeService()          # uses your saved IBM Quantum account
backend = service.least_busy(operational=True, simulator=False)

qc = QuantumCircuit(2); qc.h(0); qc.cx(0, 1); qc.measure_all()
isa = generate_preset_pass_manager(optimization_level=1, backend=backend).run(qc)

job = Sampler(mode=backend).run([isa], shots=1000)
print(job.result()[0].data.meas.get_counts())

Research level

The Estimator's resilience levels switch on mitigation such as readout correction (TREX), zero-noise extrapolation and probabilistic error amplification. These give unbiased answers only on average and cost many more shots, roughly growing exponentially with circuit size and noise. Mitigation was central to IBM's 2023 "utility" experiment on 127 qubits, which several classical methods soon reproduced, a good example of how claims are tested.

Intuition

A big future quantum computer could break today's internet locks (RSA and elliptic curves) with Shor's algorithm. The fix is new locks, still run on normal computers, based on maths problems that quantum computers are not known to solve.

Core idea

In August 2024 NIST published three standards:

Not the same as QKD. Post-quantum (quantum-safe) cryptography is ordinary software on ordinary computers, using maths problems believed hard even for quantum computers. Quantum key distribution (QKD) is different: it sends single photons over special fibre or free-space links and gets its security from physics, but needs dedicated hardware, only shares keys (it still needs authentication), and has distance limits. Agencies such as NIST and the NSA currently recommend PQC as the main migration path.

FN-DSA (from Falcon) is planned as a further standard, FIPS 206, and HQC was chosen in March 2025 as a backup key-exchange method based on codes.

Research level

The risk is "harvest now, decrypt later": data recorded today could be decrypted when a large machine exists, so migration starts now. NIST's transition plan (IR 8547, draft) aims to deprecate RSA and elliptic-curve methods around 2030 and disallow them by 2035. Symmetric ciphers are only weakened by Grover's algorithm (a square-root speed-up), so AES-256 stays safe. Resource estimates for breaking RSA-2048 keep falling, from about 20 million noisy qubits (2019) to under a million (2025), and lower with qLDPC codes.

Intuition

Alice wants to give Bob a qubit, but she cannot look at it (that would destroy it) and cannot copy it (no-cloning). If they share a Bell pair beforehand, she can do a joint measurement, phone him two ordinary bits, and he can rebuild the exact state on his side.

Core idea

Pick any state with the sliders, then step through the circuit. Watch the table of 8 amplitudes. After Alice's measurement, Bob's qubit is |ψ⟩ with an unknown X and/or Z "twist". The two bits tell him which twist to undo. Fidelity jumps to 100% only after his correction.

Before Bob gets the bits, his qubit on its own is completely random (its average state is I/2), whatever |ψ⟩ was. That is why teleportation cannot send information faster than light.

Research level

Cost: 1 Bell pair + 2 classical bits per qubit; the reverse trade is superdense coding. Gate teleportation uses the same trick to apply hard gates by consuming prepared resource states; this is exactly how magic states give T gates (card 4), and it is a model for measurement-based quantum computing. Experiments: first in photons (Bouwmeester et al., 1997), ions (2004), across 143 km of free space (2012), ground-to-satellite over 1,400 km (Micius, 2017), and deterministic teleportation between superconducting qubits on a chip (Steffen et al., 2013). In fault-tolerant machines, lattice surgery teleports logical qubits between code patches (card 5).

Intuition

News stories mix up very different kinds of "qubit". A "1,000-qubit" chip and a "1,000-qubit" plan for a logical machine are not the same thing at all.

Core idea

Theory proposal vs experiment. A proposal is a design on paper or in simulation. An experiment (preprint) has data but has not yet passed peer review. A peer-reviewed experiment has been checked by independent referees, which is better but still not proof: results can later be questioned or even retracted. A claim is strongest when other groups reproduce it.

Research level

When comparing machines, ask: how many qubits, of which kind; what are the two-qubit gate and readout error rates; how are qubits connected; how long can a circuit run; and, for logical qubits, what is the logical error per cycle and does it improve with code distance? Single headline numbers such as qubit count hide most of this.

Intuition

Imagine storing a secret by splitting it in two and hiding the halves at opposite ends of a wire. Noise that pokes one end learns nothing and changes nothing. Only an operation involving both ends can read or change it. That is the dream of a topological qubit.

Core idea

In certain superconducting nanowires (for example indium arsenide coated with aluminium, in a magnetic field), theory predicts Majorana zero modes (MZMs) at the two ends. A pair of MZMs shares one fermion "parity" (even or odd), which can store a qubit non-locally. Moving these modes around each other, called braiding, would apply gates that depend only on the braid's shape, not on small errors in how it was done. Braiding MZMs gives only Clifford gates, so a non-Clifford gate (e.g. via magic states) would still be needed.

Research level

Ideas: Kitaev's anyon model (1997) and Kitaev's Majorana wire (2000). Status is early-stage and contested. A 2018 Nature paper reporting "quantized Majorana conductance" was retracted in 2021. Microsoft reported devices passing its "topological gap protocol" (Phys. Rev. B, 2023) and, in Nature in February 2025, a single-shot interferometric parity measurement in InAs–Al devices, described in the paper as compatible with future tests of fusion rules. Nature's published peer-review material notes that the paper does not by itself establish the presence of Majorana zero modes. The same month Microsoft announced its "Majorana 1" chip; independent physicists have questioned whether its data show topological behaviour. No topological qubit with demonstrated braiding or protected logic has yet been independently confirmed.