Everything About Qubits

The fundamental unit of quantum information. From Dirac notation to physical implementations, measurement theory, and decoherence.

What Is a Qubit?

A qubit (quantum bit) is the fundamental unit of quantum information. While a classical bit can be either 0 or 1, a qubit can exist in a superposition of both states simultaneously. Mathematically, a qubit state is a unit vector in a two-dimensional complex Hilbert space.

General qubit state

|ψ⟩ = α|0⟩ + β|1⟩ where α, β ∈ ℂ and |α|² + |β|² = 1

The complex numbers α and β are called probability amplitudes. When you measure a qubit, the probability of getting outcome 0 is |α|² and the probability of getting outcome 1 is |β|². The constraint |α|² + |β|² = 1 is called the normalization condition and ensures the probabilities sum to 1.

Key differences from classical bits

Superposition: A qubit can be in a linear combination of |0⟩ and |1⟩. A classical bit is always exactly one or the other.
Interference: Amplitudes are complex numbers that can add constructively or destructively, enabling quantum algorithms.
Entanglement: Two or more qubits can share correlations with no classical analog. Measuring one instantly constrains the other, regardless of distance.
No-cloning theorem: An unknown quantum state cannot be perfectly copied. This is fundamental to quantum cryptography (BB84).
Measurement collapse: Observation irreversibly destroys the superposition, yielding a definite classical outcome.

A qubit is not "both 0 and 1 at the same time" in the sense of storing two values. It holds a single quantum state that, upon measurement, probabilistically yields one outcome. The power of quantum computing lies in manipulating amplitudes before measurement to make the desired answer overwhelmingly likely.

Dirac (Bra-Ket) Notation

Quantum mechanics uses Dirac notation, invented by Paul Dirac in 1939. It provides a clean, compact way to describe quantum states and operations.

Kets (column vectors)

A ket $|ψ⟩$ represents a quantum state as a column vector in Hilbert space.

|0⟩ = [1, 0]ᵀ (computational basis state zero) |1⟩ = [0, 1]ᵀ (computational basis state one) |ψ⟩ = α|0⟩ + β|1⟩ = [α, β]ᵀ

Bras (row vectors)

A bra $⟨ψ|$ is the conjugate transpose (adjoint) of the corresponding ket.

⟨ψ| = (|ψ⟩)† = [α*, β*]

Inner product (bra-ket)

The inner product of two states gives a complex number representing their overlap.

⟨φ|ψ⟩ = ∑ᵢ φᵢ* ψᵢ ⟨0|0⟩ = 1 ⟨1|1⟩ = 1 ⟨0|1⟩ = 0 ⟨1|0⟩ = 0

Orthonormal basis states have inner product 0 with each other and 1 with themselves.

Outer product (ket-bra)

The outer product produces a matrix (an operator).

|0⟩⟨0| = [[1,0],[0,0]] (projector onto |0⟩) |1⟩⟨1| = [[0,0],[0,1]] (projector onto |1⟩) |0⟩⟨0| + |1⟩⟨1| = I (completeness relation)

Tensor product

Multi-qubit states are written using the tensor product $\otimes$ .

|0⟩ ⊗ |1⟩ = |01⟩ = [0, 1, 0, 0]ᵀ |ψ⟩ ⊗ |φ⟩ is often written |ψφ⟩ or |ψ,φ⟩

The Bloch Sphere

Any single-qubit pure state can be visualized as a point on the Bloch sphere, a unit sphere in 3D space. This is possible because the global phase of a quantum state is unobservable, leaving two real parameters (θ and φ) to specify a state on the sphere.

Parametrization

|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ) sin(θ/2)|1⟩ θ ∈ [0, π] polar angle (latitude from north pole) φ ∈ [0, 2π) azimuthal angle (longitude)

|0⟩

|1⟩

|+⟩

|-⟩

|ψ⟩

Key points on the Bloch sphere

State	Vector	θ	φ	Position
\|0⟩	[1, 0]ᵀ	0	-	North pole (+Z)
\|1⟩	[0, 1]ᵀ	π	-	South pole (-Z)
\|+⟩	[1, 1]ᵀ/√2	π/2	0	+X axis
\|-⟩	[1, -1]ᵀ/√2	π/2	π	-X axis
\|i⟩	[1, i]ᵀ/√2	π/2	π/2	+Y axis
\|-i⟩	[1, -i]ᵀ/√2	π/2	3π/2	-Y axis

Bloch sphere and quantum gates

Every single-qubit gate corresponds to a rotation of the Bloch sphere. The Pauli X gate is a 180-degree rotation about the X axis. The Hadamard gate is a 180-degree rotation about the axis halfway between X and Z. The S gate is a 90-degree rotation about Z. The T gate is a 45-degree rotation about Z.

Mixed states (due to decoherence or entanglement with other systems) map to points inside the Bloch sphere. The maximally mixed state (I/2) sits at the center. Pure states always lie on the surface.

Single-Qubit States

The six cardinal states of the Bloch sphere form three mutually unbiased bases (MUBs). These are the most commonly used single-qubit states in quantum computing.

Computational basis (Z basis)

|0⟩ = [1, 0]ᵀ Eigenstate of Z with eigenvalue +1 |1⟩ = [0, 1]ᵀ Eigenstate of Z with eigenvalue -1

The computational basis is the standard measurement basis. All quantum computation ultimately reduces to preparing, manipulating, and measuring in this basis.

Hadamard basis (X basis)

|+⟩ = (|0⟩ + |1⟩) / √2 = [1/√2, 1/√2]ᵀ Eigenstate of X, eigenvalue +1 |-⟩ = (|0⟩ - |1⟩) / √2 = [1/√2, -1/√2]ᵀ Eigenstate of X, eigenvalue -1

Created from |0⟩ and |1⟩ by applying the Hadamard gate. The |+⟩ state is equal superposition with equal phases; |-⟩ has a relative phase flip.

Circular basis (Y basis)

|i⟩ = (|0⟩ + i|1⟩) / √2 = [1/√2, i/√2]ᵀ Eigenstate of Y, eigenvalue +1 |-i⟩ = (|0⟩ - i|1⟩) / √2 = [1/√2, -i/√2]ᵀ Eigenstate of Y, eigenvalue -1

These states have a relative phase of ±i between the |0⟩ and |1⟩ components. They correspond to right and left circular polarization in photonic qubits.

General pure state

|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ) sin(θ/2)|1⟩ Probability of measuring 0: cos²(θ/2) Probability of measuring 1: sin²(θ/2) Relative phase between components: φ

The global phase factor e^(iγ) multiplying the entire state is physically unobservable. Only the relative phase between |0⟩ and |1⟩ matters. This is why we can always choose α to be real and non-negative.

Multi-Qubit Systems

The state space of N qubits is the tensor product of individual qubit spaces, giving a 2ⁿ-dimensional complex Hilbert space. This exponential growth is the origin of quantum computing's potential power.

Tensor products

Two qubits: |ψ⟩ ⊗ |φ⟩ lives in ℂ² ⊗ ℂ² = ℂ⁴ Three qubits: ℂ² ⊗ ℂ² ⊗ ℂ² = ℂ⁸ |00⟩ = [1,0,0,0]ᵀ |01⟩ = [0,1,0,0]ᵀ |10⟩ = [0,0,1,0]ᵀ |11⟩ = [0,0,0,1]ᵀ

Product states vs entangled states

A two-qubit state $|ψ⟩$ is a product state if it can be written as $|a⟩ \otimes |b⟩$ for some single-qubit states. Otherwise, it is entangled.

Product state: |+0⟩ = (|00⟩ + |10⟩) / √2 ✓ separable Entangled: |Φ⁺⟩ = (|00⟩ + |11⟩) / √2 ✗ cannot factor

To check if a two-qubit state is entangled, compute the reduced density matrix by tracing out one qubit. If the result is a mixed state (not rank 1), the state is entangled. Equivalently, compute the Schmidt decomposition: the state is entangled if and only if it has more than one non-zero Schmidt coefficient.

Counting parameters

Qubits	Dimension	Real parameters	Classical bits to describe
1	2	2	2
2	4	6	4
3	8	14	8
10	1,024	2,046	10
50	~10¹⁵	~2 × 10¹⁵	50
300	~10⁹⁰	~2 × 10⁹⁰	300

At 300 qubits, the number of complex amplitudes exceeds the number of particles in the observable universe (~10⁸⁰). This is why simulating quantum systems on classical computers is fundamentally intractable.

Bell States

The four Bell states (or EPR pairs) are maximally entangled two-qubit states. They form an orthonormal basis for the two-qubit Hilbert space and are the fundamental resource for quantum teleportation, superdense coding, and entanglement-based cryptography (E91).

|Φ⁺⟩ = (|00⟩ + |11⟩) / √2 "Bell state" or "EPR pair" |Φ⁻⟩ = (|00⟩ - |11⟩) / √2 |Ψ⁺⟩ = (|01⟩ + |10⟩) / √2 |Ψ⁻⟩ = (|01⟩ - |10⟩) / √2 "singlet state"

Properties

Maximal entanglement: Each qubit individually is in the maximally mixed state I/2. All information is in the correlations.
Perfect correlations: In |Φ⁺⟩, measuring both qubits in the Z basis always gives the same result (both 0 or both 1). In |Ψ⁻⟩, they always differ.
Basis independence: The correlations hold in any measurement basis, not just Z. This is what Bell's theorem proves cannot be explained classically.
LOCC indistinguishable: The four Bell states cannot be perfectly distinguished using only local operations and classical communication on one copy.

Creating Bell states

Circuit: |0⟩|0⟩ → H⊗I → (|0⟩+|1⟩)|0⟩/√2 → CNOT → (|00⟩+|11⟩)/√2 = |Φ⁺⟩ All four: apply {I, X, Z, XZ} to the first qubit of |Φ⁺⟩

Applications

Quantum teleportation — transfer an unknown qubit state using one Bell pair and 2 classical bits
Superdense coding — send 2 classical bits using one qubit (with pre-shared entanglement)
Quantum key distribution (E91) — Bell inequality violations certify secure key generation
Entanglement swapping — entangle particles that have never interacted, via Bell measurement on intermediaries

GHZ and W States

For three or more qubits, there are inequivalent classes of entanglement. The two most important multi-qubit entangled states are the GHZ state (Greenberger-Horne-Zeilinger) and the W state.

GHZ state

|GHZ⟩ = (|000⟩ + |111⟩) / √2 (3-qubit) |GHZ_n⟩ = (|0⟩⊗ⁿ + |1⟩⊗ⁿ) / √2 (n-qubit generalization)

The GHZ state is a "cat state" (Schrodinger's cat): a superposition of all-zero and all-one. It is maximally entangled but fragile: tracing out any single qubit completely destroys the entanglement, leaving the remaining qubits in a separable mixed state.

W state

|W⟩ = (|001⟩ + |010⟩ + |100⟩) / √3 (3-qubit)

The W state distributes a single excitation symmetrically among all qubits. Unlike GHZ, tracing out one qubit leaves the remaining two still entangled. W states are more robust to particle loss but less useful for tasks requiring maximal multi-party correlations.

GHZ vs W: entanglement classes

Property	GHZ	W
Lose one qubit	All entanglement lost	Remaining qubits still entangled
LOCC conversion	Cannot convert to W	Cannot convert to GHZ
Bell inequality violation	Maximal (Mermin inequality)	Weaker violation
Quantum secret sharing	Ideal	Not directly useful
Robustness	Fragile	Robust to qubit loss

Physical Implementations

A qubit needs a controllable two-level quantum system. Several physical platforms compete, each with different strengths. No single technology has emerged as the definitive winner.

Superconducting qubits

Used by IBM, Google, Rigetti, and others. A superconducting circuit containing a Josephson junction creates an anharmonic oscillator whose lowest two energy levels serve as |0⟩ and |1⟩.

Transmon qubit: The dominant design. A Cooper-pair box with a large shunting capacitor that reduces charge noise sensitivity. Introduced by Koch et al. (2007).
Gate time: ~20-50 ns for single-qubit, ~100-500 ns for two-qubit (CZ/CNOT)
T1: 100-300 μs (energy relaxation), T2: 50-200 μs (dephasing)
Operating temperature: ~15 mK (millikelvin), requires dilution refrigerator
Connectivity: Fixed coupling topology (nearest-neighbor on grid/heavy-hex). SWAP operations needed for non-adjacent qubits.
Milestones: Google Sycamore (72 qubits, 2019 supremacy), IBM Eagle (127q, 2021), IBM Condor (1121q, 2023), Google Willow (105q, 2024 below-threshold QEC)

Trapped ions

Used by IonQ, Quantinuum (Honeywell), and academic labs. Individual ions (typically Yb¹⁷¹ or Ba¹³⁷) are held in electromagnetic traps and addressed with laser beams.

Qubit encoding: Two hyperfine ground states or an optical transition serve as |0⟩ and |1⟩.
Gate time: ~1-10 μs for single-qubit, ~100-600 μs for two-qubit (Molmer-Sorensen gate)
T1/T2: Seconds to minutes (orders of magnitude longer than superconducting)
Gate fidelity: 99.9%+ single-qubit, 99.5-99.8% two-qubit (highest in any platform)
Connectivity: All-to-all (any pair of ions can interact directly). Huge advantage for algorithm mapping.
Milestones: IonQ Aria (25 algorithmic qubits, 2022), Quantinuum H2 (56 qubits, highest quantum volume 2¹⁶ = 65,536, 2024)

Photonic qubits

Used by Xanadu, PsiQuantum, and academic groups. Information is encoded in properties of single photons: polarization, path, time-bin, or photon number.

Qubit encoding: Horizontal/vertical polarization, dual-rail path encoding, or squeezed states (continuous variable).
Advantages: Room-temperature operation, natural fiber-optic networking, inherent speed-of-light transmission.
Challenges: Photon loss, non-deterministic gates (two-photon interactions are weak), requires feed-forward or cluster states.
Xanadu Borealis: Gaussian boson sampling with 216 squeezed modes (2022). Demonstrated quantum advantage for a specific sampling task.
PsiQuantum: Building a million-qubit photonic chip using silicon photonics manufacturing (GlobalFoundries fab).

Neutral atoms

Used by QuEra, Pasqal, and academic labs. Individual neutral atoms (typically Rb or Cs) are trapped in optical tweezers and interact via Rydberg excitation.

Qubit encoding: Two hyperfine ground states of the atom.
Two-qubit gates: Rydberg blockade mechanism provides strong, tunable interactions.
Advantages: Reconfigurable geometry (move atoms with tweezers), scalable to thousands of qubits, long coherence times.
QuEra Aquila: 256-atom analog quantum processor available on AWS Braket.
Milestones: Harvard/MIT demonstrated 48 logical qubits with error correction on a 280-atom system (Nature, 2023).

Topological qubits

Pursued by Microsoft. The idea: encode quantum information in topological properties of exotic quasiparticles (non-Abelian anyons, specifically Majorana zero modes) that are inherently protected from local noise.

Advantage: Hardware-level error protection. Qubits would be far more stable without active error correction.
Status: Microsoft announced the first topological qubit demonstration in February 2025, using a topoconductor device. Still early-stage compared to other platforms.
Challenge: Proving the existence of the required quasiparticles and building a scalable system remain open problems.

Measurement

Quantum measurement is fundamentally different from classical observation. It is probabilistic, irreversible, and changes the state of the system.

Born rule

The Born rule (Max Born, 1926) is the fundamental postulate connecting quantum states to experimental outcomes.

Given |ψ⟩ = α|0⟩ + β|1⟩: P(outcome = 0) = |⟨0|ψ⟩|² = |α|² P(outcome = 1) = |⟨1|ψ⟩|² = |β|²

Projective measurement

A projective (von Neumann) measurement in basis {|m⟩} is described by projection operators P_m = |m⟩⟨m|.

Probability: P(m) = ⟨ψ|P_m|ψ⟩ Post-measurement state: P_m|ψ⟩ / √P(m) Example: measuring |+⟩ = (|0⟩+|1⟩)/√2 in Z basis: P(0) = |⟨0|+⟩|² = |1/√2|² = 1/2 P(1) = |⟨1|+⟩|² = |1/√2|² = 1/2 If outcome 0: state collapses to |0⟩ If outcome 1: state collapses to |1⟩

Measurement in different bases

You can measure in any orthonormal basis by first rotating to that basis, then measuring in the computational basis.

Basis	States	Pre-rotation gate	Observable
Z (computational)	\|0⟩, \|1⟩	None (identity)	Z = [[1,0],[0,-1]]
X (Hadamard)	\|+⟩, \|-⟩	H	X = [[0,1],[1,0]]
Y (circular)	\|i⟩, \|-i⟩	S†H	Y = [[0,-i],[i,0]]

POVM (generalized measurement)

A Positive Operator-Valued Measure generalizes projective measurement. A POVM is a set of positive semidefinite operators {E_m} satisfying ∑E_m = I.

P(m) = ⟨ψ|E_m|ψ⟩ Unlike projective measurement: - POVM elements need not be orthogonal - You can have more outcomes than the dimension of the space - Useful for optimal state discrimination (e.g., distinguishing non-orthogonal states)

Measurement is the only non-unitary operation in quantum computing. All other operations (gates) are unitary and reversible. This asymmetry is fundamental: quantum computation builds up superpositions through unitary gates, then extracts classical information through measurement.

Decoherence

Decoherence is the loss of quantum information due to unwanted interaction with the environment. It is the primary enemy of quantum computing and the reason we need quantum error correction.

T1: energy relaxation time

T1 measures how long a qubit stays in the excited state |1⟩ before decaying to |0⟩. This is analogous to the spontaneous emission lifetime of an atom.

P(|1⟩ at time t) = e^(-t/T1) After time T1: ~37% probability of still being in |1⟩ After 3×T1: ~5% probability

T2: dephasing time

T2 measures how long the relative phase between |0⟩ and |1⟩ remains coherent. Dephasing destroys superpositions without changing populations.

|ψ(t)⟩ = cos(θ/2)|0⟩ + e^(iφ(t)) sin(θ/2)|1⟩ φ(t) fluctuates randomly due to environment noise. T2 ≤ 2×T1 (always; dephasing cannot be slower than relaxation) T2* (Ramsey): includes static noise, often shorter T2 (echo): refocused with spin echo, filters out slow drift

Error rates by hardware platform

Platform	T1	T2	1Q gate fidelity	2Q gate fidelity	Readout fidelity
Superconducting (IBM Eagle)	100-300 μs	50-200 μs	99.9%	99.0-99.5%	98-99%
Superconducting (Google Willow)	~70 μs	~30 μs	99.85%	99.4-99.7%	~99%
Trapped ion (Quantinuum H2)	~10 s	~1 s	99.99%	99.5-99.8%	99.7%
Trapped ion (IonQ Forte)	~10 s	~1 s	99.97%	99.4-99.6%	99.6%
Neutral atom (QuEra)	~1 s	~1 s	99.5%	99.0-99.5%	97-99%
Photonic (Xanadu Borealis)	N/A (no decay)	Limited by loss	99%+	~95% (probabilistic)	~95%

Types of errors

Bit-flip (X error): |0⟩ ↔ |1⟩. Caused by T1 relaxation or transverse noise.
Phase-flip (Z error): |+⟩ ↔ |-⟩. Caused by T2 dephasing.
Depolarizing: Random application of I, X, Y, or Z. Models worst-case noise.
Amplitude damping: Decay from |1⟩ to |0⟩. Physical model of T1 process.
Leakage: Qubit population escapes the computational subspace into higher energy levels. Particularly problematic for transmons.
Crosstalk: Operations on one qubit unintentionally affect neighbors. Major issue at scale.

Coherence budget

The number of gates you can execute before errors accumulate too much is roughly T2 / gate_time. This is the "coherence budget."

Superconducting: T2 ~100 μs, gate ~50 ns → ~2,000 gates Trapped ion: T2 ~1 s, gate ~10 μs → ~100,000 gates But: trapped ion gates are 100-1000x slower, so wall-clock time per circuit is comparable.

Current error rates (~0.1-1% per gate) are too high for algorithms like Shor's that require millions of gates. This is why quantum error correction is essential for fault-tolerant quantum computing. See the QEC page for details.