Bloch Sphere
single qubitSingle-qubit state space
Any pure single-qubit state can be written as |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ} sin(θ/2)|1⟩, mapping to a point on the unit sphere (Bloch sphere) with polar angle θ ∈ [0,π] and azimuthal angle φ ∈ [0,2π]. The Bloch vector 𝐧 = (sin θ cos φ, sin θ sin φ, cos θ) represents the expectation values ⟨σx⟩, ⟨σy⟩, ⟨σz⟩.
Special states
|0⟩ (north pole): θ=0, Bloch vector = (0,0,+1)
|1⟩ (south pole): θ=π, Bloch vector = (0,0,−1)
|+⟩ = (|0⟩+|1⟩)/√2: θ=π/2, φ=0 → (+x equator)
|+i⟩ = (|0⟩+i|1⟩)/√2: θ=π/2, φ=π/2 → (+y equator)
Mixed state: ρ = I/2 → Bloch vector = origin (not on sphere)
Quantum Gates
unitary operationsUnitary transformations on the Bloch sphere
Single-qubit gates are 2×2 unitary matrices acting on |ψ⟩. Geometrically they are rotations of the Bloch sphere. Pauli gates rotate by π around x, y, or z axes. The Hadamard H swaps the |0⟩/|1⟩ basis with |±⟩. Phase gates (S, T) rotate around z.
Superposition & Interference
Mach-ZehnderMach-Zehnder Interferometer
A single photon enters a 50/50 beamsplitter (H gate), acquires a phase φ on one arm, then hits a second beamsplitter. The output probabilities are:
Output Probabilities
Measurement & Born Rule
projection, statisticsBorn rule
Measuring |ψ⟩ = cos(θ/2)|0⟩ + e^{iφ}sin(θ/2)|1⟩ in the computational basis yields |0⟩ with probability P₀ = cos²(θ/2) and |1⟩ with probability P₁ = sin²(θ/2), regardless of φ. The post-measurement state collapses to the measured outcome.
Measurement histogram
Entanglement & Bell States
non-separabilityBell states — maximally entangled two-qubit states
A state is entangled if it cannot be written as |ψ_A⟩⊗|ψ_B⟩. The four Bell states form an orthonormal basis for the two-qubit Hilbert space and are maximally entangled: measuring one qubit instantly determines the other's outcome, regardless of separation — "spooky action at a distance" (Einstein, 1935).
|Φ⁺⟩
(|00⟩ + |11⟩)/√2
|Φ⁻⟩
(|00⟩ − |11⟩)/√2
|Ψ⁺⟩
(|01⟩ + |10⟩)/√2
|Ψ⁻⟩
(|01⟩ − |10⟩)/√2
|00⟩ (separable)
|0⟩⊗|0⟩ — not entangled
|++⟩ (separable)
(|0⟩+|1⟩)/√2 ⊗ same
Rydberg Atoms & Blockade
neutral-atom QCRydberg blockade mechanism
Exciting two nearby atoms to Rydberg states shifts one atom's resonance by V(R) = C₆/R⁶ due to van der Waals interactions. When V(R) ≫ ℏΩ_R, the double excitation |rr⟩ is off-resonant — this is the Rydberg blockade. Within the blockade radius R_b, only one atom can be excited at a time, enabling two-qubit entangling gates.
V(R) = C₆/R⁶ interaction potential
Two-Qubit Gates
entangling operationsEntangling gates
Two-qubit gates act on the 4D Hilbert space spanned by |00⟩, |01⟩, |10⟩, |11⟩. CNOT, CZ, and iSWAP can generate maximal entanglement from product states — they are universal when combined with single-qubit gates.
Rabi Oscillations
coherent driveDriven two-level system
A resonant laser drive with Rabi frequency Ω causes coherent oscillations between |0⟩ and |1⟩. Off-resonance (detuning Δ), the generalised Rabi frequency is Ω' = √(Ω²+Δ²) and the oscillation amplitude is reduced to (Ω/Ω')².
$\pi$-pulse time: $t_\pi = \pi/\Omega \quad [|0\rangle \rightarrow |1\rangle$ exactly$]$
P₁(t) — population in |1⟩
Hyperfine Qubits & Atomic Clock States
AMO qubit encodingWhy atoms are excellent qubits
Neutral alkali atoms encode qubits in their hyperfine ground states — two long-lived magnetic sub-levels of the electronic ground state split by the interaction between nuclear spin I and electron spin J. The energy splitting is ΔE = Ahf · I·J, where Ahf is the magnetic dipole hyperfine constant. Clock states (|F, mF = 0⟩) are first-order insensitive to magnetic field fluctuations, enabling T₂ > 10 seconds — orders of magnitude longer than superconducting qubits (~100 μs).
Splitting: $\Delta E/h = 9{,}192{,}631{,}770$ Hz (defines the SI second!)
2nd-order Zeeman shift (clock states): $$\Delta\nu(B) \approx +427\text{ Hz/G}^2 \text{ (Cs)}, \quad +575\text{ Hz/G}^2 \text{ (Rb-87)}$$ Single-qubit gate: resonant MW or two-photon Raman pulse; $t_\pi \sim 5\,\mu\text{s}$ (Raman, $\Omega/2\pi \sim 100$ kHz)
Qubit encoding strategies
Clock qubit (mF=0 → mF'=0): best coherence,
first-order B-insensitive. Used in most neutral-atom QC experiments.
Stretched-state qubit (mF=F → mF'=F'): strongest
coupling to MW field, but first-order B-sensitive — requires ≲1 mG stability.
Optical qubit (ground ↔ metastable excited state, e.g. Sr, Yb):
very fast gates via optical fields; coherence limited by excited-state lifetime and
photon scattering.
For Rydberg gate operation, either qubit state is coupled to a Rydberg level |r⟩
via a two-photon (IR + UV) or single-photon (UV ~318 nm for Cs) transition.
The blockade between two atoms in |rr⟩ enforces a conditional phase on |11⟩.
¹³³Cs hyperfine ground structure (schematic)
| Atom | Hyperfine splitting | Clock states | T₂ (best) | Gate |
|---|---|---|---|---|
| ⁸⁷Rb | 6.835 GHz | |1,0⟩ → |2,0⟩ | ~1 min | Rydberg CZ |
| ¹³³Cs | 9.193 GHz (SI clock!) | |3,0⟩ → |4,0⟩ | ~10 s | Rydberg CZ |
| ⁶Li | 228.2 MHz | |1/2,+1/2⟩ → |1/2,−1/2⟩ | ~100 ms | Rydberg / tweezer |
| ¹⁷¹Yb | 12.643 GHz | |F=0⟩ → |F=1, m_F=0⟩ | >10 s | Rydberg CZ |
| ⁸⁸Sr (optical) | 429.2 THz (clock line) | ¹S₀ → ³P₀ | >1 min | Rydberg / optical |
Optical Pumping & State Preparation
qubit initializationDriving atoms into a single quantum state with light
Optical pumping uses circularly polarized resonant light to redistribute atomic population among magnetic sublevels mF. A σ⁺ beam drives ΔmF = +1 transitions — atoms absorb photons and reach excited states inaccessible from the highest mF level. After repeated absorption-emission cycles, population accumulates in the dark state (the sublevel that cannot absorb) or a cycling (stretch) state. This is how we initialize qubits: prepare all atoms in one specific mF before coherent manipulation.
σ⁺ optical pumping — level diagram (F=1 example)
Population dynamics — scatter photons interactively
Start: equal population in mF = −1, 0, +1. σ⁺ light pumps toward mF = +1.
Scattering rate from sublevel $m_F$ ($\sigma^+$ beam): $$R_m = \frac{\Gamma}{2} \cdot \frac{s \cdot |C_m|^2}{1 + s|C_m|^2 + (2\Delta/\Gamma)^2}$$ where $|C_m|^2$ = squared Clebsch–Gordan coeff. for $|F,m_F\rangle \rightarrow |F',m_F+1\rangle$
Dark state ($\sigma^+$): $m_F = +F$ — no allowed $\sigma^+$ absorption
Pumping time: $\tau_{pump} \approx 2F / (R_{scatter} \cdot \varepsilon_{branch})$
Clock-state preparation (|F, m_F=0⟩)
For qubit encoding in clock states (mF=0), a different strategy is used: pump first to a high-field-seeking state with σ⁺, then apply a microwave or RF π-pulse to transfer to |F=2, mF=0⟩. Alternatively, use a combination of σ⁺ + π light to create a dark state at mF=0. In practice, Rb-87 clock-state qubits are initialized with >99% fidelity in ~100 μs.
EIT & dark states in Λ systems
Coherent population trapping (CPT) is a related phenomenon: two fields in a Λ configuration drive an atom into a coherent dark state — a superposition of two ground states that has zero absorption. This underpins EIT (electromagnetically induced transparency) and STIRAP transfer. Unlike optical pumping (which is incoherent), CPT/EIT preserves quantum coherence.
Decoherence: T₁ & T₂
open quantum systemsBloch sphere collapse
Real qubits interact with their environment. T₁ (relaxation time) is how long the excited state population survives — the qubit decays from |1⟩ to |0⟩. T₂ (coherence/dephasing time) is how long superpositions survive. Always T₂ ≤ 2T₁. The off-diagonal density matrix element |ρ₀₁| decays as e^{−t/T₂}.
Population P₁(t) and Coherence |ρ₀₁(t)|
Quantum Algorithms
Grover, Deutsch-Jozsa, QAOADeutsch-Jozsa Algorithm
The first proof-of-principle quantum speedup. Determines whether a function f:{0,1}ⁿ → {0,1} is constant (same output always) or balanced (half 0, half 1) in a single oracle query, versus O(2ⁿ⁻¹+1) classically. Uses Hadamard superposition + phase kickback from the oracle, then a final Hadamard layer — all |0⟩ output means constant, any other outcome means balanced.
Grover's Search Algorithm
Finds a marked item in an unsorted database of N entries in O(√N) oracle queries — quadratic speedup over classical O(N). Each Grover iteration applies: (1) the oracle (phase flip on target state), then (2) a diffusion operator (inversion about the mean amplitude). After ≈ (π/4)√N iterations the target state has amplitude ≈ 1. Relevant for neutral-atom QC because each gate cycle is slow — minimising query count is critical.
Optimal iterations: k ≈ (π/4) √NQAOA — Quantum Approximate Optimisation
A near-term variational algorithm (NISQ-friendly) for combinatorial optimisation. Alternates between a cost unitary e−iγH_C encoding the problem and a mixing unitary e−iβH_B, with angles (γ, β) optimised classically. Neutral-atom arrays are a natural hardware match: Rydberg interactions implement the cost Hamiltonian natively, and the array geometry can be programmed to encode graph problems.
Circuit complexity & quantum advantage
Quantum advantage requires circuits deep enough that classical simulation is intractable but shallow enough to survive decoherence. Current neutral-atom processors (Lukin/Greiner/Browaeys groups) achieve ~100-qubit circuits at 99%+ two-qubit gate fidelity. Grover gives proven quadratic speedup; exponential speedups (Shor's factoring) require fault-tolerant hardware — motivating the QEC section below.
Quantum Error Correction
fault toleranceWhy QEC is essential
Physical qubits have error rates ~10⁻³–10⁻² per gate. Running a useful algorithm (e.g., Shor factoring a 2048-bit number) requires ~10⁸ logical operations — impossibly many without error correction. QEC encodes one logical qubit in many physical qubits so that errors can be detected and corrected without ever measuring the logical state directly (which would collapse it). The key insight: measuring error syndromes (parities) reveals which error occurred without revealing the encoded information.
Syndrome operators: $Z_1Z_2 \otimes I$ and $I \otimes Z_2Z_3$
Surface code (distance-$d$): encodes 1 logical qubit in $d^2$ physical qubits. $$p_L \approx A \cdot \left(\frac{p}{p_{th}}\right)^{\lceil(d+1)/2\rceil} \qquad p_{th} \approx 1\%$$ Example: $d=7$, $p=0.1\%$ physical $\Rightarrow p_L \approx (0.001/0.01)^4 \approx 10^{-8}$ per logical cycle
Key QEC concepts
Logical qubit: encoded in d² physical qubits; protected from errors
up to weight ⌊(d−1)/2⌋ — any combination of that many physical errors is correctable.
Syndrome measurement: ancilla qubits measure ZiZj
and XiXj stabilizers — parities only, never the logical state itself.
Threshold theorem: if physical error rate p < pth,
increasing code distance d exponentially suppresses logical errors —
making arbitrarily reliable computation possible from noisy components.
Transversal gates: some logical gates (Pauli, CNOT, Hadamard for certain codes)
can be applied bitwise across physical qubits without spreading errors — critical for fault tolerance.
2024–25 State of the Art
Google Willow (2024): d=5 surface code on 105-qubit superconductor —
first demonstration of exponential error suppression below threshold.
Logical error rate 4.4×10⁻⁵ per round (d=5).
Bluvstein et al. / Harvard (Nature 2024): 48 logical qubits in neutral
atom tweezer array; transversal logical gates demonstrated with fault-tolerant circuits.
Quantinuum H2 (2024): 99.9% two-qubit fidelity in trapped ions;
100× error suppression on encoded logical qubit vs raw physical qubit.
| Code | Physical / logical qubit | Threshold p_th | Status (2025) |
|---|---|---|---|
| 3-qubit bit-flip | 3 | ~50% | Corrects X errors only — not universal |
| Steane [[7,1,3]] | 7 | ~1% | All single-qubit errors; CSS code |
| Surface code d=5 | 25 | ~1% | Google Willow 2024 — below threshold ✓ |
| Surface code d=7 | 49 | ~1% | Harvard 2024 tweezer — 48 logical qubits |
| Surface code d=25 | 625 | ~1% | Target for Shor's algorithm at scale |
| Bivariate bicycle (Google) | ~10–15× | ~0.5% | More compact; demonstrated 2024 |
Analog Quantum Simulation
Hubbard model, spin models, arraysDigital vs. Analog quantum computation
Digital QC decomposes any unitary into discrete gates — fully programmable but requires error correction for deep circuits. Analog simulation engineers a physical Hamiltonian H_sim that mimics a target model H_target, then lets the system evolve. No gate decomposition needed — the physics is the algorithm. AMO platforms excel here because atom–atom interactions are highly controllable and the Hilbert space scales exponentially with atom number, making classical simulation intractable beyond ~50–60 qubits.
Bose-Hubbard & Fermi-Hubbard models
Cold atoms in optical lattices realise the Hubbard Hamiltonian:
H = −t Σ(aᵢ†aⱼ + h.c.) + (U/2) Σ nᵢ(nᵢ−1) − μ Σ nᵢwhere t is the tunnel coupling (laser intensity), U is the on-site interaction (Feshbach resonance tunable), and μ is chemical potential. At U/t ≫ 1: Mott insulator. At U/t ≪ 1: superfluid. The Fermi-Hubbard model (fermionic atoms) is believed to describe high-T_c superconductivity — directly accessible in the lab.
Rydberg spin models & quantum magnetism
Rydberg tweezer arrays naturally implement Ising and XY spin Hamiltonians:
H = Σᵢ (Ω/2)σˣᵢ − Δ nᵢ + Σᵢ<ⱼ V(rᵢⱼ) nᵢ nⱼV(r) = C₆/r⁶ for van der Waals, C₃/r³ for dipolar (dressed states). By choosing array geometry, any graph's adjacency can be encoded. Experiments have demonstrated Z₂ symmetry breaking, topological spin liquids (Semeghini et al. 2021), and frustrated magnetism in programmable arrays.