Quantum Computing Physics

Introduction to Quantum Computing

Quantum computing exploits uniquely quantum mechanical phenomena—superposition (a qubit can be |0⟩ and |1⟩ simultaneously), entanglement (correlations between qubits with no classical analogue), and interference (cancellation of computational paths leading to wrong answers)—to perform computations that would require exponential resources classically for specific problem classes. A classical bit is 0 or 1; a qubit is |ψ⟩ = alpha|0⟩ + beta|1⟩ with |α|²+|β|²=1—a unit vector on the Bloch sphere in two-dimensional complex Hilbert space. N qubits span a 2^N dimensional Hilbert space; a quantum computer operates on all 2^N basis states simultaneously in superposition, providing exponential parallelism that interference then harvests to amplify correct answers.

Key algorithmic advantages: Shor's algorithm (1994) factors N-digit integers in O(N^3) quantum operations versus O(exp(N^1/3)) classically—threatening RSA, ECC, and most public-key cryptography. Grover's algorithm provides quadratic speedup for unstructured database search (O(sqrt(N)) vs O(N)). Quantum simulation (Feynman's original motivation, 1982): simulating N-particle quantum systems requires 2^N classical bits but only N qubits—enabling exact simulation of quantum chemistry, materials, and drug molecules for drug discovery and materials design. The engineering challenge: decoherence (entanglement with environment destroys quantum information), gate errors, and measurement errors must be suppressed below ~10^-3 per gate for quantum error correction codes to provide net benefit—a threshold not achieved at scale as of 2025.

Physical Qubit Implementations

Superconducting Qubits

Superconducting qubits—the dominant platform for current quantum processors (Google, IBM, Intel, IQM, Rigetti)—use Josephson junctions (two superconducting electrodes separated by ~1 nm insulating barrier) to create anharmonic oscillator circuits with non-uniform energy level spacing, isolating the |0⟩-|1⟩ transition for qubit operation. Transmon qubit (Koch et al. 2007): shunted junction with large shunt capacitor reduces charge noise sensitivity 100-fold, making T2 coherence times of 100-500 microseconds achievable. Gate operations use microwave pulses at qubit transition frequency (~5-7 GHz)—single-qubit gates in ~20 ns with >99.9% fidelity; two-qubit (entangling) gates via tunable coupler (~100-400 ns, 99.5% fidelity in best systems). Readout: dispersive readout via microwave resonator coupled to qubit—qubit state frequency-shifts resonator enabling fast (~200 ns) quantum non-demolition measurement. Challenges: thermal photon errors (requires dilution refrigerator at 15 mK, T < 100× warmer than qubit frequency to suppress thermal occupation); frequency collisions in large arrays; and surface residual two-level systems (TLS) in oxide interfaces degrading coherence.

Trapped Ion and Neutral Atom Qubits

Trapped ion quantum computers (IonQ, Quantinuum/Honeywell, Oxford Ionics): individual ions (Yb+, Ba+, Ca+) levitated in radiofrequency Paul traps and laser-cooled to motional ground state; hyperfine or Zeeman electronic states are qubits with T2 > 10 seconds (exceptional coherence from atomic physics isolation). Two-qubit entangling gates via shared phonon bus modes (Mølmer-Sørensen or Cirac-Zoller gate)—all-to-all connectivity possible for small registers. Current record: Quantinuum H2 processor achieved quantum volume 65,536 (2024) and >99.9% two-qubit gate fidelity—best gate quality of any platform. Challenge: gate speed (trapped ion 2q gates ~100-500 microseconds, orders slower than superconducting ~200 ns); scaling to >1000 qubits in single trap requires photonic interconnects between multiple trap modules. Neutral atom arrays (QuEra, Atom Computing, Pasqal): 1000+ atoms in reconfigurable optical tweezer arrays; Rydberg atom interactions (highly excited states with strong dipole interactions ~MHz over ~μm distances) provide fast entangling gates; mid-circuit measurements and atom shuttling enable dynamic quantum circuit architecture—demonstrated 48-qubit logical qubit operation (Harvard QuEra 2023).

Quantum Error Correction

Surface Codes and Logical Qubits

Quantum error correction (QEC) encodes one logical qubit in many physical qubits, spreading quantum information non-locally so physical errors affect only a subset correctable before propagating. Surface code (Fowler et al. 2012): 2D array of physical qubits with nearest-neighbour stabiliser measurements (Z-type and X-type parity checks); threshold error rate ~1% per gate—most achievable threshold in any topological code; logical error rate suppressed exponentially with code distance d (number of physical qubits per side): p_logical ~ (p_physical/p_threshold)^((d+1)/2). Google's below-threshold demonstration (72-qubit Willow processor, Nature 2024): code distance 3, 5, 7 surface codes showing logical error rate decreasing exponentially with code distance for the first time—a milestone demonstrating QEC can suppress errors rather than just redistribute them. Physical-to-logical qubit overhead for fault tolerance: achieving p_logical ~ 10^-10 with p_physical ~ 10^-3 requires ~1000 physical qubits per logical qubit—implying ~1 million physical qubits for serious fault-tolerant computation on problems like Shor's algorithm breaking RSA-2048 (requiring ~4000 logical qubits), setting a formidable engineering target for the 2030s.

Quantum Algorithms and Applications

Quantum algorithm speedups fall into several categories. Exponential speedup: Shor's factoring (polynomial vs super-polynomial classical); hidden subgroup problems; quantum simulation of chemistry O(N^3) vs classical O(exp(N)). Quadratic speedup: Grover search O(sqrt(N)); quantum amplitude estimation (Monte Carlo speedup); quantum approximate optimisation algorithms (QAOA). Heuristic quantum advantage (uncertain): variational quantum eigensolvers (VQE) for quantum chemistry—current NISQ applicability limited by noise; quantum machine learning algorithms (HHL matrix inversion); quantum boltzmann machines. Near-term NISQ (Noisy Intermediate-Scale Quantum) algorithms tolerate noise without full QEC—VQE, QAOA, quantum neural networks—but classical simulation surpasses NISQ devices for most practical optimisation below ~50 logical qubits due to noise degradation. Fault-tolerant applications with clear advantage: quantum chemistry simulation (catalysis, drug binding, photosynthesis mechanisms); materials simulation (high-temperature superconductor mechanism, battery electrolyte stability); cryptanalysis; and potentially logistics optimisation and financial Monte Carlo—with 50-100× speedup claims pending hardware maturation.

Examples and Applications

Example 1: Google's Quantum Supremacy

Google's 53-qubit Sycamore processor achieved "quantum supremacy" in October 2019—performing a random circuit sampling task in 200 seconds that Google estimated would take classical Summit supercomputer 10,000 years. The task: sample from the output probability distribution of a random sequence of two-qubit gates (classically intractable to exactly simulate due to exponential state space). Controversy: IBM argued classical algorithms could complete the task in 2.5 days on classical hardware using optimised tensor network contraction; Chinese group (2022) further reduced classical simulation time to hours. Nonetheless, the demonstration validated scalable superconducting qubit physics and calibration at 53 qubits. Willow (2024, 105 qubits): performed random circuit sampling 10^25 times faster than estimated classical, and demonstrated below-threshold quantum error correction—a more compelling demonstration of quantum computational advantage. Quantum advantage for practically useful problems (chemistry, optimisation) at scale beyond classical reach requires error-corrected fault-tolerant processors, expected in late 2020s-2030s timeline by major players.

Example 2: Quantum Cryptography Networks

Quantum key distribution (QKD) networks extend point-to-point QKD to metropolitan and long-distance networks. China Quantum Backbone Network (Beijing-Shanghai): 2000 km fibre QKD network with trusted relay nodes every ~100 km (launched 2017)—connects banks, government agencies, and power companies. Micius satellite quantum experiments: satellite-to-ground QKD over 1200 km (2017); intercontinental secure call between Beijing and Vienna using Micius as trusted relay (2018); satellite entanglement distribution over 1200 km (2017); twin-field QKD enabling 500+ km range without trusted relays by exploiting single-photon interference between distant nodes. Post-quantum cryptography (PQC): NIST PQC standardization (completed 2024)—lattice-based algorithms CRYSTALS-Kyber (KEM) and CRYSTALS-Dilithium (signatures) are primary standards resistant to Shor's algorithm quantum attack; now mandatory for US federal government systems—a classical cryptographic response to quantum threat complementing QKD hardware solutions. Quantum internet: long term vision of global entangled quantum network for distributed sensing, blind quantum computing, and secure multi-party computation using quantum repeaters with matter quantum memories.

Example 3: Variational Quantum Eigensolvers

Variational quantum eigensolver (VQE, Peruzzo et al. 2014) is the primary NISQ-era approach to quantum chemistry: prepare a parameterised quantum state |ψ(θ)⟩ encoding a molecular wavefunction; measure expectation value ⟨H⟩ = ⟨ψ(θ)|H|ψ(θ)⟩ of the molecular Hamiltonian H on quantum hardware; classically optimise θ to minimise ⟨H⟩ (variational principle guarantees minimum is ground state energy). VQE hardware requirements scale polynomially with molecule size for fixed circuit depth—classically intractable full configuration interaction (FCI) scales exponentially. VQE demonstrated for H2 (2-qubit, 2016), BeH2 (6-qubit, 2017), and Fe2S2 cluster (active space). Challenges: deep circuits needed for good variational ansatz accumulate noise errors exceeding quantum advantage on NISQ; barren plateau problem (gradient vanishes exponentially with system size for generic circuits); measurement overhead for chemistry Hamiltonians (O(N^4) Pauli terms → thousands of circuit repetitions). Error mitigation techniques (zero-noise extrapolation, probabilistic error cancellation, measurement error mitigation) partially compensate without full QEC—extending practical VQE applicability on available 100-qubit hardware to modest systems.

Example 4: Quantum Sensing and Metrology

Quantum sensors exploit quantum coherence and entanglement to measure physical quantities beyond classical precision limits. Standard quantum limit (SQL) for N independent measurements: delta_phi = 1/sqrt(N) (shot noise). Heisenberg limit: using entangled N-particle states, delta_phi = 1/N—quadratic improvement. LIGO uses squeezed light injection to suppress photon shot noise 10 dB below SQL at high frequencies (>200 Hz) and anti-squeezed light for low frequencies in quantum non-demolition configuration. Nitrogen-vacancy (NV) centre diamonds: atomic defect in diamond lattice with spin-1 ground state; Ramsey and spin-echo sequences measure magnetic fields to ~1 nT/sqrt(Hz) at room temperature with ~10 nm spatial resolution—enables single-neuron action potential magnetic field imaging, dark matter searches via spin precession anomalies. Atomic gravimeters (cold atom fountains): resolution 1 microGal (10^-8 g)—mapping underground voids for civil engineering, hydrogeology, and dark matter detection. Quantum radar concept: entangled microwave photon pairs for low-background illumination—demonstrated low signal-to-noise advantage in laboratory but engineering challenges prevent near-term deployment.

Example 5: Topological Quantum Computing

Topological quantum computing encodes quantum information in non-local degrees of freedom—topological charges of anyons—that are intrinsically protected from local physical perturbations (errors must wind around the anyon globally, requiring macroscopic coordinated errors exponentially unlikely). Non-Abelian anyons (Majorana zero modes, parafermions, Fibonacci anyons): braiding operations represent quantum gates—the computational history is encoded in topology, not local state. Microsoft's station Q and topological qubit programme: heterostructure InAs semiconductor/Al superconductor nanowires proximitise Majorana modes at wire ends; signature is quantised zero-bias conductance peak (ZBP) in tunnelling spectroscopy. 2023 retraction: claimed "topological qubit" paper retracted from Nature due to insufficient evidence for genuine topological gap required for protected Majorana modes. 2025: Microsoft announced new topological qubit architectures claiming improved evidence—controversy reflects the difficulty of proving topological protection. Alternative non-Abelian platforms: fractional quantum Hall 5/2 state; p-wave superconducting vortex cores; topological superconductor surfaces; Kitaev honeycomb models in alpha-RuCl3—each with serious experimental progress but none demonstrated for coherent qubit operation.

Example 6: Quantum Decoherence and Error Mitigation

Decoherence—entanglement of a qubit with environmental degrees of freedom—is the central obstacle to quantum computing. Decoherence channels: T1 relaxation (energy decay from |1⟩ to |0⟩, typical superconducting T1 = 50-500 μs from quasiparticle poisoning, dielectric TLS, radiative emission); T2 dephasing (phase randomisation without energy change, T2 ≤ 2T1 from charge noise, flux noise, photon number fluctuations). Qubit Hamiltonian in Bloch sphere: dρ/dt = -i/ℏ [H,ρ] + Lindblad dissipator sum—Lindblad master equation governs open quantum system evolution. Error mitigation (NISQ era): Zero-noise extrapolation (run circuits at scaled noise levels, extrapolate to zero noise—Mitiq tool); probabilistic error cancellation (learn error model, insert inverse channels with overhead); readout error mitigation (calibration matrix inversion). Dynamical decoupling (Hahn echo, CPMG): rapid refocusing pulses rephase low-frequency noise—extending effective T2. Pauli error propagation tracking (Clifford simulation efficient on classical CPU) enables real-time syndrome measurement in QEC circuits without simulating full state vector—essential for repetition code and surface code syndrome-based error correction at chip scale.

Example 7: Quantum Simulation of Materials

Quantum simulation—using a controllable quantum system to simulate another quantum system—was Feynman's original motivation for quantum computing (1982 lecture). Cold atom quantum simulators: Hubbard model simulation in optical lattices (atoms hopping between lattice sites with on-site repulsion U and hopping t)—direct experimental realisation of models of high-temperature superconductors (cuprate d-wave pairing mechanism hypothesised from 2D Hubbard model); antiferromagnetism detected by quantum gas microscope single-site imaging. Rydberg atom arrays: Ising model simulation, quantum phase transitions, topological order—Harvard/QuEra demonstrated 256-atom programmable quantum simulation of Z2 topological spin liquid (2021). Digital quantum simulation: encode Hamiltonian as quantum circuit (Trotterisation); simulate quantum chemistry using VQE or quantum phase estimation. Quantum chemistry target: nitrogen fixation mechanism (FeMo cofactor of nitrogenase enzyme active site—requires ~100 logical qubits for accurate simulation beyond classical CASSCF), materials for room-temperature superconductivity, lithium-air battery degradation pathways—transformative impact on clean energy if realised.

Example 8: Quantum Networks and Entanglement Distribution

Quantum networks distribute entanglement between remote nodes for distributed quantum computing, quantum-secure communication, and distributed quantum sensing. Entanglement distribution over optical fibre: photon pair sources (SPDC—spontaneous parametric downconversion in nonlinear crystal BBO, KTP; or quantum dot single photon pairs); 1310/1550 nm telecom wavelength photons guided to fibre with ~0.15 dB/km attenuation limiting direct range to ~100 km. Quantum repeaters overcome distance limits: entanglement swapping (Bell measurement of two incoming photons projects distant qubits into entangled state) and quantum memories (storing photon qubit in Er3+ ions, Pr3+ ions, Rb atoms, NV centres between swapping operations) enable repeater chain extending entanglement to arbitrary distance with polynomial overhead. Quantum memory lifetime and efficiency: rare-earth doped crystal memories (AFC protocol, europium doped Y2SiO5) achieving 6-hour storage time at 3 K; warm Rb vapour memories (Duan-Lukin-Cirac-Zoller DLCZ protocol) operational at room temperature with lower storage time. SEQC (Quantum Internet Alliance EU) targets first demonstrative European quantum network by 2030; UK Quantum Network (UKQN) fibre ring connecting Cambridge, London, Bristol; Chinese quantum network extending to 4600 km backbone by 2025.

Try it live

Everything above runs in your browser — open SPH Fluid and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.