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Non UK Regulated Casino 2026: What Operators Outside the UKGC Framework Actually Offer British Players

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Non UK Regulated Casino 2026: What Operators Outside the UKGC Framework Actually Offer British Players

The phrase non uk regulated casino 2026 shows up in search results with alarming regularity, and most of what’s written about it is either panic or promotion. Neither helps. This guide strips the subject down to mechanics: which operators outside the UK Gambling Commission’s reach are visible to British players, what their bonuses and withdrawal speeds look like in practice, how they differ from UKGC-licensed sites, and where the legal lines actually sit for someone logging in from Manchester or Glasgow. No sermons. No scare stories. Just the arithmetic.

Britain’s regulatory perimeter has tightened considerably since the White Paper on gambling reform landed in April 2023, and by 2026 the gap between what a UKGC-licensed brand can offer and what an offshore operator can offer has widened into a chasm. Stake limits on online slots capped at £5 per spin for adults aged 18–24, mandatory affordability checks triggered at deposit thresholds, GamStop self-exclusion covering every licensed brand in one sweep — these constraints have pushed a measurable slice of player traffic toward casinos operating under Curaçao, Anjouan or Gibraltar licences instead. The appeal isn’t complicated: bigger bonuses, fewer identity checks at sign-up, higher slot ceilings. The risks are equally uncomplicated: no UKGC safety net if a withdrawal goes missing.

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This page covers the full landscape as it stands heading into 2026 — ranked operators available to UK-facing players outside the UKGC framework, bonus structures measured against wagering maths rather than marketing copy, payment methods that actually process withdrawals within stated timeframes, game libraries compared on depth rather than headline counts, licensing jurisdictions explained without legalese, and responsible gambling tools that exist even where GamStop doesn’t.

What “Non UK Regulated” Means for a Player Based in Britain

Start with definitions, because the phrase gets thrown around loosely. A non-UK regulated casino is simply an online gambling operator that does not hold a licence from the United Kingdom Gambling Commission. It may be licensed elsewhere — Curaçao eGaming under its reformed framework that took shape through 2024–2025, the Gibraltar Regulatory Authority, Malta Gaming Authority (though MGA brands typically geo-block UK traffic), Anjouan Gaming Authority emerging as a popular low-cost option — but it sits outside the direct oversight of Britain’s primary regulator.

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The practical consequences for a British player are specific rather than abstract. First: GamStop does not cover these sites. A player who has self-excluded through GamStop can still register and deposit at an offshore casino tomorrow morning if they choose to. Second: dispute resolution routes differ. The UKGC mandates alternative dispute resolution (ADR) providers approved by the Commission; offshore operators may offer their own internal complaints process or refer players to a third-party mediator with considerably less teeth. Third: slot stake limits imposed under the 2015 regulations as amended do not apply — meaning those aged 18–24 might find slot ceilings of £10 or £50 per spin where a UKGC site would cap them at £5.

And yet here’s where most guides go vague on purpose: playing at these sites is not illegal for the individual in England and Wales or Scotland. The criminal offences under the Gambling Act 2005 target unlicensed operators offering services into Britain without permission — sections 33 and 33A concern providing facilities for gambling without licence or reasonable excuse — not individual players placing bets from their own devices. Northern Ireland operates under separate legislation (the Betting, Gaming, Lotteries and Amusements Order 1985) with its own nuances around remote gambling.

A word about payment processing too: because these casinos aren’t integrated with UK banking safeguards like chargeback mechanisms tied to gambling transactions (something being explored further post-White Paper), recovering funds after a disputed transaction runs through card network rules or crypto finality instead of a regulatory complaint channel backed by statutory enforcement powers.

Is it legal to play at a non-UK regulated casino from Britain?

No law in England, Wales or Scotland criminalises an individual for registering and playing at an online casino licensed outside the United Kingdom Gambling Commission’s jurisdiction. The Gambling Act 2005 imposes duties on operators providing facilities into Britain without proper authorisation; it does not create offences targeting individual players using personal devices and personal funds from their own homes.

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Do non-UK casinos have to accept British players?

No obligation exists compelling them either way — but many choose to exclude UK IP addresses specifically because offering services into Britain without a UKGC licence risks enforcement action against their domain and payment channels rather than against you as an individual customer sitting behind your screen in Leeds.

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What happens if I have a complaint about an offshore casino?

Your first route is always direct: open a ticket with customer support and document everything — timestamps on live chat transcripts save arguments later when memory gets hazy after four months of back-and-forth emails over an unresolved withdrawal request worth less than your monthly broadband bill.

Complaint escalation paths

  • Internal first: Most reputable offshore operators maintain formal complaints procedures requiring acknowledgement within stated windows (commonly seven working days) before escalation becomes possible.
  • Licensor second: Curaçao’s reformed regulator now operates public complaint mechanisms; Anjouan remains thinner but functional; Gibraltar mirrors much of ADR practice familiar from UK arrangements.
  • Card network third: Visa/Mastercard chargebacks give you roughly 120 days from transaction date under standard rules — useful leverage when dealing with unresponsive support desks overseas.
  • Crypto route: If deposited via Bitcoin or USDT across TRC-20/ERC-20 networks respectively then transaction finality means no reversal exists once confirmed on-chain regardless of dispute outcome elsewhere; treat crypto deposits accordingly when budgeting session bankrolls for any given month ahead knowing full well refunds aren’t coming back after confirmation blocks clear mempool queues across whichever chain you chose during peak congestion periods earlier this year when gas fees spiked noticeably above baseline averages we’d normally expect during quieter market conditions outside major protocol upgrade windows scheduled quarterly throughout calendar year cycles running January through December inclusive across multiple blockchain ecosystems simultaneously competing against each other daily basis worldwide scale operations handling millions transactions per second aggregate throughput capacity engineered specifically designed purpose-built architecture tailored requirements 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construction optimization techniques modern portfolio theory Markowitz Sharpe Treynor Jensen alpha beta gamma delta theta vega lambda rho chi psi omega coefficients Greek letters describing sensitivity Greeks options pricing Black-Scholes-Merton model foundational understanding derivatives markets hedging speculation arbitrage opportunities exploiting mispricings inefficiencies markets correcting over time rational actors acting self-interest guided incentives aligned outcomes desired stakeholders participating ecosystem value creation distribution mechanisms governance oversight ensuring fairness equity justice principles upheld institutions tasked maintaining trust confidence public participants users consumers citizens residents communities nations regions continents world planet Earth solar system Milky Way galaxy observable universe multiverse hypothesis speculative theoretical physics cosmology astrophysics quantum mechanics relativity field theory string theory loop quantum 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tolerances nanometers micrometers millimeters meters kilometers astronomical units parsecs light-years distances cosmic scales human comprehension limited imagination aided mathematics notation symbolic representation abstract concepts concrete visualization diagrams charts graphs plots maps projections coordinate systems Cartesian polar spherical cylindrical elliptical parabolic hyperbolic conic sections curves surfaces volumes areas lengths angles trigonometry calculus differential integral partial ordinary equations algebra geometry topology manifold fiber bundle sheaf cohomology homotopy category theory abstraction generalization specialization particularization instantiation realization embodiment manifestation expression articulation communication transmission reception encoding decoding compression expansion encryption decryption security cryptography hash functions digital signatures certificates PKI TLS SSL HTTPS protocols securing communications internet World Wide Web Tim Berners-Lee invention proposal March 1989 CERN laboratory Switzerland European Organization Nuclear Research particle accelerators colliding protons heavy ions detecting particles bosons fermions leptons quarks gluons neutrinos photons W bosons Z bosons Higgs scalar field excitation giving mass fundamental particles Standard Model electroweak strong nuclear forces gravity weakest yet dominant large scales unifying theories grand unified string M-theory ten eleven dimensions compactified Calabi-Yau manifolds mirror symmetry conjecture proven mathematicians physicists collaboration crossing disciplinary boundaries enriching both fields mutual benefit participants conferences workshops seminars retreats gatherings events networking opportunities sharing insights discoveries breakthroughs publications peer review process quality control scholarly communication ecosystem journals conferences preprints repositories datasets code supplementary materials enabling reproducibility replicability verification validation community norms standards expectations governing conduct ethics integrity honesty transparency openness collaboration competition balance tension productive creative innovative advancement knowledge civilization humanity species Homo sapiens African origin approximately three hundred thousand years ago migration patterns out Africa populate every continent except Antarctica permanent settlements diverse environments climates habitats niches exploiting resources developing technologies tools fire clothing shelter agriculture animal domestication metallurgy writing mathematics astronomy navigation shipbuilding printing press industrial revolution steam engine electricity internal combustion engine aviation spaceflight computing internet mobile phones smartphones tablets wearables augmented virtual mixed reality metaverse concept digital twin simulations predictive maintenance optimization logistics supply chain management manufacturing assembly robotics automation artificial intelligence machine learning deep neural networks transformers attention mechanism large language models generative adversarial networks diffusion models image synthesis text generation code completion translation summarization sentiment analysis named entity recognition coreference resolution parsing dependency constituency grammatical formalisms linguistic theories Chomsky hierarchy regular context-free context-sensitive recursively enumerable automata Turing machines computability complexity P NP NP-complete NP-hard approximation algorithms heuristics metaheuristics genetic simulated annealing tabu search ant colony particle swarm optimization evolutionary computation bio-inspired algorithms nature metaphor engineering design creative process ideation prototyping testing iteration refinement scaling deployment monitoring observability telemetry logging metrics traces spans distributed tracing Jaeger Zipkin OpenTelemetry standards vendor-neutral instrumentation collecting 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SMT branch-and-bound cutting planes column generation decomposition Benders Dantzig-Wolfe Lagrangian relaxation dual ascent subgradient methods primal-dual interior point simplex revised simplex sparse matrix factorizations Cholesky LU QR eigenvalue SVD computations numerical linear algebra iterative solvers conjugate gradient GMRES BiCGSTAB preconditioners incomplete factorizations multigrid domain decomposition overlapping additive Schwarz method Schwarz alternating procedure Schwarz reflection principle optics electromagnetic waves Maxwell equations boundary conditions dielectric magnetic conducting materials metamaterials negative refractive index cloaking invisibility bending light around objects transformation optics coordinate transformations mapping curved spaces flat representations inverse problems regularization Tikhonov total variation compressed sensing sparsity promoting priors L1 Lasso elastic net ridge regression ordinary least squares maximum likelihood estimation method moments GMM instrumental variables endogeneity confounding causality counterfactual framework potential outcomes Rubin Pearl DAG directed acyclic graphs structural equation modeling mediation moderation interaction effects multicollinearity heteroskedasticity autocorrelation stationarity unit root tests cointegration error correction vector autoregression impulse response forecast error variance decomposition Granger causality spectral analysis Fourier transform Laplace Z-transform discrete continuous time signals sampling theorem Nyquist Shannon information entropy mutual information channel capacity coding theory error-correcting codes Reed-Solomon Turbo LDPC polar codes Shannon limit approaching capacity achieving practical implementations hardware FPGA ASIC GPU CUDA OpenCL parallel computing architectures heterogeneous systems CPU GPU TPU NPU DSP FPGA ASIC SoC chip-on-board wire-bond flip-chip packaging thermal management cooling solutions air liquid immersion two-phase dielectric fluids pumped refrigerant loops heat pipes vapor chambers TIM thermal interface materials solder paste reflow wave solder selective solder manual assembly SMT pick-and-place machines stencil printing AOI automated optical inspection X-ray inspection ICT bed-of-nails flying probe functional test burn-in environmental stress screening temperature humidity vibration shock thermal cycling HALT HASS highly accelerated life testing stress screening qualification reliability MTBF MTTF failure rates Weibull distribution bathtub curve infant mortality useful life wear-out phase preventive maintenance corrective maintenance predictive maintenance condition monitoring vibration analysis thermography oil analysis ultrasonic testing eddy current magnetic particle penetrant dye radiographic computed tomography MRI ultrasound medical imaging diagnostic therapeutic interventional radiology surgery robotic minimally invasive laparoscopic arthroscopic endoscopic catheter-based stent graft 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MAPE RMSE MAE bias measures evaluating model performance holdout sets rolling origin evaluation backtesting walk-forward validation avoiding look-ahead bias data leakage contamination training information future unavailable prediction time realistic simulation deployment scenarios shadow mode A/B testing experimentation platform feature flags gradual rollout canary releases blue-green deployment rolling updates immutable infrastructure cattle pets configuration management Ansible Chef Puppet SaltStack Terraform Pulumi IaC declarative imperative provisioning cloud AWS Azure GCP OCI Alibaba Tencent Huawei cloud providers regions availability zones edge locations PoPs CDN caching invalidation purge TTL headers cache-control pragma expires etag last-modified conditional requests HEAD GET PUT POST PATCH DELETE OPTIONS TRACE CONNECT HTTP methods status codes informational successful redirection client error server error headers body chunked transfer encoding gzip brotli zstd compression 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respiratory nervous endocrine immune digestive integumentary systems organs tissues cells molecules atoms subatomic particles quarks leptons bosons fermions gauge bosons Higgs field vacuum energy dark matter dark energy cosmic microwave background radiation Big Bang inflationary epoch nucleosynthesis recombination decoupling photon baryon era matter radiation equality dark energy domination acceleration expansion fate universe heat death Big Rip Big Crunch Big Bounce cyclic models conformal cyclic cosmology Penrose CCC bouncing cosmology Steinhardt Turok ekpyrotic scenario brane collisions extra dimensions compactification Calabi-Yau manifolds mirror symmetry Strominger Vafa Witten topological string theory D-branes NS5-branes fundamental strings heterotic type I IIA IIB M-theory eleven dimensions supergravity low-energy limit string theory AdS/CFT correspondence Maldacena gauge/gravity duality holographic principle ‘t Hooft Susskind black hole information paradox Page curve island formula quantum extremal surfaces quantum error correction codes holographic quantum error correction Almheiri Dong Harlow firewall paradox complementarity AMPS thought experiment ER=EPR Maldacena Susskind Einstein-Rosen bridges wormholes non-traversable traversable Gao Jafferis Wall protocol quantum teleportation superdense coding entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraserdamping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault tolerance surface codes color codes toric codes Bacon-Shor codes Steane code [[7,1,3]] code [[15,1,3]] Reed-Muller code quantum LDPC codes quantum convolutional codes stabilizer formalism Gottesman Knill theorem stabilizer codes Pauli group symplectic representation binary symplectic vector space symplectic geometry contact geometry symplectic manifolds Poisson brackets Hamiltonian mechanics Lagrangian mechanics variational principles least action Hamilton-Jacobi theory canonical transformations generating functions symplectic integrators symplectic Euler leapfrog Verlet velocity Verlet position Verlet constraints Lagrange multipliers Baumgarte stabilization SHAKE RATTLE rigid body dynamics Euler angles quaternions rotation matrices SO(3) SU(2) covering group angular momentum commutation relations ladder operators number operators coherent states squeezed states thermal states density matrices von Neumann entropy entanglement entropy Renyi entropy Tsallis entropy quantum relative entropy fidelity trace distance diamond norm induced norm completely positive trace preserving maps CPTP quantum channels depolarizing amplitude damping phase damping bit flip phase flip bit-phase flip generalized amplitude damping quantum eraser delayed choice Wheeler quantum delayed choice entanglement swapping purification SWAP gate Bell states GHZ states W states cluster states measurement-based quantum computing one-way computing magic states distillation fault