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Quantum Computing Research

Simulators, emulators and real quantum hardware, clearly explained: QCaaS methods, real-world use cases, benchmarks and costs.

Terms used across these articles are defined in the quantum computing dictionary

Top Quantum Computing Use Cases

Quantum computing use cases measured on QCaaS: chemistry, finance, quantum machine learning (QML), scheduling, space, logistics, manufacturing, traffic and cryptography, each with its sample run, its result and its cost.

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QCaaS: Neural States on TPU

Run neural network quantum states on a Google TPU, priced by the second, with a certified bound on every result. Why tensor networks stop, what variational Monte Carlo already measures, and the run whose estimate was wrong while its bound held.

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Logical Qubits vs Physical Qubits

A logical qubit is built from many physical ones, and the exchange rate is not a constant. Surface code simulations at distance 3, 5 and 7 under circuit-level noise, decoded by minimum-weight matching. The threshold sits between 0.5% and 0.8%, and reaching a 1e-15 logical error rate needs distance 39, or 3,041 physical qubits per logical qubit.

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Maximum Independent Set on Neutral Atoms

Maximum independent set on QuEra Aquila: 9 atoms, 1,000 shots, the optimum found, 93.2% of shots valid. The Rydberg blockade is the constraint itself, so no QUBO and no penalty terms.

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Quirk and the Browser Quantum Circuit Simulators

Where the free browser tools stop, in exact numbers. Statevector memory is 16 x 2^n bytes, which puts a hard wall about 28 qubits into a browser tab, and three certified runs at 40, 192 and 1001 qubits show what sits on the other side of it.

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We Measured a Photonic Quantum Computer

Three runs on Quandela's Belenos photonic processor: Hong-Ou-Mandel visibility of 95.4%, 97.0% and 97.7%, for about a euro each. What photons do instead of qubits, and why this device is easier to certify than any gate machine.

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Quantum Computer Shootout

Compare quantum computers. The eight machines on this service span four modalities, what each manufacturer built theirs for, and one measured run on every one of them with its cost.

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What It Costs to Rent a Quantum Computer

The real bill for renting quantum computer time on eight machines from seven manufacturers, with per-task and per-shot list prices. The same 1,000-shot experiment costs $0.725 on one machine and $80.30 on another, and around EUR 556 on a third that bills by the clock.

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Rydberg Blockade and Antiferromagnetic Order

A measured sweep of atom spacing in a seven-atom Rydberg chain, showing where antiferromagnetic order appears, where it decays, and where it collapses entirely. Seven runs, 1000 shots each, with the certificate for the best one.

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Quantum Emulator vs Simulator

The difference between a quantum emulator and a quantum simulator, why vendors use the words interchangeably, and what each one actually returns. With measured runs on stabilizer, tensor-network and exact engines, and on real hardware.

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Will a Quantum Computer Break Bitcoin?

How many qubits to break Bitcoin? 2026 estimates run from about 10,000 physical qubits over months to under 500,000 in minutes. We recovered 7-bit keys.

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Ethereum Validator Keys and Quantum Computing

Ethereum has a quantum-exposed surface Bitcoin does not: BLS12-381 validator keys securing the consensus layer, a larger target than any account. What breaking them would cost, why forging them attacks finality rather than balances, and why account abstraction still makes Ethereum the easiest major chain to migrate.

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How Many Qubits Can a Tensor Network Actually Simulate?

A measured answer. Matrix product states reach 2,000 qubits with a certified bound and 5,000 for expectation values, but bitstring sampling stops between 1,150 and 1,175 qubits for a reason that has nothing to do with entanglement.

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Qiskit, Cirq or PennyLane? Choosing a Quantum Programming Framework

Qiskit to learn, PennyLane for gradients, Cirq for device topology. What each was built for, where they overlap, and the OpenQASM 2 limitation that catches every variational algorithm.

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Neutral Atom Quantum Computing, and Why It Is Not Gate-Based

Rydberg atoms in optical tweezers reach 256 qubits today, more than any superconducting or trapped-ion device on public cloud. The catch is that the leading machine does not accept quantum gates at all.

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Why Your Quantum Circuit Was Rejected, and What to Change

Every reason a quantum circuit fails to run on a simulator or QPU: unsupported gates, memory ceilings, OpenQASM limitations, shot bounds, and error bounds too wide to be informative.

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Pauli Strings, Pauli Channels and Pauli Propagation: A Working Reference

The three distinct things called Pauli in quantum computing: strings as an operator basis, channels as a noise model, and propagation as a simulation method. What each one is and where each is used.

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Transmon or Trapped Ion? What the Price List Tells You

Superconducting and trapped-ion quantum computers compared on the numbers that decide a run: qubit count, gate fidelity, shot bounds and price per shot, which differ by a factor of 188.

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Quantum Entanglement Explained: Build a GHZ State and Run It

A hands-on guide to quantum entanglement: what a Bell state and a GHZ state are, the circuit that makes one, and a live, certified run you can reproduce.

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Grover's Search Algorithm: Find a Needle in a Quantum Haystack

How Grover's algorithm searches an unsorted space faster than any classical computer, with the full circuit, a diagram, and a live run that finds the marked item every time.

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QAOA for MaxCut: Solving Real Optimization Problems on Qubits

A practical guide to the Quantum Approximate Optimization Algorithm: how QAOA solves MaxCut, the circuit and diagram, and a live run where the optimal partitions dominate.

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VQE and the Hydrogen Molecule: Quantum Chemistry From Scratch

A practical VQE tutorial: how the variational quantum eigensolver finds the ground-state energy of H2, with the ansatz circuit, a diagram, and a live run that lands on -1.137 hartree.

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The Bernstein-Vazirani Algorithm: One Query, Whole Answer

How the Bernstein-Vazirani algorithm reads a hidden bitstring in a single query where a classical computer needs one per bit, with the circuit, a diagram, and a live run.

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Quantum Teleportation: Send a Qubit Without Sending a Qubit

How quantum teleportation moves a qubit's state across a circuit using entanglement alone, with the full circuit, a diagram, and a live run where the state arrives intact.

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Simulating Quantum Error Correction: How Logical Qubits Are Tested

How quantum error correction and logical qubits are studied on classical simulators, why stabilizer circuits make it tractable, and how tools like Stim scale to millions of qubits.

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Pauli Propagation: The Method That Simulated IBM's 127-Qubit Experiment on a Laptop

Pauli propagation explained: how tracking observables backward through a circuit simulates hundreds of qubits, and why noise makes it stronger.

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Quantum Computing as a Service (QCaaS) explained

Quantum Computing as a Service is renting quantum hardware or simulators by the task, over an API, with no machine to buy. Also written QaaS. How it is priced per task and per shot, and how to pick a backend.

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CPU vs GPU vs TPU vs QPU: Choosing a Quantum Backend

When CPU simulation, GPU acceleration, a TPU or real quantum hardware is the right and cheapest choice, and why the chip you want is decided by the method rather than the other way round.

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Quantum Machine Learning on Simulators and real Quantum hardware

Quantum machine learning trains a circuit the way you train a network: the rotation angles are the weights. Five methods measured here, a Born machine reaching total variation distance 0.0117, three of them re-run on real hardware.

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Quantum Supremacy Claims That Fell to Classical Simulation

Google 2019, IBM 2023 and more: the history of quantum advantage claims matched by classical algorithms, from tensor networks to Pauli propagation and noise exploitation.

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Grover and Shor, Simulated: Learning the Famous Quantum Algorithms

Grover is quadratic, which is why symmetric key sizes are being doubled rather than abandoned. Shor is the one that breaks things. How each works, what simulating them costs, and how far hardware still is from either.

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Post-Quantum Cryptography: A Primer on the Quantum-Safe Transition

How post-quantum cryptography works, when a quantum computer could break Bitcoin and Ethereum wallets, whether Bitcoin mining or Zcash are affected, and what governments, banks, and internet providers must do to prepare.

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How Much Does a Quantum Computer Cost? To Buy, and To Rent

Quantum computer prices in 2026, from a $5,000 desktop unit to $30M installed systems, and what renting time costs instead: per-task and per-shot QPU fees, GPU rates, and what we paid for real runs.

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Clifford Circuits: Simulating Thousands of Qubits in Milliseconds with Stim

How the Gottesman-Knill theorem lets stabilizer simulators like Stim run 5,000-qubit circuits instantly, and why error-correction research depends on it.

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QAOA at 100 Qubits: Real Benchmarks from a Laptop CPU

Reproducible benchmarks: 100-qubit QAOA MaxCut circuits simulated in seconds on consumer hardware with tensor networks, full methodology and numbers.

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Certification for Quantum Simulation and Hardware: ZCC-v0.1 and ZHF-v0.1

Quantum results, simulated or measured on real hardware, are routinely reported with no accuracy statement. This article specifies our two certification protocols: ZCC-v0.1 for simulation accuracy and ZHF-v0.1 for hardware fidelity, with code, benchmarks, and sample certificates.

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Tensor Networks Explained: How MPS Simulates 100 Qubits on a Laptop

An MPS stores one small tensor per qubit, so 100 qubits at bond dimension 128 is about 3 million numbers rather than 2^100. Bond dimension measures entanglement rather than width, and that is also where the method stops.

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The 34-Qubit Wall: Why Free Quantum Simulators Stop, and How to Go Past It

Why every free quantum simulator caps around 34 qubits, the memory math behind the wall, and the structured-circuit methods that reach 100+ qubits.

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What Is Quantum Circuit Simulation? A Practical Guide for 2026

Quantum circuit simulation is running a quantum program on a classical computer to compute what a quantum machine would output. The cost is 16 x 2^n bytes, which is why exact methods stop near 32 qubits and structured ones reach thousands.

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