Glossary

The basic unit of quantum information. Unlike a classical bit (0 or 1), a qubit can exist in a superposition of both states simultaneously. Physically realized as superconducting transmons (IBM, Tuna-9), spin qubits (QuTech Spin-2+), trapped ions, or photons.

A qubit in superposition is in a combination of |0> and |1> states, described by amplitudes alpha and beta where |alpha|^2 + |beta|^2 = 1. Measurement collapses the superposition to a definite state with probabilities given by the Born rule.

A quantum correlation between two or more qubits where the state of one cannot be described independently of the others. Measuring one instantly determines the other, regardless of distance. The resource behind quantum teleportation and many quantum algorithms.

Extracting classical information from a qubit. Measurement collapses the quantum state: a qubit in superposition becomes |0> or |1> with probabilities determined by its amplitudes (Born rule). Measurement in different bases (X, Y, Z) reveals different information.

The probability of measuring a quantum state |psi> in state |x> is |<x|psi>|^2 — the squared magnitude of the amplitude. This connects the abstract quantum state to observable outcomes.

A geometric representation of a single qubit state as a point on a unit sphere. |0> is the north pole, |1> the south pole, and superposition states lie on the surface. Single-qubit gates are rotations on this sphere.

The mathematical description of a quantum system. For n qubits, it is a vector of 2^n complex amplitudes. A 3-qubit system has 8 amplitudes. State vectors live in Hilbert space and evolve via unitary operations.

A more general representation of quantum states that can describe both pure states and mixed states (statistical ensembles). The diagonal elements give measurement probabilities; off-diagonal elements encode coherence and entanglement.

A unitary operation that transforms qubit states. Single-qubit gates (X, Y, Z, H, S, T) rotate the Bloch sphere. Multi-qubit gates (CNOT, CZ, Toffoli) create entanglement. Any quantum computation can be decomposed into single-qubit gates plus CNOT.

The three fundamental single-qubit gates. X is a bit-flip (|0> <-> |1>). Z is a phase-flip (|1> -> -|1>). Y = iXZ combines both. They form the Pauli group and are the building blocks of error correction and Hamiltonian simulation.

Creates an equal superposition: H|0> = (|0>+|1>)/sqrt(2). The workhorse gate for creating superposition. H is its own inverse (applying it twice returns to the original state). Central to nearly every quantum algorithm.

A two-qubit gate that flips the target qubit if and only if the control qubit is |1>. The standard entangling gate. Together with single-qubit rotations, CNOT forms a universal gate set — any quantum computation can be built from these.

The group generated by H, S, and CNOT gates. Clifford circuits can be efficiently simulated classically (Gottesman-Knill theorem), which makes them important for error correction but insufficient for quantum advantage. Adding the T gate breaks out of the Clifford group.

A pi/8 phase gate that, combined with Clifford gates, forms a universal gate set. T gates are the expensive resource in fault-tolerant quantum computing — they require magic state distillation, making T-count a key metric for circuit cost.

The number of sequential gate layers in a quantum circuit (gates that can run in parallel count as one layer). Deeper circuits accumulate more noise on real hardware. Reducing depth while preserving function is a key optimization challenge.

Converting a quantum circuit into the native gate set of a specific hardware backend. Different processors support different gates (e.g., IBM uses CX+sqrt(X)+Rz). Transpilation also handles qubit routing when hardware connectivity is limited.

The four maximally entangled two-qubit states: |Phi+> = (|00>+|11>)/sqrt(2), |Phi-> = (|00>-|11>)/sqrt(2), |Psi+> = (|01>+|10>)/sqrt(2), |Psi-> = (|01>-|10>)/sqrt(2). The simplest entangled states, used in teleportation, superdense coding, and as calibration benchmarks.

Greenberger-Horne-Zeilinger state: (|000...0> + |111...1>)/sqrt(2). A maximally entangled multi-qubit state where all qubits are correlated. Used for testing multipartite entanglement and as a benchmark for hardware quality.

An entangled three-qubit state: (|001>+|010>+|100>)/sqrt(3). Unlike GHZ, W state entanglement is robust — tracing out one qubit leaves the remaining two still entangled. Represents a different class of multipartite entanglement.

States that can be created from |0...0> using only Clifford gates. Efficiently described by their stabilizer group rather than the full state vector. Include computational basis states, Bell states, and GHZ states. Can be classically simulated.

A hybrid quantum-classical algorithm for finding molecular ground state energies. A parameterized quantum circuit (ansatz) prepares a trial state, the quantum computer measures the energy, and a classical optimizer adjusts the parameters. Our H2 experiments use VQE.

A variational algorithm for combinatorial optimization problems like MaxCut. Alternates between a problem Hamiltonian and a mixer Hamiltonian with tunable parameters. Performance improves with more layers (depth p) but requires more optimization. Our Tuna-9 QAOA achieved 74.1% approximation ratio on a 4-node MaxCut problem.

A quantum search algorithm that finds a marked item in an unsorted database of N items using only O(sqrt(N)) queries — a quadratic speedup over classical search. Uses amplitude amplification: the oracle marks the target, and diffusion amplifies its amplitude.

A protocol that transfers a quantum state from one qubit to another using entanglement and classical communication. Does not transmit information faster than light — the classical bits are still needed. Demonstrates the power of entanglement as a resource.

A parameterized quantum circuit used as a trial wavefunction in variational algorithms. The choice of ansatz determines what states are reachable and affects convergence. Common types: hardware-efficient, UCCSD (chemistry-inspired), and problem-specific.

The energy operator of a quantum system. In quantum chemistry, the molecular Hamiltonian encodes all electron-electron and electron-nucleus interactions. For quantum computing, it must be decomposed into Pauli strings (via Jordan-Wigner or Bravyi-Kitaev transforms).

A measure of how close a quantum state or operation is to the ideal. State fidelity F = |<psi_ideal|psi_actual>|^2 ranges from 0 (orthogonal) to 1 (identical). Gate fidelity measures how well a physical gate matches its ideal unitary. Our Bell state experiments measure fidelity across backends.

The threshold of 1 kcal/mol (0.0016 Ha / 1.6 mHa) — the accuracy needed for quantum chemistry results to be practically useful. Our emulator achieves 0.75 kcal/mol, IBM TREX achieves 0.22 kcal/mol, and Tuna-9 REM+PS achieves 0.92 kcal/mol — all within chemical accuracy.

A single-number benchmark (IBM) that captures the largest random circuit a processor can execute reliably. QV = 2^n where n is the effective number of qubits that can maintain sufficient fidelity through n layers of random 2-qubit gates. Higher is better. We measured QV=16 on Tuna-9 and QV=32 on IBM Torino and IQM Garnet.

A protocol for measuring average gate fidelity by running random sequences of Clifford gates of increasing length, followed by an inverting gate. The decay rate of the survival probability gives the error per Clifford. Robust against state preparation and measurement errors. We measured 99.82% on both Tuna-9 and IQM Garnet — but IBM reports 99.99% because their transpiler collapses Clifford sequences.

For optimization algorithms like QAOA, the ratio of the algorithm's solution quality to the optimal. An approximation ratio of 0.7 means the algorithm achieves 70% of the best possible result. Our QAOA MaxCut emulator run achieves 68.5%.

A qubit encoded in the spin state of an electron confined in a semiconductor quantum dot. Used by QuTech/Quantum Inspire (Spin-2+ processor). Note: Tuna-9 uses superconducting transmon qubits, not spin qubits. Advantages: small size, long coherence in silicon, compatibility with existing semiconductor fabrication.

A qubit made from a superconducting circuit with a Josephson junction, operated at ~15 millikelvin. Used by IBM (ibm_torino, ibm_fez, ibm_marrakesh — 133-156 qubits) and Google. Currently the most mature quantum computing platform.

The time constant for energy relaxation — how long before an excited qubit (|1>) decays to ground state (|0>). Limits the total computation time. Typical values: ~100us for superconducting qubits, variable for spin qubits.

The time constant for phase coherence — how long superposition states maintain their relative phase. Always T2 <= 2*T1. Limits the useful circuit depth. Dephasing turns pure quantum states into classical mixtures.

The process by which quantum information is lost to the environment. Includes both energy relaxation (T1) and phase randomization (T2). The fundamental challenge for quantum computing — longer algorithms need better coherence or error correction.

The probability of incorrectly measuring a qubit state. A qubit in |0> might be read as 1 (and vice versa). Typical rates: 0.5-5% depending on hardware. Can be partially corrected with readout error mitigation (measuring calibration matrices). On Tuna-9, readout error accounts for >80% of total VQE error.

A mapping from fermionic operators (electrons) to qubit operators (Pauli strings). Preserves the antisymmetry of fermions using strings of Z gates. Requires n qubits for n spin-orbitals. Our H2 VQE uses JW to get a 4-qubit Hamiltonian from the molecular problem.

An alternative fermion-to-qubit mapping that balances locality of occupation and parity information. Produces Pauli strings of length O(log n) instead of O(n) for Jordan-Wigner. Combined with tapering (symmetry reduction), can reduce qubit count — our H2 paper replication uses BK to get from 4 to 2 qubits.

Classical post-processing techniques to reduce the effect of noise without full error correction. Includes zero-noise extrapolation (run at multiple noise levels, extrapolate to zero), probabilistic error cancellation, and readout error correction. Practical for near-term devices. We tested 15+ techniques — TREX and REM+PS achieved chemical accuracy.

IBM's readout error mitigation technique. Randomizes the measurement basis across shots to average out readout bias, then classically corrects the expectation values. Available via IBM's Estimator API at resilience_level=1. Achieved 0.22 kcal/mol on H2 VQE — the best hardware result in our experiments.

Discarding measurement shots that violate a known symmetry. For H2 VQE, the ground state has odd parity (one qubit in |0>, one in |1>), so even-parity shots (|00> or |11> in Z-basis) are noise and can be thrown away. Keeps 95-97% of shots on good qubit pairs. Simple, effective, but only works when you know the symmetry.

Calibrating a confusion matrix (how often |0> is read as 1 and vice versa), then applying its inverse to correct measurement distributions. Requires separate calibration circuits. Most effective when combined with post-selection: apply REM first to correct readout bias, then post-select on parity. Our REM+PS achieved 0.92 kcal/mol on Tuna-9.

An error mitigation technique that intentionally amplifies gate noise (by repeating gates), measures at multiple noise levels, and extrapolates to the zero-noise limit. Effective when gate noise dominates. We found ZNE ineffective on both Tuna-9 and IBM for shallow VQE circuits because >80% of error is readout, not gates.

Inserting sequences of identity-equivalent gate pairs during idle periods to refocus unwanted interactions with the environment. Effective against low-frequency noise and crosstalk. On IBM, adding DD to TREX made results worse for our 3-gate VQE circuit — the overhead exceeded the benefit.

Encoding logical qubits across multiple physical qubits to detect and correct errors. The surface code is the leading approach, requiring ~1000 physical qubits per logical qubit. The threshold theorem guarantees reliability if physical error rates are below a threshold (~1%).

A symmetry reduction technique that identifies conserved quantities (like electron number or spin parity) and uses them to eliminate qubits from the Hamiltonian. Our Sagastizabal replication uses BK + tapering to reduce H2 from 4 qubits to 2.

The periodic oscillation of a qubit between |0> and |1> under a resonant driving field. The Rabi frequency depends on the drive amplitude. Observing clean Rabi oscillations is a basic calibration check for qubit control. Detuning and dephasing modify the oscillation pattern.