Fidelity

Fidelity measures how close two quantum states are; it is the primary benchmark for gate and state-preparation quality.

Measuring closeness

Fidelity quantifies how similar two quantum states are, on a scale from 0 (orthogonal, perfectly distinguishable) to 1 (identical). It is the standard yardstick for how well hardware realises an intended state or operation.

Definitions

For a pure target |psi> and an actual state rho, the fidelity is F = , the probability that rho passes a test for |psi>. For two pure states it reduces to F = ||^2, the squared overlap. For two general mixed states there is a more involved formula using the matrix square root, but the intuition is the same: overlap.

Gate fidelity

Beyond states, one measures how well an implemented operation matches an ideal gate. Average gate fidelity compares the noisy channel's output to the ideal unitary's output, averaged over input states. Reported gate fidelities of, say, 0.999 mean a 0.1% error per gate — and errors compound, so deep circuits demand many nines.

How it is measured

• Randomized benchmarking: run random gate sequences of growing length and fit the decay
• State tomography: reconstruct rho fully, then compute F (expensive, scales badly)
• Direct fidelity estimation: sample a few well-chosen observables

Randomized benchmarking is favoured because it isolates gate error from state-preparation and measurement error and scales to many qubits.

Why the number rules everything

Fidelity per gate sets whether error correction can work at all. A code has a threshold; only when physical gate fidelity exceeds it does adding qubits reduce the logical error rate. Every fraction of a nine in fidelity translates into far fewer physical qubits needed per logical qubit, which is why fidelity, not qubit count alone, is the headline metric of hardware progress.