Two-Qubit Gate

What Is a Two-Qubit Gate?

A two-qubit gate is a quantum operation that acts on two qubits at once, so that what happens to one qubit depends on the state of the other. It is described by a \(4 \times 4\) unitary matrix acting on the joint state of the pair.

Most two-qubit gates of interest are an entangling gate: given the right input, they produce entanglement, a correlation that no sequence of single-qubit gates can create. Common examples:

The controlled NOT gate (CNOT), which flips a target qubit when the control is in \(|1\rangle\).
The controlled-Z (CZ) gate, which changes the sign of the \(|11\rangle\) state only:

\[ \mathrm{CZ} = \mathrm{diag}(1, 1, 1, -1) \]

The iSWAP gate, used on some superconducting devices.

These gates are closely related. A CNOT gate equals a CZ gate with Hadamard gates applied to the target qubit before and after: \(\mathrm{CNOT} = (I \otimes H)\,\mathrm{CZ}\,(I \otimes H)\). Which one a machine uses natively depends on its physics.

Why Entangling Gates Make Computation Universal

Single-qubit gates alone cannot do much. A circuit built only from them leaves a register of unentangled qubits unentangled, and a classical computer can simulate that efficiently.

Adding one entangling gate changes this. Barenco and colleagues showed in 1995 that single-qubit gates together with CNOT can build any quantum circuit, which is why such a set is called a universal gate set. In gate-based quantum computing, algorithms are compiled into layers of single-qubit rotations and two-qubit gates.

There is a practical consequence. The two-qubit gate is usually the slowest and noisiest operation on the device, so the number of two-qubit gates in a circuit is a common measure of its cost and of how much error it will accumulate.

Two-Qubit Gate Fidelity: What It Measures

Two-qubit gate fidelity quantifies how closely the gate that was physically applied matches the ideal one. A fidelity of \(F\) corresponds to an error per gate of roughly \(\epsilon = 1 - F\). Readers will see the same idea under gate fidelity.

Common ways to estimate it:

Randomized benchmarking, which applies long random sequences and fits the decay of success probability.
Bell-state fidelity, which prepares an entangled pair with the gate and compares it to the ideal state.
Process tomography, which reconstructs the full gate but is costly and sensitive to preparation and measurement errors.

Errors compound. If each gate fails independently with probability \(\epsilon\), a circuit of \(N\) gates succeeds with probability about \(e^{-\epsilon N}\). At 99.9% fidelity, 1,000 gates give roughly a 37% chance of an error-free run. Computations with millions of operations, the scale called a megaquop, therefore rely on quantum error correction rather than on raw gate quality alone.

For reference, trapped-ion experiments by Ballance et al. and Gaebler et al. (both 2016) reported two-qubit fidelities of about 99.9%.

How Neutral Atoms Perform a Two-Qubit Gate

In a neutral-atom computer, each qubit is stored in two long-lived internal states of an atom held in an optical tweezer. Atoms do not interact strongly in their ground states, so the entangling interaction is switched on by briefly exciting them to a Rydberg atom state.

The mechanism is the Rydberg blockade. When two atoms are close enough, exciting both at once costs extra energy, roughly \(V = C_6 / r^6\), so a laser that excites one atom cannot also excite its neighbor. Jaksch and colleagues proposed in 2000 a pulse sequence that exploits this: a \(\pi\) pulse on the control, a \(2\pi\) pulse on the target, then a \(\pi\) pulse on the control. The outcome is a CZ gate up to single-qubit phases.

In 2023, Evered and colleagues reported parallel entangling gates on up to 60 atoms with about 99.5% two-qubit fidelity, using optimized laser pulses. Gemini, QuEra's gate-based system on premises since 2025, performs its entangling operations through this blockade mechanism. Because atoms can be moved, gates can in principle connect any pair, and many gates can run at once on separate pairs.

Reading a Fidelity Number Carefully

A headline figure needs context before it can be compared:

Method: Bell-state fidelity, randomized benchmarking and tomography do not measure the same thing.
Parallelism: a gate on one isolated pair differs from gates run simultaneously across an array.
Error type: an error that is detected, such as an atom lost from the trap, is easier to handle than a silent one.
Everything else: measurement, idle decoherence and single-qubit gate errors also count toward a circuit's success.

Fidelity is necessary, but a useful computer also needs enough qubits, connectivity, speed and error correction. Compare machines on the full set rather than on one number.

FAQ

What is the difference between a single-qubit gate and a two-qubit gate?

A single-qubit gate rotates one qubit independently of the others. A two-qubit gate acts on a pair, so the outcome on one qubit depends on the other. Single-qubit gates cannot create entanglement; it takes a gate that acts on two or more qubits, such as an entangling two-qubit gate.

Why is two-qubit gate fidelity usually lower than single-qubit fidelity?

Entangling gates need a physical interaction between two qubits, which exposes both to more noise and control error. They are also typically slower than single-qubit rotations, leaving more time for decoherence. For this reason they usually set the error budget of a circuit.

Is CNOT the only two-qubit gate a quantum computer needs?

No. CNOT with single-qubit gates is universal, but so are CZ and several others. Hardware usually implements whichever is most natural, such as CZ with the Rydberg blockade, and the compiler converts circuits to it.

Is 99.9% two-qubit gate fidelity enough for useful quantum computing?

On its own, no for long circuits. An error rate of 1 in 1,000 limits uncorrected circuits to roughly a thousand gates before failure becomes likely. Larger computations combine good physical gates with error correction.

Key Takeaways

A two-qubit gate acts on a pair of qubits and, when entangling, creates correlations that single-qubit gates cannot.
CNOT and CZ are interconvertible with single-qubit gates, and iSWAP can be built from them (and vice versa) with a small number of gates; a universal gate set needs only one entangling gate.
Two-qubit gate fidelity depends on how it is measured, so check the method and whether gates ran in parallel.
Neutral-atom systems such as Gemini implement entangling gates with the Rydberg blockade, and Evered et al. (2023) reported about 99.5% fidelity in parallel gates.