# Quantum logic gate

A quantum logic gate is a reversible unitary operation applied to one or more qubits. Mathematically, a single-qubit gate is a 2×2 matrix U satisfying U†U = I, where U† is the adjoint of U; remarkably, this unitarity constraint is the only constraint on quantum gates, so any unitary matrix specifies a valid quantum gate<sup>[1](https://www.michaelnielsen.org/qcqi/QINFO-book-nielsen-and-chuang-toc-and-chapter1-nov00.pdf)</sup>.

| Key fact | Detail |
|---|---|
| Defining property | Gates are unitary transformations and hence invertible; classical gates like AND are not invertible<sup>[2](https://ilorentz.org/quantumcomputers/literature/preskill_5_update.pdf)</sup> |
| Universal sets | All one-qubit gates plus CNOT is universal; finite sets such as {H, CNOT, T} are approximately universal<sup>[3](https://arxiv.org/abs/quant-ph/9503016v1)</sup><sup> • </sup><sup>[4](https://learn.microsoft.com/en-us/azure/quantum/concepts-tfactories)</sup> |
| Two-qubit fidelities (2025) | About 99.4% superconducting, 99.3–99.99% trapped ion, 99.5% neutral atom<sup>[5](https://www.mdpi.com/2076-3417/15/5/2675)</sup><sup> • </sup><sup>[6](https://www.ionq.com/blog/accelerating-towards-fault-tolerance-unlocking-99-99-two-qubit-gate)</sup> |
| Gate speeds | Superconducting 25–40 ns; trapped-ion single-qubit about 1 µs and two-qubit 10–100 µs; neutral-atom Rydberg gates 0.1–1 µs<sup>[7](https://arxiv.org/html/2507.12024)</sup><sup> • </sup><sup>[8](https://inspirehep.net/files/f446859b63ee30ad3ac8702288fc4b60)</sup> |
| Best single-qubit error | 1.5(4)×10⁻⁷ per Clifford gate on a trapped-ion clock qubit, gate times 4.4–35 µs<sup>[9](https://journals.aps.org/prl/abstract/10.1103/42w2-6ccy)</sup> |
| Error-correction milestone | Google's Willow distance-7 surface code: logical error 0.143% ± 0.003 per cycle, error suppression Λ = 2.14 ± 0.02, logical memory 2.4 ± 0.3 times the lifetime of its best physical qubit<sup>[10](https://www.nature.com/articles/s41586-024-08449-y)</sup> |
| Largest algorithm estimate | Factoring RSA-2048: about 6.5×10⁹ Toffoli gates on superconducting qubits (Gidney 2025) or 3×10⁹ on neutral atoms (Zhou 2025)<sup>[7](https://arxiv.org/html/2507.12024)</sup> |

## How quantum gates differ from classical logic gates

A classical logic gate such as AND maps input bits to an output bit irreversibly: knowing the output does not determine the input. [Quantum logic](https://www.edgepedia.org/quantum-logic) gates, by contrast, are unitary transformations and hence invertible<sup>[2](https://ilorentz.org/quantumcomputers/literature/preskill_5_update.pdf)</sup>.

Reversibility has a cost in circuit construction. Because hardware qubits are physical systems with particular coupling constraints, only certain gates can be applied to certain groups of hardware qubits<sup>[11](https://quantum.cloud.ibm.com/docs/guides/transpiler-stages)</sup>. To implement a two-qubit gate between qubits that are not directly connected, the transpiler inserts SWAP gates, each a product of three CNOT gates, an expensive operation on noisy devices<sup>[11](https://quantum.cloud.ibm.com/docs/guides/transpiler-stages)</sup>. Multi-qubit gates must also be decomposed: for every [Toffoli gate](https://www.edgepedia.org/toffoli-gate) in a circuit, the hardware may execute up to six CNOT gates and a handful of single-qubit gates, increasing depth and noise exposure<sup>[11](https://quantum.cloud.ibm.com/docs/guides/transpiler-stages)</sup>.

## Single-qubit and multi-qubit gates

[Single-qubit gates](https://www.edgepedia.org/single-qubit-gates) act on the qubit's state vector by matrix multiplication. Any single-qubit unitary operation is a valid gate because any unitary matrix is allowed<sup>[1](https://www.michaelnielsen.org/qcqi/QINFO-book-nielsen-and-chuang-toc-and-chapter1-nov00.pdf)</sup>.

Two- and three-qubit gates can create entanglement and implement conditional logic. Decomposition costs differ by hardware: the Toffoli gate can be implemented with five entangling gates instead of six CNOTs when the square root of CNOT is available, as with the trapped-ion XX gate<sup>[12](https://www.pnas.org/doi/10.1073/pnas.1618020114)</sup>.

## Universal gate sets and gate synthesis

A gate set is universal if arbitrary unitary operations on any number of qubits can be built from it. Barenco and colleagues showed in 1995 that all one-qubit gates together with the two-bit exclusive-or gate (CNOT) are universal in this sense<sup>[3](https://arxiv.org/abs/quant-ph/9503016v1)</sup>. Deutsch had earlier shown that almost every gate operating on two or more bits is a universal gate<sup>[13](https://royalsocietypublishing.org/doi/10.1098/rspa.1995.0065)</sup>.

**Exact versus approximate universality.** Exact universality with a finite set is impossible because the set of unitaries is uncountable; the set {CNOT, all single-qubit unitaries} is exactly universal, while finite sets such as {H, R(π/2), Toffoli} and {CNOT, H, R(π/4)} are approximately universal<sup>[14](https://www.cs.umd.edu/class/fall2024/cmsc657/Lecture-6.pdf)</sup>. The Hadamard, CNOT, and T gates form an approximately universal set from which any quantum computation on any number of qubits can be approximated, and H and T alone can approximate any single-qubit gate to arbitrary accuracy<sup>[4](https://learn.microsoft.com/en-us/azure/quantum/concepts-tfactories)</sup>. Preskill's notes list several finite universal sets, including {H, Λ(S)}, {H, T, Λ(X)}, and {H, S, Λ²(X)}<sup>[2](https://ilorentz.org/quantumcomputers/literature/preskill_5_update.pdf)</sup>.

The [Solovay–Kitaev theorem](https://www.edgepedia.org/solovay-kitaev-theorem) makes approximate universality practical: for any approximately universal gate set containing inverses, a classical algorithm outputs a circuit approximating a target unitary to accuracy ε using a number of gates polynomial in log(1/ε), though exponential in the number of qubits<sup>[14](https://www.cs.umd.edu/class/fall2024/cmsc657/Lecture-6.pdf)</sup>. In fault-tolerant machines, T states for the non-Clifford [T gate](https://www.edgepedia.org/t-gate) are produced by distillation factories, which run rounds taking many noisy T states encoded in smaller-distance codes and output fewer, less noisy T states in larger-distance codes; resource estimators calculate the total T states an algorithm needs, and the physical qubits and runtime of a single T factory<sup>[4](https://learn.microsoft.com/en-us/azure/quantum/concepts-tfactories)</sup>. Compilers also trade accuracy for gate count: IBM's approximation-degree parameter (0.0 to 1.0) can synthesize a circuit with one or two CX gates instead of three, and because CX is the most error-expensive gate, reducing CX count this way was used to increase quantum volume on IBM devices<sup>[15](https://quantum.cloud.ibm.com/docs/en/guides/common-parameters)</sup>.

## Physical implementation and performance

Gates are realized differently on each platform, and the platforms trade speed against fidelity and coherence. Superconducting circuits apply microwave pulses on nanosecond timescales: Willow-class parameters assume 25 ns single-qubit and 40 ns two-qubit gate times with 5×10⁻⁴ single-qubit and 2×10⁻³ two-qubit gate infidelity, against an 80 µs coherence time<sup>[7](https://arxiv.org/html/2507.12024)</sup>. Trapped ions are slower, with single-qubit gates of about 1 µs and two-qubit gates of 10–100 µs, while neutral-atom Rydberg gates are intermediate at roughly 0.1–1 µs<sup>[8](https://inspirehep.net/files/f446859b63ee30ad3ac8702288fc4b60)</sup>. Ion-trap parameters assumed in benchmarking are 1.5 s coherence, 0.5 µs single-qubit and 0.2 µs two-qubit gate times, and 1×10⁻³ and 5×10⁻³ infidelities<sup>[7](https://arxiv.org/html/2507.12024)</sup>.

A head-to-head experiment running identical algorithms on superconducting (IBM) and trapped-ion hardware found higher absolute fidelities and coherence times in the trapped-ion system but higher clock speeds in the superconducting system<sup>[12](https://www.pnas.org/doi/10.1073/pnas.1618020114)</sup>. The same algorithms compile differently: the three-qubit quantum [Fourier transform](https://www.edgepedia.org/fourier-transform) needed 42 single-qubit and 19 two-qubit gates on the star-connected superconducting device versus 8 and 3 on the ion trap<sup>[12](https://www.pnas.org/doi/10.1073/pnas.1618020114)</sup>.

For scalable error correction, physical gate error rates must be below the relevant threshold<sup>[5](https://www.mdpi.com/2076-3417/15/5/2675)</sup>. Typical targets for scalable error correction exceed 99.9% for single-qubit gates and 99% for two-qubit gates; fluxonium-qubit experiments have demonstrated single-qubit fidelities as high as 99.998%<sup>[16](https://link.springer.com/article/10.1007/s44464-026-00033-9)</sup>. For small rotation errors ε, average fidelity scales roughly as 1 − ε²/4 for a one-qubit Pauli-X gate and 1 − ε²/16 for CNOT<sup>[5](https://www.mdpi.com/2076-3417/15/5/2675)</sup>.

## By the numbers

Reported two-qubit fidelities as of 2025 cluster near 99.4% for superconducting gates, 99.3% for trapped ions, and 99.5% for neutral atoms, with single-qubit superconducting and electron-spin gates near 99.9%<sup>[5](https://www.mdpi.com/2076-3417/15/5/2675)</sup>. The best announced two-qubit fidelity is 99.99% from IonQ (via [Oxford Ionics](https://www.edgepedia.org/oxford-ionics), acquired 2024–2025), whose 2024 demonstration reached a two-qubit gate error rate of 3×10⁻⁴<sup>[6](https://www.ionq.com/blog/accelerating-towards-fault-tolerance-unlocking-99-99-two-qubit-gate)</sup>. The lowest measured single-qubit error is 1.5(4)×10⁻⁷ per Clifford gate on a ⁴³Ca⁺ hyperfine clock qubit, with gate times between 4.4 and 35 µs, calibration errors below 10⁻⁸, and a qubit T2 of about 70 seconds at room temperature without magnetic shielding<sup>[9](https://journals.aps.org/prl/abstract/10.1103/42w2-6ccy)</sup>.

At the algorithm scale, factoring RSA-2048 is estimated to require about 6.5×10⁹ Toffoli gates on superconducting qubits and about 3×10⁹ on neutral atoms<sup>[7](https://arxiv.org/html/2507.12024)</sup>.

## Recent developments

The decisive shift has been from physical-qubit records to error-corrected logical qubits operating below threshold. Google's [Willow processor](https://www.edgepedia.org/willow-processor) ran a distance-7 surface code on 101 qubits with a logical error rate of 0.143% ± 0.003 per error-correction cycle, suppressed by a factor Λ = 2.14 ± 0.02 for each distance-2 increase; the logical memory exceeded the lifetime of its best physical qubit by a factor of 2.4 ± 0.3, a beyond-breakeven logical qubit<sup>[10](https://www.nature.com/articles/s41586-024-08449-y)</sup>. Real-time decoding achieved an average decoder latency of 63 µs at distance 5 with a 1.1 µs cycle time, sustained over up to a million cycles, with rare correlated error events limiting logical performance roughly once per hour, about 3×10⁹ cycles<sup>[10](https://www.nature.com/articles/s41586-024-08449-y)</sup>.

Neutral atoms reached comparable territory with a different architecture: reconfigurable arrays of up to 448 atoms implemented key elements of a universal fault-tolerant architecture, demonstrating 2.14(13)× below-threshold error suppression in a four-round surface-code characterization circuit using atom loss detection and machine-learning decoding<sup>[17](https://www.nature.com/articles/s41586-025-09848-5)</sup>. Universal logic ran through transversal teleportation with three-dimensional [[15,1,3]] codes, with arbitrary-angle synthesis at polylogarithmic overhead via the Solovay–Kitaev approach<sup>[17](https://www.nature.com/articles/s41586-025-09848-5)</sup>. The authors estimate these teleportation-based methods are about 10–20 times slower than a conventional planar superconducting architecture with roughly 1 µs stabilizer measurement cycles<sup>[17](https://www.nature.com/articles/s41586-025-09848-5)</sup>.

## Gates in quantum algorithms

A circuit becomes algorithmic when it exploits structure that lets it avoid the exponential cost that afflicts generic unitaries: most unitaries require exponentially many gates in the number of qubits to implement, and determining which do not is the subject of quantum algorithms and complexity<sup>[14](https://www.cs.umd.edu/class/fall2024/cmsc657/Lecture-6.pdf)</sup>. An important limitation is that quantum programs made from only Clifford gates can be simulated efficiently on a classical computer; in this setting, non-Clifford gates such as T are needed to go beyond that efficient-simulation result<sup>[4](https://learn.microsoft.com/en-us/azure/quantum/concepts-tfactories)</sup>.

The gate-based circuit model is not the only route to quantum computation. Photonic proposals rely on measurement-based quantum computing, in which computation proceeds through adaptive measurements on a large, highly entangled cluster state<sup>[16](https://link.springer.com/article/10.1007/s44464-026-00033-9)</sup>. [Quantum annealing](https://www.edgepedia.org/quantum-annealing), represented by D-Wave's Advantage2 family scaling to thousands of physical qubits, targets combinatorial optimization rather than universal gate-model programmability, with limitations that include analog control error and the need for careful benchmarking against classical alternatives<sup>[16](https://link.springer.com/article/10.1007/s44464-026-00033-9)</sup>.

## References

1. [Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information, Chapter 1](https://www.michaelnielsen.org/qcqi/QINFO-book-nielsen-and-chuang-toc-and-chapter1-nov00.pdf)
2. [Preskill, J. Lecture Notes for Ph219/CS219: Quantum Information, Chapter 5](https://ilorentz.org/quantumcomputers/literature/preskill_5_update.pdf)
3. [Barenco, A. et al. (1995). Elementary gates for quantum computation. arXiv](https://arxiv.org/abs/quant-ph/9503016v1)
4. [T gates & T factories, Azure Quantum, Microsoft Learn](https://learn.microsoft.com/en-us/azure/quantum/concepts-tfactories)
5. [Demystifying Quantum Gate Fidelity for Electronics Engineers, Applied Sciences (MDPI)](https://www.mdpi.com/2076-3417/15/5/2675)
6. [IonQ: Accelerating Towards Fault Tolerance: Unlocking 99.99% Two-Qubit Gate Fidelities](https://www.ionq.com/blog/accelerating-towards-fault-tolerance-unlocking-99-99-two-qubit-gate)
7. [Benchmarking fault-tolerant quantum computing hardware via QLOPS, arXiv](https://arxiv.org/html/2507.12024)
8. [Comparative Analysis of Contemporary Quantum Computer Processors, INSPIRE-HEP](https://inspirehep.net/files/f446859b63ee30ad3ac8702288fc4b60)
9. [Single-Qubit Gates with Errors at the 10⁻⁷ Level, Physical Review Letters](https://journals.aps.org/prl/abstract/10.1103/42w2-6ccy)
10. [Quantum error correction below the surface code threshold, Nature (2024)](https://www.nature.com/articles/s41586-024-08449-y)
11. [Transpiler stages, IBM Quantum Documentation](https://quantum.cloud.ibm.com/docs/guides/transpiler-stages)
12. [Experimental comparison of two quantum computing architectures, PNAS](https://www.pnas.org/doi/10.1073/pnas.1618020114)
13. [Deutsch, D. (1989). Universality in quantum computation, Proc. R. Soc. A](https://royalsocietypublishing.org/doi/10.1098/rspa.1995.0065)
14. [Universal quantum gate sets and the Solovay–Kitaev theorem, UMD CMSC657 Lecture 6](https://www.cs.umd.edu/class/fall2024/cmsc657/Lecture-6.pdf)
15. [Commonly used parameters for transpilation, IBM Quantum Documentation](https://quantum.cloud.ibm.com/docs/en/guides/common-parameters)
16. [A study of qubit modalities in contemporary quantum computing, Springer Nature](https://link.springer.com/article/10.1007/s44464-026-00033-9)
17. [A fault-tolerant neutral-atom architecture for universal quantum computation, Nature (2025)](https://www.nature.com/articles/s41586-025-09848-5)

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*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Algorithms and computational methods › Algorithms overview*

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