In quantum computing and the quantum circuit model, a quantum gate is a basic circuit operation acting on a small number of qubits. Quantum gates are the building blocks of quantum circuits, in the same way that classical logic gates are building blocks of ordinary digital circuits.^[1]

Unlike many classical logic gates, quantum gates are reversible. They are represented mathematically by unitary matrices, so their action preserves the norm of a quantum state and has an inverse operation.^[2] This reversibility is one of the main differences between classical digital logic and quantum logic.

A gate acting on $n$ qubits is represented by a $2^{n} \times 2^{n}$ unitary matrix. The state on which it acts is a vector in a complex Hilbert space. In the usual computational basis, qubit basis states are written as $| 0 ⟩$ and $| 1 ⟩$ , while multi-qubit states are written as tensor products such as $| 00 ⟩$ , $| 01 ⟩$ , $| 10 ⟩$ , and $| 11 ⟩$ .^[3]

History

The modern notation for quantum gates was developed by many founders of quantum information science, including Adriano Barenco, Charles Bennett, Richard Cleve, David DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John Smolin, and Harald Weinfurter.^[4] Their work built on earlier ideas about quantum mechanical computers introduced by Richard Feynman.^[5]

Representation

A single qubit may be written as

| a ⟩ = v_{0} | 0 ⟩ + v_{1} | 1 ⟩ \to [\begin{matrix} v_{0} \\ v_{1} \end{matrix}] .

Here $v_{0}$ and $v_{1}$ are complex probability amplitudes. Their squared magnitudes determine the probabilities of measuring the qubit as $| 0 ⟩$ or $| 1 ⟩$ .

The computational basis states are

| 0 ⟩ = [\begin{matrix} 1 \\ 0 \end{matrix}], | 1 ⟩ = [\begin{matrix} 0 \\ 1 \end{matrix}] .

For two qubits, the state is written as

| ψ ⟩ = v_{00} | 00 ⟩ + v_{01} | 01 ⟩ + v_{10} | 10 ⟩ + v_{11} | 11 ⟩ .

A quantum gate transforms a state vector by matrix multiplication:

U | ψ_{1} ⟩ = | ψ_{2} ⟩ .

Because $U$ is unitary, the total probability is conserved.^[6]

Relation to time evolution

A quantum gate can also be interpreted as a controlled period of quantum time evolution. The Schrödinger equation describes how an unobserved quantum system evolves in time. If the Hamiltonian is constant, this evolution is represented by a unitary time-evolution operator. In circuit language, the same mathematical operation is treated as a quantum gate.^[7]

Common gates

Identity gate

The identity gate leaves a qubit unchanged:

I = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] .

It is useful when describing larger circuits or tensor products in which one qubit is left untouched.

Pauli gates

The Pauli gates $X$ , $Y$ , and $Z$ act on a single qubit and correspond to rotations about the axes of the Bloch sphere.

X = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}], Y = [\begin{matrix} 0 & - i \\ i & 0 \end{matrix}], Z = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] .

The Pauli- $X$ gate is the quantum analogue of the classical NOT gate: it maps $| 0 ⟩$ to $| 1 ⟩$ and $| 1 ⟩$ to $| 0 ⟩$ . The Pauli- $Z$ gate changes the phase of $| 1 ⟩$ while leaving $| 0 ⟩$ unchanged.

Hadamard gate

The Hadamard gate creates equal superpositions from computational basis states:

H = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}] .

It maps

| 0 ⟩ \mapsto \frac{| 0 ⟩ + | 1 ⟩}{\sqrt{2}}, | 1 ⟩ \mapsto \frac{| 0 ⟩ - | 1 ⟩}{\sqrt{2}} .

The Hadamard gate is central to many quantum algorithms because it prepares superposition states.^[8]

Phase shift gates

Phase shift gates change the relative phase of a qubit without changing the measurement probabilities in the computational basis:

P (φ) = [\begin{matrix} 1 & 0 \\ 0 & e^{i φ} \end{matrix}] .

Important examples include the $S$ gate, the $T$ gate, and the Pauli- $Z$ gate.

Controlled gates

Controlled gates act on two or more qubits. One qubit acts as a control and determines whether an operation is applied to another qubit.^[9]

The controlled-NOT gate, or CNOT gate, applies $X$ to the target qubit when the control qubit is $| 1 ⟩$ . In the basis $| 00 ⟩, | 01 ⟩, | 10 ⟩, | 11 ⟩$ , it is represented by

C N O T = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}] .

CNOT is essential for creating entanglement and for building multi-qubit algorithms.

Swap gate

The SWAP gate exchanges two qubits:

S W A P = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] .

It is useful in hardware architectures where only nearby qubits can interact.

Toffoli gate

The Toffoli gate, also called the controlled-controlled-NOT gate, acts on three qubits. It flips the third qubit if the first two qubits are both $| 1 ⟩$ . It is universal for reversible classical computation and has a direct quantum version.^[10]

Universal quantum gates

A universal quantum gate set is a set of gates from which any quantum operation can be approximated to arbitrary precision. Common universal sets include single-qubit rotations together with CNOT, or the Clifford gates supplemented by the $T$ gate.^[11]

The Solovay–Kitaev theorem shows that, under suitable conditions, arbitrary quantum gates can be efficiently approximated using a finite universal set.^[12]

Circuit composition

Quantum gates can be composed in series or in parallel.

If gate $A$ is followed by gate $B$ , the combined operation is

C = B A .

The order is reversed compared with the visual left-to-right reading of some circuit diagrams because matrices act on state vectors from the right.

Parallel composition is represented by the tensor product. If one gate acts on one qubit and another gate acts on a different qubit, the combined operation is written as

A \otimes B .

Tensor products are also used to extend a single-qubit gate so that it acts on one qubit within a larger multi-qubit state.

Measurement

Measurement is not a quantum gate because it is generally irreversible. A measurement projects a quantum state onto one of the basis states, with probabilities determined by the squared magnitudes of the probability amplitudes.^[13]

For a qubit

a | 0 ⟩ + b | 1 ⟩,

the probability of measuring $| 0 ⟩$ is $| a |^{2}$ , and the probability of measuring $| 1 ⟩$ is $| b |^{2}$ .

Measurement plays a central role in quantum algorithms, but it is separated conceptually from gate evolution.

Quantum gates and entanglement

Multi-qubit gates such as CNOT can create entanglement. For example, applying a Hadamard gate to the first qubit of $| 00 ⟩$ , followed by CNOT, gives the Bell state

\frac{| 00 ⟩ + | 11 ⟩}{\sqrt{2}} .

This state cannot be written as a tensor product of two independent single-qubit states. Entanglement produced by quantum gates is essential for quantum teleportation, quantum error correction, and many quantum algorithms.^[14]

Logic synthesis

Quantum algorithms are built from sequences of gates. More complex unitary transformations can be synthesized from elementary gates. In practice, quantum programming languages and circuit libraries provide standard gate sets and decomposition tools.^[15]

Because quantum gates are unitary, quantum functions must be reversible. Irreversible classical logic can be embedded into reversible quantum circuits by adding ancilla qubits and preserving enough information to reconstruct the input.

Applications

Quantum gates are used in:

They form the operational language of the quantum circuit model.

Table of contents (49 articles)

Index

Methods and tools

1. Mathematical methods

2. Measurement techniques

3. Experimental methods

4. Quantum information methods

Advanced methods

5. Computational methods

6. Statistical and thermodynamic methods

7. Field and many-body methods

8. Plasma and kinetic methods

Full contents

1. Mathematical methods (6) ↑ Back to index

1. Physics:Quantum methods/operator

2. Physics:Quantum methods/approximation

3. Physics:Quantum methods/linear algebra

4. Physics:Quantum methods/equation

5. Physics:Quantum methods/basis

6. Physics:Quantum methods/transformation

2. Measurement techniques (7) ↑ Back to index

7. Physics:Quantum Cascade Detector

8. Physics:Quantum methods/measurement

9. Physics:Quantum methods/observable

10. Physics:Quantum methods/projection

11. Physics:Quantum methods/uncertainty

12. Physics:Quantum methods/detector

13. Physics:Quantum methods/signal

3. Experimental methods (6) ↑ Back to index

14. Physics:Quantum methods/experiment

15. Physics:Quantum methods/optics

16. Physics:Quantum methods/laser

17. Physics:Quantum methods/interference

18. Physics:Quantum methods/beam splitter

19. Physics:Quantum methods/spectroscopy

4. Quantum information methods (6) ↑ Back to index

20. Physics:Quantum methods/information theory

21. Physics:Quantum methods/quantum gate

22. Physics:Quantum methods/quantum circuit

23. Physics:Quantum methods/entanglement

24. Physics:Quantum methods/error correction

25. Physics:Quantum methods/communication

5. Computational methods (6) ↑ Back to index

26. Physics:Quantum methods/simulation

27. Physics:Quantum methods/algorithm

28. Physics:Quantum methods/qubit

29. Physics:Quantum methods/noise

30. Physics:Quantum methods/optimization

31. Physics:Quantum methods/sampling

6. Statistical and thermodynamic methods (6) ↑ Back to index

32. Physics:Quantum methods/statistics

33. Physics:Quantum methods/distribution

34. Physics:Quantum methods/ensemble

35. Physics:Quantum methods/thermodynamics

36. Physics:Quantum methods/entropy

37. Physics:Quantum methods/equilibrium

7. Field and many-body methods (6) ↑ Back to index

38. Physics:Quantum methods/field theory

39. Physics:Quantum methods/quantization

40. Physics:Quantum methods/renormalization

41. Physics:Quantum methods/perturbation

42. Physics:Quantum methods/correlation

43. Physics:Quantum methods/many-body

8. Plasma and kinetic methods (6) ↑ Back to index

44. Physics:Quantum kinetic theory

45. Physics:Quantum Vlasov equation

46. Physics:Quantum Magnetohydrodynamics

47. Physics:Quantum Transport theory

48. Physics:Quantum Drift

49. Physics:Quantum collision operator

References

↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information (10th anniversary ed.). Cambridge University Press. ISBN 978-1-10700-217-3.
↑ Williams, Colin P. (2011). Explorations in Quantum Computing. Springer. ISBN 978-1-84628-887-6.
↑ Yanofsky, Noson S.; Mannucci, Mirco (2013). Quantum Computing for Computer Scientists. Cambridge University Press. ISBN 978-0-521-87996-5.
↑ Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A. et al. (1995). "Elementary gates for quantum computation". Physical Review A 52 (5): 3457–3467. doi:10.1103/PhysRevA.52.3457.
↑ Feynman, Richard P. (1986). "Quantum mechanical computers". Foundations of Physics 16 (6): 507–531. doi:10.1007/BF01886518.
↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 978-1-10700-217-3.
↑ Williams, Colin P. (2011). Explorations in Quantum Computing. Springer. ISBN 978-1-84628-887-6.
↑ Yanofsky, Noson S.; Mannucci, Mirco (2013). Quantum Computing for Computer Scientists. Cambridge University Press. ISBN 978-0-521-87996-5.
↑ Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A. et al. (1995). "Elementary gates for quantum computation". Physical Review A 52 (5): 3457–3467. doi:10.1103/PhysRevA.52.3457.
↑ Template:Cite arXiv
↑ Williams, Colin P. (2011). Explorations in Quantum Computing. Springer. ISBN 978-1-84628-887-6.
↑ Dawson, Christopher M.; Nielsen, Michael A. (2006). "The Solovay-Kitaev algorithm". Quantum Information and Computation 6 (1). doi:10.26421/QIC6.1-6.
↑ Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson. ISBN 978-1-107-18963-8.
↑ Yin, Juan (2017). "Satellite-based entanglement distribution over 1200 kilometers". Science 356 (6343): 1140–1144. doi:10.1126/science.aan3211.
↑ "Circuit Library". IBM Qiskit. https://docs.quantum.ibm.com/api/qiskit/circuit_library.

Author: Harold Foppele

Source attribution: Physics:Quantum methods/quantum gate

[1] Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information (10th anniversary ed.). Cambridge University Press. ISBN 978-1-10700-217-3.

[2] Williams, Colin P. (2011). Explorations in Quantum Computing. Springer. ISBN 978-1-84628-887-6.

[3] Yanofsky, Noson S.; Mannucci, Mirco (2013). Quantum Computing for Computer Scientists. Cambridge University Press. ISBN 978-0-521-87996-5.

[4] Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A. et al. (1995). "Elementary gates for quantum computation". Physical Review A 52 (5): 3457–3467. doi:10.1103/PhysRevA.52.3457.

[5] Feynman, Richard P. (1986). "Quantum mechanical computers". Foundations of Physics 16 (6): 507–531. doi:10.1007/BF01886518.

[6] Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 978-1-10700-217-3.

[7] Williams, Colin P. (2011). Explorations in Quantum Computing. Springer. ISBN 978-1-84628-887-6.

[8] Yanofsky, Noson S.; Mannucci, Mirco (2013). Quantum Computing for Computer Scientists. Cambridge University Press. ISBN 978-0-521-87996-5.

[9] Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A. et al. (1995). "Elementary gates for quantum computation". Physical Review A 52 (5): 3457–3467. doi:10.1103/PhysRevA.52.3457.

[10] Template:Cite arXiv

[11] Williams, Colin P. (2011). Explorations in Quantum Computing. Springer. ISBN 978-1-84628-887-6.

[12] Dawson, Christopher M.; Nielsen, Michael A. (2006). "The Solovay-Kitaev algorithm". Quantum Information and Computation 6 (1). doi:10.26421/QIC6.1-6.

[13] Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson. ISBN 978-1-107-18963-8.

[14] Yin, Juan (2017). "Satellite-based entanglement distribution over 1200 kilometers". Science 356 (6343): 1140–1144. doi:10.1126/science.aan3211.

[15] "Circuit Library". IBM Qiskit. https://docs.quantum.ibm.com/api/qiskit/circuit_library.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

Physics:Quantum methods/quantum gate

Contents

History

Representation

Relation to time evolution