Physics:Quantum Gates and circuits

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Quantum gates are the fundamental operations that act on qubits. They are the building blocks of quantum circuits, which define computations in a quantum computer.[1] [2]


Example of a quantum circuit composed of quantum gates acting on qubits.

Quantum operations

Quantum states evolve according to unitary transformations:[3]

|ψ⟩→U|ψ⟩,

where U is a unitary operator satisfying

U†U=I.

These transformations preserve normalization and correspond to reversible evolution in quantum mechanics.

Single-qubit gates

Single-qubit gates act on individual qubits and correspond to rotations on the Bloch sphere.

Common examples include:

  • Pauli-X gate

X=(0110)

  • Pauli-Y gate

Y=(0−ii0)

  • Pauli-Z gate

Z=(100−1)

  • Hadamard gate

H=12(111−1)

These gates create superposition and control phase relationships.

Multi-qubit gates

Multi-qubit gates act on multiple qubits and can generate entanglement.[4]

A key example is the controlled-NOT (CNOT) gate, which flips a target qubit if the control qubit is in the state |1⟩.

The CNOT gate acts on basis states as:

|00⟩→|00⟩

|01⟩→|01⟩

|10⟩→|11⟩

|11⟩→|10⟩

Together with single-qubit gates, the CNOT gate forms a universal set for quantum computation.[1]

Quantum circuits

A quantum circuit is a sequence of quantum gates applied to a set of qubits.

A typical circuit consists of:

  • initialization of qubits
  • application of unitary gates
  • measurement of the final state

Circuits are usually represented diagrammatically, with time progressing from left to right.

Universal gate sets

A set of quantum gates is called universal if any unitary operation can be approximated to arbitrary accuracy using only gates from that set.[1]

A commonly used universal set consists of:

  • all single-qubit gates
  • the controlled-NOT (CNOT) gate

In practice, finite gate sets such as {H, T, CNOT} are used, where the T gate introduces a nontrivial phase. These sets allow efficient approximation of arbitrary quantum operations.

Circuit depth and complexity

The complexity of a quantum circuit is characterized by measures such as:

  • circuit depth — the number of sequential layers of gates
  • gate count — the total number of gates used

Circuit depth is particularly important, as it determines how long a computation takes and how susceptible it is to noise.

Efficient quantum algorithms aim to minimize both gate count and depth while achieving the desired transformation.

Example: Bell state circuit

Entanglement can be generated using a simple circuit:

1. Start with |0⟩⊗|0⟩ 2. Apply a Hadamard gate to the first qubit 3. Apply a CNOT gate

This produces the entangled state

12(|00⟩+|11⟩).

Noise and decoherence

Real quantum systems are affected by noise and interactions with their environment, leading to decoherence and errors in quantum operations.

Noise limits the size and depth of quantum circuits that can be reliably executed. This is a central challenge in current quantum computing devices.

These limitations define the so-called noisy intermediate-scale quantum (NISQ) regime (see Physics:Quantum Noisy Qubits).

Measurement

Measurement converts a quantum state into classical information.

For a qubit

|ψ⟩=α|0⟩+β|1⟩,

measurement yields:

  • |0⟩ with probability |α|2
  • |1⟩ with probability |β|2

Measurement is irreversible and collapses the quantum state.

Physical significance

Quantum gates and circuits:

  • define how quantum computations are performed
  • enable superposition and entanglement to be controlled
  • provide the framework for quantum algorithms

See also

Table of content (95 articles)

Index

  1. Foundations
  2. Conceptual and interpretations
  3. Mathematical structure and systems
  4. Atomic and spectroscopy
  5. Wavefunctions and modes
  6. Quantum information and computing
  7. Quantum optics and experiments
  8. Open quantum systems
  9. Quantum field theory
  10. Statistical mechanics and kinetic theory
  11. Plasma and fusion physics
  12. Timeline
  13. Advanced and frontier topics

Full contents

1. Foundations (7)
  1. Physics:Quantum basics
  2. Physics:Quantum Postulates
  3. Physics:Quantum Hilbert space
  4. Physics:Quantum Observables and operators
  5. Physics:Quantum mechanics
  6. Physics:Quantum mechanics measurements
  7. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (4)
  1. Physics:Quantum Interpretations of quantum mechanics
  2. Physics:Quantum A Spooky Action at a Distance
  3. Physics:Quantum A Walk Through the Universe
  4. Physics:Quantum: The Secret of Cohesion: How Waves Hold Matter Together

3. Mathematical structure and systems (9)
  1. Physics:Quantum Density matrix
  2. Physics:Quantum Exactly solvable quantum systems
  3. Physics:Quantum Formulas Collection
  4. Physics:Quantum A Matter Of Size
  5. Physics:Quantum Symmetry in quantum mechanics
  6. Physics:Quantum Angular momentum operator
  7. Physics:Runge–Lenz vector
  8. Physics:Quantum Approximation Methods
  9. Physics:Quantum Matter Elements and Particles

4. Atomic and spectroscopy (5)
    Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.
    Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.
  1. Physics:Quantum Atomic structure and spectroscopy
  2. Physics:Quantum Hydrogen atom
  3. Physics:Quantum Selection rules
  4. Physics:Quantum Fermi's golden rule
  5. Physics:Quantum Spectral lines and series

5. Wavefunctions and modes (8)
    A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.
    A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.
  1. Physics:Quantum Wavefunction
  2. Physics:Quantum Superposition principle
  3. Physics:Quantum Eigenstates and eigenvalues
  4. Physics:Quantum Boundary conditions and quantization
  5. Physics:Quantum Standing waves and modes
  6. Physics:Quantum Normal modes and field quantization
  7. Physics:Number of independent spatial modes in a spherical volume
  8. Physics:Quantum Density of states

6. Quantum information and computing (6)
  1. Physics:Quantum information theory
  2. Physics:Quantum Qubit
  3. Physics:Quantum Entanglement
  4. Physics:Quantum Gates and circuits
  5. Physics:Quantum Computing Algorithms in the NISQ Era
  6. Physics:Quantum Noisy Qubits

7. Quantum optics and experiments (4)
    Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.
    Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.
  1. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
  2. Physics:Quantum optics beam splitter experiments
  3. Physics:Quantum Ultra fast lasers
  4. Physics:Quantum Experimental quantum physics
    5. Template:Quantum optics operators

8. Open quantum systems (7)
  1. Physics:Quantum Open systems
  2. Physics:Quantum Master equation
  3. Physics:Quantum Lindblad equation
  4. Physics:Quantum Decoherence
  5. Physics:Quantum Markovian dynamics
  6. Physics:Quantum Non-Markovian dynamics
  7. Physics:Quantum Trajectories

9. Quantum field theory (18)
    Structural dependency map of quantum field theory.
  1. Physics:Quantum field theory (QFT) basics
  2. Physics:Quantum field theory (QFT) core
  3. Physics:Quantum Fields and Particles
  4. Physics:Quantum Second quantization
  5. Physics:Quantum Harmonic Oscillator field modes
  6. Physics:Quantum Creation and annihilation operators
  7. Physics:Quantum vacuum fluctuations
  8. Physics:Quantum Propagators in quantum field theory
  9. Physics:Quantum Feynman diagrams
  10. Physics:Quantum Path integral formulation
  11. Physics:Quantum Renormalization in field theory
  12. Physics:Quantum Renormalization group
  13. Physics:Quantum Field Theory Gauge symmetry
  14. Physics:Quantum Non-Abelian gauge theory
  15. Physics:Quantum Electrodynamics (QED)
  16. Physics:Quantum chromodynamics (QCD)
  17. Physics:Quantum Electroweak theory
  18. Physics:Quantum Standard Model

10. Statistical mechanics and kinetic theory (9)
  1. Physics:Quantum Statistical mechanics
  2. Physics:Quantum Partition function
  3. Physics:Quantum Distribution functions
  4. Physics:Quantum Liouville equation
  5. Physics:Quantum Kinetic theory
  6. Physics:Quantum Boltzmann equation
  7. Physics:Quantum BBGKY hierarchy
  8. Physics:Quantum Transport theory
  9. Physics:Quantum Relaxation and thermalization

11. Plasma and fusion physics (9)
    Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
    Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
  1. Physics:Quantum Plasma (fusion context)
  2. Physics:Quantum Fusion reactions and Lawson criterion
  3. Physics:Quantum Magnetic confinement fusion
  4. Physics:Quantum Inertial confinement fusion
  5. Physics:Quantum Plasma instabilities and turbulence
  6. Physics:Quantum Tokamak
  7. Physics:Quantum Tokamak core plasma
  8. Physics:Quantum Tokamak edge physics and recycling asymmetries
  9. Physics:Quantum Stellarator

12. Timeline (2)
  1. Physics:Quantum mechanics/Timeline
  2. Physics:Quantum_mechanics/Timeline/Quiz/

13. Advanced and frontier topics (6)
  1. Physics:Quantum Supersymmetry
  2. Physics:Quantum Black hole thermodynamics
  3. Physics:Quantum Holographic principle
  4. Physics:Quantum gravity
  5. Physics:Quantum De Sitter invariant special relativity
  6. Physics:Quantum Doubly special relativity

References

  1. ↑ 1.0 1.1 1.2 Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. 
  2. ↑ Williams, Colin P. (2011). Explorations in Quantum Computing. Springer. ISBN 978-1-84628-887-6. 
  3. ↑ Feynman, Richard P. (1986). "Quantum mechanical computers". Foundations of Physics 16 (6): 507–531. doi:10.1007/bf01886518. ISSN 0015-9018. Bibcode: 1986FoPh...16..507F. 
  4. ↑ "UnitaryGate adjoint()". IBM. https://quantum.cloud.ibm.com/docs/api/qiskit/qiskit.circuit.library.UnitaryGate. 


Author: Harold Foppele





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