The document discusses quantum gates, which are essential components of quantum circuits used to perform operations on qubits. It categorizes quantum gates into single and multiple qubit types, detailing various gates such as Hadamard, Pauli, and controlled gates, along with their functions and mathematical representations. Additionally, the text explains specific gates and their effects on quantum states, highlighting their importance in quantum computing.

1. Quantum
Gates
By: Iqra Naz

2. • Single Qubit Representation
• Quantum Logic Gates
• Gates Notation
• Gates Matrices
• Gate’s Bloch Sphere Representation
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4. A quantum circuit is a computational routine
consisting of coherent quantum operations
on quantum data, such as qubits. It is an
ordered sequence of quantum gates,
measurements, and resets, which may be
conditioned on real-time classical
computation.
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5. • They are the building blocks of quantum
circuits.
• They are basic quantum circuits operating on
a small number of qubits.
• Quantum logic gates are reversible.
• Quantum logic gates are represented
by unitary matrices.
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7. We can categorize Quantum Gates in to 2
categories:
• Single Qubit Quantum Gates
• Multiple Qubits Quantum Gates
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8. • Hadamard Gate
• Pauli Gates
• Phase Shifter (Rɸ) Gates
• I, S, T Gates
• General U Gates
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9. • The Hadamard gate is particularly important.
• It can be used to create a superposition of the
|0〉 and |1〉 states.
• It creates a rotation of around the x-axis by π
radians (180°) followed by a (clockwise)
rotation around the y-axis by π/2 radians (90°)
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11. • The three most basic single-qubit gates
are:
• Pauli-X Gate
• Pauli-Y Gate
• Pauli-Z Gate
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12. • The X gate is directly analogous to the classical
NOT gate.
• It transforms |0〉 to |1〉 and |1〉 to |0〉.
• In terms of the Bloch Sphere, this is equivalent
to rotating around the x axis by π radians
(180°).
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13. • Similarly to Pauli-X gate, the Y gate represents
a rotation of around the y axis by π radians.
• It transforms |0〉 to i|1〉 and |1〉 to -i|0〉.
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14. • The Z gate is actually a special case of the
phase shift gate where 𝜙 = π = 180°.
• It represents a rotation of around the z axis
by π radians.
• It has no effect on |0〉 but transforms |1〉 to -
|1〉.
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15. • This is a family of single-qubit gates that leave the
basis state |0> unchanged and map |1> to
e^iɸ|1>.
• The probability of measuring a|0> or |1> is
unchanged after applying this gate, however it
modifies the phase of the quantum state.
• This is equivalent to tracing a horizontal circle (a
line of latitude) on the Bloch sphere by ɸ radians.
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16. • This is simply a gate that does nothing. Its matrix
is the identity matrix.
• circuit should have no effect on the qubit state.
• It is often used in calculations, for example:
proving the X-gate is its own inverse:
• It is considering real hardware to specify a ‘do-
nothing’ or ‘none’ operation.
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17. • The Phase gate ( S gate) is a single-qubit
operation.
• The S gate is also known as the phase gate or
the Z90 gate, because it represents a 90-
degree rotation around the z-axis.
• The conjugate transpose of S gate is S† gate.
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18. • The T-gate is a very commonly used gate, it is
an Rϕ-gate with ɸ = π/4.
• As with the S-gate, the T-gate is sometimes
also known as (Z)^(1/4) gate.
• The conjugate transpose of T gate is T† gate.
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19. • I, Z, S & T-gates were all special cases of the
more general Rϕ-gate.
• The U3-gate is the most general of all single-
qubit quantum gates.
• It is a parameterized gate of the form:
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20. • Swap Gate
• Controlled Gates
• Toffoli (CCNOT) Gate
• Fredkin (CSWAP) gate
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21. • The swap gate swaps two qubits.
• With respect to the basis |00> |01>, |10>,
|11>.
• It is represented by the matrix:
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22. • Controlled gates require at least one control
and one target qubit the gate in question will
only operate on the target qubit if the control
qubit is in a specific state.
• These are:
• Cx Gate
• Cy Gate
• Cz Gate
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23. • Cx leaves the control qubit unchanged and
performs a Pauli-X gate on the target qubit
when the control qubit is in state |1⟩, leaves
the target qubit unchanged when the control
qubit is in state ∣0⟩.
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24. • The controlled or Cy gate is another well-used
two-qubit gate.
• Just as the CNOT applies an Y to its target
qubit whenever its control is in state |1⟩, the
controlled-Y applies a Y in the same case.
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25. • The controlled-Z or cz gate is another well-
used two-qubit gate.
• Just as the CNOT applies an X to its target
qubit whenever its control is in state |1⟩, the
controlled-Z applies a Z in the same case.
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26. • The Toffoli gate is a three qubit gate with two
controls and one target.
• It performs an X on the target only if both
controls are in the state |1⟩.
• The final state of the target is then equal to either
the AND or the NAND of the two controls,
depending on whether the initial state of the
target was |0⟩ or |1⟩.
• A Toffoli can also be thought of as a controlled-
controlled-NOT, and is also called the CCX gate.
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27. • It is controlled-SWAP gate.
• It only swaps the values of the second and
third bits only if the first bit is set to 1.
• It is also a reversible gate.
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29. • (https://qiskit.org/textbook/ch-algorithms/defining-quantum-
circuits.html)
• Jun Zhang, Jiri Vala, Shankar Sastry, and K. Birgitta Whaley.
Optimal quantum circuit synthesis from controlled-unitary
gates.
• J. A. Jones, R. H. Hansen, and M. Mosca. Quantum logic gates
and nuclear magnetic resonance pulse sequences. Journal of
Magnetic Resonance
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