A brief description about Quantum Gates, their specification, their behavior, matrices, notation and their application.

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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