Qiskit - Revision history

Mol: /* Theory */

2020-11-27T14:16:40Z

Theory

← Older revision		Revision as of 17:16, 27 November 2020
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	[[Grover's Algorithm]]		[[Grover's Algorithm]]

			[[qRAM]]

	== Exercises ==		== Exercises ==

Mol: /* Theory */

2020-11-26T13:55:30Z

Theory

← Older revision		Revision as of 16:55, 26 November 2020
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	Clifford Gates		Clifford Gates

	[Grover's Algorithm]		[[Grover's Algorithm]]

	== Exercises ==		== Exercises ==

Mol: /* Theory */

2020-11-26T13:51:59Z

Theory

← Older revision		Revision as of 16:51, 26 November 2020
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	[https://www.cod3v.info/index.php?title=Quantum_Gates Quantum Gates ]		[https://www.cod3v.info/index.php?title=Quantum_Gates Quantum Gates ]

	~~===~~ Clifford Gates ~~===~~		Clifford Gates

	~~<syntaxhighlight>~~		[Grover's Algorithm]
	~~</syntaxhighlight>~~

	~~<syntaxhighlight>~~
	~~</syntaxhighlight>~~

	~~<syntaxhighlight>~~
	~~</syntaxhighlight>~~

	~~<syntaxhighlight>~~
	~~</syntaxhighlight>~~

	~~<syntaxhighlight>~~
	~~</syntaxhighlight>~~

	== Exercises ==		== Exercises ==

Mol: /* Theory */

2020-11-26T08:34:12Z

Theory

← Older revision		Revision as of 11:34, 26 November 2020
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	Quantum operations are reversible, thus the reversible computing. That makes some ''complications'' to the gate design.		Quantum operations are reversible, thus the reversible computing. That makes some ''complications'' to the gate design.

	[[https://www.cod3v.info/index.php?title=Quantum_Gates \| Quantum Gates ]]		[https://www.cod3v.info/index.php?title=Quantum_Gates Quantum Gates ]

	=== Clifford Gates ===		=== Clifford Gates ===

Mol: /* Theory */

2020-11-26T08:33:59Z

Theory

← Older revision		Revision as of 11:33, 26 November 2020
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	Quantum operations are reversible, thus the reversible computing. That makes some ''complications'' to the gate design.		Quantum operations are reversible, thus the reversible computing. That makes some ''complications'' to the gate design.

	https://www.cod3v.info/index.php?title=Quantum_Gates		[[https://www.cod3v.info/index.php?title=Quantum_Gates \| Quantum Gates ]]

	=== Clifford Gates ===		=== Clifford Gates ===

Mol: /* Theory */

2020-11-26T08:33:36Z

Theory

← Older revision		Revision as of 11:33, 26 November 2020
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	Quantum operations are reversible, thus the reversible computing. That makes some ''complications'' to the gate design.		Quantum operations are reversible, thus the reversible computing. That makes some ''complications'' to the gate design.

			https://www.cod3v.info/index.php?title=Quantum_Gates

	=== Clifford Gates ===		=== Clifford Gates ===

Mol: /* Three Qubits */

2020-11-26T08:33:11Z

Three Qubits

← Older revision		Revision as of 11:33, 26 November 2020
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	Quantum operations are reversible, thus the reversible computing. That makes some ''complications'' to the gate design.		Quantum operations are reversible, thus the reversible computing. That makes some ''complications'' to the gate design.

	~~=== Three Qubits ===~~

	~~[[File:Toffoli.svg\|thumb\|Toffoli gate made using CNOTs.]]~~


	~~For universal computations we need more qubits. Eg. the AND gate is not reversible, and thus we need eg. Toffoli (CCNOT) gate.~~

	~~Toffoli gate performs <math>X</math> on target qubit if both control cubits are set to state <math>\|1\rangle</math>.~~

	~~<syntaxhighlight>~~
	~~qc = QuantumCircuit(3)~~
	~~a = 0~~
	~~b = 1~~
	~~t = 2~~
	~~# Toffoli with control qubits a and b and target t~~
	~~qc.ccx(a,b,t)~~
	~~qc.draw()~~
	~~</syntaxhighlight>~~


	~~Toffoli using CNOTs uses fewer gates.~~
	~~<syntaxhighlight>~~
	~~qc = QuantumCircuit(3)~~
	~~qc.ch(a,t)~~
	~~qc.cz(b,t)~~
	~~qc.ch(a,t)~~
	~~qc.draw()~~
	~~</syntaxhighlight>~~

	~~AND gate is Toffoli gate with . . .~~

	~~[[File:Nand.svg\|thumb\|The nand gate]]~~

	~~<math>\text{CCNOT}(x,y,z) = (x,y,( x \and y ) \otimes z )</math> gives the reversible NAND <math>\text{NAND}(x,y) = \text{CCNOT}(x,y,1) = (x,y,( x \and y ) \otimes 1 )</math>~~


	~~NAND gate is~~

	=== Clifford Gates ===		=== Clifford Gates ===

Mol: /* Two Qubit Quantum Gates */

2020-11-26T08:32:24Z

Two Qubit Quantum Gates

← Older revision		Revision as of 11:32, 26 November 2020
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	Quantum operations are reversible, thus the reversible computing. That makes some ''complications'' to the gate design.		Quantum operations are reversible, thus the reversible computing. That makes some ''complications'' to the gate design.

	~~=== Two Qubit Quantum Gates ===~~

	~~The reversibel gates are eg. ''identity'', or CNOT.~~

	~~Eg. <math>X\otimes H = \begin{bmatrix} 0 &H\\H&0\end{bmatrix}</math>.~~
	~~<syntaxhighlight>~~
	~~backend = Aer.get_backend('unitary_simulator')~~
	~~unitary = execute(qc,backend).result().get_unitary()~~
	#
	~~# In Jupyter Notebooks we can display this nicely using Latex.~~
	~~# If not using Jupyter Notebooks you may need to remove the~~
	~~# array_to_latex function and use print(unitary) instead.~~
	~~from qiskit_textbook.tools import array_to_latex~~
	~~array_to_latex(unitary, pretext="\\text{Circuit = }\n")~~
	~~</syntaxhighlight>~~

	~~[[File:Cnot.svg\|thumb\|CNOT gate as a pictoram.]]~~


	~~Eg. CNOT is a conditional gate that performs an X-gate on the second qubit, if the state of the first qubit (control) is <math>\|1\rangle</math>. <math>CNOT= \begin{bmatrix} 1 & 0 & 0 & 0 \\~~
	~~0 & 0 & 0 & 1 \\~~
	~~0 & 0 & 1 & 0 \\~~
	~~0 & 1 & 0 & 0 \\~~
	~~\end{bmatrix}~~
	~~</math>. This matrix swaps the amplitudes of \|01⟩ and \|11⟩ in the statevector. <math>\text{CNOT}(x,y) = (x, x\otimes y)</math>.~~

	~~CNOT if a control qubit is on the superposition:~~

	~~<math>\text{CNOT}\|{-}0\rangle = \|{-}0\rangle</math>~~

	~~<math>\text{CNOT}\|{-}1\rangle = -\|{-}1\rangle</math>~~

	~~<math>\text{CNOT}\|0{+}\rangle = \tfrac{1}{\sqrt{2}}(\|00\rangle + \|11\rangle)</math>, which is ''Bell State''. Entanglement, but [https://arxiv.org/abs/quant-ph/0212023 no-communication theorem].~~

	~~<math>\text{CNOT}\|{+}{+}\rangle = \|{+}{+}\rangle)</math>. Unchanged.~~

	~~<math>\text{CNOT}\|{-}{+}\rangle = \tfrac{1}{\sqrt{2}}(\|{-}0\rangle -\|{-}1\rangle) = \|{-}{-} \rangle</math>~~

	~~<math>\text{CNOT}\|{+}{-}\rangle = </math>.~~

	~~<math>\text{CNOT}\|{-}{-}\rangle = \tfrac{1}{2}(\|00\rangle - \|01\rangle - \|10\rangle + \|11\rangle) = \|{-}{-}\rangle </math>. Affects the state of the control qubit, only.~~

	~~<syntaxhighlight>~~
	~~qc = QuantumCircuit(2)~~
	~~# Apply H-gate to the first:~~
	~~qc.h(0)~~
	~~# Apply a CNOT:~~
	~~qc.cx(0,1)~~
	~~qc.draw('mpl')~~
	#
	~~# Let's see the result:~~
	~~backend = Aer.get_backend('statevector_simulator')~~
	~~final_state = execute(qc,backend).result().get_statevector()~~
	~~# Print the statevector neatly:~~
	~~array_to_latex(final_state, pretext="\\text{Statevector = }")~~
	#
	~~results = execute(qc,backend).result().get_counts()~~
	~~plot_histogram(results)~~
	~~</syntaxhighlight>~~



	~~Any controlled quantum gate is <math>\text{Controlled-U} = \begin{bmatrix}I &0\\0&U\end{bmatrix}</math> and in Qiskit formalism is written in matrix as~~
	~~<math>\text{Controlled-U} =~~
	~~\begin{bmatrix}~~
	~~1 & 0 & 0 & 0 \\~~
	~~0 & u_{00} & 0 & u_{01} \\~~
	~~0 & 0 & 1 & 0 \\~~
	~~0 & u_{10} & 0 & u_{11}\\~~
	~~\end{bmatrix}~~
	~~</math>~~

	~~Controlled-Z. Because <math>H X H = Z</math> and <math>H Z H = X</math> we can write~~

	~~<syntaxhighlight>~~
	~~qc = QuantumCircuit(2)~~
	~~# also a controlled-Z~~
	~~qc.h(t)~~
	~~qc.cx(c,t)~~
	~~qc.h(t)~~
	~~qc.draw('mpl')~~
	~~</syntaxhighlight>~~

	~~Controlled-Y is~~

	~~<syntaxhighlight>~~
	~~qc = QuantumCircuit(2)~~
	~~# a controlled-Y~~
	~~qc.sdg(t)~~
	~~qc.cx(c,t)~~
	~~qc.s(t)~~
	~~qc.draw('mpl')~~
	~~</syntaxhighlight>~~

	~~or Controlled-H is~~

	~~<syntaxhighlight>~~
	~~qc = QuantumCircuit(2)~~
	~~# a controlled-H~~
	~~qc.ry(pi/4,t)~~
	~~qc.cx(c,t)~~
	~~qc.ry(-pi/4,t)~~
	~~qc.draw('mpl')~~
	~~</syntaxhighlight>~~

	~~Swap gate~~

	~~An arbitrary controlled-controlled-U for any single-qubit rotation U. We need <math>V = \sqrt U</math> and <math>V^\dagger</math>~~

	~~<syntaxhighlight>~~
	~~#The controls are qubits a and b, and the target is qubit t.~~
	~~#Subroutines cu1(theta,c,t) and cu1(-theta,c,t) need to be defined~~
	~~qc = QuantumCircuit(3)~~
	~~qc.cu1(theta,b,t)~~
	~~qc.cx(a,b)~~
	~~qc.cu1(-theta,b,t)~~
	~~qc.cx(a,b)~~
	~~qc.cu1(theta,a,t)~~
	~~qc.draw('mpl')~~
	~~</syntaxhighlight>~~

	=== Three Qubits ===		=== Three Qubits ===

Mol: /* Quantum Gates of One Qubit */

2020-11-26T08:31:32Z

Quantum Gates of One Qubit

← Older revision		Revision as of 11:31, 26 November 2020
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	Quantum operations are reversible, thus the reversible computing. That makes some ''complications'' to the gate design.		Quantum operations are reversible, thus the reversible computing. That makes some ''complications'' to the gate design.

	~~=== Quantum Gates of One Qubit ===~~

	~~There are only two reversible gates, also ''identity'' (return the input unchanged) and NOT (return the opposite of the input), but neither is universal.~~

	~~Identity gate.~~

	~~Pauli X gate.~~
	~~<math>\sigma_x = X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \|0\rangle\langle1\| + \|1\rangle\langle0\|</math>~~

	~~Pauli Y gate~~
	~~<math>\sigma_y = Y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} = -i\|0\rangle\langle1\| + i\|1\rangle\langle0\| </math>~~

	~~Pauli Z gate~~
	~~<math>\sigma_z = Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \|0\rangle\langle0\| - \|1\rangle\langle1\|</math>~~

	[https://en.wikipedia.org/wiki/Hadamard_transform#Hadamard_gate_operations\| Hadamard gate] <math>H = \tfrac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} = \|+\rangle\langle0\| + \|-\rangle\langle1\| </math>

	~~R gate <math>R_\phi = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\phi} \end{bmatrix}</math>~~

	~~S gate or <math>\sqrt Z</math> gate <math>S = \begin{bmatrix} 1 & 0 \\ 0 & e^{\frac{i\pi}{2}} \end{bmatrix}</math>~~

	~~T gate <math>T = \begin{bmatrix} 1 & 0 \\ 0 & e^{\frac{i\pi}{4}} \end{bmatrix}</math>~~

	~~U1 gate: <math>~~
	~~U_1 = U_3(0, 0, \lambda) = U_1 = \begin{bmatrix} 1 & 0 \\~~
	~~0 & e^{i\lambda}\\~~
	~~\end{bmatrix}~~
	~~</math>~~

	~~U2 gate: <math>~~
	~~U_2 = U_3(\tfrac{\pi}{2}, \phi, \lambda) = \tfrac{1}{\sqrt{2}}\begin{bmatrix} 1 & -e^{i\lambda} \\~~
	~~e^{i\phi} & e^{i\lambda+i\phi}~~
	~~\end{bmatrix}~~
	~~</math>~~

	~~<syntaxhighlight>~~
	~~qc = QuantumCircuit(1)~~
	~~qc.x(0)~~
	~~#qc.y(0) # Y-gate on qubit 0~~
	~~#qc.z(0) # Z-gate on qubit 0~~
	~~#qc.rz(pi/4, 0)~~
	~~#qc.s(0) # Apply S-gate to qubit 0~~
	~~#qc.sdg(0) # Apply Sdg-gate to qubit 0~~
	~~qc.t(0) # Apply T-gate to qubit 0~~
	~~qc.tdg(0) # Apply Tdg-gate to qubit 0~~

	~~qc.draw('mpl')~~
	~~# Let's see the result~~
	~~backend = Aer.get_backend('statevector_simulator')~~
	~~out = execute(qc,backend).result().get_statevector()~~
	~~plot_bloch_multivector(out)~~
	~~</syntaxhighlight>~~

	=== Two Qubit Quantum Gates ===		=== Two Qubit Quantum Gates ===

Mol: /* Two Qubit Quantum Gates */

2020-11-24T17:09:29Z

Two Qubit Quantum Gates

← Older revision		Revision as of 20:09, 24 November 2020
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	# Apply a CNOT:		# Apply a CNOT:
	qc.cx(0,1)		qc.cx(0,1)
	qc.draw()		qc.draw('mpl')
	#		#
	# Let's see the result:		# Let's see the result:
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	qc.cx(c,t)		qc.cx(c,t)
	qc.h(t)		qc.h(t)
	qc.draw()		qc.draw('mpl')
	</syntaxhighlight>		</syntaxhighlight>

Line 218:		Line 218:
	qc.cx(c,t)		qc.cx(c,t)
	qc.s(t)		qc.s(t)
	qc.draw()		qc.draw('mpl')
	</syntaxhighlight>		</syntaxhighlight>

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	qc.cx(c,t)		qc.cx(c,t)
	qc.ry(-pi/4,t)		qc.ry(-pi/4,t)
	qc.draw()		qc.draw('mpl')
	</syntaxhighlight>		</syntaxhighlight>

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	qc.cx(a,b)		qc.cx(a,b)
	qc.cu1(theta,a,t)		qc.cu1(theta,a,t)
	qc.draw()		qc.draw('mpl')
	</syntaxhighlight>		</syntaxhighlight>