Quantum Computing Circuits

Quantum Computing Circuits
by José Luis Gómez-Muñoz
http://homepage.cem.itesm.mx/lgomez/quantum/
[email protected]
With contributions by Francisco Delgado-Cepeda and Sergio E. Martínez-Casas
Introduction
This is a tutorial on the use of Quantum`Computing` Mathematica add-on to generate and work with Quantum Computing
Circuits.
Load the Package
First load the Quantum`Computing` package. Write:
Needs["Quantum`Computing`"];
then press at the same time the keys ˜-Û to evaluate. Mathematica will load the package. The semicolon prevents
Mathematica from printing the welcome message:
Needs@"Quantum`Computing`"D
Quantum`Computing` Version 2.2.0. HJune 2010L
A Mathematica package for Quantum Computing
in Dirac bra−ket notation and plotting of quantum circuits
by José Luis Gómez−Muñoz
Execute SetComputingAliases@D in order to use
the keyboard to enter quantum objects in Dirac's notation
SetComputingAliases@D must be executed again in each new notebook that is created
In order to use the keyboard to enter quantum objects write:
SetComputingAliases[ ];
then press at the same time the keys ˜-Û to evaluate. The semicolon prevents Mathematica from printing the help
message. Remember that SetComputingAliases[ ] must be evaluated again in each new notebook:
SetComputingAliases@D;
Simple Quantum Computing Circuits
In order to plot a simple quantum circuit, press the keys:
QuantumPlot[ [ESC]cnot[ESC] [ESC]on[ESC] [ESC]hg[ESC] ]
Then press the [TAB] key several times to select the first "place holder" Ñ, and press the keys:
1[TAB]2[TAB]1
then press at the same time the keys ˜-Û to evaluate:
2
v7qcircuits.nb
ˆ
QuantumPlotB81< Aˆ2 E ⋅ ˆ1 F
1
2
Quantum Circuits can be used exactly the same way as the Dirac expressions that were used to generate them.
As an example, write:
QuantumEvaluate[ ]
then copy-paste the circuit inside the brackets. Place the cursor to the right of the circuit and press the keys:
[ESC]on[ESC] [ESC]qqket[ESC]
Then press the [TAB] key one or several times to select the first "place holder" Ñ, and press the keys:
0[TAB]1[TAB]0[TAB]2
then press at the same time the keys ˜-Û to evaluate:
1
⋅
QuantumEvaluateB
0ˆ1 , 0ˆ2 ]F
2
0ˆ1 , 0ˆ2 ]
+
2
1ˆ1 , 1ˆ2 ]
2
It is the same result that is obtained with the original Dirac expression, press the keys:
QuantumEvaluate[ [ESC]cnot[ESC] [ESC]on[ESC] [ESC]hg[ESC] [ESC]on[ESC][ESC]qqket[ESC] ]
Then press the [TAB] key several times to select the first "place holder" Ñ, and press the keys:
1[TAB]2[TAB]1[TAB]0[TAB]1[TAB]0[TAB]2
then press at the same time the keys ˜-Û to evaluate:
ˆ
QuantumEvaluateB81< Aˆ2 E ⋅ ˆ1 ⋅
0ˆ1 , 0ˆ2 ]
2
+
0ˆ1 , 0ˆ2 ]F
1ˆ1 , 1ˆ2 ]
2
In order to plot in 3D a simple quantum circuit, press the keys:
QuantumPlot3D[ [ESC]cnot[ESC] [ESC]on[ESC] [ESC]hg[ESC] ]
Then press the [TAB] key several times to select the first "place holder" Ñ, and press the keys:
1[TAB]2[TAB]1
then press at the same time the keys ˜-Û to evaluate:
v7qcircuits.nb
ˆ
QuantumPlot3DB81< Aˆ2 E ⋅ ˆ1 F
Quantum Circuits can be used exactly the same way as the Dirac expressions that were used to generate them.
As an example, write:
QuantumMatrixForm[ ]
then copy-paste the circuit from the Mathematica output inside the brackets. Finally press at the same time the keys
˜-Û to evaluate:
QuantumMatrixFormB
1
1
0
2
0
0
0
2
1
1
0
2
1
−
0
2
1
2
1
2
0
2
−
1
0
2
The same result is obtained from the original Dirac expression:
F
3
4
v7qcircuits.nb
ˆ
QuantumMatrixFormB81< Aˆ2 E ⋅ ˆ1 F
1
1
0
2
0
2
1
0
0
1
0
2
1
−
0
2
1
2
1
2
1
−
0
2
0
2
Remember that QuantumMatrixForm is only for display purposes. For actual Mathematica calculations, QuantumMatrix
must be used:
ˆ
QuantumMatrixB81< Aˆ2 E ⋅ ˆ1 F
::
1
1
, 0,
2
, 0>, :0,
2
1
1
, 0,
2
2
>, :0,
1
2
, 0, −
1
2
>, :
1
, 0, −
2
1
, 0>>
2
This is a Determinant, calculated from QuantumMatrix, not from QuantumMatrixForm
ˆ
DetBQuantumMatrixB81< Aˆ2 E ⋅ ˆ1 FF
−1
This is a normal power of a Quantum Circuit. Press [ESC]po[ESC] for the normal power template and copy-paste the
circuit from a previous Mathematica output. Write a 3 in the exponent:
3
1
F
QuantumPlotB
2
1
2
This is a tensor power of a Quantum Circuit. Press [ESC]tpow[ESC] for the tensor power template and copy-paste the
circuit from the Mathematica output. Write a 3 in the exponent:
v7qcircuits.nb
⊗3
1
F
QuantumPlotB
2
1
2
3
4
The option QuantumGateShiftingØFalse can be used to show the circuit in another, equivalent form (the arrow Ø can be
entered by pressing [ESC]->[ESC])
⊗3
1
QuantumPlotB
,
2
QuantumGateShifting → FalseF
1
2
3
4
Here is another tensor power of the circuit, where the "exponent" is actually a list that defines the first qubit for each
"factor":
5
6
v7qcircuits.nb
⊗ 81,3,5<
1
F
QuantumPlotB
2
1
2
3
4
5
6
This is the same circuit obtained from the Dirac expression:
v7qcircuits.nb
ˆ
QuantumPlotBJ81< Aˆ2 E ⋅ ˆ1 N
⊗ 81,3,5<
F
1
2
3
4
5
6
This circuit is similar to the previous one, but it looks different with the default qubit order:
7
8
v7qcircuits.nb
ˆ
ˆ E ⋅ ˆ N
QuantumPlotBJ81< A101
1
⊗3
F
1
2
3
101
102
103
Using the option QubitList we can rearrange the circuit:
v7qcircuits.nb
ˆ
ˆ E ⋅ ˆ N
QuantumPlotBJ81< A101
1
⊗3
,
QubitList → 81, 101, 2, 102, 3, 103<F
1
101
2
102
3
103
The qubit labels can be hidden using the option QubitLabels:
9
10
v7qcircuits.nb
ˆ
ˆ E ⋅ ˆ N
QuantumPlotBJ81< A101
1
⊗3
,
QubitList → 81, 101, 2, 102, 3, 103<,
QubitLabels → FalseF
In order to plot a quantum circuit with initial (nonentangled) values for the qubits, press the keys:
QuantumPlot[ [ESC]cnot[ESC] [ESC]on[ESC] [ESC]hg[ESC] [ESC]on[ESC][ESC]qqket[ESC] ]
Then press the [TAB] key several times to select the first "place holder" Ñ, and press the keys:
1[TAB]2[TAB]1[TAB]0[TAB]1[TAB]0[TAB]2
then press at the same time the keys ˜-Û to evaluate:
ˆ
QuantumPlotB81< Aˆ2 E ⋅ ˆ1 ⋅
0ˆ1 , 0ˆ2 ]F
1
0\
2
0\
Qubit labels and initial qubit values do not need to be numbers:
v7qcircuits.nb
ˆ
ˆF⋅ˆ ⋅
QuantumPlotB8q1< Bq2
q1
ˆ , y ˆ ^F
xq1
q2
q1
x\
q2
y\
Use the option QubitLabels->False in order to have the plot without the qubit labels:
ˆ
ˆF⋅ˆ ⋅
QuantumPlotB8q1< Bq2
q1
x\
ˆ , y ˆ ^, QubitLabels → FalseF
xq1
q2
y\
You can also have the Dirac notation for the circuit:
ˆ
QuantumEvaluateB81< Aˆ2 E ⋅ ˆ1 F
0ˆ1 , 0ˆ2 ] ⋅ Y0ˆ1 , 0ˆ2
1ˆ1 , 1ˆ2 ] ⋅ Y0ˆ1 , 0ˆ2
+
2
0ˆ1 , 1ˆ2 ] ⋅ Y0ˆ1 , 1ˆ2
+
2
0ˆ1 , 0ˆ2 ] ⋅ Y1ˆ1 , 0ˆ2
2
1ˆ1 , 1ˆ2 ] ⋅ Y1ˆ1 , 0ˆ2
−
2
1ˆ1 , 0ˆ2 ] ⋅ Y0ˆ1 , 1ˆ2
+
2
0ˆ1 , 1ˆ2 ] ⋅ Y1ˆ1 , 1ˆ2
+
2
−
1ˆ1 , 0ˆ2 ] ⋅ Y1ˆ1 , 1ˆ2
2
2
TraditionalForm gives a format closer to the the format used in papers and textbooks:
ˆ
TraditionalFormBQuantumEvaluateB81< Aˆ2 E ⋅ ˆ1 FF
00\X00
2
+
11\X00
2
+
01\X01
2
+
10\X01
2
+
00\X10
2
-
11\X10
2
+
01\X11
2
-
10\X11
2
+
11
12
v7qcircuits.nb
In order to plot the circuit of a tensor product of FREDKIN gates, press the keys:
QuantumPlot[ [ESC]tprod[ESC] ]
Then press the [TAB] key one or more times to select the first "place holder" Ñ, and press the keys:
j[TAB]1[TAB]3[TAB]
Then press the keys:
[ESC]fred[ESC]
Then press the [TAB] key one or more times to select the first "place holder" Ñ, and press the keys:
j [TAB] j+1[TAB]j+2
then press at the same time the keys ˜-Û to evaluate:
3
QuantumPlotB
⊗ ˆj,j+1ˆ ,j+2ˆ F
j=1
1
2
3
4
5
Use the option QubitList in order to have the plot with some (or all) wires in a differente order. Press the keys [ESC]qb[ESC]
in order to enter the qubit template:
v7qcircuits.nb
ˆ ˆ
3
QuantumPlotB
⊗ ˆj,j+1ˆ ,j+2ˆ , QubitList → :4, 3>F
j=1
4
3
1
2
5
ˆ ˆ for Plotting Quantum Gates in the Same Column
The Notation :q1
,q2>
Notice that the three Hadamard gates are not in a column:
ˆ
QuantumPlotBIˆ1 M ⊗ ⋅ 81< Aˆ2 EF
3
1
2
3
Notice that the Hadamard gates are not in a column:
13
14
v7qcircuits.nb
ˆ
QuantumPlotBIˆ1 M ⊗ ⋅ 81< Aˆ2 E, QuantumGateShifting → FalseF
3
1
2
3
There is a notation that allows to have all the gates in the same column.
Press the keys [ESC]hggg[ESC] for the template:
ˆ
QuantumPlotB:1ˆ,2ˆ,3ˆ> ⋅ 81< Aˆ2 EF
1
2
3
These two notations evaluate to the same Dirac expression, the only diference is whether gates will be plotted in the same
column or not in a quantum circuit:
3
QuantumEvaluateB:1ˆ,2ˆ,3ˆ> F QuantumEvaluateBIˆ1 M ⊗ F
True
There is a notation that allows to have all the gates in the same column.
Press the keys [ESC]xggg[ESC] for the template:
v7qcircuits.nb
15
ˆ
QuantumPlotB:1ˆ,2ˆ,3ˆ> ⋅ 81< Aˆ2 EF
1
2
3
These two notations evaluate to the same Dirac expression, the only diference is whether gates will be plotted in the same
column or not in a quantum circuit:
QuantumEvaluateB:1ˆ,2ˆ,3ˆ> F QuantumEvaluateBIˆ1 M ⊗ F
3
True
Textbook Examples
It is a textbook exercise to show that the following circuit is equivalente to a Toffoli (controlled-controlled-not) gate. Press
[ESC]her[ESC] for the "Hermitian Conjugate" HL† template:
ˆ
ˆ
QuantumPlotBˆ1 ⋅ ˆ2 ⋅ 81< Aˆ2 E ⋅ Iˆ2 M ⋅ 81< Aˆ2 E ⋅ Iˆ2 M ⋅ ˆ3 ⋅ ˆ3 ⋅
ˆ
†
ˆ
†
ˆ
ˆ
81< Aˆ3 E ⋅ Iˆ3 M ⋅ 82< Aˆ3 E ⋅ ˆ3 ⋅ 81< Aˆ3 E ⋅ Iˆ3 M ⋅ 82< Aˆ3 E ⋅ ˆ3 F
†
1
2
3
†
†
†
†
†
Use the option ImageSize Æ {250, Automatic} to have a smaller version of the circuit.
Press the keys [ESC]->[ESC] ("Escape","Minus","Grater Than","Escape") in order to enter the arrow Ø
16
v7qcircuits.nb
ˆ
ˆ
ˆ
QuantumPlotBˆ1 ⋅ ˆ2 ⋅ 81< Aˆ2 E ⋅ Iˆ2 M ⋅ 81< Aˆ2 E ⋅ Iˆ2 M ⋅ ˆ3 ⋅ ˆ3 ⋅ 81< Aˆ3 E ⋅ Iˆ3 M ⋅
ˆ
†
ˆ
†
†
ˆ
82< Aˆ3 E ⋅ ˆ3 ⋅ 81< Aˆ3 E ⋅ Iˆ3 M ⋅ 82< Aˆ3 E ⋅ ˆ3 , ImageSize → 8250, Automatic<F
†
1
2
3 †
†
†
†
This is the Truth-Table of the circuit (write QuantumTableForm[ ] and copy-paste the circuit from the previous calculation
inside the brackets). Notice that only the two last rows of the table are different from the table of an identity:
1
QuantumTableFormB 2
†
3 †
Input
Output
0
0ˆ1 , 0ˆ2 , 0ˆ3 ]
0ˆ1 , 0ˆ2 , 0ˆ3 ]
1
0ˆ1 , 0ˆ2 , 1ˆ3 ]
0ˆ1 , 0ˆ2 , 1ˆ3 ]
2
0ˆ1 , 1ˆ2 , 0ˆ3 ]
0ˆ1 , 1ˆ2 , 0ˆ3 ]
3
0ˆ1 , 1ˆ2 , 1ˆ3 ]
0ˆ1 , 1ˆ2 , 1ˆ3 ]
4
1ˆ1 , 0ˆ2 , 0ˆ3 ]
1ˆ1 , 0ˆ2 , 0ˆ3 ]
5
1ˆ1 ,
1ˆ3 ]
1ˆ1 , 0ˆ2 , 1ˆ3 ]
6
1ˆ1 , 1ˆ2 , 0ˆ3 ]
1ˆ1 , 1ˆ2 , 1ˆ3 ]
7
1ˆ1 , 1ˆ2 , 1ˆ3 ]
1ˆ1 , 1ˆ2 , 0ˆ3 ]
0ˆ2 ,
†
†
F
We can see that is identical the Truth-Table of the Toffoli gate:
QuantumTableForm[ [ESC]toff[ESC] ]
Then press the [TAB] key one or more times to select the first "place holder" Ñ, and press the keys:
1 [TAB] 2 [TAB] 3
then press at the same time the keys ˜-Û to evaluate. Notice that only the two last rows of the table are different from
the table of an identity:
QuantumTableFormAˆ1,2ˆ,3ˆ E
Input
Output
0
0ˆ1 , 0ˆ2 , 0ˆ3 ]
0ˆ1 , 0ˆ2 , 0ˆ3 ]
1
0ˆ1 ,
1ˆ3 ]
0ˆ1 , 0ˆ2 , 1ˆ3 ]
2
0ˆ1 , 1ˆ2 , 0ˆ3 ]
0ˆ1 , 1ˆ2 , 0ˆ3 ]
3
0ˆ1 , 1ˆ2 , 1ˆ3 ]
0ˆ1 , 1ˆ2 , 1ˆ3 ]
4
1ˆ1 , 0ˆ2 , 0ˆ3 ]
1ˆ1 , 0ˆ2 , 0ˆ3 ]
5
1ˆ1 , 0ˆ2 , 1ˆ3 ]
1ˆ1 , 0ˆ2 , 1ˆ3 ]
6
1ˆ1 , 1ˆ2 , 0ˆ3 ]
1ˆ1 , 1ˆ2 , 1ˆ3 ]
7
1ˆ1 , 1ˆ2 , 1ˆ3 ]
1ˆ1 , 1ˆ2 , 0ˆ3 ]
0ˆ2 ,
v7qcircuits.nb
17
Another textbook example. In order to enter the "controlled-controlled not" gates press the keys [ESC]ccnot[ESC]. For the
"tensor product" press the keys [ESC]tprod[ESC]. For an arbitrary controlled gate press the keys [ESC]cgate[ESC]. For the
arbitrary gate w, press the keys [ESC]qg[ESC]
SetQuantumGate@w, 1D;
QuantumPlotB
ˆ ˆ
81,2< Aˆ6 E ⋅
3
ˆ
ˆ
⊗ 8j+2,j+5<Bj+6ˆ F
j=1
ˆ
ˆE⋅
⋅ 89< Aw10
3
ˆ
ˆ
ˆ F
⊗ 86−j,9−j<B10−j
j=1
ˆ ˆ
⋅ 81,2< Aˆ6 EF
1
2
3
4
5
6
7
8
9
10
w
Quantum Register Notation
Press [ESC]hn[ESC] [TAB] [ESC]qr[ESC] to use the "quantum register" template in the Hadamard gate:
18
v7qcircuits.nb
QuantumPlotA@3,"q"D E
q1
q2
q3
Press [ESC]hn[ESC] [TAB] [ESC]qr[ESC] to use the "quantum register" template in the Hadamard gate:
v7qcircuits.nb
19
QuantumPlotA@10,22,3,"Q"D E
Q10
Q13
Q16
Q19
Q22
Powers of Quantum Gates in Circuits
The power of a gate is by default shown as a single block. Press the keys [ESC]qg[ESC] in order to obtain the gate template
ˆ
SetQuantumGate@w, 1D;
QuantumPlotAwˆa 4 E
a
w4
Integer powers of gates are shown as several gates if the option GatePowers is set to True. Press the keys [ESC]qg[ESC] in
order to obtain the gate template ˆ
20
v7qcircuits.nb
SetQuantumGate@w, 1D;
QuantumPlotAwˆa 4 , QuantumGatePowers → FalseE
a
w
w
w
w
Gates of two qubits
Here is a gate of two qubits. Press the keys [ESC]qqg[ESC] in order to obtain the gate template ˆ,ˆ
SetQuantumGate@mygate, 2D;
QuantumPlotAmygateˆa,bˆ E
a
b
This is a controlled gate. Press the keys [ESC]cgate[ESC] in order to obtain the controlled gate template. Press the keys
[ESC]qqg[ESC] in order to obtain the gate template ˆ,ˆ
v7qcircuits.nb
SetQuantumGate@mygate, 2D;
QuantumPlotB8â< Amygateˆb,cˆ EF
a
b
c
This is a controlled gate. Press the keys [ESC]ccgate[ESC] in order to obtain the controlled-controlled gate template
ˆ ˆ
8,< @D. Press the keys [ESC]qqg[ESC] in order to obtain the gate template ˆ,ˆ
21
22
v7qcircuits.nb
SetQuantumGate@mygate, 2D;
ˆ
:â,b>
QuantumPlotB
Amygateˆc,dˆ EF
a
b
c
d
This circuit shows controlled gates of two qbits, powers of gates, and initial values for some of the qubits:
SetQuantumGate@u, 2D;
QuantumPlotB
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅
j2
j2
j2
j1
j1
j0
ˆ
ˆ
ˆ
ˆ ˆ E ⋅ 8j1< Au ˆ ˆ 2 E ⋅ 8j0< Au ˆ ˆ 4 E ⋅ ˆ ⋅ ˆ ⋅ ˆ ⋅
8j2< Aus0
,s1
s0,s1
s0,s1
j2
j1
j0
0\
0\
0\
j0
j1
j2
ˆ , 0 ˆ , 0 ˆ ^F
0j0
j1
j2
s0
s1
Use the option ImageSize Æ {500, Automatic} to have a larger version of the circuit:
v7qcircuits.nb
SetQuantumGate@u, 2D;
QuantumPlotB
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅ 8j1< Au ˆ ˆ 2 E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
s0,s1
ˆ
ˆ ˆ 4E ⋅ ˆ ⋅ ˆ ⋅ ˆ ⋅
8j0< Aus0
,s1
j2
j1
j0
ˆ , 0 ˆ , 0 ˆ ^, ImageSize → 8500, Automatic<F
0j0
j1
j2
j0
0\
j1
0\
j2
0\
s0
s1
A 3D-version of the circuit
SetQuantumGate@u, 2D;
QuantumPlot3DB
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅
j2
j2
j2
j1
j1
j0
ˆ
ˆ
ˆ
ˆ ˆ E ⋅ 8j1< Au ˆ ˆ 2 E ⋅ 8j0< Au ˆ ˆ 4 E ⋅ ˆ ⋅ ˆ ⋅ ˆ ⋅
8j2< Aus0
,s1
s0,s1
s0,s1
j2
j1
j0
Same circuit with QuantumGateShiftingØFalse
ˆ , 0 ˆ , 0 ˆ ^F
0j0
j1
j2
23
24
v7qcircuits.nb
SetQuantumGate@u, 2D;
QuantumPlotB
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅ 8j1< Au ˆ ˆ 2 E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
s0,s1
ˆ
ˆ ˆ 4E ⋅ ˆ ⋅ ˆ ⋅ ˆ ⋅
8j0< Aus0
,s1
j2
j1
j0
0\
0\
0\
j0
j1
ˆ , 0 ˆ , 0 ˆ ^, QuantumGateShifting → FalseF
0j0
j1
j2
j2
s0
s1
Same circuit with QuantumGateShiftingØFalse AND the use of the template 9ˆ,ˆ,ˆ= for plotting the three Hadamard
gates in the same column, see the gates next to the ket and compare with the circuit above. The 9ˆ,ˆ,ˆ= template can be
entered y pressing the keys [ESC]hggg[ESC]
SetQuantumGate@u, 2D;
QuantumPlotB
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅ 8j1< Au ˆ ˆ 2 E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
s0,s1
ˆ
ˆ ˆ 4E ⋅ ˆ ˆ ˆ ⋅
8j0< Aus0
,s1
:j0,j1,j2>
0\
0\
0\
j0
j1
j2
ˆ , 0 ˆ , 0 ˆ ^, QuantumGateShifting → FalseF
0j0
j1
j2
s0
s1
Same circuit with QuantumGatePowersØFalse
SetQuantumGate@u, 2D;
QuantumPlotB
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅ 8j1< Au ˆ ˆ 2 E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
s0,s1
ˆ
ˆ ˆ 4E ⋅ ˆ ⋅ ˆ ⋅ ˆ ⋅
8j0< Aus0
,s1
j2
j1
j0
0\
0\
0\
j0
j1
j2
s0
s1
ˆ , 0 ˆ , 0 ˆ ^, QuantumGatePowers → FalseF
0j0
j1
j2
v7qcircuits.nb
25
Same circuit with qubit order specified by QubitList
SetQuantumGate@u, 2D;
QuantumPlotB
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅ 8j1< Au ˆ ˆ 2 E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
s0,s1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ , 0 ˆ , 0 ˆ ^, QubitList → :s0, s1, j0, j1, j2>F
0j0
j1
j2
ˆ
ˆ ˆ 4E ⋅ ˆ ⋅ ˆ ⋅ ˆ ⋅
8j0< Aus0
,s1
j2
j1
j0
s0
s1
0\
0\
0\
j0
j1
j2
Measuring Meters
Next circuit shows how plot Measuring Meters in a quantum circuit.
Notice that QubitMeasurement has not the same syntaxis as a quantum gate:
SetQuantumGate@u, 2D;
QuantumPlotBQubitMeasurementB
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
ˆ
ˆ
ˆ ˆ 2 E ⋅ 8j0< Au ˆ ˆ 4 E ⋅ ˆ ⋅ ˆ ⋅ ˆ ⋅
8j1< Aus0
,s1
s0,s1
j2
j1
j0
0\
0\
0\
j0
j1
j2
ˆ
ˆ
ˆ
ˆ , 0 ˆ , 0 ˆ ^, :j0, j1, j2>FF
0j0
j1
j2
s0
s1
Same circuit with QuantumGateShiftingØFalse AND the use of the template 9ˆ,ˆ,ˆ= for plotting the three Hadamard
gates in the same column, see the gates next to the ket and compare with the circuit above. The 9ˆ,ˆ,ˆ= template can be
entered y pressing the keys [ESC]hggg[ESC]
26
v7qcircuits.nb
SetQuantumGate@u, 2D;
QuantumPlotBQubitMeasurementB
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅ 8j1< Au ˆ ˆ 2 E ⋅ 8j0< Au ˆ ˆ 4 E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
s0,s1
s0,s1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
0j0 , 0j1 , 0j2 ^, :j0, j1, j2>F, QuantumGateShifting → FalseF
:j0,j1,j2> ⋅
0\
0\
0\
j0
j1
j2
s0
s1
Controlled gates commute with measurements in their control qubit. This means these two circuits are equivalent:
ˆ
:QuantumPlotB QubitMeasurementB81< Aˆ2 E ⋅ ˆ1 ⋅
ˆ
QuantumPlotB 81< Aˆ2 E ⋅ QubitMeasurementBˆ1 ⋅
ˆ
,1ˆ,2ˆ ], 81<F F,
ˆ
,1ˆ,2ˆ ], 81<F F>
1
:
,
2
1
>
2
It can be seen that both circuits are equivalent by writting QuantumEvaluate instead of QuantumPlot:
v7qcircuits.nb
ˆ
:QuantumEvaluateB QubitMeasurementB81< Aˆ2 E ⋅ ˆ1 ⋅
ˆ
QuantumEvaluateB 81< Aˆ2 E ⋅ QubitMeasurementBˆ1 ⋅
Probability Measurement
:
1
2
990ˆ1 ==
1
2
991ˆ1 ==
0ˆ ]
0ˆ1 ] ⊗
2
State
991ˆ1 ==
0ˆ ]
2
+
1ˆ ]
+
1ˆ ]
2
2
0ˆ ]
2
,
2
2
2
1ˆ1 ] ⊗ −
Probability Measurement
2
2
Probability Measurement
1
2
1ˆ ]
2
0ˆ ]
1ˆ1 ] ⊗ −
0ˆ1 ] ⊗
+
2
2
State
990ˆ1 ==
ˆ
,1ˆ,2ˆ ], 81<F F>
State
Probability Measurement
1
2
ˆ
,1ˆ,2ˆ ], 81<F F,
+
2
1ˆ ]
>
2
2
State
Use of Circuit Options
SetQuantumGate@u, 2D;
QuantumPlotB
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅ 8j1< Au ˆ ˆ 2 E ⋅ 8j0< Au ˆ ˆ 4 E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
s0,s1
s0,s1
ˆ ⋅ˆ ⋅ˆ ⋅
j2
j1
j0
ˆ , 0 ˆ , 0 ˆ ^, QuantumConnectionStyle → Directive@Red, ThinD,
0j0
j1
j2
QuantumGateStyle → Directive@Yellow, EdgeForm@8Blue<DDF
0\
0\
0\
j0
j1
j2
s0
s1
27
28
v7qcircuits.nb
SetQuantumGate@u, 2D;
QuantumPlotB
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅
j2
j2
j2
j1
j1
j0
ˆ
ˆ
ˆ
ˆ ˆ E ⋅ 8j1< Au ˆ ˆ 2 E ⋅ 8j0< Au ˆ ˆ 4 E ⋅ ˆ ⋅ ˆ ⋅ ˆ ⋅
8j2< Aus0
,s1
s0,s1
s0,s1
j2
j1
j0
ˆ , 0 ˆ , 0 ˆ ^,
0j0
j1
j2
QuantumControlStyle → Directive@Yellow, EdgeForm@8Red<DD
F
0\
0\
0\
j0
j1
j2
s0
s1
SetQuantumGate@u, 2D;
QuantumPlot3DB
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅ 8j1< Au ˆ ˆ 2 E ⋅ 8j0< Au ˆ ˆ 4 E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
s0,s1
s0,s1
ˆ ⋅ˆ ⋅ˆ ⋅
j2
j1
j0
ˆ , 0 ˆ , 0 ˆ ^, QuantumConnectionStyle → Directive@Red, ThinD,
0j0
j1
j2
QuantumGateStyle → Directive@Yellow, EdgeForm@8Blue, Thick<DD
F
v7qcircuits.nb
SetQuantumGate@u, 2D;
QuantumPlot3DB
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
ˆ
ˆ
ˆ ˆ 2 E ⋅ 8j0< Au ˆ ˆ 4 E ⋅ ˆ ⋅ ˆ ⋅ ˆ ⋅
8j1< Aus0
,s1
s0,s1
j2
j1
j0
ˆ , 0 ˆ , 0 ˆ ^,
0j0
j1
j2
QuantumBackground → Black,
QuantumGateStyle → Directive@Glow@BlueDD,
QuantumConnectionStyle −> Directive@Glow@YellowDD,
QuantumWireStyle → Directive@Glow@Darker@MagentaDDD,
QuantumControlStyle → Directive@Glow@GreenDD,
QuantumTextStyle → Red
F
SetQuantumGate@u, 2D;
QuantumPlot3DB
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
ˆ
ˆ
ˆ ˆ 2 E ⋅ 8j0< Au ˆ ˆ 4 E ⋅ ˆ ⋅ ˆ ⋅ ˆ ⋅
8j1< Aus0
,s1
s0,s1
j2
j1
j0
ˆ , 0 ˆ , 0 ˆ ^,
0j0
j1
j2
QuantumBackground → Black,
QuantumGateStyle → Directive@Glow@BlueDD,
QuantumConnectionStyle −> Directive@Glow@YellowDD,
QuantumWireStyle → Directive@Glow@Darker@MagentaDDD,
QuantumControlStyle → Directive@Glow@GreenDD,
QuantumTextStyle → Red,
Lighting → None
F
29
30
v7qcircuits.nb
QuantumPlot3DB
ˆ
ˆ2,3ˆ ⋅ 81< Aˆ2 E ⋅ ˆ1 ,
QuantumBackground → Black,
QuantumGateStyle → Directive@Glow@BlueDD,
QuantumConnectionStyle −> Directive@Glow@MagentaDD,
QuantumControlStyle → Directive@Glow@YellowDD,
QuantumNotStyle → Directive@Glow@BrownDD,
QuantumWireStyle → Directive@Glow@RedDD,
QuantumTextStyle → Green,
Lighting → NoneF
SetQuantumGate@u, 2D;
QuantumPlot3DB
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
ˆ
ˆ
ˆ ˆ 2 E ⋅ 8j0< Au ˆ ˆ 4 E ⋅ ˆ ⋅ ˆ ⋅ ˆ ⋅
8j1< Aus0
,s1
s0,s1
j2
j1
j0
ˆ , 0 ˆ , 0 ˆ ^,
0j0
j1
j2
QuantumConnectionStyle → Brown, QuantumGateStyle →
Directive@LightYellow, EdgeForm@GreenDD,
QuantumWireStyle → Directive@Blue, ThickD,
QuantumBackground → LightGreen,
QuantumTextStyle → Directive@Red, FontWeight → Bold, FontFamily → "Verdana"D
F
v7qcircuits.nb
31
SetQuantumGate@u, 2D;
QuantumPlot3DB
ˆ
ˆ
ˆ
ˆ
ˆ ⋅ 8j1< B ˆ F ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j0< B ˆ F ⋅ ˆ ⋅ 8j2< Au ˆ ˆ E ⋅
j2
j2
j2
j1
j1
j0
s0,s1
ˆ
ˆ
ˆ ˆ 2 E ⋅ 8j0< Au ˆ ˆ 4 E ⋅ ˆ ⋅ ˆ ⋅ ˆ ⋅
8j1< Aus0
,s1
s0,s1
j2
j1
j0
ˆ , 0 ˆ , 0 ˆ ^,
0j0
j1
j2
QuantumTextStyle → Directive@Black, Medium, FontWeight → BoldD,
QuantumWireStyle → Directive@Red, ThickD,
QuantumBackground → Yellow
F
QuantumPlot3DB
ˆ
ˆ2,3ˆ ⋅ 81< Aˆ2 E ⋅ ˆ1 ,
QuantumSwapStyle → Directive@Red, [email protected],
QuantumGateStyle → Directive@[email protected]
F
QuantumPlot has its own options, plus all the options of the standard Mathematica commands Plot and Plot3D. Following
commands show a list of the options that are exclusive of QuantumPlot, together with their default value:
32
v7qcircuits.nb
Select@Options@QuantumPlotD, Characters@SymbolName@@@1DDDD@@1DD "Q" &D
:QubitList → 8<, QubitLabels → True,
QuantumGatePowers → True, QuantumGateShifting → True,
QuantumBackground → [email protected], 0.9, 0.9D, QuantumMeterStyle →
4
DirectiveBRGBColorB0, 0, F, [email protected], Arrowheads@SmallDF,
9
4
QuantumWireStyle → DirectiveBRGBColorB0, 0, F, [email protected],
9
4
QuantumConnectionStyle → DirectiveBRGBColorB , 0, 0F, [email protected],
9
2
QuantumNotStyle → DirectiveBRGBColorB0, , 0F, [email protected],
3
QuantumControlStyle →
4
DirectiveBRGBColor@0, 1, 0D, EdgeFormB:RGBColorB , 0, 0F, [email protected]>FF,
9
2
2
QuantumSwapStyle → DirectiveBRGBColorB , 0, F, [email protected],
3
3
QuantumGateStyle →
4
DirectiveBRGBColor@1, 1, 0.85D, EdgeFormB:[email protected], RGBColorB , 0, 0F>FF,
9
4
QuantumTextStyle → DirectiveBSmall, RGBColorB0, 0, F, FontFamily → VerdanaF,
9
4
QuantumVerticalTextStyle → DirectiveBRGBColorB , 0, 0F, FontFamily → VerdanaF,
9
QuantumPlot3D → False>
Use of Standard Graphics Options
Standard Mathematica Graphics options can be used in a Quantum Circuit using the syntax:
QuantumPlot[expression,options]
QuantumPlotB
ˆ
ˆ2,3ˆ ⋅ 81< Aˆ2 E ⋅ ˆ1 , Background → Yellow, Frame → True, FrameStyle → BlueF
1
-1.0
-1.5
2
-2.0
-2.5
3
-3.0
0
1
2
3
4
v7qcircuits.nb
Standard Mathematica Graphics3D options can be used in a Quantum Circuit using the syntax:
QuantumPlot3D[expression,options]
QuantumPlot3DB
ˆ
ˆ2,3ˆ ⋅ 81< Aˆ2 E ⋅ ˆ1 ,
Background → Yellow, Boxed → True, Axes → TrueF
Standard Mathematica Graphics options can be used in a Quantum Circuit using the syntax:
QuantumPlot[expression,options]
QuantumPlotB
ˆ
ˆ2,3ˆ ⋅ 81< Aˆ2 E ⋅ ˆ1 ,
Background → LightPink, Frame → True, FrameTicks → None, FrameStyle → ThickF
1
2
3
Standard Mathematica primitives can be combined Quantum Circuit using the syntax:
Show[QuantumPlot[expression],Graphics[{primitives}],options]
33
34
v7qcircuits.nb
ShowBQuantumPlotB
ˆ
ˆ2,3ˆ ⋅ 81< Aˆ2 E ⋅ ˆ1 F,
Graphics@8Text@Style@"My First Circuit", LargeD, 82, − 0.4<D<D, Frame → FalseF
My First Circuit
1
2
3
Standard Mathematica primitives can be combined Quantum Circuit using the syntax:
Show[QuantumPlot[expression],Graphics[{primitives}],options]
ShowB
QuantumPlotB
ˆ
ˆ2,3ˆ ⋅ 81< Aˆ2 E ⋅ ˆ1 F,
Graphics@8
[email protected], − 1.3<, 82.1, − 1.1<<D,
Text@Style@"control", LargeD, 82.5, − 1.3<, 8− 1.1, 0<D
<D, Frame → TrueF
1
-1.0
control
-1.5
2
-2.0
-2.5
3
-3.0
0
1
2
3
4
Standard Mathematica primitives can be combined Quantum Circuit using the syntax:
Show[QuantumPlot[expression],Graphics[{primitives}],options]
v7qcircuits.nb
ShowB
QuantumPlotB
ˆ
ˆ2,3ˆ ⋅ 81< Aˆ2 E ⋅ ˆ1 F,
Graphics@8
[email protected], − 1.3<, 82.1, − 1.1<<D,
Text@Style@"control", LargeD, 82.5, − 1.3<, 8− 1.1, 0<D
<D, Frame → FalseF
1
control
2
3
Standard Mathematica primitives can be combined Quantum Circuit using the syntax:
Show[QuantumPlot[expression],Graphics[{primitives}],options]
35
36
v7qcircuits.nb
ShowB
QuantumPlotB
ˆ
ˆ2,3ˆ ⋅ 81< Aˆ2 E ⋅ ˆ1 F,
Graphics@8
[email protected], − 1.3<, 82.1, − 1.1<<D,
Text@Style@"control", LargeD, 82.5, − 1.3<, 8− 1.1, 0<D
<D, Frame → False, ImageSize → 8600, Automatic<F
1
control
2
3
Standard Mathematica primitives can be combined Quantum Circuit using the syntax:
Show[QuantumPlot3D[expression],Graphics3D[{primitives}],options]
v7qcircuits.nb
ShowB
QuantumPlot3DB
ˆ
ˆ2,3ˆ ⋅ 81< Aˆ2 E ⋅ ˆ1 F,
Graphics3D@8
[email protected], 0.3, 0<, 82.1, − 0.9, 0<<D,
Text@Style@"control", LargeD, 82.6, 0.3, 0<, 8− 1.1, 0<D
<DF
by José Luis Gómez-Muñoz
http://homepage.cem.itesm.mx/lgomez/quantum/
[email protected]
37