CS计算机代考程序代写 algorithm MULT20015 Elements of Quantum Computing

MULT20015 Elements of Quantum Computing
Lecture 8

Subject outline

Lecture topics (by week)

1 – Introduction to quantum computing and maths basics
2 – Single qubit representations and logic operations
3 – Two qubit states and logic gates
4 – Multi-qubit states and quantum arithmetic
5 – Classical complexity and simple quantum algorithms
6 – Cryptography and Shor’s quantum factoring algorithm
7 – Post quantum cryptography and quantum key distribution
8 – Quantum search algorithms
9 – Quantum algorithms for option pricing
10 – Optimisation problems on quantum computers
11 – Portfolio optimisation using quantum computers
12 – Quantum machine learning

Assignment schedule:
#1: Hand out in Week 2
#2: Hand out in Week 8

MULT20015 Elements of Quantum Computing
Lecture 8

Week 4

Lecture 7
7.1 Binary notation with multiple qubits
7.2 Quantum Registers
7.3 Hadamard gates and equal superposition
7.4 Quantum “Parallelism”
7.5 Multiqubit measurement
7.6 Toffoli gate

Lecture 8
8.1 Classical digital logic and universality
8.2 Reversible logic
8.3 Arithmetic operations on a quantum computer
8.4 Universality in quantum computing

Practice class 4
Multi-qubit states and operations

MULT20015 Elements of Quantum Computing
Lecture 8

Recap: Lecture 7

|xi
AAACAXicbVDJSgNBEK2JW4xb1KOXxiB4CjMumGPAi8cIZoFkCD2dmqSZnoXuHjEMOfkFXvULvIlXv8QP8D/sJHMwiQ8KHu9VUVXPSwRX2ra/rcLa+sbmVnG7tLO7t39QPjxqqTiVDJssFrHseFSh4BE2NdcCO4lEGnoC215wO/XbjygVj6MHPU7QDekw4j5nVBup3QtQkyfSL1fsqj0DWSVOTiqQo9Ev//QGMUtDjDQTVKmuYyfazajUnAmclHqpwoSygA6xa2hEQ1RuNjt3Qs6MMiB+LE1FmszUvxMZDZUah57pDKkeqWVvKv7ndVPt19yMR0mqMWLzRX4qiI7J9Hcy4BKZFmNDKJPc3ErYiErKtEloYYsnqUlmYnJxllNYJa2LqnNZvb6/qtRreUJFOIFTOAcHbqAOd9CAJjAI4AVe4c16tt6tD+tz3lqw8pljWID19QtShZdt

|yi
AAACAHicbVDJSgNBEK2JW4xb1KOXxiB4CjMumGPAi8cIZoFkCD2dnqRNz0J3jTAMufgFXvULvIlX/8QP8D/sJHMwiQ8KHu9VUVXPi6XQaNvfVmFtfWNzq7hd2tnd2z8oHx61dJQoxpsskpHqeFRzKULeRIGSd2LFaeBJ3vbGt1O//cSVFlH4gGnM3YAOQ+ELRtFIrd6YI0n75YpdtWcgq8TJSQVyNPrln94gYknAQ2SSat117BjdjCoUTPJJqZdoHlM2pkPeNTSkAdduNrt2Qs6MMiB+pEyFSGbq34mMBlqngWc6A4ojvexNxf+8boJ+zc1EGCfIQzZf5CeSYESmr5OBUJyhTA2hTAlzK2EjqihDE9DCFk9RE8zE5OIsp7BKWhdV57J6fX9VqdfyhIpwAqdwDg7cQB3uoAFNYPAIL/AKb9az9W59WJ/z1oKVzxzDAqyvX/n0l0Q=

|zi
AAACAHicbVDLSgNBEOz1GeMr6tHLYBA8hV0fmGPAi8cI5gHJEmYnk2TM7Owy0yvEJRe/wKt+gTfx6p/4Af6Hk2QPJrGgoajqprsriKUw6Lrfzsrq2vrGZm4rv72zu7dfODismyjRjNdYJCPdDKjhUiheQ4GSN2PNaRhI3giGNxO/8ci1EZG6x1HM/ZD2legJRtFK9faQI3nqFIpuyZ2CLBMvI0XIUO0UftrdiCUhV8gkNabluTH6KdUomOTjfDsxPKZsSPu8ZamiITd+Or12TE6t0iW9SNtSSKbq34mUhsaMwsB2hhQHZtGbiP95rQR7ZT8VKk6QKzZb1EskwYhMXiddoTlDObKEMi3srYQNqKYMbUBzWwJNbTBjm4u3mMIyqZ+XvIvS1d1lsVLOEsrBMZzAGXhwDRW4hSrUgMEDvMArvDnPzrvz4XzOWlecbOYI5uB8/QL7i5dF

| i = |xi |yi |zi
AAACHHicbVDLSsNAFL3xWesr6tLNYBFclcQHdiMU3LisYB/QhDKZTtqhkwczEzGGbv0Mv8CtfoE7cSv4Af6H0yYL23pguIdz7uXeOV7MmVSW9W0sLa+srq2XNsqbW9s7u+befktGiSC0SSIeiY6HJeUspE3FFKedWFAceJy2vdH1xG/fUyFZFN6pNKZugAch8xnBSks9EzkjqpATS4aucv6QlzQvjz2zYlWtKdAisQtSgQKNnvnj9COSBDRUhGMpu7YVKzfDQjHC6bjsJJLGmIzwgHY1DXFApZtNfzJGx1rpIz8S+oUKTdW/ExkOpEwDT3cGWA3lvDcR//O6ifJrbsbCOFE0JPkiP+FIRWgSC+ozQYniqSaYCKZvRWSIBSZKhzezxRNYBzPWudjzKSyS1mnVPqte3J5X6rUioRIcwhGcgA2XUIcbaEATCDzBC7zCm/FsvBsfxmfeumQUMwcwA+PrF4YjoRA=

| i = |xi ⌦ |yi ⌦ |zi
AAACLHicbVDLSsNAFJ34rPUVdelmsAiuSuIDuxEKblxWsA9oQplMJ+3QyYOZGzGG/oaf4Re41S9wI+JSv8NpmoVtPTBw7rn3cu4cLxZcgWV9GEvLK6tr66WN8ubW9s6uubffUlEiKWvSSESy4xHFBA9ZEzgI1oklI4EnWNsbXU/67XsmFY/CO0hj5gZkEHKfUwJa6pmWM2KAnVhxfIVz/oCdCHjA1LRMZ8vHnlmxqlYOvEjsglRQgUbP/Hb6EU0CFgIVRKmubcXgZkQCp4KNy06iWEzoiAxYV9OQaC83y382xsda6WM/kvqFgHP170ZGAqXSwNOTAYGhmu9NxP963QT8mpvxME6AhXRq5CcCQ4QnMeE+l4yCSDUhVHJ9K6ZDIgkFHeaMiyeJDmasc7HnU1gkrdOqfVa9uD2v1GtFQiV0iI7QCbLRJaqjG9RATUTRE3pBr+jNeDbejU/jazq6ZBQ7B2gGxs8vcHenyg==

or

Quantum Registers: Group qubits together

H⊗n|0i

Hadamard gates: Create an equal
superposition state at the start of the
algorithm

| i =
1

2n/2

2n�1X

i=0

|ii
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

|ai
AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKez6wBwDXjxGMA9IljA76SRjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSy4Nq777eTW1jc2t/LbhZ3dvf2D4uFRQ0eJYlhnkYhUK6AaBZdYN9wIbMUKaRgIbAaj26nffEKleSQfzDhGP6QDyfucUWOlRmeEhtBuseSW3RnIKvEyUoIMtW7xp9OLWBKiNExQrdueGxs/pcpwJnBS6CQaY8pGdIBtSyUNUfvp7NoJObNKj/QjZUsaMlP/TqQ01HocBrYzpGaol72p+J/XTky/4qdcxolByeaL+okgJiLT10mPK2RGjC2hTHF7K2FDqigzNqCFLYGiNpiJzcVbTmGVNC7K3mX5+v6qVK1kCeXhBE7hHDy4gSrcQQ3qwOARXuAV3pxn5935cD7nrTknmzmGBThfv9PMlyw=

|bi
AAACAHicbVDLSgNBEJyNrxhfUY9eBoPgKez6wBwDXjxGMA9IljA76SRjZmeXmV4hLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSyFQdf9dnJr6xubW/ntws7u3v5B8fCoYaJEc6jzSEa6FTADUiioo0AJrVgDCwMJzWB0O/WbT6CNiNQDjmPwQzZQoi84Qys1OiNAGnSLJbfszkBXiZeREslQ6xZ/Or2IJyEo5JIZ0/bcGP2UaRRcwqTQSQzEjI/YANqWKhaC8dPZtRN6ZpUe7UfalkI6U/9OpCw0ZhwGtjNkODTL3lT8z2sn2K/4qVBxgqD4fFE/kRQjOn2d9oQGjnJsCeNa2FspHzLNONqAFrYEmtlgJjYXbzmFVdK4KHuX5ev7q1K1kiWUJyfklJwTj9yQKrkjNVInnDySF/JK3pxn5935cD7nrTknmzkmC3C+fgHVY5ct

|ci
AAACAHicbVDJSgNBFHwTtxi3qEcvjUHwFGZcMMeAF48RzALJEHo6b5I2PQvdPUIYcvELvOoXeBOv/okf4H/Yk8zBJBY0FFXvUa/LiwVX2ra/rcLa+sbmVnG7tLO7t39QPjxqqSiRDJssEpHseFSh4CE2NdcCO7FEGngC2974NvPbTygVj8IHPYnRDegw5D5nVBup1RujJqxfrthVewaySpycVCBHo1/+6Q0ilgQYaiaoUl3HjrWbUqk5Ezgt9RKFMWVjOsSuoSENULnp7NopOTPKgPiRNC806Zn6dyOlgVKTwDOTAdUjtexl4n9eN9F+zU15GCcaQzYP8hNBdESyr5MBl8i0mBhCmeTmVsJGVFKmTUELKZ6kppip6cVZbmGVtC6qzmX1+v6qUq/lDRXhBE7hHBy4gTrcQQOawOARXuAV3qxn6936sD7nowUr3zmGBVhfv9b6ly4=

|c0i
AAACAnicbVDJSgNBEK2JW4xb1KOXxiB6CjMumGPAi8cIZoFkCD2dnqRJz0J3jRCG3PwCr/oF3sSrP+IH+B/2JHMwiQ8KHu9VUVXPi6XQaNvfVmFtfWNzq7hd2tnd2z8oHx61dJQoxpsskpHqeFRzKULeRIGSd2LFaeBJ3vbGd5nffuJKiyh8xEnM3YAOQ+ELRtFInd6YY8rOp/1yxa7aM5BV4uSkAjka/fJPbxCxJOAhMkm17jp2jG5KFQom+bTUSzSPKRvTIe8aGtKAazed3TslZ0YZED9SpkIkM/XvREoDrSeBZzoDiiO97GXif143Qb/mpiKME+Qhmy/yE0kwItnzZCAUZygnhlCmhLmVsBFVlKGJaGGLp6iJJsvFWU5hlbQuq85V9ebhulKv5QkV4QRO4QIcuIU63EMDmsBAwgu8wpv1bL1bH9bnvLVg5TPHsADr6xe8A5hB

|ai
AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKez6wBwDXjxGMA9IljA76SRjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSy4Nq777eTW1jc2t/LbhZ3dvf2D4uFRQ0eJYlhnkYhUK6AaBZdYN9wIbMUKaRgIbAaj26nffEKleSQfzDhGP6QDyfucUWOlRmeEhtBuseSW3RnIKvEyUoIMtW7xp9OLWBKiNExQrdueGxs/pcpwJnBS6CQaY8pGdIBtSyUNUfvp7NoJObNKj/QjZUsaMlP/TqQ01HocBrYzpGaol72p+J/XTky/4qdcxolByeaL+okgJiLT10mPK2RGjC2hTHF7K2FDqigzNqCFLYGiNpiJzcVbTmGVNC7K3mX5+v6qVK1kCeXhBE7hHDy4gSrcQQ3qwOARXuAV3pxn5935cD7nrTknmzmGBThfv9PMlyw=

|bi
AAACAHicbVDLSgNBEJyNrxhfUY9eBoPgKez6wBwDXjxGMA9IljA76SRjZmeXmV4hLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSyFQdf9dnJr6xubW/ntws7u3v5B8fCoYaJEc6jzSEa6FTADUiioo0AJrVgDCwMJzWB0O/WbT6CNiNQDjmPwQzZQoi84Qys1OiNAGnSLJbfszkBXiZeREslQ6xZ/Or2IJyEo5JIZ0/bcGP2UaRRcwqTQSQzEjI/YANqWKhaC8dPZtRN6ZpUe7UfalkI6U/9OpCw0ZhwGtjNkODTL3lT8z2sn2K/4qVBxgqD4fFE/kRQjOn2d9oQGjnJsCeNa2FspHzLNONqAFrYEmtlgJjYXbzmFVdK4KHuX5ev7q1K1kiWUJyfklJwTj9yQKrkjNVInnDySF/JK3pxn5935cD7nrTknmzkmC3C+fgHVY5ct

Toffoli gate: Doubly controlled NOT gate

| i = a |00i+ b |01i+ c |10i+ d |11i
AAACNXicbZDLSgMxFIYzXmu9jbp0EyyCIJSMCFZBKLhxWcGxhc5QMpm0Dc1cSDJCGeZdfAyfwK1u3bhSt76CmcvCtv4Q+PKfczjJ78WcSYXQu7G0vLK6tl7bqG9ube/smnv7DzJKBKE2iXgkeh6WlLOQ2oopTnuxoDjwOO16k5u83n2kQrIovFfTmLoBHoVsyAhW2hqYV86EqtSJJcvgNcTFDaEMnkKvZCtnUrBV+H7JVjYwG6iJCsFFsCpogEqdgfnl+BFJAhoqwrGUfQvFyk2xUIxwmtWdRNIYkwke0b7GEAdUumnxxwwea8eHw0joEypYuH8nUhxIOQ083RlgNZbztdz8r9ZP1LDlpiyME0VDUi4aJhyqCOaBQZ8JShSfasBEMP1WSMZYYKJ0rDNbPIF1NHku1nwKi2CfNS+b6O680W5VAdXAITgCJ8ACF6ANbkEH2ICAJ/ACXsGb8Wx8GJ/Gd9m6ZFQzB2BGxs8v5lqqfA==AAACNXicbZDLSgMxFIYzXmu9jbp0EyyCIJSMCFZBKLhxWcGxhc5QMpm0Dc1cSDJCGeZdfAyfwK1u3bhSt76CmcvCtv4Q+PKfczjJ78WcSYXQu7G0vLK6tl7bqG9ube/smnv7DzJKBKE2iXgkeh6WlLOQ2oopTnuxoDjwOO16k5u83n2kQrIovFfTmLoBHoVsyAhW2hqYV86EqtSJJcvgNcTFDaEMnkKvZCtnUrBV+H7JVjYwG6iJCsFFsCpogEqdgfnl+BFJAhoqwrGUfQvFyk2xUIxwmtWdRNIYkwke0b7GEAdUumnxxwwea8eHw0joEypYuH8nUhxIOQ083RlgNZbztdz8r9ZP1LDlpiyME0VDUi4aJhyqCOaBQZ8JShSfasBEMP1WSMZYYKJ0rDNbPIF1NHku1nwKi2CfNS+b6O680W5VAdXAITgCJ8ACF6ANbkEH2ICAJ/ACXsGb8Wx8GJ/Gd9m6ZFQzB2BGxs8v5lqqfA==AAACNXicbZDLSgMxFIYzXmu9jbp0EyyCIJSMCFZBKLhxWcGxhc5QMpm0Dc1cSDJCGeZdfAyfwK1u3bhSt76CmcvCtv4Q+PKfczjJ78WcSYXQu7G0vLK6tl7bqG9ube/smnv7DzJKBKE2iXgkeh6WlLOQ2oopTnuxoDjwOO16k5u83n2kQrIovFfTmLoBHoVsyAhW2hqYV86EqtSJJcvgNcTFDaEMnkKvZCtnUrBV+H7JVjYwG6iJCsFFsCpogEqdgfnl+BFJAhoqwrGUfQvFyk2xUIxwmtWdRNIYkwke0b7GEAdUumnxxwwea8eHw0joEypYuH8nUhxIOQ083RlgNZbztdz8r9ZP1LDlpiyME0VDUi4aJhyqCOaBQZ8JShSfasBEMP1WSMZYYKJ0rDNbPIF1NHku1nwKi2CfNS+b6O680W5VAdXAITgCJ8ACF6ANbkEH2ICAJ/ACXsGb8Wx8GJ/Gd9m6ZFQzB2BGxs8v5lqqfA==

| 0i =
a |00i+ b |01i
p

|a|2 + |b|2
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

Multiqubit measurement: Apply the
collapse principle, and renormalize the
state.

MULT20015 Elements of Quantum Computing
Lecture 8

8.1 Classical Logic and Universality

MULT20015
Lecture 8

MULT20015 Elements of Quantum Computing
Lecture 8

AND Gate

An example of a classical digital logic operation is the AND gate:

A B A B

0 0 0

0 1 0

1 0 0

1 1 1

Output

A

B
AB

Inputs

MULT20015 Elements of Quantum Computing
Lecture 8

OR Gate

Another example is the OR gate:

A B A OR B

0 0 0

0 1 1

1 0 1

1 1 1

OutputInputs

A

B
A OR B

MULT20015 Elements of Quantum Computing
Lecture 8

NOT Gate

Our final example of a digital logic gate:

A NOT A

0 1

1 0

OutputInput

A NOT A

MULT20015 Elements of Quantum Computing
Lecture 8

Implementing Boolean Functions

AND

OR

NOTAND

Combine several logic gates to create more complex Boolean functions.

A set of (classical) gates is said to be functionally complete (or
“universal”) if every possible truth table (ie. Boolean function) can be
expressed using members of the set.

MULT20015 Elements of Quantum Computing
Lecture 8

Universality in classical logic

A set of (classical) gates is said to be functionally complete (or
“universal”) if every possible truth table (ie. Boolean function) can be
expressed using members of the set.

For example, in classical logic: {AND, OR, NOT} is universal.

AND OR NOT

Every logical circuit can be made from combinations of these gates. In fact,
the NAND gate alone is universal. However, we cannot implement these
gates directly in a quantum computer.

MULT20015 Elements of Quantum Computing
Lecture 8

8.2 Reversible Logic Gates

MULT20015
Lecture 8

MULT20015 Elements of Quantum Computing
Lecture 8

Quantum Computers are Reversible

All the operations we have seen a quantum computer can perform are
reversible.

Eg. CNOT, X, Hadamard.

The only operation we have seen which a quantum computer performs
which is not reversible is measurement.

So quantum computers cannot directly perform AND, OR gates.

MULT20015 Elements of Quantum Computing
Lecture 8

Irreversible Functions

AND or OR are irreversible.

A B A B

0 0 0

0 1 0

1 0 0

1 1 1

We cannot determine the
inputs from the output.

Irreversible functions are cannot be implemented
directly on a quantum computer.

A

B
AB

MULT20015 Elements of Quantum Computing
Lecture 8

Reversible Logic

Classically, if we would like to calculate some Boolean function, f, we can
construct a circuit out of AND, OR and NOT gates. However, AND and OR
gates are not reversible, and so can’t be implemented in a quantum
computer.

|1i
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NOT a

But by use of reversible circuits
and additional bits/qubits, we can
express everything in terms of
reversible gates (such as Toffoli):

MULT20015 Elements of Quantum Computing
Lecture 8

Reversible Logic and the Toffoli Gate

|ai
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|0i
AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2N0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkRwbVz32yltbG5t75R3K3v7B4dH1eOTjo5TxbDNYhGrx4BqFFxi23Aj8DFRSKNAYDeY3OZ+9wmV5rF8MNME/YiOJA85o8ZK3f4ETebOBtWaW3fnIOvEK0gNCrQG1Z/+MGZphNIwQbXueW5i/Iwqw5nAWaWfakwom9AR9iyVNELtZ/NzZ+TCKkMSxsqWNGSu/p3IaKT1NApsZ0TNWK96ufif10tN2PAzLpPUoGSLRWEqiIlJ/jsZcoXMiKkllClubyVsTBVlxia0tCVQ1CaT5+KtprBOOld1z61799e1ZqNIqAxncA6X4MENNOEOWtAGBhN4gVd4c56dd+fD+Vy0lpxi5hSW4Hz9AgLOl9Y=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2N0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkRwbVz32yltbG5t75R3K3v7B4dH1eOTjo5TxbDNYhGrx4BqFFxi23Aj8DFRSKNAYDeY3OZ+9wmV5rF8MNME/YiOJA85o8ZK3f4ETebOBtWaW3fnIOvEK0gNCrQG1Z/+MGZphNIwQbXueW5i/Iwqw5nAWaWfakwom9AR9iyVNELtZ/NzZ+TCKkMSxsqWNGSu/p3IaKT1NApsZ0TNWK96ufif10tN2PAzLpPUoGSLRWEqiIlJ/jsZcoXMiKkllClubyVsTBVlxia0tCVQ1CaT5+KtprBOOld1z61799e1ZqNIqAxncA6X4MENNOEOWtAGBhN4gVd4c56dd+fD+Vy0lpxi5hSW4Hz9AgLOl9Y=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2N0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkRwbVz32yltbG5t75R3K3v7B4dH1eOTjo5TxbDNYhGrx4BqFFxi23Aj8DFRSKNAYDeY3OZ+9wmV5rF8MNME/YiOJA85o8ZK3f4ETebOBtWaW3fnIOvEK0gNCrQG1Z/+MGZphNIwQbXueW5i/Iwqw5nAWaWfakwom9AR9iyVNELtZ/NzZ+TCKkMSxsqWNGSu/p3IaKT1NApsZ0TNWK96ufif10tN2PAzLpPUoGSLRWEqiIlJ/jsZcoXMiKkllClubyVsTBVlxia0tCVQ1CaT5+KtprBOOld1z61799e1ZqNIqAxncA6X4MENNOEOWtAGBhN4gVd4c56dd+fD+Vy0lpxi5hSW4Hz9AgLOl9Y=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2N0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkRwbVz32yltbG5t75R3K3v7B4dH1eOTjo5TxbDNYhGrx4BqFFxi23Aj8DFRSKNAYDeY3OZ+9wmV5rF8MNME/YiOJA85o8ZK3f4ETebOBtWaW3fnIOvEK0gNCrQG1Z/+MGZphNIwQbXueW5i/Iwqw5nAWaWfakwom9AR9iyVNELtZ/NzZ+TCKkMSxsqWNGSu/p3IaKT1NApsZ0TNWK96ufif10tN2PAzLpPUoGSLRWEqiIlJ/jsZcoXMiKkllClubyVsTBVlxia0tCVQ1CaT5+KtprBOOld1z61799e1ZqNIqAxncA6X4MENNOEOWtAGBhN4gVd4c56dd+fD+Vy0lpxi5hSW4Hz9AgLOl9Y=

|bi
AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=

|ai
AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=

|bi
AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=

a AND b

|a ^ bi
AAACC3icbZDLSsNAFIYn9VbrpVGXbgaL4KokIthlwY3LCvYCTSgnk0k7dDIJMxOhhD6CT+BWn8CduPUhfADfw0mbhW394cDPf87hHL4g5Uxpx/m2KlvbO7t71f3aweHRcd0+Oe2pJJOEdknCEzkIQFHOBO1qpjkdpJJCHHDaD6Z3Rb//RKViiXjUs5T6MYwFixgBbaKRXfemVOeAPQ4ixMF8ZDecprMQ3jRuaRqoVGdk/3hhQrKYCk04KDV0nVT7OUjNCKfzmpcpmgKZwpgOjRUQU+Xni8fn+NIkIY4SaUpovEj/buQQKzWLAzMZg56o9V4R/tcbZjpq+TkTaaapIMtDUcaxTnBBAYdMUqL5zBggkplfMZmABKINq5UrgQRDqODirlPYNL3rpus03YebRrtVEqqic3SBrpCLblEb3aMO6iKCMvSCXtGb9Wy9Wx/W53K0YpU7Z2hF1tcvOoSbJQ==AAACC3icbZDLSsNAFIYn9VbrpVGXbgaL4KokIthlwY3LCvYCTSgnk0k7dDIJMxOhhD6CT+BWn8CduPUhfADfw0mbhW394cDPf87hHL4g5Uxpx/m2KlvbO7t71f3aweHRcd0+Oe2pJJOEdknCEzkIQFHOBO1qpjkdpJJCHHDaD6Z3Rb//RKViiXjUs5T6MYwFixgBbaKRXfemVOeAPQ4ixMF8ZDecprMQ3jRuaRqoVGdk/3hhQrKYCk04KDV0nVT7OUjNCKfzmpcpmgKZwpgOjRUQU+Xni8fn+NIkIY4SaUpovEj/buQQKzWLAzMZg56o9V4R/tcbZjpq+TkTaaapIMtDUcaxTnBBAYdMUqL5zBggkplfMZmABKINq5UrgQRDqODirlPYNL3rpus03YebRrtVEqqic3SBrpCLblEb3aMO6iKCMvSCXtGb9Wy9Wx/W53K0YpU7Z2hF1tcvOoSbJQ==AAACC3icbZDLSsNAFIYn9VbrpVGXbgaL4KokIthlwY3LCvYCTSgnk0k7dDIJMxOhhD6CT+BWn8CduPUhfADfw0mbhW394cDPf87hHL4g5Uxpx/m2KlvbO7t71f3aweHRcd0+Oe2pJJOEdknCEzkIQFHOBO1qpjkdpJJCHHDaD6Z3Rb//RKViiXjUs5T6MYwFixgBbaKRXfemVOeAPQ4ixMF8ZDecprMQ3jRuaRqoVGdk/3hhQrKYCk04KDV0nVT7OUjNCKfzmpcpmgKZwpgOjRUQU+Xni8fn+NIkIY4SaUpovEj/buQQKzWLAzMZg56o9V4R/tcbZjpq+TkTaaapIMtDUcaxTnBBAYdMUqL5zBggkplfMZmABKINq5UrgQRDqODirlPYNL3rpus03YebRrtVEqqic3SBrpCLblEb3aMO6iKCMvSCXtGb9Wy9Wx/W53K0YpU7Z2hF1tcvOoSbJQ==AAACC3icbZDLSsNAFIYn9VbrpVGXbgaL4KokIthlwY3LCvYCTSgnk0k7dDIJMxOhhD6CT+BWn8CduPUhfADfw0mbhW394cDPf87hHL4g5Uxpx/m2KlvbO7t71f3aweHRcd0+Oe2pJJOEdknCEzkIQFHOBO1qpjkdpJJCHHDaD6Z3Rb//RKViiXjUs5T6MYwFixgBbaKRXfemVOeAPQ4ixMF8ZDecprMQ3jRuaRqoVGdk/3hhQrKYCk04KDV0nVT7OUjNCKfzmpcpmgKZwpgOjRUQU+Xni8fn+NIkIY4SaUpovEj/buQQKzWLAzMZg56o9V4R/tcbZjpq+TkTaaapIMtDUcaxTnBBAYdMUqL5zBggkplfMZmABKINq5UrgQRDqODirlPYNL3rpus03YebRrtVEqqic3SBrpCLblEb3aMO6iKCMvSCXtGb9Wy9Wx/W53K0YpU7Z2hF1tcvOoSbJQ==

|ai
AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=

|bi
AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=

|ai
AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=

a XOR b

|ai
AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=

|bi
AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=

|ai
AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAGZsNqjW37s5B14lXkBop0BpUf/rDmKcRKOSSGdPz3AT9jGkUXMKs0k8NJIxP2Ah6lioWgfGz+bkzemGVIQ1jbUshnat/JzIWGTONAtsZMRybVS8X//N6KYYNPxMqSREUXywKU0kxpvnvdCg0cJRTSxjXwt5K+ZhpxtEmtLQl0Mwmk+firaawTjpXdc+te/fXtWajSKhMzsg5uSQeuSFNckdapE04mZAX8krenGfn3flwPhetJaeYOSVLcL5+AVDmmAc=

|bi
AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=AAACAXicbVBNS8NAEN3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2J0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJx8Sp5tDmsYz1Y8AMSKGgjQIlPCYaWBRI6AaT29zvPoE2IlYPOE3Aj9hIiVBwhlbq9ieAWTAbVGtu3Z2DrhOvIDVSoDWo/vSHMU8jUMglM6bnuQn6GdMouIRZpZ8aSBifsBH0LFUsAuNn83Nn9MIqQxrG2pZCOlf/TmQsMmYaBbYzYjg2q14u/uf1UgwbfiZUkiIovlgUppJiTPPf6VBo4CinljCuhb2V8jHTjKNNaGlLoJlNJs/FW01hnXSu6p5b9+6va81GkVCZnJFzckk8ckOa5I60SJtwMiEv5JW8Oc/Ou/PhfC5aS04xc0qW4Hz9AlJ+mAg=

(NOT a) OR (NOT b)

|1i
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|1i
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|a� bi
AAACDHicbZBLSgNBEIZ74ivG16hLN41BcBVmRDDLgBuXEcwDMkPo6VSSJj3dQ3dPIAy5gidwqydwJ269gwfwHvYkszCJPxT8/FVFFV+UcKaN5307pa3tnd298n7l4PDo+MQ9PWtrmSoKLSq5VN2IaOBMQMsww6GbKCBxxKETTe7zfmcKSjMpnswsgTAmI8GGjBJjo77rBhMwGcGBTHiqcTTvu1Wv5i2EN41fmCoq1Oy7P8FA0jQGYSgnWvd8LzFhRpRhlMO8EqQaEkInZAQ9awWJQYfZ4vM5vrLJAA+lsiUMXqR/NzISaz2LIzsZEzPW6708/K/XS82wHmZMJKkBQZeHhinHRuIcAx4wBdTwmTWEKmZ/xXRMFKHGwlq5EiliEeVc/HUKm6Z9U/O9mv94W23UC0JldIEu0TXy0R1qoAfURC1E0RS9oFf05jw7786H87kcLTnFzjlakfP1C1Gqm8M=AAACDHicbZBLSgNBEIZ74ivG16hLN41BcBVmRDDLgBuXEcwDMkPo6VSSJj3dQ3dPIAy5gidwqydwJ269gwfwHvYkszCJPxT8/FVFFV+UcKaN5307pa3tnd298n7l4PDo+MQ9PWtrmSoKLSq5VN2IaOBMQMsww6GbKCBxxKETTe7zfmcKSjMpnswsgTAmI8GGjBJjo77rBhMwGcGBTHiqcTTvu1Wv5i2EN41fmCoq1Oy7P8FA0jQGYSgnWvd8LzFhRpRhlMO8EqQaEkInZAQ9awWJQYfZ4vM5vrLJAA+lsiUMXqR/NzISaz2LIzsZEzPW6708/K/XS82wHmZMJKkBQZeHhinHRuIcAx4wBdTwmTWEKmZ/xXRMFKHGwlq5EiliEeVc/HUKm6Z9U/O9mv94W23UC0JldIEu0TXy0R1qoAfURC1E0RS9oFf05jw7786H87kcLTnFzjlakfP1C1Gqm8M=AAACDHicbZBLSgNBEIZ74ivG16hLN41BcBVmRDDLgBuXEcwDMkPo6VSSJj3dQ3dPIAy5gidwqydwJ269gwfwHvYkszCJPxT8/FVFFV+UcKaN5307pa3tnd298n7l4PDo+MQ9PWtrmSoKLSq5VN2IaOBMQMsww6GbKCBxxKETTe7zfmcKSjMpnswsgTAmI8GGjBJjo77rBhMwGcGBTHiqcTTvu1Wv5i2EN41fmCoq1Oy7P8FA0jQGYSgnWvd8LzFhRpRhlMO8EqQaEkInZAQ9awWJQYfZ4vM5vrLJAA+lsiUMXqR/NzISaz2LIzsZEzPW6708/K/XS82wHmZMJKkBQZeHhinHRuIcAx4wBdTwmTWEKmZ/xXRMFKHGwlq5EiliEeVc/HUKm6Z9U/O9mv94W23UC0JldIEu0TXy0R1qoAfURC1E0RS9oFf05jw7786H87kcLTnFzjlakfP1C1Gqm8M=AAACDHicbZBLSgNBEIZ74ivG16hLN41BcBVmRDDLgBuXEcwDMkPo6VSSJj3dQ3dPIAy5gidwqydwJ269gwfwHvYkszCJPxT8/FVFFV+UcKaN5307pa3tnd298n7l4PDo+MQ9PWtrmSoKLSq5VN2IaOBMQMsww6GbKCBxxKETTe7zfmcKSjMpnswsgTAmI8GGjBJjo77rBhMwGcGBTHiqcTTvu1Wv5i2EN41fmCoq1Oy7P8FA0jQGYSgnWvd8LzFhRpRhlMO8EqQaEkInZAQ9awWJQYfZ4vM5vrLJAA+lsiUMXqR/NzISaz2LIzsZEzPW6708/K/XS82wHmZMJKkBQZeHhinHRuIcAx4wBdTwmTWEKmZ/xXRMFKHGwlq5EiliEeVc/HUKm6Z9U/O9mv94W23UC0JldIEu0TXy0R1qoAfURC1E0RS9oFf05jw7786H87kcLTnFzjlakfP1C1Gqm8M=

|1i
AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2N0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkRwbVz32yltbG5t75R3K3v7B4dH1eOTjo5TxbDNYhGrx4BqFFxi23Aj8DFRSKNAYDeY3OZ+9wmV5rF8MNME/YiOJA85o8ZK3f4ETebNBtWaW3fnIOvEK0gNCrQG1Z/+MGZphNIwQbXueW5i/Iwqw5nAWaWfakwom9AR9iyVNELtZ/NzZ+TCKkMSxsqWNGSu/p3IaKT1NApsZ0TNWK96ufif10tN2PAzLpPUoGSLRWEqiIlJ/jsZcoXMiKkllClubyVsTBVlxia0tCVQ1CaT5+KtprBOOld1z61799e1ZqNIqAxncA6X4MENNOEOWtAGBhN4gVd4c56dd+fD+Vy0lpxi5hSW4Hz9AgRml9c=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2N0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkRwbVz32yltbG5t75R3K3v7B4dH1eOTjo5TxbDNYhGrx4BqFFxi23Aj8DFRSKNAYDeY3OZ+9wmV5rF8MNME/YiOJA85o8ZK3f4ETebNBtWaW3fnIOvEK0gNCrQG1Z/+MGZphNIwQbXueW5i/Iwqw5nAWaWfakwom9AR9iyVNELtZ/NzZ+TCKkMSxsqWNGSu/p3IaKT1NApsZ0TNWK96ufif10tN2PAzLpPUoGSLRWEqiIlJ/jsZcoXMiKkllClubyVsTBVlxia0tCVQ1CaT5+KtprBOOld1z61799e1ZqNIqAxncA6X4MENNOEOWtAGBhN4gVd4c56dd+fD+Vy0lpxi5hSW4Hz9AgRml9c=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2N0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkRwbVz32yltbG5t75R3K3v7B4dH1eOTjo5TxbDNYhGrx4BqFFxi23Aj8DFRSKNAYDeY3OZ+9wmV5rF8MNME/YiOJA85o8ZK3f4ETebNBtWaW3fnIOvEK0gNCrQG1Z/+MGZphNIwQbXueW5i/Iwqw5nAWaWfakwom9AR9iyVNELtZ/NzZ+TCKkMSxsqWNGSu/p3IaKT1NApsZ0TNWK96ufif10tN2PAzLpPUoGSLRWEqiIlJ/jsZcoXMiKkllClubyVsTBVlxia0tCVQ1CaT5+KtprBOOld1z61799e1ZqNIqAxncA6X4MENNOEOWtAGBhN4gVd4c56dd+fD+Vy0lpxi5hSW4Hz9AgRml9c=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2k3bpZhN2N0IJPfkLvOov8CZe/SX+AP+HmzYH2/pg4PHeDDPzgkRwbVz32yltbG5t75R3K3v7B4dH1eOTjo5TxbDNYhGrx4BqFFxi23Aj8DFRSKNAYDeY3OZ+9wmV5rF8MNME/YiOJA85o8ZK3f4ETebNBtWaW3fnIOvEK0gNCrQG1Z/+MGZphNIwQbXueW5i/Iwqw5nAWaWfakwom9AR9iyVNELtZ/NzZ+TCKkMSxsqWNGSu/p3IaKT1NApsZ0TNWK96ufif10tN2PAzLpPUoGSLRWEqiIlJ/jsZcoXMiKkllClubyVsTBVlxia0tCVQ1CaT5+KtprBOOld1z61799e1ZqNIqAxncA6X4MENNOEOWtAGBhN4gVd4c56dd+fD+Vy0lpxi5hSW4Hz9AgRml9c=

|¬a _ ¬bi
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

The Toffoli gate is universal and reversible. In principle every classical Boolean
function can be written in terms of reversible gates (such as Toffoli) which can
be implemented on a quantum computer.

a b a ⋀ b

0 0 0

0 1 0

1 0 0

1 1 1

a b a ⊕ b

0 0 0

0 1 1

1 0 1

1 1 0

a b ¬a ⋁ ¬b

0 0 1

0 1 1

1 0 1

1 1 0

MULT20015 Elements of Quantum Computing
Lecture 8

Implementing irreversible functions

Another strategy which you often see to make an irreversible function
reversible is simply to propagate the inputs:

Uf

|xi
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|xi
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You will see this pattern in several of the quantum algorithms we will study.

Input Repeated at
output

|0i
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|f(x)i
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MULT20015 Elements of Quantum Computing
Lecture 8

Implementing irreversible functions

Another strategy which you often see to make an irreversible function
reversible is simply to propagate the inputs:

Uf

|xi
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|xi
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|y � f(x)i
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

Add (bit-by-bit modulo 2)
input and calculated f(x)

Must deal with all possible
inputs, not just 0.

You will see this pattern in several of the quantum algorithms we will study.

Input Repeated at
output

MULT20015 Elements of Quantum Computing
Lecture 8

8.3 Arithmetic on a Quantum Computer

MULT20015
Lecture 8

MULT20015 Elements of Quantum Computing
Lecture 8

One Bit Adder

We can, for example, implement a one bit adder using only reversible gates:

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AAACAHicbVDLSgNBEJyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmV4hLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSKFQdf9dgobm1vbO8Xd0t7+weFR+fikZeJUc2jyWMa6EzADUihookAJnUQDiwIJ7WB8O/PbT6CNiNUDThLwIzZUIhScoZVavTEgDfrlilt156DrxMtJheRo9Ms/vUHM0wgUcsmM6Xpugn7GNAouYVrqpQYSxsdsCF1LFYvA+Nn82im9sMqAhrG2pZDO1b8TGYuMmUSB7YwYjsyqNxP/87ophjU/EypJERRfLApTSTGms9fpQGjgKCeWMK6FvZXyEdOMow1oaUugmQ1manPxVlNYJ62rqudWvfvrSr2WJ1QkZ+ScXBKP3JA6uSMN0iScPJIX8krenGfn3flwPhetBSefOSVLcL5+AdMflyY=AAACAHicbVDLSgNBEJyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmV4hLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSKFQdf9dgobm1vbO8Xd0t7+weFR+fikZeJUc2jyWMa6EzADUihookAJnUQDiwIJ7WB8O/PbT6CNiNUDThLwIzZUIhScoZVavTEgDfrlilt156DrxMtJheRo9Ms/vUHM0wgUcsmM6Xpugn7GNAouYVrqpQYSxsdsCF1LFYvA+Nn82im9sMqAhrG2pZDO1b8TGYuMmUSB7YwYjsyqNxP/87ophjU/EypJERRfLApTSTGms9fpQGjgKCeWMK6FvZXyEdOMow1oaUugmQ1manPxVlNYJ62rqudWvfvrSr2WJ1QkZ+ScXBKP3JA6uSMN0iScPJIX8krenGfn3flwPhetBSefOSVLcL5+AdMflyY=AAACAHicbVDLSgNBEJyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmV4hLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSKFQdf9dgobm1vbO8Xd0t7+weFR+fikZeJUc2jyWMa6EzADUihookAJnUQDiwIJ7WB8O/PbT6CNiNUDThLwIzZUIhScoZVavTEgDfrlilt156DrxMtJheRo9Ms/vUHM0wgUcsmM6Xpugn7GNAouYVrqpQYSxsdsCF1LFYvA+Nn82im9sMqAhrG2pZDO1b8TGYuMmUSB7YwYjsyqNxP/87ophjU/EypJERRfLApTSTGms9fpQGjgKCeWMK6FvZXyEdOMow1oaUugmQ1manPxVlNYJ62rqudWvfvrSr2WJ1QkZ+ScXBKP3JA6uSMN0iScPJIX8krenGfn3flwPhetBSefOSVLcL5+AdMflyY=AAACAHicbVDLSgNBEJyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmV4hLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSKFQdf9dgobm1vbO8Xd0t7+weFR+fikZeJUc2jyWMa6EzADUihookAJnUQDiwIJ7WB8O/PbT6CNiNUDThLwIzZUIhScoZVavTEgDfrlilt156DrxMtJheRo9Ms/vUHM0wgUcsmM6Xpugn7GNAouYVrqpQYSxsdsCF1LFYvA+Nn82im9sMqAhrG2pZDO1b8TGYuMmUSB7YwYjsyqNxP/87ophjU/EypJERRfLApTSTGms9fpQGjgKCeWMK6FvZXyEdOMow1oaUugmQ1manPxVlNYJ62rqudWvfvrSr2WJ1QkZ+ScXBKP3JA6uSMN0iScPJIX8krenGfn3flwPhetBSefOSVLcL5+AdMflyY=

|ai
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We will now explain this circuit and in the lab we will extend
this to a two bit adder.

MULT20015 Elements of Quantum Computing
Lecture 8

0 + 0 = 0

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002=0

0 + 0 = 0
|0i

AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=

MULT20015 Elements of Quantum Computing
Lecture 8

0 + 1 = 1

|0i
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|0i
AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=

|0i
AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=

|0i
AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=

|0i
AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=

012 = 1

|1i
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|1i
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0 + 1 = 1

MULT20015 Elements of Quantum Computing
Lecture 8

1 + 0 = 1

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|0i
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|0i
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|0i
AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=

012 = 1

|1i
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|1i
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|1i
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1 + 0 = 1

MULT20015 Elements of Quantum Computing
Lecture 8

1 + 1 = 2

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AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=

|0i
AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=

102 = 2

|1i
AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=

|1i
AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=

|1i
AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=

1 + 1 = 2
|1i

AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=

|1i
AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=

MULT20015 Elements of Quantum Computing
Lecture 8

1+1 Quantum Style

Here is what happens when we add together numbers in superposition:

|0i+ |1i
p
2

AAACHXicbZDLSsNAFIYn9VbrLerSzWgRBKEkRbDLghuXFewFmlAm00k7dHJx5kQoIWsfwydwq0/gTtyKD+B7OGmzsK0/DPz85xzOmc+LBVdgWd9GaW19Y3OrvF3Z2d3bPzAPjzoqSiRlbRqJSPY8opjgIWsDB8F6sWQk8ATrepObvN59ZFLxKLyHaczcgIxC7nNKQEcD89TxJaGpM2GALXyJZ8bGWeqoBwlpPcsGZtWqWTPhVWMXpooKtQbmjzOMaBKwEKggSvVtKwY3JRI4FSyrOIliMaETMmJ9bUMSMOWms69k+FwnQ+xHUr8Q8Cz9O5GSQKlp4OnOgMBYLdfy8L9aPwG/4aY8jBNgIZ0v8hOBIcI5FzzkklEQU20IlVzfiumYaDag6S1s8STRkHIu9jKFVdOp12yrZt9dVZuNglAZnaAzdIFsdI2a6Ba1UBtR9IRe0Ct6M56Nd+PD+Jy3loxi5hgtyPj6BQdJoec=AAACHXicbZDLSsNAFIYn9VbrLerSzWgRBKEkRbDLghuXFewFmlAm00k7dHJx5kQoIWsfwydwq0/gTtyKD+B7OGmzsK0/DPz85xzOmc+LBVdgWd9GaW19Y3OrvF3Z2d3bPzAPjzoqSiRlbRqJSPY8opjgIWsDB8F6sWQk8ATrepObvN59ZFLxKLyHaczcgIxC7nNKQEcD89TxJaGpM2GALXyJZ8bGWeqoBwlpPcsGZtWqWTPhVWMXpooKtQbmjzOMaBKwEKggSvVtKwY3JRI4FSyrOIliMaETMmJ9bUMSMOWms69k+FwnQ+xHUr8Q8Cz9O5GSQKlp4OnOgMBYLdfy8L9aPwG/4aY8jBNgIZ0v8hOBIcI5FzzkklEQU20IlVzfiumYaDag6S1s8STRkHIu9jKFVdOp12yrZt9dVZuNglAZnaAzdIFsdI2a6Ba1UBtR9IRe0Ct6M56Nd+PD+Jy3loxi5hgtyPj6BQdJoec=AAACHXicbZDLSsNAFIYn9VbrLerSzWgRBKEkRbDLghuXFewFmlAm00k7dHJx5kQoIWsfwydwq0/gTtyKD+B7OGmzsK0/DPz85xzOmc+LBVdgWd9GaW19Y3OrvF3Z2d3bPzAPjzoqSiRlbRqJSPY8opjgIWsDB8F6sWQk8ATrepObvN59ZFLxKLyHaczcgIxC7nNKQEcD89TxJaGpM2GALXyJZ8bGWeqoBwlpPcsGZtWqWTPhVWMXpooKtQbmjzOMaBKwEKggSvVtKwY3JRI4FSyrOIliMaETMmJ9bUMSMOWms69k+FwnQ+xHUr8Q8Cz9O5GSQKlp4OnOgMBYLdfy8L9aPwG/4aY8jBNgIZ0v8hOBIcI5FzzkklEQU20IlVzfiumYaDag6S1s8STRkHIu9jKFVdOp12yrZt9dVZuNglAZnaAzdIFsdI2a6Ba1UBtR9IRe0Ct6M56Nd+PD+Jy3loxi5hgtyPj6BQdJoec=AAACHXicbZDLSsNAFIYn9VbrLerSzWgRBKEkRbDLghuXFewFmlAm00k7dHJx5kQoIWsfwydwq0/gTtyKD+B7OGmzsK0/DPz85xzOmc+LBVdgWd9GaW19Y3OrvF3Z2d3bPzAPjzoqSiRlbRqJSPY8opjgIWsDB8F6sWQk8ATrepObvN59ZFLxKLyHaczcgIxC7nNKQEcD89TxJaGpM2GALXyJZ8bGWeqoBwlpPcsGZtWqWTPhVWMXpooKtQbmjzOMaBKwEKggSvVtKwY3JRI4FSyrOIliMaETMmJ9bUMSMOWms69k+FwnQ+xHUr8Q8Cz9O5GSQKlp4OnOgMBYLdfy8L9aPwG/4aY8jBNgIZ0v8hOBIcI5FzzkklEQU20IlVzfiumYaDag6S1s8STRkHIu9jKFVdOp12yrZt9dVZuNglAZnaAzdIFsdI2a6Ba1UBtR9IRe0Ct6M56Nd+PD+Jy3loxi5hgtyPj6BQdJoec=

“+” |1iAAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=

|0i+ |1i
p
2

AAACHXicbZDLSsNAFIYn9VbrLerSzWgRBKEkRbDLghuXFewFmlAm00k7dHJx5kQoIWsfwydwq0/gTtyKD+B7OGmzsK0/DPz85xzOmc+LBVdgWd9GaW19Y3OrvF3Z2d3bPzAPjzoqSiRlbRqJSPY8opjgIWsDB8F6sWQk8ATrepObvN59ZFLxKLyHaczcgIxC7nNKQEcD89TxJaGpM2GALXyJZ8bGWeqoBwlpPcsGZtWqWTPhVWMXpooKtQbmjzOMaBKwEKggSvVtKwY3JRI4FSyrOIliMaETMmJ9bUMSMOWms69k+FwnQ+xHUr8Q8Cz9O5GSQKlp4OnOgMBYLdfy8L9aPwG/4aY8jBNgIZ0v8hOBIcI5FzzkklEQU20IlVzfiumYaDag6S1s8STRkHIu9jKFVdOp12yrZt9dVZuNglAZnaAzdIFsdI2a6Ba1UBtR9IRe0Ct6M56Nd+PD+Jy3loxi5hgtyPj6BQdJoec=AAACHXicbZDLSsNAFIYn9VbrLerSzWgRBKEkRbDLghuXFewFmlAm00k7dHJx5kQoIWsfwydwq0/gTtyKD+B7OGmzsK0/DPz85xzOmc+LBVdgWd9GaW19Y3OrvF3Z2d3bPzAPjzoqSiRlbRqJSPY8opjgIWsDB8F6sWQk8ATrepObvN59ZFLxKLyHaczcgIxC7nNKQEcD89TxJaGpM2GALXyJZ8bGWeqoBwlpPcsGZtWqWTPhVWMXpooKtQbmjzOMaBKwEKggSvVtKwY3JRI4FSyrOIliMaETMmJ9bUMSMOWms69k+FwnQ+xHUr8Q8Cz9O5GSQKlp4OnOgMBYLdfy8L9aPwG/4aY8jBNgIZ0v8hOBIcI5FzzkklEQU20IlVzfiumYaDag6S1s8STRkHIu9jKFVdOp12yrZt9dVZuNglAZnaAzdIFsdI2a6Ba1UBtR9IRe0Ct6M56Nd+PD+Jy3loxi5hgtyPj6BQdJoec=AAACHXicbZDLSsNAFIYn9VbrLerSzWgRBKEkRbDLghuXFewFmlAm00k7dHJx5kQoIWsfwydwq0/gTtyKD+B7OGmzsK0/DPz85xzOmc+LBVdgWd9GaW19Y3OrvF3Z2d3bPzAPjzoqSiRlbRqJSPY8opjgIWsDB8F6sWQk8ATrepObvN59ZFLxKLyHaczcgIxC7nNKQEcD89TxJaGpM2GALXyJZ8bGWeqoBwlpPcsGZtWqWTPhVWMXpooKtQbmjzOMaBKwEKggSvVtKwY3JRI4FSyrOIliMaETMmJ9bUMSMOWms69k+FwnQ+xHUr8Q8Cz9O5GSQKlp4OnOgMBYLdfy8L9aPwG/4aY8jBNgIZ0v8hOBIcI5FzzkklEQU20IlVzfiumYaDag6S1s8STRkHIu9jKFVdOp12yrZt9dVZuNglAZnaAzdIFsdI2a6Ba1UBtR9IRe0Ct6M56Nd+PD+Jy3loxi5hgtyPj6BQdJoec=AAACHXicbZDLSsNAFIYn9VbrLerSzWgRBKEkRbDLghuXFewFmlAm00k7dHJx5kQoIWsfwydwq0/gTtyKD+B7OGmzsK0/DPz85xzOmc+LBVdgWd9GaW19Y3OrvF3Z2d3bPzAPjzoqSiRlbRqJSPY8opjgIWsDB8F6sWQk8ATrepObvN59ZFLxKLyHaczcgIxC7nNKQEcD89TxJaGpM2GALXyJZ8bGWeqoBwlpPcsGZtWqWTPhVWMXpooKtQbmjzOMaBKwEKggSvVtKwY3JRI4FSyrOIliMaETMmJ9bUMSMOWms69k+FwnQ+xHUr8Q8Cz9O5GSQKlp4OnOgMBYLdfy8L9aPwG/4aY8jBNgIZ0v8hOBIcI5FzzkklEQU20IlVzfiumYaDag6S1s8STRkHIu9jKFVdOp12yrZt9dVZuNglAZnaAzdIFsdI2a6Ba1UBtR9IRe0Ct6M56Nd+PD+Jy3loxi5hgtyPj6BQdJoec=

|1i
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Let’s do the walkthrough…

MULT20015 Elements of Quantum Computing
Lecture 8

One Qubit adder – example

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2

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MULT20015 Elements of Quantum Computing
Lecture 8

One Qubit adder – example

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| i =
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p
2

⌦ |1i ⌦ |0i ⌦ |0i
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flip

no flip

| i =
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p
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MULT20015 Elements of Quantum Computing
Lecture 8

One Qubit adder – example

|0i+ |1i
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| i =
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p
2

⌦ |1i ⌦ |0i ⌦ |0i
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flip flip

| i =
|0100i+ |1101i

p
2

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| i =
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p
2

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MULT20015 Elements of Quantum Computing
Lecture 8

| i =
|0101i+ |1100i

p
2

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

One Qubit adder – example

|0i+ |1i
p
2

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|1i
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|0i
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|0i
AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=

| i =
|0i+ |1i

p
2

⌦ |1i ⌦ |0i ⌦ |0i
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

no flip

flip

| i =
|0100i+ |1101i

p
2

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

| i =
|0101i+ |1110i

p
2

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

MULT20015 Elements of Quantum Computing
Lecture 8

One Qubit adder – example

|0i+ |1i
p
2

AAACHXicbZDLSsNAFIYn9VbrLerSzWgRBKEkRbDLghuXFewFmlAm00k7dHJx5kQoIWsfwydwq0/gTtyKD+B7OGmzsK0/DPz85xzOmc+LBVdgWd9GaW19Y3OrvF3Z2d3bPzAPjzoqSiRlbRqJSPY8opjgIWsDB8F6sWQk8ATrepObvN59ZFLxKLyHaczcgIxC7nNKQEcD89TxJaGpM2GALXyJZ8bGWeqoBwlpPcsGZtWqWTPhVW MXpooKtQbmjzOMaBKwEKggSvVtKwY3JRI4FSyrOIliMaETMmJ9bUMSMOWms69k+FwnQ+xHUr8Q8Cz9O5GSQKlp4OnOgMBYLdfy8L9aPwG/4aY8jBNgIZ0v8hOBIcI5FzzkklEQU20IlVzfiumYaDag6S1s8STRkHIu9jKFVdOp12yrZt9dVZuNglAZnaAzdIFsdI2a6Ba1UBtR9IRe0Ct6M56Nd+PD+Jy3loxi5hgtyPj6BQdJoec=AAACHXicbZDLSsNAFIYn9VbrLerSzWgRBKEkRbDLghuXFewFmlAm00k7dHJx5kQoIWsfwydwq0/gTtyKD+B7OGmzsK0/DPz85xzOmc+LBVdgWd9GaW19Y3OrvF3Z2d3bPzAPjzoqSiRlbRqJSPY8opjgIWsDB8F6sWQk8ATrepObvN59ZFLxKLyHaczcgIxC7nNKQEcD89TxJaGpM2GALXyJZ8bGWeqoBwlpPcsGZtWqWTPhVW MXpooKtQbmjzOMaBKwEKggSvVtKwY3JRI4FSyrOIliMaETMmJ9bUMSMOWms69k+FwnQ+xHUr8Q8Cz9O5GSQKlp4OnOgMBYLdfy8L9aPwG/4aY8jBNgIZ0v8hOBIcI5FzzkklEQU20IlVzfiumYaDag6S1s8STRkHIu9jKFVdOp12yrZt9dVZuNglAZnaAzdIFsdI2a6Ba1UBtR9IRe0Ct6M56Nd+PD+Jy3loxi5hgtyPj6BQdJoec=AAACHXicbZDLSsNAFIYn9VbrLerSzWgRBKEkRbDLghuXFewFmlAm00k7dHJx5kQoIWsfwydwq0/gTtyKD+B7OGmzsK0/DPz85xzOmc+LBVdgWd9GaW19Y3OrvF3Z2d3bPzAPjzoqSiRlbRqJSPY8opjgIWsDB8F6sWQk8ATrepObvN59ZFLxKLyHaczcgIxC7nNKQEcD89TxJaGpM2GALXyJZ8bGWeqoBwlpPcsGZtWqWTPhVW MXpooKtQbmjzOMaBKwEKggSvVtKwY3JRI4FSyrOIliMaETMmJ9bUMSMOWms69k+FwnQ+xHUr8Q8Cz9O5GSQKlp4OnOgMBYLdfy8L9aPwG/4aY8jBNgIZ0v8hOBIcI5FzzkklEQU20IlVzfiumYaDag6S1s8STRkHIu9jKFVdOp12yrZt9dVZuNglAZnaAzdIFsdI2a6Ba1UBtR9IRe0Ct6M56Nd+PD+Jy3loxi5hgtyPj6BQdJoec=AAACHXicbZDLSsNAFIYn9VbrLerSzWgRBKEkRbDLghuXFewFmlAm00k7dHJx5kQoIWsfwydwq0/gTtyKD+B7OGmzsK0/DPz85xzOmc+LBVdgWd9GaW19Y3OrvF3Z2d3bPzAPjzoqSiRlbRqJSPY8opjgIWsDB8F6sWQk8ATrepObvN59ZFLxKLyHaczcgIxC7nNKQEcD89TxJaGpM2GALXyJZ8bGWeqoBwlpPcsGZtWqWTPhVW MXpooKtQbmjzOMaBKwEKggSvVtKwY3JRI4FSyrOIliMaETMmJ9bUMSMOWms69k+FwnQ+xHUr8Q8Cz9O5GSQKlp4OnOgMBYLdfy8L9aPwG/4aY8jBNgIZ0v8hOBIcI5FzzkklEQU20IlVzfiumYaDag6S1s8STRkHIu9jKFVdOp12yrZt9dVZuNglAZnaAzdIFsdI2a6Ba1UBtR9IRe0Ct6M56Nd+PD+Jy3loxi5hgtyPj6BQdJoec=

|1i
AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=AAACAXicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEsMeCF48VbCu0oWy2m3bJZhN2J0IJPfkLvOov8CZe/SX+AP+H2zYH2/pg4PHeDDPzglQKg6777ZQ2Nre2d8q7lb39g8Oj6vFJxySZZrzNEpnox4AaLoXibRQo+WOqOY0DybtBdDvzu09cG5GoB5yk3I/pSIlQMIpW6vYjjsQjg2rNrbtzkHXiFaQGBVqD6k9/mLAs5gqZpMb0PDdFP6caBZN8WulnhqeURXTEe5YqGnPj5/Nzp+TCKkMSJtqWQjJX/07kNDZmEge2M6Y4NqveTPzP62UYNvxcqDRDrthiUZhJggmZ/U6GQnOGcmIJZVrYWwkbU00Z2oSWtgSa2mSmNhdvNYV10rmqe27du7+uNRtFQmU4g3O4BA9uoAl30II2MIjgBV7hzXl23p0P53PRWnKKmVNYgvP1C98Klx8=

|0i
AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=

|0i
AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=

Or, considering the last two registers as a two-qubit register (binary notation):

0+1=1 1+1=2

Note: this is an
entangled state.| i =

|0i |1i |1i+ |1i |1i |2i
p
2

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

| i =
|0101i+ |1110i

p
2

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

| i =
|0101i+ |1110i

p
2

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

MULT20015 Elements of Quantum Computing
Lecture 8

Implementing irreversible functions

We cannot compute irreversible functions directly (not unitary). One strategy
which you often see to make an irreversible function reversible is simply to
propagate the input to the output:

Uf
|0i

AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=AAACAHicbVDLSgNBEOyNrxhfUY9eBoPgKeyKYI4BLx4jmAckS5id9CZjZmeXmVkhLLn4BV71C7yJV//ED/A/nCR7MIkFDUVVN91dQSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzCRBP6JDyUPOqLFSqzdGQ9x+ueJW3TnIOvFyUoEcjX75pzeIWRqhNExQrbuemxg/o8pwJnBa6qUaE8rGdIhdSyWNUPvZ/NopubDKgISxsiUNmat/JzIaaT2JAtsZUTPSq95M/M/rpias+RmXSWpQssWiMBXExGT2OhlwhcyIiSWUKW5vJWxEFWXGBrS0JVDUBjO1uXirKayT1lXVc6ve/XWlXssTKsIZnMMleHADdbiDBjSBwSO8wCu8Oc/Ou/PhfC5aC04+cwpLcL5+AYOhlvQ=

|xi
AAACAHicbVDLSgNBEJz1GeMr6tHLYBA8hV0RzDHgxWME84BkCbOT3mTM7Owy0yuGJRe/wKt+gTfx6p/4Af6Hk2QPJrGgoajqprsrSKQw6Lrfztr6xubWdmGnuLu3f3BYOjpumjjVHBo8lrFuB8yAFAoaKFBCO9HAokBCKxjdTP3WI2gjYnWP4wT8iA2UCAVnaKVmdwRIn3qlsltxZ6CrxMtJmeSo90o/3X7M0wgUcsmM6Xhugn7GNAouYVLspgYSxkdsAB1LFYvA+Nns2gk9t0qfhrG2pZDO1L8TGYuMGUeB7YwYDs2yNxX/8zophlU/EypJERSfLwpTSTGm09dpX2jgKMeWMK6FvZXyIdOMow1oYUugmQ1mYnPxllNYJc3LiudWvLurcq2aJ1Qgp+SMXBCPXJMauSV10iCcPJAX8krenGfn3flwPueta04+c0IW4Hz9AvYZlzw=AAACAHicbVDLSgNBEJz1GeMr6tHLYBA8hV0RzDHgxWME84BkCbOT3mTM7Owy0yuGJRe/wKt+gTfx6p/4Af6Hk2QPJrGgoajqprsrSKQw6Lrfztr6xubWdmGnuLu3f3BYOjpumjjVHBo8lrFuB8yAFAoaKFBCO9HAokBCKxjdTP3WI2gjYnWP4wT8iA2UCAVnaKVmdwRIn3qlsltxZ6CrxMtJmeSo90o/3X7M0wgUcsmM6Xhugn7GNAouYVLspgYSxkdsAB1LFYvA+Nns2gk9t0qfhrG2pZDO1L8TGYuMGUeB7YwYDs2yNxX/8zophlU/EypJERSfLwpTSTGm09dpX2jgKMeWMK6FvZXyIdOMow1oYUugmQ1mYnPxllNYJc3LiudWvLurcq2aJ1Qgp+SMXBCPXJMauSV10iCcPJAX8krenGfn3flwPueta04+c0IW4Hz9AvYZlzw=AAACAHicbVDLSgNBEJz1GeMr6tHLYBA8hV0RzDHgxWME84BkCbOT3mTM7Owy0yuGJRe/wKt+gTfx6p/4Af6Hk2QPJrGgoajqprsrSKQw6Lrfztr6xubWdmGnuLu3f3BYOjpumjjVHBo8lrFuB8yAFAoaKFBCO9HAokBCKxjdTP3WI2gjYnWP4wT8iA2UCAVnaKVmdwRIn3qlsltxZ6CrxMtJmeSo90o/3X7M0wgUcsmM6Xhugn7GNAouYVLspgYSxkdsAB1LFYvA+Nns2gk9t0qfhrG2pZDO1L8TGYuMGUeB7YwYDs2yNxX/8zophlU/EypJERSfLwpTSTGm09dpX2jgKMeWMK6FvZXyIdOMow1oYUugmQ1mYnPxllNYJc3LiudWvLurcq2aJ1Qgp+SMXBCPXJMauSV10iCcPJAX8krenGfn3flwPueta04+c0IW4Hz9AvYZlzw=AAACAHicbVDLSgNBEJz1GeMr6tHLYBA8hV0RzDHgxWME84BkCbOT3mTM7Owy0yuGJRe/wKt+gTfx6p/4Af6Hk2QPJrGgoajqprsrSKQw6Lrfztr6xubWdmGnuLu3f3BYOjpumjjVHBo8lrFuB8yAFAoaKFBCO9HAokBCKxjdTP3WI2gjYnWP4wT8iA2UCAVnaKVmdwRIn3qlsltxZ6CrxMtJmeSo90o/3X7M0wgUcsmM6Xhugn7GNAouYVLspgYSxkdsAB1LFYvA+Nns2gk9t0qfhrG2pZDO1L8TGYuMGUeB7YwYDs2yNxX/8zophlU/EypJERSfLwpTSTGm09dpX2jgKMeWMK6FvZXyIdOMow1oYUugmQ1mYnPxllNYJc3LiudWvLurcq2aJ1Qgp+SMXBCPXJMauSV10iCcPJAX8krenGfn3flwPueta04+c0IW4Hz9AvYZlzw=

|xi
AAACAHicbVDLSgNBEJz1GeMr6tHLYBA8hV0RzDHgxWME84BkCbOT3mTM7Owy0yuGJRe/wKt+gTfx6p/4Af6Hk2QPJrGgoajqprsrSKQw6Lrfztr6xubWdmGnuLu3f3BYOjpumjjVHBo8lrFuB8yAFAoaKFBCO9HAokBCKxjdTP3WI2gjYnWP4wT8iA2UCAVnaKVmdwRIn3qlsltxZ6CrxMtJmeSo90o/3X7M0wgUcsmM6Xhugn7GNAouYVLspgYSxkdsAB1LFYvA+Nns2gk9t0qfhrG2pZDO1L8TGYuMGUeB7YwYDs2yNxX/8zophlU/EypJERSfLwpTSTGm09dpX2jgKMeWMK6FvZXyIdOMow1oYUugmQ1mYnPxllNYJc3LiudWvLurcq2aJ1Qgp+SMXBCPXJMauSV10iCcPJAX8krenGfn3flwPueta04+c0IW4Hz9AvYZlzw=AAACAHicbVDLSgNBEJz1GeMr6tHLYBA8hV0RzDHgxWME84BkCbOT3mTM7Owy0yuGJRe/wKt+gTfx6p/4Af6Hk2QPJrGgoajqprsrSKQw6Lrfztr6xubWdmGnuLu3f3BYOjpumjjVHBo8lrFuB8yAFAoaKFBCO9HAokBCKxjdTP3WI2gjYnWP4wT8iA2UCAVnaKVmdwRIn3qlsltxZ6CrxMtJmeSo90o/3X7M0wgUcsmM6Xhugn7GNAouYVLspgYSxkdsAB1LFYvA+Nns2gk9t0qfhrG2pZDO1L8TGYuMGUeB7YwYDs2yNxX/8zophlU/EypJERSfLwpTSTGm09dpX2jgKMeWMK6FvZXyIdOMow1oYUugmQ1mYnPxllNYJc3LiudWvLurcq2aJ1Qgp+SMXBCPXJMauSV10iCcPJAX8krenGfn3flwPueta04+c0IW4Hz9AvYZlzw=AAACAHicbVDLSgNBEJz1GeMr6tHLYBA8hV0RzDHgxWME84BkCbOT3mTM7Owy0yuGJRe/wKt+gTfx6p/4Af6Hk2QPJrGgoajqprsrSKQw6Lrfztr6xubWdmGnuLu3f3BYOjpumjjVHBo8lrFuB8yAFAoaKFBCO9HAokBCKxjdTP3WI2gjYnWP4wT8iA2UCAVnaKVmdwRIn3qlsltxZ6CrxMtJmeSo90o/3X7M0wgUcsmM6Xhugn7GNAouYVLspgYSxkdsAB1LFYvA+Nns2gk9t0qfhrG2pZDO1L8TGYuMGUeB7YwYDs2yNxX/8zophlU/EypJERSfLwpTSTGm09dpX2jgKMeWMK6FvZXyIdOMow1oYUugmQ1mYnPxllNYJc3LiudWvLurcq2aJ1Qgp+SMXBCPXJMauSV10iCcPJAX8krenGfn3flwPueta04+c0IW4Hz9AvYZlzw=AAACAHicbVDLSgNBEJz1GeMr6tHLYBA8hV0RzDHgxWME84BkCbOT3mTM7Owy0yuGJRe/wKt+gTfx6p/4Af6Hk2QPJrGgoajqprsrSKQw6Lrfztr6xubWdmGnuLu3f3BYOjpumjjVHBo8lrFuB8yAFAoaKFBCO9HAokBCKxjdTP3WI2gjYnWP4wT8iA2UCAVnaKVmdwRIn3qlsltxZ6CrxMtJmeSo90o/3X7M0wgUcsmM6Xhugn7GNAouYVLspgYSxkdsAB1LFYvA+Nns2gk9t0qfhrG2pZDO1L8TGYuMGUeB7YwYDs2yNxX/8zophlU/EypJERSfLwpTSTGm09dpX2jgKMeWMK6FvZXyIdOMow1oYUugmQ1mYnPxllNYJc3LiudWvLurcq2aJ1Qgp+SMXBCPXJMauSV10iCcPJAX8krenGfn3flwPueta04+c0IW4Hz9AvYZlzw=

|f(x)i
AAACBHicbVDLSsNAFJ34rPVVdekmWIS6KYkIdllw47KCfWAbymR60w6dTMLMjVhCt36BW/0Cd+LW//AD/A8nbRa29cCFwzn3cu89fiy4Rsf5ttbWNza3tgs7xd29/YPD0tFxS0eJYtBkkYhUx6caBJfQRI4COrECGvoC2v74JvPbj6A0j+Q9TmLwQjqUPOCMopEeemPANKg8XUz7pbJTdWawV4mbkzLJ0eiXfnqDiCUhSGSCat11nRi9lCrkTMC02Es0xJSN6RC6hkoagvbS2cVT+9woAzuIlCmJ9kz9O5HSUOtJ6JvOkOJIL3uZ+J/XTTCoeSmXcYIg2XxRkAgbIzt73x5wBQzFxBDKFDe32mxEFWVoQlrY4itqwslycZdTWCWty6rrVN27q3K9lidUIKfkjFSIS65JndySBmkSRiR5Ia/kzXq23q0P63PeumblMydkAdbXLxCumPM=AAACBHicbVDLSsNAFJ34rPVVdekmWIS6KYkIdllw47KCfWAbymR60w6dTMLMjVhCt36BW/0Cd+LW//AD/A8nbRa29cCFwzn3cu89fiy4Rsf5ttbWNza3tgs7xd29/YPD0tFxS0eJYtBkkYhUx6caBJfQRI4COrECGvoC2v74JvPbj6A0j+Q9TmLwQjqUPOCMopEeemPANKg8XUz7pbJTdWawV4mbkzLJ0eiXfnqDiCUhSGSCat11nRi9lCrkTMC02Es0xJSN6RC6hkoagvbS2cVT+9woAzuIlCmJ9kz9O5HSUOtJ6JvOkOJIL3uZ+J/XTTCoeSmXcYIg2XxRkAgbIzt73x5wBQzFxBDKFDe32mxEFWVoQlrY4itqwslycZdTWCWty6rrVN27q3K9lidUIKfkjFSIS65JndySBmkSRiR5Ia/kzXq23q0P63PeumblMydkAdbXLxCumPM=AAACBHicbVDLSsNAFJ34rPVVdekmWIS6KYkIdllw47KCfWAbymR60w6dTMLMjVhCt36BW/0Cd+LW//AD/A8nbRa29cCFwzn3cu89fiy4Rsf5ttbWNza3tgs7xd29/YPD0tFxS0eJYtBkkYhUx6caBJfQRI4COrECGvoC2v74JvPbj6A0j+Q9TmLwQjqUPOCMopEeemPANKg8XUz7pbJTdWawV4mbkzLJ0eiXfnqDiCUhSGSCat11nRi9lCrkTMC02Es0xJSN6RC6hkoagvbS2cVT+9woAzuIlCmJ9kz9O5HSUOtJ6JvOkOJIL3uZ+J/XTTCoeSmXcYIg2XxRkAgbIzt73x5wBQzFxBDKFDe32mxEFWVoQlrY4itqwslycZdTWCWty6rrVN27q3K9lidUIKfkjFSIS65JndySBmkSRiR5Ia/kzXq23q0P63PeumblMydkAdbXLxCumPM=AAACBHicbVDLSsNAFJ34rPVVdekmWIS6KYkIdllw47KCfWAbymR60w6dTMLMjVhCt36BW/0Cd+LW//AD/A8nbRa29cCFwzn3cu89fiy4Rsf5ttbWNza3tgs7xd29/YPD0tFxS0eJYtBkkYhUx6caBJfQRI4COrECGvoC2v74JvPbj6A0j+Q9TmLwQjqUPOCMopEeemPANKg8XUz7pbJTdWawV4mbkzLJ0eiXfnqDiCUhSGSCat11nRi9lCrkTMC02Es0xJSN6RC6hkoagvbS2cVT+9woAzuIlCmJ9kz9O5HSUOtJ6JvOkOJIL3uZ+J/XTTCoeSmXcYIg2XxRkAgbIzt73x5wBQzFxBDKFDe32mxEFWVoQlrY4itqwslycZdTWCWty6rrVN27q3K9lidUIKfkjFSIS65JndySBmkSRiR5Ia/kzXq23q0P63PeumblMydkAdbXLxCumPM=

Propagate
inputs to
output

Calculated f(x)

MULT20015 Elements of Quantum Computing
Lecture 8

One Bit Adder

The adder is an example. Given a + b, we can’t uniquely determine a and b
So we used this trick:

|ai
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|bi
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|carryi
AAACEXicbVDLSsNAFJ3UV62vqLhyM1gEVyURwS4LblxWsA9oQrmZTtuhM0mYmQgh5Cv8Arf6Be7ErV/gB/gfTtosbOuBC4dz7uUeThBzprTjfFuVjc2t7Z3qbm1v/+DwyD4+6aookYR2SMQj2Q9AUc5C2tFMc9qPJQURcNoLZneF33uiUrEofNRpTH0Bk5CNGQFtpKF95s2oxpknQE+lyAhImeb50K47DWcOvE7cktRRifbQ/vFGEUkEDTXhoNTAdWLtZyA1I5zmNS9RNAYygwkdGBqCoMrP5vFzfGmUER5H0kyo8Vz9e5GBUCoVgdksYqpVrxD/8waJHjf9jIVxomlIFo/GCcc6wkUXeMQkJZqnhgCRzGTFZAoSiDaNLX0JJJieil7c1RbWSfe64ToN9+Gm3mqWDVXRObpAV8hFt6iF7lEbdRBBGXpBr+jNerberQ/rc7FascqbU7QE6+sXPQOegQ==AAACEXicbVDLSsNAFJ3UV62vqLhyM1gEVyURwS4LblxWsA9oQrmZTtuhM0mYmQgh5Cv8Arf6Be7ErV/gB/gfTtosbOuBC4dz7uUeThBzprTjfFuVjc2t7Z3qbm1v/+DwyD4+6aookYR2SMQj2Q9AUc5C2tFMc9qPJQURcNoLZneF33uiUrEofNRpTH0Bk5CNGQFtpKF95s2oxpknQE+lyAhImeb50K47DWcOvE7cktRRifbQ/vFGEUkEDTXhoNTAdWLtZyA1I5zmNS9RNAYygwkdGBqCoMrP5vFzfGmUER5H0kyo8Vz9e5GBUCoVgdksYqpVrxD/8waJHjf9jIVxomlIFo/GCcc6wkUXeMQkJZqnhgCRzGTFZAoSiDaNLX0JJJieil7c1RbWSfe64ToN9+Gm3mqWDVXRObpAV8hFt6iF7lEbdRBBGXpBr+jNerberQ/rc7FascqbU7QE6+sXPQOegQ==AAACEXicbVDLSsNAFJ3UV62vqLhyM1gEVyURwS4LblxWsA9oQrmZTtuhM0mYmQgh5Cv8Arf6Be7ErV/gB/gfTtosbOuBC4dz7uUeThBzprTjfFuVjc2t7Z3qbm1v/+DwyD4+6aookYR2SMQj2Q9AUc5C2tFMc9qPJQURcNoLZneF33uiUrEofNRpTH0Bk5CNGQFtpKF95s2oxpknQE+lyAhImeb50K47DWcOvE7cktRRifbQ/vFGEUkEDTXhoNTAdWLtZyA1I5zmNS9RNAYygwkdGBqCoMrP5vFzfGmUER5H0kyo8Vz9e5GBUCoVgdksYqpVrxD/8waJHjf9jIVxomlIFo/GCcc6wkUXeMQkJZqnhgCRzGTFZAoSiDaNLX0JJJieil7c1RbWSfe64ToN9+Gm3mqWDVXRObpAV8hFt6iF7lEbdRBBGXpBr+jNerberQ/rc7FascqbU7QE6+sXPQOegQ==AAACEXicbVDLSsNAFJ3UV62vqLhyM1gEVyURwS4LblxWsA9oQrmZTtuhM0mYmQgh5Cv8Arf6Be7ErV/gB/gfTtosbOuBC4dz7uUeThBzprTjfFuVjc2t7Z3qbm1v/+DwyD4+6aookYR2SMQj2Q9AUc5C2tFMc9qPJQURcNoLZneF33uiUrEofNRpTH0Bk5CNGQFtpKF95s2oxpknQE+lyAhImeb50K47DWcOvE7cktRRifbQ/vFGEUkEDTXhoNTAdWLtZyA1I5zmNS9RNAYygwkdGBqCoMrP5vFzfGmUER5H0kyo8Vz9e5GBUCoVgdksYqpVrxD/8waJHjf9jIVxomlIFo/GCcc6wkUXeMQkJZqnhgCRzGTFZAoSiDaNLX0JJJieil7c1RbWSfe64ToN9+Gm3mqWDVXRObpAV8hFt6iF7lEbdRBBGXpBr+jNerberQ/rc7FascqbU7QE6+sXPQOegQ==

Propagated
inputs to
output

Evaluated function

MULT20015 Elements of Quantum Computing
Lecture 8

8.4 Universality in Quantum Computing

MULT20015
Lecture 8

MULT20015 Elements of Quantum Computing
Lecture 8

Universality

In classical computing the NAND gate is universal, i.e. every Boolean function can be
implemented as a sequence of NAND (NOT AND) gates:

In quantum computing every quantum circuit can be expressed as a sequence of:

CNOT

hXi

hY i

hZi

|0i

|1i

Single Qubit Rotations

MULT20015 Elements of Quantum Computing
Lecture 8

Universality

In classical computing the NAND gate is universal, i.e. every Boolean function can be
implemented as a sequence of NAND (NOT AND) gates:

In quantum computing every quantum circuit can be expressed as a sequence of:

CNOT Hadamard

H T

T Gate
(pi/8)

MULT20015 Elements of Quantum Computing
Lecture 8

Subject outline

Lecture topics (by week)

1 – Introduction to quantum computing and maths basics
2 – Single qubit representations and logic operations
3 – Two qubit states and logic gates
4/5 – Multi-qubit states and quantum arithmetic
5/6 – Classical complexity and simple quantum algorithms
6/7 – Cryptography and Shor’s quantum factoring algorithm
7 – Post quantum cryptography and quantum key distribution
8 – Quantum search algorithms
9 – Quantum algorithms for option pricing
10 – Optimisation problems on quantum computers
11 – Portfolio optimisation using quantum computers
12 – Quantum machine learning

Assignment schedule:
#1: Hand out in Week 2
#2: Hand out in Week 8