Quantum Computation 101 (Almost)

Combinatorial 101
• You have a bucket of blue balls (50%) and red balls (50%).
• Probability ?
• Uncertainty ?

Combinatorial 101
• You have blue human balls (50%) and red human balls (50%) to rip of.

Combinatorial 101
• You also know that some balls
have longer sticks, some don’t.

Combinatorial 101
• YOU HATE REDbull.

Combinatorial 101
• YOU are OK with ASIAN.

• ASIAN.
Combinatorial 101

• You encounter light beams.
Combinatorial 102

• You know some sweet physics.
Combinatorial 102

• You know some sweet physics…?
Combinatorial 102

• You know some sweat physics…..?
Combinatorial 102

Yeah, yeah….Wave…cosine….2#$@$#!#%@
• Two different model of uncertainty…How different?
• Observation affects the experimental outcome, aka the distribution.

Yeah, yeah….Wave…cosine….2#$@$#!#%@
• Two different model of uncertainty…How different?
• Observation affects the experimental outcome, aka the distribution.
• Wave particle duality.
• Intrinsic randomness vs classical randomness.

Outline
• A quantum intuition. (Observed !)

Let’s get our hand dirty
• Quantum physics are stated as a series of three axioms.
• Axiom 1: An isolated quantum system has associated with a Hilbert space H.
The system is completely determined by a unit vector in H
• Ex: [1, 0], [0, 0, -i], [0.5,
3
2
] are all unit vectors in H.
• We can write [0.5,
3
2
] in a cocky way: 0.5 |0> +
3
2
|1>
• |0>, |1> are called base states.

Let’s get our hand dirty (Axiom 1)
• What is ‘state’ ?
• Can be anything that matters in the system !
• Ex: |0> : |1>:
• Ex: |0>: laser (inactivated) |1>: laser (activated) |2>: laser (emitted)
• Ex: |0>: False |1>: True
• Axiom 1: System can always be described by a vector. We view the
‘meaning’ of the vector basis as state.
• Ex: The state of a bit storage with quantum system can be described as
a|0> + b|1>, with |a|2+|b|2 = 1. We usually call it ‘qubit’.
• What about the ‘coefficient’ of the basis ? Later on this.

Let’s get our hand dirty (Axiom 1.5)
• Wait…what about a system to describe ( ) ?
• A ball can have two different state
• |0> : |1>:
• State of A ball can be described by a length two vector.
• w = [a, b] a.k.a a|0> + b|1>, with |a|2+|b|2 = 1.
• Two balls can have four different state
• |00>: |01>: |10>: |11>:
• State of two balls can be described by a length four vector.
• Composite system (Axiom 1.5)
• A composite quantum system has state space given by the tensor product of the
consistent quantum subsystem.

• Tensor product: Natural product of vectors
• w1 = a|0> + b|1>, w2 = c|0> + d|1>
• How to compute w1 ‘times’ w2 ?
• Phen Pei Lu, duh….
• w1⨂ w2 = (a|0> + b|1>) ⨂ (c|0> + d|1>)
= a c |0> ⨂ |0> + a d|0> ⨂ |1> + b c |1> ⨂ |0 > + b d |1> ⨂ |1 >
= ac |00 > + ad |01> + bc |10 > + bd |11>
• Properties of tensor product
• (w1⨂ w2 ) ∙ (w3⨂ w4 ) =(w1 ∙ w3)(w2 ∙ w4)
• Preserve axiom 1 !!
• Why tensor product ? -> More on this later.

• Axiom 2: A physical system evolves according to a linear unitary
transform.
• Unitary transform U: U*U = UU* = I
• Think of ‘*’ as transpose + conjugate.
• Ex: [0, i]* = [[-i], [0]]
• Property of ‘*’: (AB)* = B*A*

• Why unitary transform?
• Unitary transform U: U*U = UU* = I
• Preserve unit length of state after evolution ! (Axiom 1)
• Check it out:
• w is a state (column vector), from axiom 1 we have w* w = 1.
• Let w’ = U w, we see that w’* w’ = w* U*U w = w* w = 1.
• Why linear ? -> Schrodinger equation.

• Back to single qubit
• Reminder: qubit can be represented as [a, b] a.k.a a|0> + b|1>, with
|a|2+|b|2 = 1.
• Special case: a = 1 (Logic false) and b = 1 (Logic true)
• Unitary transform on single qubit (Quantum gates)
X
0 1
1 0
Y
0 −𝑖
𝑖 0
Z
1 0
0 −1
H
1
2
1
2
1
2
−
1
2

• Unitary transform on two qubits (Quantum gates)
• How to realize AND ?
• How to realize OR ??
• Wait… AND/OR are irreversible operations !
• Nature permits only unitary operation.
• Quantum gates are reversible !
AND
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?
OR
? ?
? ?
? ?
? ?
? ?
? ?
? ?
? ?

• Unitary transform on two qubits (A smart way around)
• NAND gates are universal. -> Want to implement this.
• The Toffoli gate
• Quantum gate design trick 101: add an acilla bit and XOR it.
X
A
C
A
C ⨁(𝐴⋀𝐵)
BB

• So far we’ve talked about the deterministic part of quantum physics.
• Remember we’ve mentioned intrinsic randomness and experimental effect ?
• Axiom 3 (Verbally)
• The information you can get from a quantum state is through measurement.
• Each possible experimental outcomes is associated with an operator.
• Probability of obtaining an experimental outcome is determined by state and
the operator associated with that outcome.
• Quantum state, after the measurement, collapses to a state associated with
the experimental outcomes.

• Axiom 3 (Measurement)
• A quantum measurement is characterized by a collection of
operators.
• Operators: 𝑀 𝛼1
, 𝑀 𝛼2
, ⋯ 𝑀 𝛼 𝑘
.
• 𝛼1, 𝛼2, ⋯ 𝛼 𝑘 are indications of the outcomes.
• Operators obey completeness property: 𝑖=1
𝑘
𝑀 𝛼 𝑖
∗ 𝑀 𝛼 𝑖
= 𝐼
• Given state w, the probability P[X = 𝛼𝑖] = w*𝑀 𝛼 𝑖
∗ 𝑀 𝛼 𝑖
w
• After the measurement, quantum state collapses to:
𝑀 𝛼 𝑖
w
P[X = 𝛼 𝑖]

• What the heck are these operators ?
• Why completeness property: 𝑖=1
𝑘
𝑀 𝛼 𝑖
∗
𝑀 𝛼 𝑖
= 𝐼 ??
• Preserve outcome probability: 𝑖=1
𝑘
P[X = 𝛼𝑖] = 1
• Check it out: 𝑖=1
𝑘
P[X = 𝛼𝑖] = 𝑖=1
𝑘
w∗𝑀 𝛼 𝑖
∗ 𝑀 𝛼 𝑖
w
= w∗ 𝑖=1
𝑘
𝑀 𝛼 𝑖
∗ 𝑀 𝛼 𝑖
w
= w∗ w = 1 (Axiom 1)
• But… What ARE these operators ?
• Ex: w =
𝑎
𝑏
, 𝑀0 =
1 0
0 0
, 𝑀1 =
0 0
0 1
• Meaning of coefficients: Probability amplitude.

• Remember the natural tensor product ?
• w1 = a|0> + b|1>, w2 = c|0> + d|1>
• w1⨂ w2 = (a|0> + b|1>) ⨂ (c|0> + d|1>)
= ac |00 > + ad |01> + bc |10 > + bd |11>
• Remember Probability amplitude ?
• |a|2 : Probability of observing |0> with measurement {M0, M1} on w1 -> P[X1 = 0]
• |c|2 : Probability of observing |0> with measurement {M0, M1} on w2 -> P[X2 = 0]
• Probability of observing |0>|0> with measurement {…} on w1⨂ w2 ? -> P[X1 = 0] P[X2 = 0]
• Tensor product reveals the fact that two consistent quantum states are
independent !
• But not every 4-dim state can be written in tensor product.
• These states are called unseparable -> Quantum correlation.

Let’s get our hand dirty (Recap)
• Axiom 1: Everything (isolated) is a unit length vector.
• Axiom 1.5: Two things not quantum correlated (independent), write
them down as their tensor product.
• Axiom 2: Nature only permits unitary operations.
• Axiom 3: Information from the state can be accessed through
measurement, where the outcomes exhibits intrinsic randomness.

Outline
• Axiomize quantum physics. (Observed !)
• Break ?
• Applications : Sth you might raise your eyebrows.
• A quantum algorithm: Grover’s search.
• A quantum algorithm: Shor’s factorizing algorithm.

Quantum parallelism : Good stuff 
• Think about a state a1 |000> + a2 |001> +… a8 |111>
• Quiz 1: what are the meaning of [|000> … |111> ] ?
• Quiz 2: what are the meaning of [a1 … a8] ?
• Quiz 3: what are the permitted operations ?
• Now think about operation again.
• What does it mean to perform ‘flip the last bit’ on this state ?
• Operation on a state -> operation on all possible classical configurations.
• This is where quantum computation gain advantage from.

Quantum parallelism : Bad stuff ?
• Yey! Then why don’t we replace classical bits (or other computing
thiny) with quantum state ?
• At a first glance: classic : |0> or |1> ---> quantum : a|0> + b|1>
• More degree of freedom, parallelism…
• But…how to construct a state a|0> + b|1> ?
• Classical : How to construct a bit string 0001010100 -> stupid question.
• Classical (2) : Hey, here’s a bit string (in the memory), make a copy of it ->
stupid question again.
• Quantum: How to construct a state a|00> + b|10> -> stupid question ?
• Quantum (2): Hey, here’s a state, make a copy of it -> ????

Quantum parallelism: No-cloning theorem
• There is no universal unitary U that can clone a state, i.e., there’s no
unitary U such that for all |𝜑 > and ancilla |𝑎 >
𝑈|𝜑 > |𝑎 >= |𝜑 > |𝜑 >

Quantum teleportation
• We can’t CLONE a state by unitary transform
• Can we do something with measurement ?
• But wait…After measurement, the state collapses.
• So we shall ask: can we at least carry these measurement result to
somewhere and do something ?
• We CAN !
• The protocol is called quantum teleportation.
• Why teleportation -> Imagine transporting a person in a speed of light ?

Quantum teleportation
|𝜑 >1 = 𝑎 0 > +𝑏 1 >
|𝜑 >23 =
1
2
00 > +
1
2
11 >
H M
{|00>, |01>
|10>, |11>}
X Z
b1
b2
|𝜑 >3 = 𝑎 0 > +𝑏 1 >

Quantum superdense coding
• Hmm…It seems that each qubit need to be described in two bits…
• Can we reverse it -> send two classical bits by only one qubit ?
• WE CAN !
X Z
|𝜑 > 𝐴
|𝜑 > 𝐴𝐵 =
1
2
00 > +
1
2
11 >
𝑠𝑒𝑛𝑑
M (b1,b2)
b1 b2

Quantum might raise your eyebrow (Recap)
• Quantum parallelism
• No cloning theorem
• Application of quantum entanglement
• Quantum teleportation
• Quantum superdense coding

Outline
• Applications : Sth you might raise your eyebrows. (Observed !)
• A quantum algorithm: Grover’s search.

Quantum algorithm
• Remember quantum parallelism ?
• This insights leads to development quantum algorithms !
• Currently most famous quantum algorithms (based on parallelism)
• Grover’s search.
• Shor’s factorization algorithm.
• HHL (Harrow, Hassidim, Lloyd) linear equations solver.
• Let’s jump into it.

Black box search problem
• The problem is simple: Find the red ball
• Permitted operation: Take out and check.
• Imagine there’s only 1 red ball out of N balls.
• Classical algorithm ? O(N).
• Why does this problem matter ?
• Find a assignment for SAT problem.
• Find a element in a array equals to xxx
• Find maximum value/ minimum value/ … etc

Black box search problem: Needle in haystack
• General formulation: we have a function f: {0,1}n -> {0, 1}, want to find the
assignment x∈ {0,1}n such that f(x) = 1.
• Total number of possibility: 2n
• Classical algorithm:
• Choose (sample) one solution, ex: x = |01>
• Compute f, store the value, verify |01>|f(x)>.
• Classical object only permit one check at a time !
• Quantum algorithm:
• Construct a uniform state: |𝜑 > = |𝑠 >=
1
2
|00 > +
1
2
01 > +
1
2
10 > +
1
2
|11 >
• Verify it, …… f(| 𝜑 > ) =?????

Quantum search algorithm (1)
• We want to take advantage of quantum parallelism.
• i.e. we want to have by some operation:
|𝜑 > |0 >→
1
2
|00 > |𝑓(00) > +
1
2
|10 > |𝑓(10) > +
1
2
|01 > |𝑓(01) > +
1
2
|11 > |𝑓(11) >
• Why do we want it -> We compute f for all possible input in just 1 operation !
• Is this operation possible -> Yes ! It’s unitary !
• Remember the ancilla trick mentioned in the Tofolli gate slide ?
X
A A
BB
C⨁(𝐴⋀𝐵)C

|𝜑 > |0 >→
1
2
|00 > |𝑓(00) > +
1
2
|10 > |𝑓(10) > +
1
2
|01 > |𝑓(01) > +
1
2
|11 > |𝑓(11) >
X
A A
BB
f
C⨁(𝐴⋀𝐵)C

|𝜑 > |0 >→
1
2
|00 > |𝑓(00) > +
1
2
|10 > |𝑓(10) > +
1
2
|01 > |𝑓(01) > +
1
2
|11 > |𝑓(11) >
X
A A
BB
f
C⨁𝑓(𝐴, 𝐵)C

|𝜑 > |0 >→
1
2
|00 > |𝑓(00) > +
1
2
|10 > |𝑓(10) > +
1
2
|01 > |𝑓(01) > +
1
2
|11 > |𝑓(11) >
X
A A
BB
f
0⨁𝑓(𝐴, 𝐵)0

|𝜑 > |0 >→
1
2
|00 > |𝑓(00) > +
1
2
|10 > |𝑓(10) > +
1
2
|01 > |𝑓(01) > +
1
2
|11 > |𝑓(11) >
• Quantum boolean gate.
X
A A
BB
f
0⨁𝑓(𝐴, 𝐵)
Uf
0

• What’s next ?
• We have: 1
2
|00 > |𝑓(00) > +
1
2
|10 > |𝑓(10) > +
1
2
|01 > |𝑓(01) > +
1
2
|11 > |𝑓(11) >
• Remember the meaning of the coefficient ? -> Square is the probability.
• How to sample a x -> Measurement !
• Wait…Nothing change ! Only have ¼ probability to get the search target !
• If only…we can do something to the state
• Make the coefficient of our target to be one!
• Need to construct unitary transformation…
• Grover’s idea: Rotate the state !

• Suppose |00> is the answer, let’s revisit what we have…
• |𝜑 >=
1
2
|00 > +
1
2
|10 > +
1
2
|01 > +
1
2
|11 >=
1
4
|00 > +
3
4
|00 >⊥
|𝜑 >=
1
4
|00 > +
3
4
|00 >⊥
|00 >
|00 >⊥

• |𝜑 >=
1
2
|00 > +
1
2
|10 > +
1
2
|01 > +
1
2
|11 >=
1
4
|00 > +
3
4
|00 >⊥
|00 >
|00 >⊥
|𝜑 >= |00 >

• |𝜑 >=
1
2
|00 > +
1
2
|10 > +
1
2
|01 > +
1
2
|11 >=
1
4
|00 > +
3
4
|00 >⊥
|𝜑 >=
1
4
|00 > +
3
4
|00 >⊥
|00 >
|00 >⊥
𝜃 = 30

• |𝜑 >=
1
2
|00 > +
1
2
|10 > +
1
2
|01 > +
1
2
|11 >=
1
4
|00 > +
3
4
|00 >⊥
|00 >
|00 >⊥
|𝜑 > = −
1
4
|00 > +
3
4
|00 >⊥

• |𝜑 >=
1
2
|00 > +
1
2
|10 > +
1
2
|01 > +
1
2
|11 >=
1
4
|00 > +
3
4
|00 >⊥
• Only two steps are enough !
|00 >
|00 >⊥
|𝜑 >= |00 >

• |𝜑 >=
1
2
|00 > +
1
2
|10 > +
1
2
|01 > +
1
2
|11 >=
1
4
|00 > +
3
4
|00 >⊥
• Done with N=4 example. Consider N to be very large…
𝜃 ≈
1
𝑁
|𝑡𝑎𝑟𝑔𝑒𝑡 >⊥
|𝑡𝑎𝑟𝑔𝑒𝑡 >

• |𝜑 >=
1
2
|00 > +
1
2
|10 > +
1
2
|01 > +
1
2
|11 >=
1
4
|00 > +
3
4
|00 >⊥
𝜃 ≈
3
𝑁

• |𝜑 >=
1
2
|00 > +
1
2
|10 > +
1
2
|01 > +
1
2
|11 >=
1
4
|00 > +
3
4
|00 >⊥
𝜃 ≈
5
𝑁

• |𝜑 >=
1
2
|00 > +
1
2
|10 > +
1
2
|01 > +
1
2
|11 >=
1
4
|00 > +
3
4
|00 >⊥
𝜃 ≈
2𝑘 + 1
𝑁
=
𝜋
2
𝑘 = Ο( 𝑁)
Quadratic speedup !!!

Quantum search algorithm
• Let’s implement each iteration
• How to flip the sign only at the target spot ?
• The ancilla trick ver 2.0
|𝜑 >=
1
4
|00 > +
3
4
|00 >⊥ → −
1
4
|00 > +
3
4
|00 >⊥
X
A A
BB Uf
C C⨁𝑓(𝐴, 𝐵)

|𝜑 >=
1
4
|00 > +
3
4
|00 >⊥ → −
1
4
|00 > +
3
4
|00 >⊥
X
A A
BB Uf
C
1
2
|0 > −
1
2
|1 > =
|0 >=
|0 >=

|𝜑 >=
1
4
|00 > +
3
4
|00 >⊥ → −
1
4
|00 > +
3
4
|00 >⊥
X
A A
BB Uf
C
1
2
|0 > −
1
2
|1 > =
|0 >=
|0 >=
C⨁𝑓(0,0)

|𝜑 >=
1
4
|00 > +
3
4
|00 >⊥ → −
1
4
|00 > +
3
4
|00 >⊥
X
A A
BB Uf
C
1
2
|0 > −
1
2
|1 > =
|0 >=
|0 >=
-C

|𝜑 >=
1
4
|00 > +
3
4
|00 >⊥ → −
1
4
|00 > +
3
4
|00 >⊥
X
A A
BB Uf
C
1
2
|0 > −
1
2
|1 > =
|0 >=
|0 >=
-C
𝑈𝑓 |𝜑 > |𝐶 > =
1
4
𝑈𝑓 |00 > |𝐶 > +
3
4
𝑈𝑓 |00 >⊥|𝐶 > = −
1
4
|00 > |𝐶 > +
3
4
|00 >⊥|𝐶 >

• Discard the ancilla bit, we get what we want !!
|𝜑 >=
1
4
|00 > +
3
4
|00 >⊥ → −
1
4
|00 > +
3
4
|00 >⊥
X
A A
BB Uf
C
1
2
|0 > −
1
2
|1 > =
|0 >=
|0 >=
-C
𝑈𝑓 |𝜑 > |𝐶 > =
1
4
𝑈𝑓 |00 > |𝐶 > +
3
4
𝑈𝑓 |00 >⊥|𝐶 > = −
1
4
|00 > |𝐶 > +
3
4
|00 >⊥|𝐶 >

• How to flip the state with respect to the uniform state |s> ?
• Uniform state |𝑠 > = 𝑖=1
𝑁 1
𝑁
|𝑖 >
• I’ll give you the expression of this operation A: (Note: N=2n)
𝐴 = 2 𝑠 >< 𝑠 − 𝐼 = 𝐻⨂𝑛
2 0 >< 0 − 𝐼 𝐻⨂𝑛
𝐻⨂2
𝑍
𝐻⨂2

Grover’s search
• Step 1: Initialize |𝜑 > with a uniform state n qubits such that N=2n
• Step 2: (iteration) For k in range ( 𝑁):
• |𝜑′ > |𝐶 > = 𝐺 |𝜑 > |𝐶 >
• Step 3: Measure the state
• Obtain final outcome.
𝐻⨂2
𝑍
𝐻⨂2
Uf

Grover’s search: Recap
• Query complexity: 𝑁 𝑣𝑠 𝑁
• How many functional call do we need ?
• Again: Quantum parallelism !
Uf
c

Grover’s search: Further reading
• Can you think of a quantum algorithm that find maximum in a list ?
• Can you initialize with arbitrary quantum state?
• Interesting References:
• Amplitude amplification : Boost up a quantum algo only need O(
1
𝛿
)
repetition.
• Grover search is optimal !
• Quantum walk via reflection between subspace.

Outline
• A quantum algorithm: Grover’s search. (Observed !)

Recap: Quantum Boolean circuit
• What’s the critical step in Grover’s search ?
• Quantum parallelism for sure…
• A smart way to extract the output of the function ! -> Encode to amplitude
• We’ve considered binary output…
• What if the function takes more than 2 outcome?
• E.g. |𝜑 > = 00 >→ 𝑐 𝑓 0,0 00 >
• What shall this c be ?
• A natural extension: Phase encoding!
X
A A
BB Uf
C
1
2
|0 > −
1
2
|1 > =
|0 >=
|0 >=
𝐶⨁𝑓 𝐴, 𝐵
= −𝐶

Phase encoding
• Let’s apply a two-outcome function to a uniform state
• Example of such function: Indicator function of the search problem
• The phase encoding (What we want)
1
2
|00 > +
1
2
|10 > +
1
2
|01 > +
1
2
|11 >→ −
1
2
|00 > +
1
2
|10 > +
1
2
|01 > +
1
2
|11 >
• How do we do that: We pick a ancilla C =
1
2
|0 > −
1
2
|1 > -> Eigenvector !
• Let’s apply a four-outcome function to a uniform state.
• Example of such function: f(a,b) = (2*a + b) mod 4
• The phase encoding (What we want)
1
2
00 > +
1
2
10 > +
1
2
01 > +
1
2
11 >→ 𝑒
2𝜋𝑖 𝑓 0,0
4
1
2
00 > +𝑒
2𝜋𝑖 𝑓 1,0
4
1
2
10 > +𝑒
2𝜋𝑖 𝑓 0,1
4
1
2
01 > +𝑒
2𝜋𝑖 𝑓 1,1
4
1
2
11 >
• How do we do that: We pick a ancilla C = 𝑘=1
4 𝑒2𝜋𝑖𝑘/4
4
|𝑘 > -> Eigenvector !

Quantum period finding
• What’s next -> How to use the encoding
• In search problem, we use the encoding to ‘squeeze’ the quantum state.
• Introduce a new problem: Find the period of the function f
• Period: f(x) = f(x ⨁ r). Find r.
• Why important -> You’ll see, hehehee..
• E.g. f(x) = f(x ⨁ 4 )
|𝜑 > = 𝑒
2𝜋𝑖 ∗ 0
4
1
2
00 > +𝑒
2𝜋𝑖 ∗ 1
4
1
2
10 > +𝑒
2𝜋𝑖 ∗ 2
4
1
2
01 > +𝑒
2𝜋𝑖 ∗3
4
1
2
11 >
• How to extract ‘4’ ? -> Inverse Fourier transform !!

Quantum Fourier transform
• Trust me : Discrete Fourier transform is unitary.
• Trust me: Quantum Fourier transform can be done in O((logN)2) gates.
• Trust me:
|𝜑 > = 𝑒
2𝜋𝑖 ∗ 0
4
1
2
00 > +𝑒
2𝜋𝑖 ∗ 1
4
1
2
10 > +𝑒
2𝜋𝑖 ∗ 2
4
1
2
01 > +𝑒
2𝜋𝑖 ∗3
4
1
2
11 >→ |01 >

Quantum period finding (Details)
• In general case…
• We know 0 < r < 2L
• But we don’t know how to construct C -> wait…wha…?
• Actually, we can ‘solve’ it !
• Here’s the sketch of the actual algorithm
• Step 1: construct a uniform state with (L + log(
1
𝜀
)) qubits : |s>
• Step 2: Apply quantum Boolean circuit Uf to |s>|0> : Uf |s>|0>
• Step 3: Apply QIFT to the first register.
• Step 4: Measure the first register, and do continuous fraction algorithm to find r.

Quantum period finding (Complexity)
• Query complexity: 1 !!
• Other operation: O(L2), with 0 < r < 2L
• Time to Shor up !

Quantum order finding
• Definition: let x < N and (x, N) are coprime. The order of x modulo N is
defined to be the least positive integer r, such that
𝑥 𝑟
≡ 1 𝑚𝑜𝑑 𝑁
• Property: 𝑟 ≤ 𝑁
• Property: the function f(k) = xk mod N is periodic with period r !!
• How to find the order-> Quantum period finding. The end.
• Query: 1
• Gate complexity: O(log 2 (N))

Two lemmas I haven’t thought through yet
Lemma 1: (How to find non-trivial factor)
Suppose N is an L bit composite number, and x is an non trivial solution to
P1 : 𝑥2 ≡ 1 𝑚𝑜𝑑 𝑁,
which means 𝑥 ≠ 1 𝑚𝑜𝑑 𝑁 nor 𝑥 ≠ 𝑁 − 1 ≡ −1 𝑚𝑜𝑑 𝑁 . Then, at least one of
gcd(x-1 ,N), gcd(x+1, N) is a non-trivial factor of N that can be computed in O(L3)
Lemma 2: (It’s really likely to find such ‘x’)
Suppose 𝑁 = 𝑝1
𝛼1
𝑝2
𝛼2
⋯ 𝑝 𝑚
𝛼 𝑚
and let x be chosen uniformly at random from
the set {0<t<N, (t,N) is coprime}. Let r be the order of x modulo N, then
Pr 𝑟 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑥
𝑟
2 𝑖𝑠 𝑛𝑜𝑛𝑡𝑟𝑖𝑣𝑖𝑎𝑙 𝑡𝑜 𝑃1 ≥ 1 −
1
2 𝑚

Shor’s factoring algorithm (Find a nontrivial
factor)
• Input: a composite number N
• Step 1: Randomly sample 0<x<N, check if gcd(x,N)>1
• If so, you are lucky -> return gcd(x,N).
• If not, don’t panic, quantum has your back -> go to step 2.
• Step 2: Apply quantum order finding algorithm to x.
• Check 𝑟 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑥
𝑟
2 𝑖𝑠 𝑛𝑜𝑛𝑡𝑟𝑖𝑣𝑖𝑎𝑙 𝑡𝑜 𝑃1. If so, compute gcd(xr/2-1 ,N),
gcd(xr/2+1, N) , test if one of them is a non-trivial factor. Output it.
• If not, 雖小。 But that’s fine-> run the algo again !
• Performance: O(log3 N) with O(1) success probability.

Offline.
• A quantum algorithm: Grover’s search. (Observed !)
• A quantum algorithm: Shor’s factorizing algorithm. (Observed !)

Recommended reference
• Quantum computation bible -> Nielsen & Chuang, quantum
computation and quantum information.
• Linear algebraic reference -> The theory of quantum information,
Waltrous.
• Quantum tale -> Alice in quantumland.

Quantum Computation 101 (Almost)

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