INTRODUCTION TO THE THEORY OF QUANTUM COMPUTING

INTRODUCTION TO THE THEORY OF QUANTUM COMPUTING

Daniel K. Park | dkp.quantum@gmail.com

NITHeP Mini-school on quantum computing

Outline

Part I: What & Why

• Introduction & Background Part II: How

• Quantum Circuit

• Quantum Algorithms

• Quantum Error Correction

| i

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Quantum Computing Is Hard in Practice

• Classical digital computation: Very robust to noise (bit-flip error).

• Quantum: Protect not only against bit-flip errors, but also from the environment constantly interacting to the quantum system.

• But qubits need to interact strongly and accurately with each other.

• At first glance, quantum computers resembles classical analog computers .

• Let , then

|ψ⟩ = cos (θ/2)|0⟩ + e ^iϕ sin (θ/2)|1⟩

Pr(gate) = 1 − p

# of gates Success

Prob. (1 − p) ⁿ

Density Matrix Formalism

• Quantum mechanics can also be formulated with density operator or density matrix, which is mathematically equivalent to the state vector approach.

• Useful for describing quantum systems whose state is not completely known

• For an ensemble of pure states , the density matrix is defined by

• If , it is a pure state. Otherwise, it is a mixed state.

• and (equal iff is a pure state).

{p _i , |ψ _i ⟩}

ρ = ∑

i

p _i |ψ _i ⟩⟨ψ _i | ρ = |ψ⟩⟨ψ|

⟨ϕ|ρ|ϕ⟩ ≥ 0, Tr(ρ) = 1 Tr(ρ ² ) ≤ 1 ρ

Revisit Bloch Sphere

• Four Pauli matrices form a basis over Real.

•

•

➞

{I, X, Y, Z}

ρ = 1

2 ( I + a _x X + a _y Y + a _z Z), a ^x , a _y , a _z ∈ ℝ, a _x ² + a _y ² + a _z ² ≤ 1.

|ψ⟩ = cos ( θ

2 ) |0⟩ + e ^iϕ sin ( θ

2 ) |1⟩

ρ = 1

2 ( I + sin(θ)cos(ϕ)X + sin(θ)sin(ϕ)Y + cos(θ)Z)

|ψ ₁ ⟩

|ψ ₂ ⟩ X

Y Z

I/2 p ₁ p ₂

p ₁ |ψ ₁ ⟩⟨ψ ₁ | + p ₂ |ψ⟩⟨ψ|

General Quantum Operation

A quantum operation: map that sends a density matrix to a density matrix.

• Linear:

• Preserve trace:

• Preserve positivity:

• Completely positive:

Λ(pρ + qσ) = pΛ(ρ) + qΛ(σ) Tr (Λ(ρ)) = Tr (ρ)

ρ ≥ 0, Λ(ρ) ≥ 0

(Λ ⊗ I) ρ ≥ 0

General Quantum Operation

The following are equivalent:

• is completely-positive, trace-preserving (CPTP).

• such that

• . is called a Kraus operator.

Λ

∃ U Λ (ρ) = Tr ^E (Uρ ⊗ σ E U ^† ) Λ(ρ) = ∑

i

A _i ρA _i ^† , ∑

i

A _i ^† A _i = I A _i

*Motivates the use of unitary circuits

Depolarizing Channel

• .

• Kraus operators: .

• Can also be written as .

• Any single qubit error can be transformed to depolarizing error.

Λ(ρ) = (1 − p)ρ + p

3 ( XρX + YρY + ZρZ) 1 − pI, p

3 X, p

3 Y, p

3 Z Λ(ρ) = (1 − p′ )ρ + p′ I

2

Y Z

X I/2

Dephasing Channel

•

• An example of how this map can arise:

•

Λ(ρ) = (1 − p)ρ + pZρZ

Λ(ρ) = ∫dϕ f(ϕ)R ^z (ϕ)ρR _z ^† (ϕ) =

[

ρ ₀₀ e ^{−t/T 2} ρ ₀₁

e ^{−t/T 2} ρ ₁₀ ρ ₁₁ ]

p = 1 − exp (−t/T ² ) 2

x

y z

t=0

x

y z

t=t 1

Y Z

I/2

X

Amplitude Damping

•

• For purely amplitude damping channel, .

• In general, .

• ➞ Often used for state initialization.

Λ(ρ) =

[

(ρ ⁰⁰ − ρ ₀₀ ^eq ) e ^{−t/T 1} + ρ ₀₀ ^eq ρ ₀₁ e ^{−t/2T 1}

ρ ₁₀ e ^{−t/2T 1} (ρ ¹¹ − ρ ₁₁ ^eq ) e ^{−t/T 1} + ρ ₁₁ ^eq ]

T ₂ = 2T ₁ T ₂ ≤ 2T ₁

t→∞ lim Λ(ρ) = ρ ^eq

Y Z

X I/2

Active Quantum Error Correction

The goal of quantum error correction is to use redundancy and

correction to realize logical qubits with logical error rates below the error rate of the elementary constituent qubits. .

*Recall: Λ (ρ) = Tr ^E (Uρ ⊗ σ ^E U ^† )

Classical Repetition Code

• Probability to fail is changed from to : Improvement as long as .

• Can add more bits (redundancy) to correct more errors.

• Quantum case is not as simple!

p 3p(1 − p) ² + p ³ p < 1/2

3-bits encoding &

majority vote (1-p) ³ no error O(1)

3p(1-p) ² 1 error O(p) 3(1-p)p ² 2 errors O(p ² )

p ³ 3 errors O(p ³ )

000 001 010 100

000

111 110 101 011

} } ^1→111 111

0→000

QEC Has To Overcome…

• Measurement destroys superposition.

• No cloning theorem prohibits repetition.

• Must correct multiple types of errors, i.e., bit-flip and phase-flip.

• Continuous errors.

Digitization of Noise

Similar Pauli-expansion holds for n-qubits: |ψ _n ⟩ ⊗ |E⟩ → ∑

j

ε _j |ψ _n ⟩ ⊗ |E _j ⟩

Design QECC so that a subset of Pauli errors ℰ ⊆ {I, X, Y, Z} ^⊗n can be detected

|0⟩|E⟩ → |0⟩|E ₀₀ ⟩ + |1⟩|E ₀₁ ⟩ |1⟩|E⟩ → |0⟩|E ₁₀ ⟩ + |1⟩|E ₁₁ ⟩

(α|0⟩ + β|1⟩)|E⟩ → α (|0⟩|E ⁰⁰ ⟩ + |1⟩|E ₀₁ ⟩) + β (|0⟩|E ¹⁰ ⟩ + |1⟩|E ₁₁ ⟩)

= (α|0⟩ + β|1⟩) ⊗ (|E ⁰⁰ ⟩ + |E ₁₁ ⟩)/2 +(α|0⟩ − β|1⟩) ⊗ (|E ⁰⁰ ⟩ − |E ₁₁ ⟩)/2 +(α|1⟩ + β|0⟩) ⊗ (|E ⁰¹ ⟩ + |E ₁₀ ⟩)/2 +(α|1⟩ − β|0⟩) ⊗ (|E ⁰¹ ⟩ − |E ₁₀ ⟩)/2

= I|ψ⟩|E _I ⟩ + Z|ψ⟩|E _Z ⟩ + X|ψ⟩|E _X ⟩ + XZ|ψ⟩|E _XZ ⟩

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Smallest Code: a Bit-Flip Error

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^L = ↵ |000i + |111i

Encoding

Redundancy, not copying

Measure the error, not the data Error Syndrome

Encoding Decoding Recovery

M z

M z

X | i

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