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Amplitude Amplification

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Table of Contents

  1. Introduction
  2. Motivation and Background
  3. What Is Amplitude Amplification?
  4. Relationship to Grover’s Algorithm
  5. General Form of Amplitude Amplification
  6. Oracle and Reflection Operators
  7. Mathematical Foundation
  8. Amplitude Amplification Operator
  9. The Grover Iterate as a Special Case
  10. Quadratic Speedup over Classical Methods
  11. Algorithm Steps
  12. Geometric Interpretation
  13. Success Probability Analysis
  14. Amplitude Estimation Overview
  15. Applications of Amplitude Amplification
  16. Generalization Beyond Grover
  17. Constructing Custom Oracles
  18. Controlled Amplification
  19. Quantum Counting Integration
  20. Robustness and Error Considerations
  21. Comparison with Classical Repetition
  22. Complexity and Resource Requirements
  23. Circuit Example in Qiskit
  24. Use in Quantum Machine Learning
  25. Conclusion

1. Introduction

Amplitude amplification is a key quantum algorithmic technique that generalizes the core idea behind Grover’s search. It increases the probability of measuring desired states in a quantum system — providing quadratic speedup for a wide class of problems.


2. Motivation and Background

Classical search and sampling methods rely on repeated random trials. In contrast, quantum amplitude amplification boosts success probabilities using interference — enabling fewer repetitions.


3. What Is Amplitude Amplification?

Given a quantum state \( |\psi\rangle \) and a procedure to mark “good” states, amplitude amplification systematically increases the amplitude of those good states.


4. Relationship to Grover’s Algorithm

Grover’s algorithm is a special case of amplitude amplification:

  • Grover searches for a marked item
  • Amplitude amplification works with any initial distribution

5. General Form of Amplitude Amplification

Let \( \mathcal{A} \) be a quantum algorithm that prepares \( |\psi\rangle = \mathcal{A}|0\rangle \). Let \( \mathcal{P} \) be a projection onto good states. The goal is to amplify their amplitudes.


6. Oracle and Reflection Operators

Define:

  • \( Q = -\mathcal{A} S_0 \mathcal{A}^{-1} S_f \)
  • \( S_f = I – 2|\text{good}\rangle\langle\text{good}| \)
  • \( S_0 = I – 2|0\rangle\langle 0| \)

The operator \( Q \) rotates the state towards the good subspace.


7. Mathematical Foundation

Let:
\[
|\psi\rangle = \sin(\theta)|\text{good}\rangle + \cos(\theta)|\text{bad}\rangle
\]

After \( k \) iterations:

\[
Q^k |\psi\rangle = \sin((2k+1)\theta)|\text{good}\rangle + \cos((2k+1)\theta)|\text{bad}\rangle
\]


8. Amplitude Amplification Operator

The operator \( Q \) performs a rotation in the 2D subspace spanned by good and bad components. Each iteration amplifies the amplitude of the good states.


9. The Grover Iterate as a Special Case

Grover’s algorithm is:

\[
Q = D \cdot U_f
\]

where:

  • \( U_f \) marks the solution
  • \( D \) is the diffusion operator

10. Quadratic Speedup over Classical Methods

If classical success probability is \( p \), expected repetitions are \( \mathcal{O}(1/p) \). Amplitude amplification achieves it in \( \mathcal{O}(1/\sqrt{p}) \) iterations.


11. Algorithm Steps

  1. Prepare initial state \( |\psi\rangle = \mathcal{A}|0\rangle \)
  2. Define good states via oracle
  3. Apply Grover-like iterate \( Q \)
  4. Measure after optimal number of iterations

12. Geometric Interpretation

Each iteration rotates the state vector toward the “good” axis. After \( \mathcal{O}(1/\sqrt{p}) \) steps, the angle aligns with good states — maximizing success probability.


13. Success Probability Analysis

After \( k \) iterations:

\[
\text{Success probability} = \sin^2((2k+1)\theta)
\]

Choose \( k \approx \frac{\pi}{4\theta} – 1/2 \) for optimal amplification.


14. Amplitude Estimation Overview

Amplitude estimation builds on amplitude amplification:

  • Instead of sampling directly
  • Uses phase estimation to determine \( p \)

15. Applications of Amplitude Amplification

  • Search problems
  • Decision procedures
  • Quantum machine learning
  • Quantum simulation
  • Cryptographic verification

16. Generalization Beyond Grover

Amplitude amplification allows:

  • Non-uniform initial states
  • Non-binary success criteria
  • Composable subroutines

17. Constructing Custom Oracles

The marking operation can be designed using:

  • CNOT, Toffoli gates
  • Phase oracles
  • Comparators and encoders

18. Controlled Amplification

Controlled amplitude amplification allows conditional probability boosts — important for quantum walks and decision-based algorithms.


19. Quantum Counting Integration

Combining with quantum counting, one can estimate the number of solutions before applying amplitude amplification.


20. Robustness and Error Considerations

  • Sensitive to over-rotation
  • Needs precise control of oracle and \( \mathcal{A} \)
  • Robust to moderate noise under proper gate optimization

21. Comparison with Classical Repetition

MethodComplexity
Classical Repeats\( \mathcal{O}(1/p) \)
Amplitude Amplification\( \mathcal{O}(1/\sqrt{p}) \)

22. Complexity and Resource Requirements

Gate depth depends on:

  • Oracle complexity
  • Implementation of \( \mathcal{A} \)
  • Number of iterations \( \mathcal{O}(1/\sqrt{p}) \)

23. Circuit Example in Qiskit

from qiskit import QuantumCircuit, Aer, execute

qc = QuantumCircuit(3)
qc.h([0, 1])  # Initial superposition

# Oracle marking |11⟩
qc.cz(0, 1)

# Diffusion
qc.h([0, 1])
qc.x([0, 1])
qc.h(1)
qc.cx(0, 1)
qc.h(1)
qc.x([0, 1])
qc.h([0, 1])

qc.measure_all()
backend = Aer.get_backend('qasm_simulator')
result = execute(qc, backend, shots=1024).result()
print(result.get_counts())

24. Use in Quantum Machine Learning

  • Boosts correct classification states
  • Reduces number of quantum samples
  • Key for subroutines in QML models

25. Conclusion

Amplitude amplification is a versatile and powerful quantum primitive that extends the Grover framework to a wider set of applications. It embodies the quantum principle of exploiting interference to boost the probability of success — achieving quadratic speedups and enabling efficient quantum subroutines across computing, simulation, and learning tasks.


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Today in History – 30 October

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1502

Vasco da Gama returns to Calicut, India for the second time.

1883

Swami Dayanand Saraswati, founder of Arya Samaj, died in Ajmer.

1961

Soviet Union tests a 58 megaton hydrogen bomb named Tsar Bomba, the most powerful nuclear weapon ever detonated.

1966

First meeting of the Shiv Sena at Shivaji Park.

1990

Several people died and many injured as security forces opened fire at a crowd marching towards Ram Janmabhumi/Babari Masjid to perform ‘karseva’.

1997

Argentine soccer star Diego Maradona announces his retirement from football on his 37th birthday

1999

A super cyclone struck Odisha, causing widespread destruction, with at least 10,000 lives lost and an estimated 1.5 million people rendered homeless.

Phase Estimation Algorithm

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Table of Contents

  1. Introduction
  2. Motivation and Applications
  3. Problem Statement
  4. Mathematical Background
  5. Quantum Phase Estimation: Goal
  6. Overview of the Algorithm
  7. Unitary Operators and Eigenstates
  8. Phase Encoding via Controlled Operations
  9. Quantum Fourier Transform in Phase Estimation
  10. Register Initialization
  11. Applying Hadamards
  12. Controlled-\(U^{2^j}\) Operations
  13. Inverse Quantum Fourier Transform
  14. Measurement and Phase Extraction
  15. Precision and Error Bounds
  16. Example with 3 Qubits
  17. Geometric Interpretation
  18. Relationship to Eigenvalue Estimation
  19. Phase Estimation in Shor’s Algorithm
  20. Applications in Quantum Chemistry
  21. Hamiltonian Simulation Use Case
  22. Variants of the Algorithm
  23. Circuit Construction in Qiskit
  24. Limitations and Practical Considerations
  25. Conclusion

1. Introduction

The Phase Estimation Algorithm (PEA) is one of the most powerful quantum algorithms, allowing for the extraction of eigenvalues of a unitary operator. It underlies several other algorithms including Shor’s factoring and quantum simulation techniques.


2. Motivation and Applications

Phase estimation is essential in:

  • Shor’s Algorithm (order finding)
  • Quantum chemistry (eigenvalue problems)
  • Hamiltonian simulation
  • Quantum machine learning

3. Problem Statement

Given:

  • A unitary operator \( U \)
  • An eigenstate \( |\psi\rangle \) such that \( U|\psi\rangle = e^{2\pi i \phi}|\psi\rangle \)

The goal is to estimate the unknown phase \( \phi \in [0,1) \).


4. Mathematical Background

Any unitary matrix \( U \) has eigenvalues on the complex unit circle. The goal is to extract \( \phi \) from \( e^{2\pi i \phi} \).


5. Quantum Phase Estimation: Goal

Prepare a quantum system in which \( \phi \) is encoded in the amplitudes of a state. Then, use quantum interference (QFT) to reveal \( \phi \) through measurement.


6. Overview of the Algorithm

Steps:

  1. Prepare \( t \) qubit control register and target eigenstate \( |\psi\rangle \)
  2. Apply Hadamard gates to control register
  3. Apply controlled-\(U^{2^j}\) gates
  4. Apply inverse QFT
  5. Measure control register

7. Unitary Operators and Eigenstates

We assume that:
\[
U|\psi\rangle = e^{2\pi i \phi}|\psi\rangle
\]
We don’t know \( \phi \), but we want to estimate it to precision \( 1/2^t \).


8. Phase Encoding via Controlled Operations

Apply controlled operations \( CU^{2^j} \) for each qubit in the control register. This encodes \( \phi \) as a phase:

\[
|j\rangle \rightarrow e^{2\pi i 2^j \phi}|j\rangle
\]


9. Quantum Fourier Transform in Phase Estimation

Apply inverse QFT on control register to decode the phase into the computational basis:

\[
\text{QFT}^{-1}\left( \sum_k e^{2\pi i k \phi}|k\rangle \right) \rightarrow |\phi\rangle
\]


10. Register Initialization

Initial state:

\[
|0\rangle^{\otimes t} \otimes |\psi\rangle
\]

After Hadamards:

\[
\frac{1}{\sqrt{2^t}} \sum_{k=0}^{2^t – 1} |k\rangle \otimes |\psi\rangle
\]


11. Applying Hadamards

Each control qubit is placed in superposition via Hadamard:

\[
H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
\]

Entire control register becomes:

\[
\frac{1}{\sqrt{2^t}} \sum_{k=0}^{2^t-1} |k\rangle
\]


12. Controlled-\(U^{2^j}\) Operations

Each control qubit \( j \) controls the operation \( U^{2^j} \):

\[
\sum_{k=0}^{2^t-1} |k\rangle U^k|\psi\rangle
\Rightarrow
\sum_{k=0}^{2^t-1} e^{2\pi i k \phi}|k\rangle |\psi\rangle
\]


13. Inverse Quantum Fourier Transform

Applying \( QFT^{-1} \) to the control register transforms the relative phases into measurable probability peaks around \( \phi \).


14. Measurement and Phase Extraction

Measuring the control register gives a binary approximation of \( \phi \). If \( \phi \) has a binary expansion with \( t \) digits, the algorithm recovers it exactly.


15. Precision and Error Bounds

The probability of success increases with more qubits. For \( t \) qubits, the algorithm approximates \( \phi \) to \( t \) bits of precision with high probability.


16. Example with 3 Qubits

Let \( \phi = 0.625 = 101_2/8 \)

  • The QFT circuit transforms the control register
  • Measurement yields \( 101 \), or 0.625 with high probability

17. Geometric Interpretation

The amplitudes rotate based on \( \phi \), and QFT aligns these rotations to reveal the phase through constructive interference.


18. Relationship to Eigenvalue Estimation

PEA is equivalent to estimating eigenvalues \( e^{2\pi i \phi} \) of \( U \), giving access to eigenenergies and dynamics of physical systems.


19. Phase Estimation in Shor’s Algorithm

In Shor’s algorithm, phase estimation is used to find the period \( r \) of \( a^x \mod N \), where the phase is \( \phi = k/r \).


20. Applications in Quantum Chemistry

Used to compute eigenvalues of molecular Hamiltonians, giving accurate energy levels — a key application in quantum simulation.


21. Hamiltonian Simulation Use Case

In combination with Trotterization and Hamiltonian exponentiation, PEA can extract energy levels of quantum systems.


22. Variants of the Algorithm

  • Iterative Phase Estimation (IPEA): requires fewer qubits
  • Kitaev’s algorithm: early approach using entanglement
  • Robust PEA: improves fault tolerance and precision

23. Circuit Construction in Qiskit

from qiskit import QuantumCircuit, Aer, execute
from qiskit.circuit.library import QFT

n = 3
qc = QuantumCircuit(n+1, n)

# Hadamard on control register
qc.h(range(n))

# Simulated U = Z gate, controlled-U^2^j
for j in range(n):
    qc.cp(2 * 3.1415 / (2**(j+1)), j, n)

# Inverse QFT
qc.append(QFT(num_qubits=n, inverse=True, do_swaps=True), range(n))

# Measure
qc.measure(range(n), range(n))

24. Limitations and Practical Considerations

  • Requires controlled-\(U^{2^j}\) implementations
  • Needs large depth for high precision
  • Sensitive to noise and decoherence

25. Conclusion

The Phase Estimation Algorithm is a foundational quantum algorithm with wide-ranging applications, from factoring to simulating quantum systems. It leverages quantum superposition, interference, and Fourier transforms to extract hidden phase information with exponential precision.


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Today in History – 29 October

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Today in History - 29 October

Today in History - 29 October

539 BC

King Cyrus The Great of Persia marches into the city of Babylon, releasing the Jews from almost 70 years of exile and making the first Human Rights Declaration.

1390

First trial for witchcraft in Paris.

1851

British Indian Association was established in Bengal with Radha Kanta Dev as president and Devendranath Tagore as the Secretary.

1859

Spain declares war on Morocco

1881

Judge (U.S. magazine) first published.

1894

First election of Hawaiian Republic

1920

Jamia Milliya Islamia was founded by Dr. Jakir Husain and his collegues at Aligarh. ‘Khilafat’ and the Non- Cooperation Movement were an integral part of policy animating.

1923

Turkey declares independence (successor state to Ottoman Empire)

1930

Pandit Jawaharlal Nehru is sent back to prison for two years following a speech he made at Allahabad the day after his release from jail.

1945

First ballpoint pen goes on sale, manufactured by Biro

1952

Amalgamation of the Steel Corporation of Bengal with Indian Iron and Steel Company Ltd. to be effective from 1st January 1953 under the latter’s name.

1961

New All India Muslim League was established.

1962

China attacked India.

1964

The United Republic of Tanganyika & Zanzibar renamed The United Republic of Tanzania

1966

National Organization of Women founded

1990

Vinod Mehra, famous Hindi film actor, died.

1992

Komar Singh, the first Indian Ambassador to Israel, presented credentials to President Chain Herzog in Jerusalem.

1994

National Museum of American Indian opens (NYC)

2005

Delhi bombings kill more than 60.

2015

China announces the end of their one-child policy after 35 years

Also Read:

Today in History – 28 October

Today in History – 27 October

Today in History – 26 October

Today in History – 25 October

Quantum Fourier Transform (QFT)

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Table of Contents

  1. Introduction
  2. Classical Fourier Transform Background
  3. Motivation for Quantum Fourier Transform
  4. Definition of QFT
  5. Mathematical Formulation
  6. Action on Quantum Basis States
  7. Inverse Quantum Fourier Transform
  8. Comparison to Classical FFT
  9. Circuit Construction
  10. Hadamard and Controlled Phase Gates
  11. Step-by-Step Circuit Decomposition
  12. QFT for 2 and 3 Qubits
  13. QFT Matrix Representation
  14. Inverse QFT Circuit
  15. Measurement and Extraction
  16. Efficiency and Complexity
  17. Applications of QFT
  18. QFT in Shor’s Algorithm
  19. Phase Estimation and QFT
  20. Precision and Error Analysis
  21. Example: Applying QFT on Basis States
  22. Geometric Interpretation
  23. Tools for Simulating QFT
  24. Limitations in Physical Implementation
  25. Conclusion

1. Introduction

The Quantum Fourier Transform (QFT) is a quantum analog of the classical discrete Fourier transform (DFT). It plays a central role in many quantum algorithms, most notably Shor’s factoring algorithm and Quantum Phase Estimation.


2. Classical Fourier Transform Background

The classical discrete Fourier transform transforms a vector of complex numbers into its frequency representation:

\[ \text{DFT}(x)k = \sum{j=0}^{N-1} x_j e^{2\pi i jk / N} \]

This transformation reveals frequency components of data.


3. Motivation for Quantum Fourier Transform

The QFT allows a quantum computer to extract periodic structure from superpositions. It is exponentially faster than classical DFT, with \(\mathcal{O}(n^2)\) gate complexity for \( n \)-qubit inputs.


4. Definition of QFT

Let \( |x\rangle \) be a computational basis state of \( n \) qubits. The QFT maps this to:

\[
\text{QFT}|x\rangle = \frac{1}{\sqrt{2^n}} \sum_{y=0}^{2^n – 1} e^{2\pi i x y / 2^n} |y\rangle
\]


5. Mathematical Formulation

Given:

  • \( x = x_0 x_1 \dots x_{n-1} \)
  • \( y = y_0 y_1 \dots y_{n-1} \)

QFT maps:

\[
|x\rangle \rightarrow \frac{1}{\sqrt{2^n}} \sum_{y=0}^{2^n – 1} e^{2\pi i x y / 2^n} |y\rangle
\]

Each amplitude encodes a phase dependent on \( x \cdot y \).


6. Action on Quantum Basis States

For example, for \( n = 2 \), and \( x = 01 \):

\[
\text{QFT}|01\rangle = \frac{1}{2} (|0\rangle + i|1\rangle – |2\rangle – i|3\rangle)
\]


7. Inverse Quantum Fourier Transform

The inverse QFT is defined by taking the complex conjugate of the exponent:

\[
\text{QFT}^{-1}|y\rangle = \frac{1}{\sqrt{2^n}} \sum_{x=0}^{2^n – 1} e^{-2\pi i x y / 2^n} |x\rangle
\]


8. Comparison to Classical FFT

  • Classical FFT: \( \mathcal{O}(N \log N) \) operations
  • QFT: \( \mathcal{O}(n^2) \) quantum gates, where \( N = 2^n \)

Exponential improvement for large inputs.


9. Circuit Construction

The QFT circuit uses:

  • Hadamard gates
  • Controlled phase gates (e.g., \( R_k \))

10. Hadamard and Controlled Phase Gates

For each qubit \( q_j \), apply:

  1. Hadamard gate
  2. Controlled \( R_k \) gates with \( k = 2, 3, …, n-j \)

Where:

\[
R_k = \begin{bmatrix}
1 & 0 \
0 & e^{2\pi i / 2^k}
\end{bmatrix}
\]


11. Step-by-Step Circuit Decomposition

For 3-qubit QFT:

  • H on qubit 0
  • \( R_2 \) between qubits 1 and 0
  • \( R_3 \) between qubits 2 and 0
  • H on qubit 1
  • \( R_2 \) between qubits 2 and 1
  • H on qubit 2
  • Finally, swap qubits to reverse order

12. QFT for 2 and 3 Qubits

2-qubit QFT:

q0: ──H────■───────────X──
           │           │
q1: ───────R2────H─────X──

3-qubit QFT:

q0: ──H──■────■────────────X──────────
         │    │            │
q1: ─────R2───■────H───────┼──────X───
              │            │      │
q2: ──────────R3───R2──────H──────X───

13. QFT Matrix Representation

For \( n = 2 \), QFT matrix is:

\[
\frac{1}{2}
\begin{bmatrix}
1 & 1 & 1 & 1 \
1 & i & -1 & -i \
1 & -1 & 1 & -1 \
1 & -i & -1 & i
\end{bmatrix}
\]


14. Inverse QFT Circuit

Same as QFT but:

  • Reverse the gate order
  • Use complex conjugate of phase gates \( R_k^\dagger \)

15. Measurement and Extraction

The result after QFT encodes the frequency of periodic functions. Measuring yields values correlated to the period or eigenvalue of operators in phase estimation.


16. Efficiency and Complexity

Gate complexity: \( \mathcal{O}(n^2) \)
Can be approximated to \( \mathcal{O}(n \log n) \) by dropping small-angle gates for faster circuits.


17. Applications of QFT

  • Shor’s Algorithm (period finding)
  • Quantum Phase Estimation
  • Quantum counting
  • Hidden subgroup problems

18. QFT in Shor’s Algorithm

Used to extract the period \( r \) from \( e^{2\pi i k r / Q} \) via measurement of the QFT output.


19. Phase Estimation and QFT

The QFT is the core of phase estimation, allowing estimation of eigenvalues of unitary operators to exponential precision.


20. Precision and Error Analysis

The QFT spreads amplitude over many basis states. Precision depends on:

  • Number of qubits
  • Accuracy of \( R_k \) gates
  • Noise and decoherence

21. Example: Applying QFT on Basis States

Let’s apply QFT to \( |01\rangle \):

\[
\text{QFT}|01\rangle = \frac{1}{2}(|0\rangle + i|1\rangle – |2\rangle – i|3\rangle)
\]

The result is a state encoding the relative phase of the binary index.


22. Geometric Interpretation

QFT rotates the quantum state into the Fourier basis, revealing periodicity encoded in phase relationships. The angles between basis vectors become structured after transformation.


23. Tools for Simulating QFT

  • Qiskit: qiskit.circuit.library.QFT
  • Cirq: custom gate construction
  • Q#: built-in QFT
  • Python simulation (NumPy, QuTiP) for visualization

24. Limitations in Physical Implementation

  • Requires many controlled-phase gates
  • Precision limited by gate fidelity
  • Qubits need high coherence for deeper circuits

25. Conclusion

The Quantum Fourier Transform is a cornerstone of quantum computing. It enables powerful algorithms for factoring, phase estimation, and periodicity detection. Though challenging to implement physically, it exemplifies quantum computing’s strength in manipulating phase and interference.


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