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Grover’s Search Algorithm

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Table of Contents

  1. Introduction
  2. The Unstructured Search Problem
  3. Classical Complexity
  4. Quantum Speedup with Grover’s Algorithm
  5. Problem Statement
  6. Oracle Design in Grover’s Algorithm
  7. Amplitude Amplification Intuition
  8. High-Level Steps of the Algorithm
  9. Initial State Preparation
  10. Applying the Oracle
  11. Diffusion Operator Explained
  12. Geometric Interpretation
  13. Number of Iterations
  14. Mathematical Formulation
  15. Grover Operator Definition
  16. Proof of Quadratic Speedup
  17. Circuit Diagram Overview
  18. Example: Searching in 4 Elements
  19. Visualization on the Bloch Sphere
  20. Implementation in Qiskit
  21. Limitations and Assumptions
  22. Grover’s Algorithm for Multiple Solutions
  23. Comparison with Classical Random Search
  24. Generalizations and Real-World Impact
  25. Conclusion

1. Introduction

Grover’s Algorithm is one of the most significant quantum algorithms, providing a quadratic speedup for unstructured search problems. It’s applicable to any scenario where we need to search for a marked item among many without knowing the structure of the data.


2. The Unstructured Search Problem

Suppose we are given a function \( f : \{0,1\}^n \rightarrow \{0,1\} \) such that:

  • \( f(x) = 1 \) for a single (or few) unknown input(s) \( x_0 \)
  • \( f(x) = 0 \) otherwise

The goal is to find \( x_0 \).


3. Classical Complexity

A classical algorithm requires \( \mathcal{O}(2^n) \) queries in the worst case to find the marked item. With randomness, expected queries are \( \mathcal{O}(2^n) \) as well.


4. Quantum Speedup with Grover’s Algorithm

Grover’s algorithm finds the marked input in \( \mathcal{O}(\sqrt{2^n}) \) queries — a quadratic speedup.


5. Problem Statement

Given black-box access to \( f(x) \), find the unique \( x_0 \) such that \( f(x_0) = 1 \), assuming such \( x_0 \) exists.


6. Oracle Design in Grover’s Algorithm

The oracle flips the phase of the marked state:

\[
U_f |x\rangle = (-1)^{f(x)} |x\rangle
\]

So the marked item acquires a negative amplitude, while all others remain unchanged.


7. Amplitude Amplification Intuition

Grover’s algorithm amplifies the amplitude of the solution state using constructive interference, and reduces the amplitude of non-solution states via destructive interference.


8. High-Level Steps of the Algorithm

  1. Prepare uniform superposition of all \( 2^n \) inputs
  2. Apply oracle \( U_f \)
  3. Apply diffusion operator
  4. Repeat \( \mathcal{O}(\sqrt{2^n}) \) times
  5. Measure and retrieve the solution

9. Initial State Preparation

Start with:

\[
|0\rangle^{\otimes n} \xrightarrow{H^{\otimes n}} \frac{1}{\sqrt{2^n}} \sum_x |x\rangle
\]

This is the equal superposition over all inputs.


10. Applying the Oracle

Oracle flips the sign of the marked state \( |x_0\rangle \):

\[
|x\rangle \rightarrow
\begin{cases}
-|x\rangle & \text{if } x = x_0 \
|x\rangle & \text{otherwise}
\end{cases}
\]


11. Diffusion Operator Explained

The diffusion operator reflects the state about the average amplitude:

\[
D = 2|\psi\rangle\langle \psi| – I
\]

This step increases the amplitude of the solution state.


12. Geometric Interpretation

Each iteration is a rotation in 2D space:

  • One axis for solution \( |x_0\rangle \)
  • One axis for non-solutions
  • Each iteration brings the state closer to \( |x_0\rangle \)

13. Number of Iterations

Optimal number of iterations:

\[
k \approx \frac{\pi}{4} \sqrt{N} \quad \text{where } N = 2^n
\]

Going beyond the optimum reduces success probability.


14. Mathematical Formulation

Let \( |\alpha\rangle \) be the solution state, and \( |\beta\rangle \) the equal superposition of the rest. Then Grover iteration rotates the state vector closer to \( |\alpha\rangle \).


15. Grover Operator Definition

\[
G = D \cdot U_f
\]

Where:

  • \( U_f \): oracle (phase flip)
  • \( D \): diffusion operator (inversion about mean)

16. Proof of Quadratic Speedup

After \( \mathcal{O}(\sqrt{N}) \) applications of \( G \), amplitude of solution approaches 1, and measurement yields \( x_0 \) with high probability.


17. Circuit Diagram Overview

|0⟩ — H —■— D —■— ... — M
         │     │
         f     f

Here:

  • H: Hadamard
  • f: oracle
  • D: diffusion
  • M: measurement

18. Example: Searching in 4 Elements

Let \( f(x) = 1 \) for \( x = 10 \)

  • Start in equal superposition of 4 states
  • Apply Grover operator once
  • Measurement yields \( x = 10 \) with high probability

19. Visualization on the Bloch Sphere

The state evolves on a 2D plane toward the marked state, with each Grover iteration rotating it closer via amplitude amplification.


20. Implementation in Qiskit

from qiskit import QuantumCircuit, Aer, execute
from qiskit.circuit.library import ZGate, MCXGate

n = 2
qc = QuantumCircuit(n, n)
qc.h(range(n))

# Oracle for x=2 (binary 10)
qc.cz(0, 1)  # simple placeholder

# Diffusion operator
qc.h(range(n))
qc.x(range(n))
qc.h(n-1)
qc.mcx(list(range(n-1)), n-1)
qc.h(n-1)
qc.x(range(n))
qc.h(range(n))

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

21. Limitations and Assumptions

  • Oracle must be available
  • Assumes one or few marked states
  • Too many iterations will overshoot

22. Grover’s Algorithm for Multiple Solutions

For \( t \) solutions:

\[
k \approx \frac{\pi}{4} \sqrt{\frac{N}{t}}
\]

If \( t \) is unknown, amplitude estimation or quantum counting is used.


23. Comparison with Classical Random Search

  • Classical: \( \mathcal{O}(N) \)
  • Grover: \( \mathcal{O}(\sqrt{N}) \)

Applicable in brute-force search, cryptanalysis, NP-complete problems (in principle).


24. Generalizations and Real-World Impact

Grover’s framework applies to:

  • Constraint satisfaction
  • Cryptography (e.g., key search)
  • Data mining
  • Machine learning models

25. Conclusion

Grover’s Algorithm is a powerful demonstration of quantum advantage, offering a quadratic speedup for unstructured search problems. Its concepts of amplitude amplification and quantum iteration are foundational in quantum algorithm design and optimization.


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Today in History – 25 October

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Today in History - 25 October

 

1296

Saint Gyaneshwar passed away. (Samadhee).

1854

During the Crimean War, a brigade of British light infantry is destroyed by Russian artillery as they charge down a narrow corridor in full view of the Russians.

1916

German pilot Rudolf von Eschwege shoots down his first enemy plane, a Nieuport 12 of the Royal Naval Air Service over Bulgaria.

1932

Pearless’ investment company was established.

1940

German troops capture Kharkov and launch a new drive toward Moscow.

1950

Chinese Communist Forces launch their first-phase offensive across the Yalu River into North Korea.

1951

First General Election of India begins.

1954

President Eisenhower conducts the first televised Cabinet meeting.

1964

First indigenous tank was manufactured in Awdi factory and was named ‘Vijayant’.

1971

National Science and Technology Committee established.

1983

Technical Cell constituteded with a view to promote the use of official language Hindi on mechanical or electronic devices.

1990

JD and BJP withdrew support to Rajasthan and Gujarat governments respectively.

1990

Captain Sangma, first Chief Minister of Meghalaya, died.

1997

180mw Lower Periyar Hydroelectric Project in Kerala commissioned after 13 long years.

1999

Pramod Mahajan, Minister for Parliamentary Affairs, says, “”No objection to Bofors discussio

2009

Terrorist bombings in Baghdad kill over 150 and wound over 700.

Also Read:

Today in History – 24 October

Today in History – 22 October

Today in History – 21 October

Today in History – 20 October

Simon’s Algorithm

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Table of Contents

  1. Introduction
  2. Problem Overview
  3. Classical vs Quantum Approach
  4. Mathematical Formulation
  5. The Oracle Promise
  6. Simon’s Problem Statement
  7. High-Level Idea of Simon’s Algorithm
  8. Oracle Structure and Functionality
  9. Input and Output Qubit Registers
  10. Quantum Circuit Overview
  11. Step-by-Step Execution
  12. Initial State and Superposition
  13. Oracle Application and Entanglement
  14. Measurement of Output Register
  15. Collapse and Hidden Information
  16. Final Hadamard Transform on Input
  17. Interference and Orthogonality
  18. Solving the Linear Equations
  19. Number of Queries Required
  20. Quantum Speedup Analysis
  21. Geometric and Algebraic Intuition
  22. Simon’s Algorithm vs Bernstein-Vazirani
  23. Qiskit Implementation Example
  24. Limitations and Real-World Impact
  25. Conclusion

1. Introduction

Simon’s Algorithm was the first to demonstrate an exponential separation between classical and quantum computation. It laid the groundwork for later breakthroughs such as Shor’s Algorithm and provided early evidence of quantum advantage.


2. Problem Overview

Simon’s problem asks us to find a hidden binary string \( s \in \{0,1\}^n \) that characterizes a special function’s behavior.


3. Classical vs Quantum Approach

  • Classical deterministic approach: Requires \( \Omega(2^{n/2}) \) queries
  • Quantum approach: Solves with O(n) queries

This exponential gap makes Simon’s algorithm a landmark in quantum computing.


4. Mathematical Formulation

Let \( f: \{0,1\}^n \rightarrow \{0,1\}^n \) be a function such that:

\[
\forall x, x’: f(x) = f(x’) \iff x = x’ \oplus s
\]

That is, the function is 2-to-1, and \( s \) is the period or hidden string we want to find.


5. The Oracle Promise

The oracle \( f \) is guaranteed to satisfy:

  • If \( s = 0^n \): function is one-to-one
  • Otherwise: \( f(x) = f(x \oplus s) \) and only these two share output

This “promise” distinguishes Simon’s algorithm from general function learning.


6. Simon’s Problem Statement

Find the secret string \( s \) given query access to a black-box function \( f \) satisfying the above conditions.


7. High-Level Idea of Simon’s Algorithm

The algorithm uses:

  • Superposition to query multiple inputs at once
  • Interference to encode \( s \)
  • Measurement + linear algebra to extract \( s \) from sampled bitstrings orthogonal to it

8. Oracle Structure and Functionality

The oracle \( U_f \) acts as:

\[
U_f |x\rangle|0\rangle = |x\rangle|f(x)\rangle
\]

It entangles input and output registers based on the secret period \( s \).


9. Input and Output Qubit Registers

  • Input: \( n \) qubits initialized to \( |0\rangle \)
  • Output: \( n \) qubits initialized to \( |0\rangle \)

Total: \( 2n \) qubits used.


10. Quantum Circuit Overview

  1. Apply Hadamard to input qubits
  2. Apply oracle \( U_f \)
  3. Measure output register
  4. Apply Hadamard again to input
  5. Measure input register
  6. Repeat to gather \( n-1 \) independent equations

11. Step-by-Step Execution

Initial state:

\[
|\psi_0\rangle = |0\rangle^{\otimes n} \otimes |0\rangle^{\otimes n}
\]


12. Initial State and Superposition

After applying Hadamard to the input:

\[
|\psi_1\rangle = \frac{1}{\sqrt{2^n}} \sum_{x} |x\rangle |0\rangle
\]


13. Oracle Application and Entanglement

Apply the oracle:

\[
|\psi_2\rangle = \frac{1}{\sqrt{2^n}} \sum_x |x\rangle|f(x)\rangle
\]

Each input becomes entangled with its output.


14. Measurement of Output Register

Measurement collapses the output state to some \( f(x_0) \). The input register is now in a superposition of:

\[
\frac{1}{\sqrt{2}}(|x_0\rangle + |x_0 \oplus s\rangle)
\]


15. Collapse and Hidden Information

The measurement restricts the input state to a 2-dimensional subspace defined by the period \( s \).


16. Final Hadamard Transform on Input

Apply Hadamard again to input:

\[
H^{\otimes n} \left( \frac{|x\rangle + |x \oplus s\rangle}{\sqrt{2}} \right)
\]

Using identity:

\[
H^{\otimes n} |x\rangle = \frac{1}{\sqrt{2^n}} \sum_y (-1)^{x \cdot y} |y\rangle
\]

Results in:

\[
\frac{1}{\sqrt{2^{n+1}}} \sum_y \left[ (-1)^{x \cdot y} + (-1)^{(x \oplus s) \cdot y} \right] |y\rangle
\]


17. Interference and Orthogonality

Only those \( y \) for which \( s \cdot y = 0 \mod 2 \) remain with non-zero amplitude. Each measurement yields a \( y \) such that:

\[
y \cdot s = 0 \mod 2
\]


18. Solving the Linear Equations

After collecting \( n – 1 \) independent such \( y \), solving the linear system over \( \mathbb{F}_2 \) yields the secret string \( s \).


19. Number of Queries Required

Expected number of iterations to find \( n – 1 \) linearly independent vectors is O(n).


20. Quantum Speedup Analysis

  • Quantum: O(n)
  • Classical: Requires \( \Omega(2^{n/2}) \) queries

This provides exponential speedup.


21. Geometric and Algebraic Intuition

Each query reveals a hyperplane orthogonal to \( s \). With enough hyperplanes, the only vector satisfying all orthogonality constraints is \( s \).


22. Simon’s Algorithm vs Bernstein-Vazirani

  • BV: Dot product function \( f(x) = a \cdot x \)
  • Simon: 2-to-1 mapping with hidden XOR mask
  • BV needs 1 run, Simon needs O(n) runs but solves exponentially harder problem

23. Qiskit Implementation Example

from qiskit import QuantumCircuit, Aer, execute
from qiskit.visualization import plot_histogram

n = 3
qc = QuantumCircuit(2*n, n)

# Step 1: Hadamard on input
qc.h(range(n))

# Step 2: Oracle
# Sample oracle with s = 101: f(x) = f(x ⊕ 101)
qc.cx(0, n+0)
qc.cx(2, n+0)

# Step 3: Hadamard on input
qc.h(range(n))

# Step 4: Measure input
qc.measure(range(n), range(n))

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

24. Limitations and Real-World Impact

Simon’s problem is contrived and has limited direct application. However, it:

  • Demonstrates exponential quantum speedup
  • Inspired Shor’s factoring algorithm

25. Conclusion

Simon’s Algorithm is a pivotal quantum algorithm that showcases exponential advantage over classical methods. It uses clever entanglement and measurement strategies to infer hidden structure from a black-box function — a fundamental example of the quantum computing paradigm shift.


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Today in History – 24 October

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Today in History - 24 October

1505

Don Francis-Di-Almeda of Portugal arrived at Cochin as Viceroy of India.

1579

Jesuit father and first Englishman S.J.Thomas Stephens arrived at Goa in a Portuguese ship. He settled here and died in 1619.

1605

Jahangir became the fourth Emperor of Mughal Empire at Agra.

1648

The signing of the Treaty of Westphalia ends the German Thirty Years’ War.

1657

Kalyan and Bhiwandi came under the rule of Chhatrapati Shivaji Maharaj.

1827

Viceroy Lord Rippan (1880-1884) was born.

1851

First official telegraph line was opened between Calcutta and Daimond Harbour spanning 33.8 km.

1856

Veer Narayan Singh, freedom fighter, was arrested by the British Government at Sambalpur for distributing grains from the warehouse to the people. He was sent to jail at Raipur.

1897

The first comic strip appears in the Sunday color supplement of the New York Journal called the ‘Yellow Kid.’

1904

Lalchand Hirachand, famous industialist, was born.

1929

Black Thursday–the first day of the stock market crash which began the Great Depression.

1934

Mahatma Gandhi quits Indian Indian National Congress in struggle over tactics.

1938

The Fair Labor Standards Act becomes law, establishing the 40-hour work week.

1945

UNO Day.

1953

Selection of the Dassault Ouragan fighter from France at this time reflected the decision to initiate diversification of supply sources. The first four of over 100 Ouragans, or ‘Toofanis’ as they were to become known in the IAF, reached Palam Airport from France and this type re-equipped Nos.8, 3 and 4 Squadrons in that order.

1973

Yom Kippur War ends.

1975

Bounded labour system was abolished by Ordinance.

1984

First Metro Train (Underground Train) in India started between Esplanade and Bhowanipore in Calcutta city.

1992

Narmada, World Bank sets time to correct defects.

1997

Kerala bans ragging in educational institutions by an ordinance.

2008

Many stock exchanges worldwide suffer the steepest declines in their histories; the day becomes known as “Bloody Friday.”

Also Read:

Today in History – 22 October

Today in History – 21 October

Today in History – 20 October

Today in History -19 October

Bernstein-Vazirani Algorithm

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Table of Contents

  1. Introduction
  2. Problem Statement
  3. Classical Complexity
  4. Quantum Speedup Overview
  5. Mathematical Formulation
  6. Structure of the Oracle
  7. Quantum Circuit Layout
  8. Initial State Preparation
  9. Hadamard Transform
  10. Oracle Encoding of the Secret String
  11. Final Hadamard and Measurement
  12. Output Interpretation
  13. Example with n = 3
  14. Understanding Phase Kickback
  15. Why It Works: Interference Explained
  16. Comparison to Deutsch-Jozsa Algorithm
  17. Gate-Level Decomposition
  18. Circuit Diagram Overview
  19. Geometric Interpretation on the Bloch Sphere
  20. Algebraic Breakdown
  21. Use in Learning Algorithms
  22. Implementation in Qiskit
  23. Scalability and Limitations
  24. Educational Value
  25. Conclusion

1. Introduction

The Bernstein-Vazirani algorithm is a foundational quantum algorithm that showcases how quantum computation can solve certain problems exponentially faster than classical approaches. It finds a hidden binary string using just one query to a quantum oracle.


2. Problem Statement

Given a function \( f: \{0,1\}^n \rightarrow \{0,1\} \) defined as:

\[
f(x) = a \cdot x \mod 2
\]

where \( a \in \{0,1\}^n \) is an unknown hidden bit string, the goal is to determine \( a \) using only one query to \( f \).


3. Classical Complexity

Classically, determining all bits of \( a \) requires \( n \) evaluations of \( f \), each with a unique basis vector \( x \).


4. Quantum Speedup Overview

The Bernstein-Vazirani Algorithm solves this problem with a single query using quantum parallelism and phase interference — demonstrating linear to constant query complexity reduction.


5. Mathematical Formulation

The oracle is defined as:

\[
U_f |x\rangle|y\rangle = |x\rangle|y \oplus f(x)\rangle
\]

The function evaluates \( f(x) = a \cdot x \mod 2 \), where \( a \cdot x \) is the bitwise dot product.


6. Structure of the Oracle

The oracle applies a phase flip if \( a \cdot x = 1 \):

\[
U_f |x\rangle \left( \frac{|0\rangle – |1\rangle}{\sqrt{2}} \right) = (-1)^{a \cdot x} |x\rangle \left( \frac{|0\rangle – |1\rangle}{\sqrt{2}} \right)
\]

This encoding moves \( f(x) \) into the phase of the quantum state.


7. Quantum Circuit Layout

Steps:

  1. Prepare input qubits in \( |0\rangle^{\otimes n} \)
  2. Output qubit in \( |1\rangle \)
  3. Apply Hadamard to all qubits
  4. Apply the oracle \( U_f \)
  5. Apply Hadamard to input qubits again
  6. Measure input qubits

8. Initial State Preparation

\[
|\psi_0\rangle = |0\rangle^{\otimes n} \otimes |1\rangle
\]

After Hadamard transforms:

\[
|\psi_1\rangle = \left( \frac{1}{\sqrt{2^n}} \sum_{x=0}^{2^n – 1} |x\rangle \right) \otimes \left( \frac{|0\rangle – |1\rangle}{\sqrt{2}} \right)
\]


9. Hadamard Transform

Applies a uniform superposition over all \( x \in \{0,1\}^n \), while output qubit becomes a phase-encoding ancilla.


10. Oracle Encoding of the Secret String

\[
U_f: |x\rangle \rightarrow (-1)^{a \cdot x} |x\rangle
\]

The output qubit is unchanged and can be ignored. The phase encodes information about \( a \).


11. Final Hadamard and Measurement

Applying Hadamard gates to each input qubit:

\[
H^{\otimes n} \sum_x (-1)^{a \cdot x} |x\rangle = |a\rangle
\]

This constructive interference causes all amplitudes to cancel except at the position corresponding to \( a \).


12. Output Interpretation

Measuring the input qubits yields the hidden string \( a \) with 100% probability — in just one oracle call.


13. Example with n = 3

Let \( a = 101 \)

  • \( f(x) = x_0 \oplus x_2 \)
  • Circuit yields \( |101\rangle \) with certainty
  • Classically: requires 3 evaluations

14. Understanding Phase Kickback

The oracle’s output qubit acts as a phase reservoir, kicking back the function value as a phase factor onto the input register.


15. Why It Works: Interference Explained

Each \( x \) accumulates a phase \( (-1)^{a \cdot x} \), and the final Hadamard maps this interference uniquely to \( |a\rangle \).


16. Comparison to Deutsch-Jozsa Algorithm

  • Both algorithms use a final Hadamard to extract phase-encoded data
  • Deutsch-Jozsa distinguishes balanced vs constant
  • Bernstein-Vazirani extracts a binary string

17. Gate-Level Decomposition

The oracle can be implemented using:

  • CNOT gates for each bit \( a_i = 1 \)
  • Controlled-Z or phase flip if implementing in phase

18. Circuit Diagram Overview

q0: ──H──────■────H────M──
q1: ──H───────┼────H────M──
q2: ──H──────■────H────M──
            │
anc: ──X────H──────────────

This assumes \( a = 101 \), where qubits 0 and 2 are connected to the output ancilla.


19. Geometric Interpretation on the Bloch Sphere

Each Hadamard prepares an equator superposition. Oracle flips phase across Bloch sphere equator. Final Hadamard projects to Z-axis basis — revealing \( a \).


20. Algebraic Breakdown

Final amplitude of outcome \( z \):

\[
\frac{1}{2^n} \sum_x (-1)^{a \cdot x + x \cdot z}
\]

Constructive interference at \( z = a \), destructive elsewhere.


21. Use in Learning Algorithms

A toy model for parity function learning. It underlies concepts in:

  • Quantum machine learning
  • Query complexity
  • Fourier sampling

22. Implementation in Qiskit

from qiskit import QuantumCircuit, Aer, execute

a = '101'
n = len(a)
qc = QuantumCircuit(n+1, n)
qc.x(n)
qc.h(range(n+1))
for i, bit in enumerate(reversed(a)):
    if bit == '1':
        qc.cx(i, n)
qc.h(range(n))
qc.measure(range(n), range(n))

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

23. Scalability and Limitations

Scales well for demonstration purposes but assumes:

  • Perfect oracle
  • No noise
  • Non-adaptive query model

Still valuable pedagogically and for benchmark testing.


24. Educational Value

Helps learners grasp:

  • Phase kickback
  • Quantum parallelism
  • Exact extraction of encoded information

25. Conclusion

The Bernstein-Vazirani Algorithm is a landmark quantum algorithm that elegantly demonstrates how phase-based encoding and interference allow quantum computers to outperform classical ones in structured problems. It remains one of the best examples for teaching quantum speedup and oracle-based computation.


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