Deutsch algorithm explained in Quantum Computing

Deutsch algorithm

David Deutsch initially presented a quantum method in 1985 called the Deutsch algorithm. Using quantum parallelism to assess a certain attribute of a function with less operations than needed traditionally shows how promising quantum computation is to surpass conventional computation.

The method primarily aims to find if a given binary function f(x) is balanced (that returns 0 for one input and 1 for the other) or constant (the same for all inputs). Classical computing calls for evaluating f(x) for both inputs, 0 and 1; the Deutsch method does this with a single evaluation utilizing a quantum computer.

Problem Statement

Consider a binary function f(x):{0,1}→{0,1} The task is to determine whether f(x) is:

1. Constant: Returns the same value for all inputs (i.e., f(0)=f(1)).
2. Balanced: Returns 0 for one input and 1 for the other (i.e., f(0)≠f(1)).

Classical Approach

To determine whether f(x) is constant or balanced, a classical computer must evaluate f(x) for both x=0 and x=1. This requires two function evaluations.

Quantum Approach

Using the Deutsch algorithm, this classification can be achieved with just one evaluation of f(x), exploiting the principles of quantum superposition and interference.

Quantum Foundations

Superposition

In quantum computing, a qubit (quantum bit) can exist in a linear combination of basis states ∣0⟩ and ∣1⟩, described as:

∣ψ⟩=α∣0⟩+β∣1⟩, where ∣α∣²+∣β∣²=1.

Entanglement and Interference

The Deutsch algorithm leverages entanglement and quantum interference to encode and manipulate function information across multiple states simultaneously.

Oracle Representation

A quantum oracle is a black-box operation that encodes the function f(x). The oracle for f(x) is represented by a unitary transformation U_f, defined as:

U_f∣x⟩∣y⟩=∣x⟩∣y⊕f(x)⟩

where x is the input, y is an auxiliary qubit, and ⊕ denotes addition modulo 2.

Deutsch algorithm circuit diagram

The Deutsch algorithm can be visualized as a quantum circuit with the following components:

|0⟩ ──H───●───H───┬───|ψ₀⟩
│ │
|1⟩ ──H────⊕───────┴───|ψ₁⟩

Explanation of the Circuit Components

Input Qubits:

• Top wire (|0⟩): Represents the first qubit, initialized to ∣0⟩, which stores the input x.
• Bottom wire (|1⟩): Represents the auxiliary (ancilla) qubit, initialized to ∣1⟩.

Hadamard Gates (H):

Applied to both qubits at the start. The Hadamard gates create a superposition:

∣0⟩→(∣0⟩+∣1⟩)/ √2, ∣1⟩→(∣0⟩−∣1⟩)/ √2

This enables the algorithm to evaluate the function f(x) on both possible inputs (x=0 and x=1) simultaneously.

Oracle U_f (Black Box):

• Represented by the controlled operation (● and ⊕).
• It encodes the function f(x) into the quantum state by flipping the phase of the auxiliary qubit based on f(x):

U_f∣x⟩∣y⟩=∣x⟩∣y⊕f(x)⟩

Second Hadamard Gate:

Applied to the first qubit (|0⟩) after the oracle. This causes interference that extracts information about whether f(x) is constant or balanced.

Measurement:

The final state of the first qubit is measured:

• If the result is ∣0⟩, f(x) is constant.
• If the result is ∣1⟩, f(x) is balanced.

Deutsch Algorithm working Step by step

Initialization

Prepare two qubits in the state:

∣ψ₀⟩=∣0⟩⊗∣1⟩

The first qubit represents the input x, while the second qubit is an auxiliary qubit initialized to ∣1⟩.

Hadamard Gate Application:

Apply a Hadamard gate to each qubit. The Hadamard gate creates a superposition, transforming the qubits as follows:

∣0⟩→(∣0⟩+∣1⟩)/ √2, ∣1⟩→(∣0⟩−∣1⟩)/ √2

After applying Hadamard gates, the system is in the state:

∣ψ₁⟩=1/√2(∣0⟩+∣1⟩)⊗1/√2(∣0⟩−∣1⟩)

Oracle Query

The oracle U_f acts on the qubits, modifying the auxiliary qubit based on the function f(x). The state becomes:

∣ψ₂⟩=1/√2 $\sum _{\mathrm{x=0}}^1$ (-1)^f(x) ∣x⟩⊗1/√2(∣0⟩−∣1⟩)

Here, the function f(x) introduces a phase factor (−1)^f(x) that encodes the function’s properties.

Interference

Apply a Hadamard gate to the first qubit again, enabling constructive or destructive interference. The result is:

∣ψ₃⟩=1/2 $\sum _{\mathrm{x=0}}^1$ $\sum _{\mathrm{z=0}}^1$ (-1)^f(x)(-1)^x.z ∣z⟩ ⊗(∣0⟩−∣1⟩)

This interference pattern determines whether f(x) is constant or balanced.

Measurement

Measure the first qubit: If the result is ∣0⟩, f(x) is constant. If the result is ∣1⟩, f(x) is balanced.

Limitations

The Deutsch algorithm solves a unnatural problem that isn’t directly applicable to practical scenarios. However, its value lies in its role as a proof of concept, showcasing the computational power of quantum mechanics. It is an excellent instructive tool for understanding core quantum principles.

Advantages of Deutsch Algorithm in Qunatum Computing

Less Research Questions

The capacity of the Deutsch method to solve a given issue with fewer searches to an oracle than a classical algorithm is its main benefit. In particular, it may ascertain if a function is balanced or constant with just one question; a conventional method calls for two.

Quantum Parallelism

Using quantum parallelism—where the quantum computer lives in several states concurrently—the technique evaluates the function for several inputs at once. The Deutsch method concurrently computes f(0) and f(1) by first running the function f on a superposition of inputs.

Quantum interference

The Deutsch method derives a global property of the function f by means of quantum interference. Following the application of the function to a superposition, the method modulates the quantum state to expose whether f is constant or balanced by use of relative phase between the states.

Demonstration of Key Quantum Concepts

The fundamental ideas of quantum computing—that of superposition, quantum parallelism, and quantum interference—are shown by the Deutsch algorithm. It offers a basic and unambiguous illustration of how these quantum events may be used to address issues in alternative ways than conventional approaches.

Foundation for Other Algorithms

A forerunner of more intricate quantum algorithms such the Deutsch-Jozsa algorithm and Shor’s algorithm, the Deutsch algorithm was the first quantum method to show a definite advantage over the best conceivable conventional algorithm. It reflects fundamental features of effective quantum algorithms, including unitary transformations and superposed state preparation.

Relation Between Deutsch and Deutsch-Jozsa Algorithms in Quantum Computing

Origin

Designed to solve a basic issue highlighting the potential of quantum computing, David Deutsch developed the Deutsch algorithm in 1985 as the first quantum algorithm. Finds if, one query to the quantum oracle, a 1-bit function f(x):{0,1}→{0,1} is constant or balanced.

Introduced by David Deutsch and Richard Jozsa in 1992, the Deutsch-Jozsa method is a modification of the original Deutsch algorithm to n-bit functions, therefore addressing a more difficult variation of the same kind of issue.
Extends this approach to n-bit functions f(x):{0,1}ⁿ→{0,1} deciding if the function is constant or balanced.

 Core Problem Structure

Both algorithms aim to classify functions based on their behavior:

• A constant function outputs the same value (either 0 or 1) for all inputs.
• A balanced function outputs 0 for half of the inputs and 1 for the other half.

Problem Size:

• Deutsch Algorithm: Deals with a single-bit input function (f(0) and f(1)).
• Deutsch-Jozsa Algorithm: Generalizes to n-bit input functions with 2ⁿ possible inputs.

Quantum Speedup and Efficiency

Deutsch Algorithm:

• To find if f(x) is constant or balanced, classical method calls two evaluations (f(0) and f(1)).
• The quantum method achieves a speed-up for this particular work by only one evaluation.

Deutsch-Jozsa Algorithm:

• To separate constant from balanced functions, classical method calls for up to 2ⁿ⁻¹+1 evaluations in the worst scenario.
• Demonstrating exponential speedup as n increases, the quantum method gets the result with a single oracle query.

Deutsch Algorithm vs Deutsch-Jozsa Algorithm

Aspect	Deutsch Algorithm	Deutsch-Jozsa Algorithm
Problem Scope	1-bit function (2 inputs)	n-bit function (2n inputs)
Goal	Constant vs. Balanced	Constant vs. Balanced
Efficiency	1 query vs. 2 classically	1 query vs. 2n−1+1 classically
Significance	First quantum algorithm	Generalization with exponential speedup
Concepts Used	Superposition, Interference, Oracle	Same as Deutsch Algorithm