Quantum Hardware Readiness for Two-Step Quantum Search Algorithm

By IEEE Computer Society Team on

June 18, 2025

Theoretical quadratic speedup over classical brute-force methods

Reduced quantum resource requirements compared to existing quantum approaches

Practical circuit designs that could work on near-term quantum devices

The Algorithm's Core Innovation

Previous quantum search algorithms for TSP faced a fundamental chicken-and-egg problem: they needed to start with a superposition of all valid solutions, but couldn't efficiently create that starting state. The TSQS algorithm solves this through a clever two-phase approach:

Phase 1: Solution Preparation

Creates an equal superposition of all feasible tour routes
Uses quantum search to identify valid solutions from the encoding space
Eliminates invalid configurations automatically

Phase 2: Optimization

Searches within the prepared solution space for optimal routes
Amplifies minimum-cost tours using quantum interference
Achieves quadratic speedup in the optimization phase

Hardware Requirements: The Reality Check

Qubit Efficiency Breakthrough

The researchers made a crucial encoding choice that dramatically reduces hardware demands:

HOBO vs. QUBO Encoding Comparison:

HOBO encoding: Requires approximately n·log₂(n) qubits for n cities
QUBO encoding: Requires n² qubits for the same problem
Real-world impact: A 16-city problem needs ~64 qubits (HOBO) vs. 256 qubits (QUBO)

Circuit Depth and Noise Considerations

The research team conducted extensive simulations analyzing performance under realistic noise conditions:

Key Findings:

- TSQS shows better noise tolerance than single-step alternatives
- Shallower circuit depth makes it suitable for noisy intermediate-scale quantum devices
- Performance degrades gracefully as error rates increase

Classical vs. Quantum Performance Analysis

The paper provides crucial context by comparing quantum advantages against established classical methods:

Theoretical Speedup:

- Classical heuristics: O(n!) complexity for exact solutions
- TSQS algorithm: O(√n!) theoretical improvement
- Quantum-inspired classical: Can handle larger instances today, but lack asymptotic advantages

Small TSP instances (4-6 cities) on existing 50-100 qubit systems
Algorithm validation and optimization studies

Proof-of-concept demonstrations for logistics companies

Current Challenges:

Circuit depth increases dramatically with problem size

Algorithm amplifies both optimal and worst-case solutions simultaneously

Classical post-processing required to extract final answers

Medium-Term Outlook (5-10 years)
The path to practical quantum advantage requires:
Hardware Milestones:

Fault-tolerant quantum computers with hundreds of logical qubits

Extended coherence times supporting deeper circuits

Improved error correction enabling complex algorithm execution

Broader Implications for Quantum Computing
This research addresses fundamental questions about quantum algorithm hardware feasibility that extend beyond the TSP:
Scientific Impact:

Demonstrates practical quantum circuit design for NP-hard problems

Provides benchmarking framework for quantum optimization algorithms

Establishes encoding strategies applicable to other combinatorial problems

Industrial Relevance:

Logistics and supply chain optimization represent trillion-dollar markets

Network routing and resource allocation benefit from similar approaches

Conclusion
The TSQS algorithm represents meaningful progress toward practical quantum optimization. While current hardware limits immediate large-scale applications, the research provides a clear roadmap for quantum advantage in combinatorial optimization. The research paper "Two-Step Quantum Search Algorithm for Solving Traveling Salesman Problems" represents an important advancement in quantum algorithms for combinatorial optimization. To explore the detailed circuit implementations, performance analysis, and complete technical insights, download the full article below.
Download "Two-Step Quantum Search Algorithm for Solving Traveling Salesman Problems" Article