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Solving Segment Display Problems Using Quantum Grover's Search Algorithm

Shanyan Chen, Ali Al-Bayaty, Xiaoyu Song, Marek Perkowski

TL;DR

This work addresses solving Segment Display Problems (SDPs) with quantum search by mapping SDPs to a Boolean oracle suitable for Grover's algorithm. It introduces a two-level oracle design: a high-level decomposition into segment-code verifiers, decoders, and arithmetic units, and a layout-aware, circuit-level synthesis aided by the Combination Sequence of Exclusive Sums (CSES) to minimize quantum cost. Central contributions include formalizing SDP encoding as a triple $\langle Q,D,C\rangle$, detailing constraints as inner and amid geometric and cryptarithmetic categories, and presenting concrete circuit components (SC verifier, SC-BCD, TDN generator) built with Stesso for efficient $C^nX$ gate synthesis. A case study solving a matchstick SDP on a noisy Qiskit simulator demonstrates feasibility, achieving correct solutions with a modest ancilla budget and illustrating potential scalability to broader CSPs. The approach lays groundwork for a reusable quantum library and automated solver workflows applicable to alphabet-based and other constraint-rich problems in quantum hardware.

Abstract

This paper introduces a new Boolean-based methodology for constructing Segment Display Problems (SDPs) in the quantum domain and solving them using Grover's quantum search algorithm. In the classical domain, the SDPs are typically solved using various techniques, such as human deduction, heuristic search, and methods for solving Boolean satisfiability (SAT) and constraint satisfaction problems (CSPs) that are based on different problem design models. In this paper, our newly introduced methodology proposes a quantum-based approach for solving such SDPs, by building their quantum oracle using binary reversible circuits and our previously proposed step-decreasing structures shaped operators (Stesso). To demonstrate the usability of this proposed method, we experimentally solve an SDP instance of the matchstick problem using Grover's algorithm with a noisy simulated quantum computer implemented in Qiskit.

Solving Segment Display Problems Using Quantum Grover's Search Algorithm

TL;DR

This work addresses solving Segment Display Problems (SDPs) with quantum search by mapping SDPs to a Boolean oracle suitable for Grover's algorithm. It introduces a two-level oracle design: a high-level decomposition into segment-code verifiers, decoders, and arithmetic units, and a layout-aware, circuit-level synthesis aided by the Combination Sequence of Exclusive Sums (CSES) to minimize quantum cost. Central contributions include formalizing SDP encoding as a triple , detailing constraints as inner and amid geometric and cryptarithmetic categories, and presenting concrete circuit components (SC verifier, SC-BCD, TDN generator) built with Stesso for efficient gate synthesis. A case study solving a matchstick SDP on a noisy Qiskit simulator demonstrates feasibility, achieving correct solutions with a modest ancilla budget and illustrating potential scalability to broader CSPs. The approach lays groundwork for a reusable quantum library and automated solver workflows applicable to alphabet-based and other constraint-rich problems in quantum hardware.

Abstract

This paper introduces a new Boolean-based methodology for constructing Segment Display Problems (SDPs) in the quantum domain and solving them using Grover's quantum search algorithm. In the classical domain, the SDPs are typically solved using various techniques, such as human deduction, heuristic search, and methods for solving Boolean satisfiability (SAT) and constraint satisfaction problems (CSPs) that are based on different problem design models. In this paper, our newly introduced methodology proposes a quantum-based approach for solving such SDPs, by building their quantum oracle using binary reversible circuits and our previously proposed step-decreasing structures shaped operators (Stesso). To demonstrate the usability of this proposed method, we experimentally solve an SDP instance of the matchstick problem using Grover's algorithm with a noisy simulated quantum computer implemented in Qiskit.
Paper Structure (15 sections, 2 theorems, 7 equations, 7 figures, 5 tables)

This paper contains 15 sections, 2 theorems, 7 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Segment display problems
  4. Classical approaches for solving SDPs
  5. Quantum approaches for specific problem solving
  6. Methods for a High-Level Oracle Design
  7. Encoding an SDP
  8. Defining the correctness of the solution to the SDP
  9. Designing a Boolean oracle
  10. Methods for a Layout-Aware Logic Design
  11. Segment code verifier
  12. Segment code to binary code decoder
  13. Two-digit number generator
  14. Case: a matchstick problem
  15. Conclusions and Future Work

Key Result

Theorem 1

The inner geometric constraints $C_{gin}:D_{seg} \bigtimes \{ \ket{0}, \ket{1}\} \bigtimes D_{dig} \rightarrow \{ 0, 1\}$ always can be described by the function $f_{gin}$ that is a sparse function related to $D_{seg}$ and $\{ \ket{0}, \ket{1}$ or a bijection function related to $D_{seg}$ and $D_{di

Figures (7)

  • Figure 1: Group relationships among constraint satisfaction problems (CSPs), segment display problems (SDPs), and cryptarithmetic problems, along with some examples.
  • Figure 2: An example of an SDP for adding two one-digit numbers. A seven-segment display presents a one-digit number using a 7-bit binary vector $[a,b,c,d,e,f,g]$, while a fourteen-segment display presents an operation using a fourteen-bit binary vector $[a,b,c,d,e,f,g_1,g_2,h,i,j,k,l,m]$.
  • Figure 3: The relationship between the model and its Oracle for an SDP.
  • Figure 4: The SC-State of Number 9, which is an 8-bit Toffoli gate, where $a$, $b$, $c$, $d$, $e$, $f$, $g$ are controls, $c$, $d$, $e$, $f$, and the first ancilla qubit are the step-outputs, and the last ancilla qubit is the target.
  • Figure 5: Compositional parts of the SC Verifier of seven-bit binary segment variable $q_s = abcdefg$, i.e., seven input qubits $a,b,c,d,e,f,g$, one step-output qubit, and one output qubit $v_1$.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Theorem 1
  • ...and 8 more