We present the resource estimation of how known-constant arithmetic changes the cost of a Grover oracle for Minecraft 1.13.1 seed recovery, an application of quantum search to procedural content generation. The oracle developed here combines a structure-placement predicate with a reversible inverse transform of the linear congruential generator for the 32-bit residual search. Reduced-width arithmetic, phase marking, uncomputation, and complete Grover loops are verified against an independent Java reference, while production-width circuits are structurally audited. Direct Fourier-basis constant arithmetic reduces the placement-predicate width from 6n+3 to 4n+3 qubits and the structural scaling of the two-qubit gate counts from Theta(n^3) to O(n^2). Under backend-agnostic common-basis Qiskit level-0 {u, CX} accounting, the representative nonzero-r folded residual-search construction uses 275 circuit qubits, 48,896 CX gates, and depth 4,482, versus 467 qubits, 2,038,804 CX gates, and depth 186,328 for the original register-to-register baseline. Multi-placement conjunctions expose a width-recomputation tradeoff, and exact enumeration identifies a plateau in candidate reduction for the benchmark. We executed reduced-width placement circuits within complete Grover loops on IBM processors and Quantinuum H2-1 and show larger one-iteration increases in marked-state probability for the specialized construction than for the baseline. These results quantify the resource benefits of arithmetic specialization and show substantial routing sensitivity in the reduced-width IBM hardware experiments. Together, the results quantify the effect of specializing known-constant arithmetic within a PCG-derived oracle and show that its compiled hardware behavior strongly depends on routing and execution context.
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The paper asks what Grover search costs when its oracle implements a software-derived Minecraft seed-recovery predicate. It searches a $2^{32}$ residual space, given a classically recovered 16-bit auxiliary value, by reversibly reconstructing a 48-bit structure seed and checking Shipwreck placement constraints.
The construction combines inverse-LCG arithmetic, affine jump-ahead, and known-constant accumulation directly in the Fourier basis. Relative to an initial register-to-register implementation, placement-predicate width falls from $6n+3$ to $4n+3$ qubits and two-qubit scaling from $\Theta(n^3)$ to $O(n^2)$. For a representative nonzero region constant, the complete single-placement oracle uses 275 qubits, 48,896 CX gates, and depth 4,482, versus 467 qubits, 2,038,804 CX gates, and depth 186,328 under backend-agnostic Qiskit level-0 accounting.
Concurrent and workspace-reusing conjunction schedules quantify width–recomputation tradeoffs. Exact classical enumeration finds eight surviving residuals after six constraints, unchanged by the seventh and eighth; six constraints require 18,198 Grover iterations and approximately $4.22\times10^9$ or $8.22\times10^9$ oracle CX gates. Reduced-width semantics are checked against an independent Java reference; production circuits receive structural audits. Hardware tests search eight candidates: on H2-1, one-iteration success is 0.728 for specialized arithmetic versus 0.540 for the baseline. IBM contrasts vary substantially with routing.
The estimates exclude diffuser, routing, and fault-tolerance costs. Arithmetic and schedules are not claimed optimal, survivor counts are fixture-specific, and hardware demonstrations are reduced-width. Structured classical recovery is not benchmarked; full classical enumeration takes about five seconds on four CPU threads.
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32 pages, 10 figures. Supplementary material included after the appendices
We present the resource estimation of how known-constant arithmetic changes the cost of a Grover oracle for Minecraft 1.13.1 seed recovery, an application of quantum search to procedural content generation. The oracle developed here combines a structure-placement predicate with a reversible inverse transform of the linear congruential generator for the 32-bit residual search. Reduced-width arithmetic, phase marking, uncomputation, and complete Grover loops are verified against an independent Java reference, while production-width circuits are structurally audited. Direct Fourier-basis constant arithmetic reduces the placement-predicate width from 6n+3 to 4n+3 qubits and the structural scaling of the two-qubit gate counts from Theta(n^3) to O(n^2). Under backend-agnostic common-basis Qiskit level-0 {u, CX} accounting, the representative nonzero-r folded residual-search construction uses 275 circuit qubits, 48,896 CX gates, and depth 4,482, versus 467 qubits, 2,038,804 CX gates, and depth 186,328 for the original register-to-register baseline. Multi-placement conjunctions expose a width-recomputation tradeoff, and exact enumeration identifies a plateau in candidate reduction for the benchmark. We executed reduced-width placement circuits within complete Grover loops on IBM processors and Quantinuum H2-1 and show larger one-iteration increases in marked-state probability for the specialized construction than for the baseline. These results quantify the resource benefits of arithmetic specialization and show substantial routing sensitivity in the reduced-width IBM hardware experiments. Together, the results quantify the effect of specializing known-constant arithmetic within a PCG-derived oracle and show that its compiled hardware behavior strongly depends on routing and execution context.
The paper asks what Grover search costs when its oracle implements a software-derived Minecraft seed-recovery predicate. It searches a 232 residual space, given a classically recovered 16-bit auxiliary value, by reversibly reconstructing a 48-bit structure seed and checking Shipwreck placement constraints.
The construction combines inverse-LCG arithmetic, affine jump-ahead, and known-constant accumulation directly in the Fourier basis. Relative to an initial register-to-register implementation, placement-predicate width falls from 6n+3 to 4n+3 qubits and two-qubit scaling from Θ(n3) to O(n2). For a representative nonzero region constant, the complete single-placement oracle uses 275 qubits, 48,896 CX gates, and depth 4,482, versus 467 qubits, 2,038,804 CX gates, and depth 186,328 under backend-agnostic Qiskit level-0 accounting.
Concurrent and workspace-reusing conjunction schedules quantify width–recomputation tradeoffs. Exact classical enumeration finds eight surviving residuals after six constraints, unchanged by the seventh and eighth; six constraints require 18,198 Grover iterations and approximately 4.22×109 or 8.22×109 oracle CX gates. Reduced-width semantics are checked against an independent Java reference; production circuits receive structural audits. Hardware tests search eight candidates: on H2-1, one-iteration success is 0.728 for specialized arithmetic versus 0.540 for the baseline. IBM contrasts vary substantially with routing.
The estimates exclude diffuser, routing, and fault-tolerance costs. Arithmetic and schedules are not claimed optimal, survivor counts are fixture-specific, and hardware demonstrations are reduced-width. Structured classical recovery is not benchmarked; full classical enumeration takes about five seconds on four CPU threads.
Grover’s algorithm reduces the number of oracle evaluations required for unstructured search, but spatial queries require explicit data loading and reversible geometric computation whose costs are hidden by oracle-level analysis. We implement reversible quantum circuits for range, distance, and pairwise proximity search, including QROM coordinate loading, geometric predicate evaluation, phase marking, and uncomputation. The circuits are transpiled under topology constraints and evaluated using Grover oracle calls, CX-gate count, circuit depth, and mapped qubits. For range and distance search, oracle-call scaling exhibits an empirical exponent of 0.617, while pairwise search reaches 1.018 because its candidate space grows as N2. In contrast, CX-count and depth exponents range from 1.416 to 2.067. Increasing coordinate precision also substantially raises circuit resources at fixed dataset size. These results demonstrate that favorable query-level scaling can diverge substantially from circuit-level scaling, highlighting spatial representation, data access, and reversible geometry as first-class costs in quantum spatial computing.
Modular arithmetic is central to quantum algorithms for cryptographic problems, including Shor's algorithm and Grover-based cryptanalysis, with modular reduction contributing substantially to circuit cost. Pseudo-Mersenne moduli q=2n−c allow classical Crandall reduction to replace division with folding and constant arithmetic, providing a structural opportunity for more efficient quantum modular reduction than Barrett reduction. We translate this advantage into a reversible quantum setting by deriving explicit folding and normalization conditions for 2n-bit inputs. To the best of our knowledge, this constitutes the first exact reversible quantum circuit formulation of Crandall reduction. Based on this formulation, we develop two variants: Crandall reduction-1 is designed to minimize execution cost through one-step normalization, whereas Crandall reduction-2 uses two-step normalization to support a wider range of c with limited overhead. Logical resource estimates show that both variants require fewer qubits and lower T-count and T-depth than optimized folding Barrett reduction. At n=10, Crandall reduction-1 reduces both T-count and T-depth by approximately 46.9% relative to optimized folding Barrett reduction. Surface-code analysis further shows that, at n=20 under the Sparse Blossom decoder, the estimated runtimes of the two variants are 30.05 ms and 35.39 ms, respectively, compared with 53.77 ms for optimized folding Barrett reduction. These results demonstrate the practical value of exploiting modulus-specific arithmetic structure in fault-tolerant quantum circuit design.
This paper presents a quantum search approach to combinatorial constraint satisfaction problems, demonstrated through the generation of magic squares. We reformulate magic square construction as a quantum search problem in which a reversible, constraint-sensitive oracle marks valid configurations for amplitude amplification via Grover's algorithm. Classical pre-processing using the Siamese construction and partial constraint checks generates a compact candidate domain before quantum encoding. Rather than integrating classical and quantum solvers in an iterative loop, this work uses the classical component for structured initialisation and the quantum component for search, and benchmarks the quantum approach against classical brute-force enumeration and backtracking. Our Qiskit implementation demonstrates the design of multi-register modular arithmetic circuits, oracle logic, and diffusion operators. Experiments are conducted on small grid instances, as larger grids are intractable on classical statevector simulators due to exponential memory growth. The results validate the correctness of the proposed quantum search pipeline and confirm the theoretical quadratic query advantage over classical search.
The Quantum Pi Benchmark separates oracle-construction provenance (P0–P4), controller blindness (B0 or B1), and a non-total capability grid (R). Quantum search over digit patterns requires an oracle that preserves an address superposition while computing address-dependent data. This paper describes testing of that requirement for low hexadecimal digits of Pi using Bailey–Borwein–Plouffe (BBP)-derived reversible arithmetic on IBM Heron r2 QPU, Kingston. Two direct multiplexed constructions fail at 738–1038 routed native CZ gates despite exact noiseless validation. A prefix compute–uncompute echo localizes a selected implementation’s collapse to the second multiplexed stage. Reachable-cone synthesis restricts exactness to declared protected trajectories and reduces the core to 47 routed native CZ gates. Across eighteen runs, per-branch digit success is 0.69–0.84, cross-pair support remains below 0.02, and roundtrip return is 0.56–0.74; these jointly support a preserved address-conditioned structure, while the correlation gap alone is not a coherence proof. A P0/B0/R2 compiled Grover layer amplifies Pi-table offsets: sixteen-candidate bands return the mark at 0.618–0.688, and ten 64-candidate single-iteration runs return it at 2.2–2.9 times the uniform reference. A separate 34-decision outer loop correctly classifies occupied versus empty tiles, but fails its original earliest-position objective, and is therefore reported as tile-occurrence classification. At a fixed four-position tile, one declared comparator consumes a BBP-computed digit; target rotation and Ramsey controls support P3/B0/R5. The final thirteen-tile campaign uses execution-locked BBP bodies and target-independent phase calibration, however, known codewords select or omit its discriminator and only planner-positive tiles receive confirmation. The final campaign supports a bounded P4/B0/R6 range validation.
Abstract This guide is based on the IBM Quantum course Phase Estimation and Factoring and focuses on Grover’s algorithm, a quantum algorithm for unstructured search problems that provides a quadratic speedup over classical approaches. In an unstructured search problem, we are given a function (often represented as an oracle) that marks one or more “correct” solutions among NNN possible candidates. A classical algorithm requires on the order of O(N)O(N)O(N) evaluations in the worst case. In contrast, Grover’s algorithm finds a solution using only O(N)O(\sqrt{N})O(N) oracle queries. This quadratic improvement implies: The quantum algorithm requires roughly the square root of the number of operations needed classically. Any classical algorithm for unstructured search must have cost at least quadratic relative to Grover’s quantum procedure. Core Idea of the Algorithm Grover’s algorithm relies on: Preparation of a uniform superposition (typically using Hadamard gates). An oracle that marks valid solutions via phase inversion. The Grover diffusion operator, which amplifies the probability amplitude of the marked states through interference. By repeatedly applying the Grover operator, the amplitude of the desired solution grows while others diminish, allowing measurement to reveal the correct answer with high probability. Practical Considerations Despite its theoretical elegance and wide applicability, the quadratic speedup of Grover’s algorithm is unlikely to yield near-term practical advantage. Modern classical hardware operates at extremely high clock rates and is vastly more mature than current quantum devices. For problem sizes accessible to today’s quantum computers, classical systems remain dominant. However, history shows that even sub-quadratic classical improvements — such as: Fast Fourier Transform (FFT) Efficient sorting algorithms like quicksort and mergesort have had transformative practical impact. The crucial distinction is that Grover’s algorithm depends on fundamentally different hardware — quantum processors — which are still in early development. Long-Term Perspective As quantum hardware advances in qubit count, coherence time, and error correction, Grover’s quadratic advantage may become practically significant for: Cryptographic key search Optimization heuristics Combinatorial search problems Amplitude amplification subroutines in broader quantum algorithms Through hands-on experimentation on IBM Q, this lesson demonstrates how theoretical amplitude amplification translates into executable quantum circuits, reinforcing the relationship between interference, probability amplitudes, and computational complexity.
Abstract This laboratory work presents a practical implementation of Grover’s search algorithm using the Qiskit framework. The material is based on educational resources developed by the International Quantum Center at CERN (Switzerland) and is intended to provide hands-on experience with amplitude amplification and oracle-based quantum search. In Laboratory Work No. 7, participants construct Grover’s algorithm step by step. The implementation begins with defining a function that initializes the quantum registers and prepares the initial stage of the circuit. The output qubit associated with the oracle is set to the state ∣1⟩|1\rangle∣1⟩ to enable phase inversion through the phase kickback mechanism. Hadamard gates are then applied to create a uniform superposition over the search space. The oracle is designed to mark the target state by applying a phase shift, effectively flipping the sign of the amplitude corresponding to the solution. This is followed by the diffusion operator (inversion about the mean), which amplifies the probability amplitude of the marked state. The Grover iteration—composed of the oracle and diffusion operator—is repeated an optimal number of times to maximize the probability of measuring the correct result. The laboratory exercise emphasizes the principles of quantum amplitude amplification, circuit modularization, oracle construction, and measurement-based verification. By executing the algorithm on simulators and cloud-accessible quantum hardware, participants observe the quadratic speedup of Grover’s algorithm compared to classical linear search methods. This practical guide bridges theoretical understanding of quantum search with executable quantum circuits, reinforcing the operational aspects of quantum algorithm design within modern quantum computing