A Fast Measurement based fixed-point Quantum Search Algorithm

Generic quantum search algorithm searches for target entity in an unsorted database by repeatedly applying canonical Grover’s quantum rotation transform to reach near the vicinity of the target entity represented by a basis state in the Hilbert space associated with the qubits. Thus, when qubits are measured, there is a high probability of finding the target entity. However, the number of times quantum rotation transform is to be applied for reaching near the vicinity of the target is a function of the number of target entities present in the unsorted database, which is generally unknown. A wrong estimate of the number of target entities can lead to overshooting or undershooting the targets, thus reducing the success probability. Some proposals have been made to overcome this limitation. These proposals either employ quantum counting to estimate the number of solutions or fixed point schemes. This paper proposes a new scheme for stopping the application of quantum rotation transformation on reaching near the targets by measurement and subsequent processing to estimate the distance of the state vector from the target states. It ensures a success probability, which is at least greater than half for all the ratios of the number of target entities to the total number of entities in a database, which are less than half. The search problem is trivial for remaining possible ratios. The proposed scheme is simpler than quantum counting and more efficient than the known fixed-point schemes. It has same order of computational complexity as canonical Grover’s search algorithm but is slow by a factor of two and requires an additional ancilla qubit.

💡 Research Summary

The paper addresses a fundamental limitation of Grover’s quantum search algorithm: the optimal number of Grover iterations (rotations) depends on the unknown number of target items m (or the ratio p = m/N). If the iteration count is mis‑estimated, the algorithm either undershoots or overshoots the target subspace, causing the success probability to drop dramatically. Existing remedies include quantum counting to estimate m, heuristic guesses, and fixed‑point search schemes. Quantum counting adds circuit depth and extra queries, while known fixed‑point algorithms lose the quadratic speed‑up, running in O(N) time.

The authors propose a measurement‑based fixed‑point approach that retains the O(√N) asymptotic complexity of Grover’s algorithm but adds only a single ancilla qubit and a modest constant overhead. The circuit uses two ancilla registers, |OQ₁⟩ and |OQ₂⟩, together with two classical counters C₀ and C₁. The algorithm proceeds as follows:

1. Initialise C₀ = C₁ = 0.
2. Prepare the search register |s⟩ and ancilla |OQ₁⟩ in the uniform superposition via Hadamard gates; set |OQ₂⟩ to |0⟩.
3. Apply an oracle‑like unitary Uf₁ on |s⟩ and |OQ₂⟩. This operation flips |OQ₂⟩ to |1⟩ with probability proportional to the current overlap of |s⟩ with the target subspace.
4. Apply the standard Grover diffusion operator G on |s⟩ (controlled by |OQ₁⟩) to amplify the target amplitude.
5. Measure |OQ₂⟩. Increment C₁ if the outcome is |1⟩, otherwise increment C₀.
6. Compute the ratio R = C₁/C₀. If R ≥ Set_Val, stop the iteration and measure the search register |s⟩; otherwise return to step 3.
7. If the final measurement of |s⟩ does not yield a target, restart the whole procedure.

The key insight is that the expected value of R is a monotonic function of the success probability g_r(p) = sin²