[Paper Review] The Precise Formula in a Sine Function Form of the norm of the Amplitude and the Necessary and Sufficient Phase Condition for Any Quantum Algorithm with Arbitrary Phase Rotations

Dafa Li|Jul 3, 2001

Quantum Computing Algorithms and Architecture10 references7 citations

TL;DR

This paper derives a precise sine-function formula for the amplitude norm in Grover-type quantum search algorithms with arbitrary phase rotations, enabling a necessary and sufficient phase condition for successful state amplification. It proves that identical rotation angles (θ = φ) are sufficient but not necessary for achieving full success probability, resolving contradictions in prior work and enabling broader algorithmic design.

ABSTRACT

In this paper we derived the precise formula in a sine function form of the norm of the amplitude in the desired state, and by means of he precise formula we presented the necessary and sufficient phase condition for any quantum algorithm with arbitrary phase rotations. We also showed that the phase condition: identical rotation angles, is a sufficient but not a necessary phase condition.

Motivation & Objective

• To derive an exact analytical formula for the norm of the amplitude in the desired state after k iterations of a generalized quantum search algorithm with arbitrary phase rotations.
• To resolve the long-standing open problem of determining the necessary and sufficient phase condition for any quantum algorithm with arbitrary phase rotations, as posed by Grover.
• To challenge and correct prior claims that identical rotation angles (θ = φ) are both necessary and sufficient for successful search, by showing they are only sufficient.
• To provide a precise optimal iteration count k₀ for achieving the desired state with certainty, applicable to any unitary operation and arbitrary phase angles.
• To establish general properties such as periodicity and monotonicity of the amplitude norm as a function of iteration count k.

Proposed method

• Derives a closed-form expression for the amplitude norm in the desired state as a sine function of the form |bₖ| = |sin(kΔ + ψ)| / √(1 + |β|²), where Δ and β are functions of the phase angles θ, φ and the overlap p = |⟨γ|τ⟩|.
• Uses a two-dimensional invariant subspace analysis to model the evolution of the quantum state under repeated application of the generalized Grover operator Q = -I_γ^(θ) U⁻¹ I_τ^(φ) U.
• Applies complex analysis and trigonometric identities to derive the phase condition |cos(θ - φ) - 2p² cosθ cosφ| ≤ 1, which ensures the amplitude norm remains bounded and oscillatory.
• Establishes the necessary and sufficient condition for achieving unit success probability as sinΔ ≤ |β|, where Δ and β are derived from the phase and overlap parameters.
• Uses recursion relations and eigenvalue analysis to derive the optimal number of iterations k₀ = ⌊π/(2Δ)⌋ for reaching the desired state with certainty.
• Analyzes special cases (e.g., θ = φ) to recover known results and compare with prior approximations, validating the exactness of the derived formulas.

Experimental results

Research questions

• RQ1 What is the exact analytical form of the amplitude norm in the desired state for a generalized quantum search algorithm with arbitrary phase rotations?
• RQ2 What is the necessary and sufficient condition on the phase angles θ and φ that guarantees the algorithm can find the desired state with certainty?
• RQ3 Is the condition θ = φ both necessary and sufficient for successful search, or can other phase combinations achieve full success probability?
• RQ4 What is the precise optimal number of iterations k₀ required to reach the desired state with certainty, given arbitrary phase rotations?
• RQ5 How do the amplitude norm and its behavior (e.g., monotonicity, periodicity) depend on the iteration count k and the phase parameters?

Key findings

• The norm of the amplitude in the desired state is exactly expressible as |bₖ| = |sin(kΔ + ψ)| / √(1 + |β|²), where Δ and β are derived from the phase angles and the overlap p = |⟨γ|τ⟩|.
• The necessary and sufficient condition for achieving unit success probability is sinΔ ≤ |β|, where Δ and β are functions of the phase angles and the overlap.
• The condition θ = φ is sufficient but not necessary for achieving full success probability, contradicting prior claims in [5] and [10] that it was both necessary and sufficient.
• The optimal number of iterations k₀ to achieve the desired state with certainty is k₀ = ⌊π/(2Δ)⌋, derived from the exact phase condition.
• The amplitude norm |bₖ| is periodic and monotonic increasing in the interval [0, k₀], with |bₖ| reaching its maximum at k₀.
• For the special case of θ = φ, the formula reduces to the standard Grover form, and k₀ is shown to be smaller than for non-identical phase rotations, indicating higher efficiency in that case.