Standard Grover’s Algorithm[1]:

Iterations: K ≈ ( π / 4 ) N Fixed operator G applied repeatedlyArithmetic series amplitude growth: r(k) ≈ 2k + 1

Exponential Speedup Algorithm[2]:

Iterations: K = n/2 = (1/2) log₂N (for β = 2)Iteration-dependent operators U₀, U₁, ∙∙∙, U_K−1Geometric series amplitude growth: r(k) = β^k

Single-Rotation Quantum Search: The Single-Rotation Quantum Search circuit was first constructed by the author in 2024 [3] for a special case where only one qubit in the marked state is 1. This paper works for general cases and has many new results than that of [3].

This Work (Single-Rotation Algorithm):

Iterations: K = 1,Single rotation: Δθ = π/4,Direct amplification: θ₀ ≈ 0 → θ₁ = π/4.

Relationship to Exponential Speedup Algorithm:The Single-Rotation Algorithm is the limiting case of the Exponential Speedup Algorithm [2] as β → N :

As β increases, fewer iterations are needed,In the limit β → N , exactly one iteration suffices,The single rotation Δθ = π/4 directly achieves the target angle.

This paper establishes:

1) Theoretical Minimum: One oracle call plus one π/4 rotation is necessary and sufficient for amplitude amplification (Section 3).

2) Explicit Numerical Matrices: Complete N × N unitary matrix construction with closed-form numerical values for the π/4 rotation (Section 5).

3) Physical Realizability: The rotation operator is unitary and can be implemented with log(N) steps (Section 6).

4) Implementation: The π/4 rotation is implemented by one oracle call, one diffusion call and log(N) reflections (Sections 7 and 8).

5) Lower Bounds: The standard lower bounds for unstructured quantum search still apply (Section 9).

This work provides a possible quantum search algorithm—a single π/4 rotation.

Throughout this paper:

n Number of qubits,N = 2ⁿSearch space size,K Iteration count,|α⟩ the single marked state,|β⟩ uniform superposition of N − 1 unmarked states,a₁ Amplitude of marked state (assumed to be state |α⟩),a₀ Amplitude of each unmarked state,θ Angle in geometric representation, relative to |β⟩,Δθ Rotation angle implemented by U.

Section 1 provides motivation and context. Section 2 compares rotation vs. inversion-about-average. Section 3 proves that one rotation by π/4 is necessary and sufficient. Section 4 analyzes the variations. Section 5 constructs explicit N × N unitary matrices with numerical values. Section 6 proves physical realizability. Section 7 introduces implementation strategy. Section 8 implements the Single-Iteration algorithm. Section 9 discusses time complexity. Section 10 discusses inexplicit assumptions used in the proposed algorithm. Section 11 concludes.

The diffusion operator acts as a reflection in the N-dimensional Hilbert space, which combined with the oracle creates a rotation in the 2D subspace {|α⟩, |β⟩} [4]. The superposition over all 2ⁿ computational states is given in Equation (1):

Note:

Throughout this paper, let the marked state be |1⟩ for convenience of discussion whenever necessary, assumed without loss of generality [2]. The quantum state can be visualized as a vector in a 2-dimensional subspace spanned by the marked state in Equation (3):

and uniform superposition of unmarked states in Equation (4)

Mathematical form of the diffusion operator is:

where |ψ⟩ is the uniform superposition. Physical realizability requires that only unitary operators can be implemented in quantum hardware.

The diffusion operator is often pedagogically described [4]-[6] as a process:

1) Invert the amplitude of the marked state (oracle)

2) Compute mean amplitude: 〈 a 〉 = ( 1 / N ) Σ i a i

3) Shift coordinate system to center at mean: a i → a i − 〈 a 〉

4) Reflect all amplitudes about zero: a i − 〈 a 〉 → − ( a i − 〈 a 〉 )

5) Shift back: − ( a i − 〈 a 〉 ) → 2 〈 a 〉 − a i

Net effect: Each amplitude transforms as a i → 〈 a 〉 − a i (reflection about the mean). Inversion About Average is Not physically realizable step-by-step (see Section 2.3).

THEOREM 2.1 (Non-Unitarity of Intermediate Steps): The “shift” and “reflect” operations in the “inversion about average” description do not preserve quantum state normalization and therefore cannot represent physical quantum states.

Proof by Example:Consider N = 8, let the marked state be |1⟩, starting from uniform superposition in Equation (1).

Initial state:

All amplitudes: a = 1 / 8 = 0.3536 All probabilities: a 2 = 1 / 8 = 0.125 Sum of probabilities: 1.0 ✓ (normalized)

After oracle(invert state 1):

State 1 amplitude: −0.3536Other amplitudes: +0.3536All probabilities: 0.125 Sum of probabilities: 1.0 ✓ (normalized-oracle is unitary)

After shifting(center at mean = 0.2652):

State 1: −0.3536 − 0.2652 = −0.6187 → probability = 0.3828Other states: +0.3536 − 0.2652 = +0.0884 → probability = 0.0078 eachSum of probabilities: 0.3828 + 7(0.0078) = 0.4375 (NOT normalized!)

After reflecting:

State 1: +0.6187 → probability = 0.3828Other states: −0.0884 → probability = 0.0078 eachSum of probabilities: 0.4375 (still not normalized!)

After shifting back (add mean):

State 1: +0.6187 + 0.2652 = 0.8839 → probability = 0.7813Other states: −0.0884 + 0.2652 = 0.1768 → probability = 0.0313 eachSumof probabilities: 0.7813 + 7(0.0313) = 1.0 ✓ (normalized again)

Conclusion: The intermediate states after “shifting” and “reflecting” have Σ | a i | 2 = 0.4375 ≠ 1 , violating the fundamental requirement for quantum states. These are unphysical mathematical intermediates, not realizable quantum states. Table 1 lists sums of probabilities. □

Table 1.Normalization throughout inversion about average process.

THEOREM 2.2 (Single Implementation Path): Both Grover’s algorithm and the Single-Rotation Algorithm (to be proposed in this paper) have exactly one physically realizable implementation: rotation in Hilbert space.

Proof: For Grover’s algorithm:

Description 1 (rotation): ✓ Unitary operator, preserves normalizationDescription 2 (inversion about average): ✗ Intermediate steps are unphysical (Theorem 2.1)Conclusion: Only rotation is implementable

For Single-Rotation Algorithm:

Single π/4 rotation: ✓ Unitary operator, preserves normalization throughoutConclusion: Rotation is the only option

Therefore, both algorithms have exactly one implementation pathway: unitary rotation in the {|α⟩, |β⟩} subspace. □

COROLLARY 2.3 (Equal Implementation Complexity): Grover’s algorithm and the Single-Rotation Algorithm have identical implementation requirements.

Both require:

1) Oracle access: To identify the marked state |α⟩

Grover: Used in every iteration (K ≈ N times)Single-Rotation: Used once

2) Rotation in {|α⟩, |β⟩} subspace:

Grover: Small angle Δθ [2], repeated K times Single-Rotation: Large angle Δθ = π/4, once

No implementation advantage exists for either algorithm in terms of the fundamental operations required. □

THEOREM 2.4 (Exponential Time Complexity): The “inversion about average” has an exponential Time Complexity.

Proof.The time complexities are:

1) Invert the amplitude of the marked state (oracle): NA

2) Compute mean-amplitude: 〈 a 〉 = ( 1 / N ) Σ i a i , O(N)

3) Shift coordinate system to center at mean: a i → a i − 〈 a 〉 , O(N)

4) Reflect all amplitudes about zero: a i − 〈 a 〉 → − ( a i − 〈 a 〉 ) , O(N)

5) Shift back: − ( a i − 〈 a 〉 ) → 2 〈 a 〉 − a i , O(N)

Therefore, T = O(N) = O(2ⁿ). □

Common Misconception:“Grover’s algorithm is easier to implement because the ‘inversion about average’ provides a simple step-by-step recipe: compute mean, shift, reflect, shift back. The Single-Rotation Algorithm requires identifying a direction in Hilbert space by oracle, |α⟩, ‘constructing’ of |β⟩, and rotating |β⟩ toward |α⟩, which is more complex.”

Refutation:This argument is invalid because:

1) The “inversion about average” is not implementable step-by-step (Theorem 2.1)

The intermediate states are unphysicalOnly the overall unitary operator D can be implementedExponential complexity

2) Bothalgorithms require the same fundamental operation: rotation in the {|α⟩, |β⟩} subspace (Theorem 2.2), geometrically,

Both need oracle to identify |α⟩Both need to “construct” |β⟩Both need to rotate toward |α⟩ in this 2D subspace

3) The Single-Rotation Algorithm is simpler:

Grover: K iterations of rotationSingle-Rotation: 1 iteration of rotationBoth use the same per-iteration mechanics

Conclusion: Neither algorithm has an “ease of implementation” advantage. The Single-Rotation Algorithm is strictly simpler, requiring fewer iterations of the same fundamental operation. □

Key Results:

Theorem 2.1: The “inversion about average” intermediate steps are unphysicalTheorem 2.2: Both algorithms have only one implementation: rotation in Hilbert spaceCorollary 2.3: Implementation requirements are identicalTheorem 2.4: The “inversion about average” is exponentialRefutation: No “easier implementation” advantage exists for Grover’s algorithm

The playing field is level: Both algorithms require unitary rotation in a 2D subspace. The Single-Rotation Algorithm simply requires fewer iterations of the same operation. This establishes that the comparison between algorithms reduces purely to iteration count and total complexity, with no advantage in fundamental implementation difficulty for either approach.

ALGORITHM 1 (Single-Rotation Quantum Search):

Input:

Search space size N = 2ⁿOracle O that marks the target state

Output:

Marked state with probability 1/2

Steps:

1) Initialize: Create uniform superposition in Equation (1)

2) Oracle:oracle, O = I − 2|α⟩⟨α|, identifies marked state, |α⟩

3) Apply Single Rotation: Rotate by π/4 from |β⟩ to |α⟩ in {|α⟩, |β⟩} plane

where U(π/4) is the rotation operator (defined in Section 6)

4) Measure: Measure in computational basis

5) Verify: Check if measured state is marked

6) Repeat if necessary: If measurement fails, repeat from Step 1. Expected attempts: ≤2.

Following [4], we work in the 2-dimensional subspace spanned by {|α⟩, |β⟩}. The marked state is |α⟩. The unmarked superposition is given in Equation (4). Any state at iteration k can be written in Equation (7):

Equation (7) can be written in Equation (8):

where θ_k is the angle from the |β⟩ axis st iteration k and the rotation is from |β⟩ axis to |α⟩ axis. Amplitude of the marked state is given in Equation (9):

The amplitude of unmarked superposition is given in Equation (10):

Initial angle (k = 0), from the initial superposition in Equation (1), is given in Equation (11):

Target angle (k = 1) is given in Equation (12):

Required rotation is given in Equation (13):

THEOREM 3.1 (Single Rotation Optimality): For unstructured search over N items with one marked item, a single rotation by angle Δθ = π/4 − arcsin( 1 / N ) is both necessary and sufficient to achieve success probability P(marked) = 1/2.

Proof: Part 1 (Sufficiency): A single rotation by Δθ achieves the threshold.

Starting from θ₀ = arcsin( 1 / N ), after rotation by Δθ:

Amplitude of marked state:

Success probability:

P 1 = | a 1 ( 1 ) | 2 = 1 / 2 ✓

Part 2 (Necessity): No rotation smaller than Δθ can achieve P ≥ 1/2.

For P(marked) ≥ 1/2, we need | a 1 | 2 ≥ 1/2 , which requires:

This is satisfied when θ ≥ π/4 (for 0 ≤ θ ≤ π/2).

Since θ₀ < π/4 (for all N > 1), we must rotate to at least θ = π/4, requiring:

Therefore, Δθ = π/4 − arcsin( 1 / N ) is the minimum rotation needed. □

THEOREM 3.2 (Large N Limit): For large N, the required rotation angle approaches π/4:

Proof:For large N:

Therefore:

Δ θ = π / 4 − arcsin ( 1 / N ) ≈ π / 4 − 0 = π / 4 □

Example 3.1:Δθ vs. N is given in Table 2.

Table 2.Δθ vs. N.

COROLLARY 3.3 (π/4 Rotation for any N): For any N, the single rotation can be approximated as exactly π/4.

Proof: From Theorem 3.1, Δθ = π/4 − arcsin( 1 / N ) is the minimum rotation needed. Δθ = π/4 ≥ π/4 − arcsin( 1 / N ) is larger than required minimum rotation. □

Practical Implication: For all realistic applications, we can implement the rotation as exactly U(π/4) without computing the arcsin correction term in Equation (13).

This subsection establishes that |β⟩ need not be explicitly constructed, avoiding potential exponential overhead.

THEOREM 3.4 (Implicit |β⟩ Representation): The state |β⟩ in Equation (4) can be represented as:

where |ψ₀⟩ is the initial uniform superposition in Equation (1) and |α⟩ is the marked state.

Proof:The initial uniform superposition in Equation (1) can be decomposed:

By definition of Equation (4):

Solving for |β⟩ will give Equation (15) □

Following [2], the rotation operator for angle θ is:

For θ = π/4, we have numerical values:

The matrix elements of the operator U are defined in Equation (19):

The matrix elements of U(θ), for |α⟩ = |1⟩, are:

THEOREM 5.1 (Matrix Elements for π/4 Rotation): The matrix elements of U(π/4), for |α⟩ = |1⟩, are:

Proof: Substituting cos(π/4) = sin(π/4) = 1 / 2 into Equation (20) will prove (21) [2]. □

Example 5.1 CaseN= 2 (n= 1):

Example 5.2 CaseN= 4 (n= 2):

Example 5.3 CaseN= 8 (n= 3):

THEOREM 5.2 (Matrix Structure): For arbitrary N = 2ⁿ, the matrix U(π/4), for |α⟩ = |1⟩, has the structure:

Row i = 1 (marked state row):

Columnj= 1 (marked state column):

Unmarked block (i ≠ 1, j ≠ 1):

Proof: Direct substitution of θ = π/4 into Equation (21) will do [2]. □

THEOREM 6.1 (Unitarity of U(π/4)): The operator U(π/4) defined in Equation (16) is unitary for all N = 2ⁿ.

Proof 1:From [2], the general rotation operator U(θ) is unitary for any angle θ. Since U(π/4) is a special case with θ = π/4, it inherits unitarity. □

The state |β⟩ serves two distinct purposes:

Geometric Role:

Forms orthonormal basis {|α⟩, |β⟩} for the 2D amplification subspace.Clarifies the rotation mechanism (from θ₀ to θ₀ + π/4).Explains amplitude growth through sin(θ) increase.Connects to Grover’s geometric interpretation [4].

Computational Role:

Need NOT be explicitly constructed as a quantum state.Represented implicitly via Theorem 3.4.All operations involving |β⟩ rewritten using |ψ₀⟩ and |α⟩ in Equation (15).Avoids exponential construction overhead.Approximated by |ψ₀⟩ for large N.

The implicit representation in Equation (15) is the key to polynomial-time implementation.

THEOREM 6.2 (Projector Expansion): The projector |β⟩⟨β| can be expanded as:

Proof: From Equation (15), direct expansion of |β⟩⟨β|:

| β 〉 〈 β | = 1 N − 1 ( N ⋅ | ψ 0 〉 − | α 〉 ) ⋅ 1 N − 1 ( N ⋅ 〈 ψ 0 | − 〈 α | ) . □

THEOREM 6.3 (Off-Diagonal Terms): The operators |α⟩⟨β| and |β⟩⟨α| can be expanded as:

Proof: Direct substitutions of Equation (15) into |α⟩⟨β| and |β⟩⟨α| complete proof. □

THEOREM 6.4 (Rotation Operator): The complete rotation operator U(π/4) can be expanded using only |ψ₀⟩ and |α⟩:

where

|β⟩⟨β| is given in Equation (26),

|α⟩⟨β| is given in Equation (27), and

|β⟩⟨α| is given in Equation (28).

Proof. Substituting Theorem 6.2 and 6.3 into the standard rotation operator formula in Equation (16) will prove the theorem. □

THEOREM 6.5 (Rotation Operator): The complete rotation operator U(π/4) can be expanded using only |ψ₀⟩ and |α⟩:

where c = cos(θ) and s = sin(θ).

Proof. Expandingβ⟩⟨β|, |α⟩⟨β| and |β⟩⟨α| in Theorem 6.4 will prove the theorem. □

THEOREM 6.6 (Rotation Operator for π/4): The complete rotation operator U(π/4) can be expanded using only |ψ₀⟩ and |α⟩:

Proof.Using π/4, organizing Theorem 6.5 by I, |ψ₀⟩⟨ψ₀|, |ψ₀⟩⟨α|, |α⟩⟨ψ₀|, and |α⟩⟨α| will prove this theorem. □

THEOREM 6.7 (Rotation Operator for π/4): The complete rotation operator U(π/4) can be expanded using only |ψ₀⟩ and |α⟩:

where:

A = ( 1 / 2 − 1 ) ⋅ N / ( N − 1 ) B = − N ⋅ [ ( 1 / 2 − 1 ) / ( N − 1 ) + ( 1 / 2 ) / N − 1 ] C = − N ⋅ [ ( 1 / 2 − 1 ) / ( N − 1 ) − ( 1 / 2 ) / N − 1 ]

Proof.Organizing Theorem 6.6 by A, B and C will prove this theorem. □

Every term in these expansions involves only Identity I, |ψ₀⟩⟨ψ₀|, |ψ₀⟩⟨α|, |α⟩⟨ψ₀|, |α⟩⟨α|.

This subsection provides the explicit formula for the rotation operator of π/2. Theorem 6.4 and 6.5 can be applied to any angle.

THEOREM 6.8 (π/2 Rotation Case): For θ = π/2,

Proof: we have cos(π/2) = 0 and sin(π/2) = 1, Equation (31) gives:

Simplifying:

For the |ψ₀⟩⟨α| coefficient:

For the |α⟩⟨ψ₀| coefficient:

Final form is Equation (32). □

THEOREM 6.9 (π/2 Rotation Case): For θ = π/2,

where:

A = − N / ( N − 1 ) B = N / ( N − 1 ) ⋅ [ 1 − N − 1 ] = N / ( N − 1 ) − N / N − 1 C = N / ( N − 1 ) ⋅ [ 1 + N − 1 ] = N / ( N − 1 ) + N / N − 1

For π/2, the off-diagonal coefficients B and C grow as ± N , while for π/4 they remain O(1). This suggests the π/2 rotation involves more significant amplitude transfer between |α⟩ and |ψ₀⟩ components, thus, the π/2 rotation is not a good choice.

THEOREM 6.10 (π/4 Rotation Coefficient Scaling): For the π/4 rotation operator U(π/4) with large N, all coefficients scale as O(1):

where:

Proof:For θ = π/4, we have cos(π/4) = sin(π/4) = 1 / 2 . The coefficients are:

For large N, N/(N − 1) → 1, therefore:

Next,

For large N:

Therefore:

Next

For large N:

Therefore:

All three coefficients are bounded constants for large N. □

THEOREM 6.11 (π/2 Rotation Coefficient Scaling): For the π/2 rotation operator U(π/2) with large N, the diagonal coefficients scale as O(1) while the off-diagonal coefficients scale as O( N ):

where:

Proof:For θ = π/2, we have cos(π/2) = 0 and sin(π/2) = 1. The coefficients are

For large N, N/(N − 1) → 1, therefore:

Next,

For large N:

Therefore:

Next,

Therefore:

The off-diagonal coefficients, B and C, grow asymptotically as N , while the diagonal coefficient, A, remains bounded. □

For large-scale quantum search problems (large N), the π/4 rotation is preferable to the π/2 rotation despite the π/2 rotation achieving higher success probability. Reasons are: 1) Bounded coefficients; 2) Numerical stability: For N = 2ⁿ with large n, the π/2 coefficients become extremely large; 3. Potential error sensitivity: the extremely large coefficients in π/2 rotation may introduce:

Numerical precision issues in classical parameter computation.Potential error amplification if quantum gate errors scale with parameter magnitude.Implementation challenges on quantum hardware with finite control precision.

Conclusion: For large N, the π/4 rotation is better due to being bound by O(1) coefficients, numerical stability, and robustness to potential implementation imperfections.

Both π/4 and π/2 rotations have the same gate complexity structure.

THEOREM 6.12 (Universal Simplified Form for Practical Applications):For all practical quantum search applications with n ≥ 7 (N ≥ 128), the π/4 rotation operator with less than 5% error is:

Proof. From Table 3, for n ≥ 7 or N ≥ 128, less than 5% error can be achieved.

Table 3.Exact vs. simplified coefficients.

As an example of Cartan-Dieudonné theorem in Mathematics, consider the usual 2-D x-y coordinate system, let:

Β= (1, 0) → pointing along the x-axisα= (cosθ, sinθ) → rotated by angle θ

Furthermore, let θ = 30˚, so:

α= (cos30˚, sin30˚)β= (1, 0)

First reflection: Reflect across the line perpendicular to α (this corresponds to R_α). If you start at β = (1, 0), after reflecting across α, you land on the other side of α, symmetric with respect to α (240˚):

α= (cos30˚, sin30˚)β= (cos240˚, sin240˚)

Second reflection: Now reflect across the line perpendicular to β. Since β = (1, 0), this is just Reflection across the y-axis, so (x, y) → (−x, y):

α= (cos30˚, sin30˚)β= (−cos240˚, sin240˚)

Note, cos(2 × 30˚) = −cos240˚, sin(2 × 30˚) = sin(240˚), we have:

α= (cos30˚, sin30˚)β= (cos(2 × 30˚), sin(2 × 30˚))

This demonstrates “Two flips = one rotation”:

The rotation R(2θ) can be implemented using two reflections of θ. The basic ideal is the key identity [7]:

Where U prepares the state: U|0⟩ = |ψ⟩. Instead of implementing:

Which looks exponential, we do:

1) Uncompute |ψ⟩ to |0⟩: |ψ⟩ = U|0⟩

2) Apply a simple phase flip on |0⟩: |0⟩ → −|0⟩

3) Compute back: U|0⟩ = |ψ⟩

So:

Three steps for reflections are:

Step 1: Apply U^†

cost (U^†)

Step 2: Flip phase of |0⟩

This is |0ⁿ⟩ → −|0ⁿ⟩, implemented by:

multi-controlled Z gate cost: O(n)

Step 3: Apply U

cost (U)

We will use Cartan-Dieudonnétheoremin Mathematics, which essentially states that any rotation can be “factored” into a pair of reflections. The core rule is the Double Angle Rule: the angle of the resulting rotation is exactly twice the angle between the two lines (or planes) of reflections.

However, the direct application of Cartan-Dieudonnétheoremwith one rotation will not work because the cost is exponential, this problem can be resolved by multiple rotations.

Construction of rotations forθ:To specifically rotate a vector, we can follow these two-step reflections:

1) First Reflection: Choose the first line (or plane) to pass directly through the vector |β⟩. When you reflect |β⟩ across a line that it already lies on, nothing happens—the vector stays exactly where it is.

2) Second Reflection: Choose the second line to be the angle bisector between |β⟩ and |α⟩. When you reflect the vector |β⟩ (which is still at position |β⟩ across this bisector, it will “jump” across the line and land perfectly on |α⟩.

If we denote the reflection operation as R_β, R_v, the combined operation is:

Because the angle between |β⟩ and the bisector is half of the total angle between |β⟩ and |α⟩, the composition results in a total rotation of θ. To find the angle bisector, one must normalize them first:

Standard reflection uses:

New rotation operator is

In general, the state preparation, |v⟩, in Equation (41) is NOT poly (n). In this section, we have shown that when you reflect |β⟩ across a line that it already lies on, nothing happens—the vector stays exactly where it is. Now by a proper choice, two reflections above are reduced to one reflection.

When rotating from |β⟩ to |α⟩ by 2θ within 2-D plane, {|α⟩, |β⟩}, a single reflection around θ-line is enough. If geometric series for Rotation Angles is:

then the number of rotations is K = log(N).

THEOREM 8.1The number of reflections from |β⟩ to |α⟩, R(π/4), is log(N).

Proof. We will apply the sequence of oracle, diffusion, reflection, reflection, …

Step 0. From Equation (8),

From Equation (11), initial angle (k = 0), from initial superposition in Equation (8), is given:

This gives the first term in the geometric series for rotation angles:

Step 1. From Equation (44), after the oracle operator and diffusion operator,

which is normalized. Ignore higher order terms O(1/N),

This gives the second term in the geometric series for rotation angles:

Step 2. Reflection cross line | ψ 1 〉 . Standard reflection operator uses:

This line is 3 θ 0 = 3 / N from |β⟩-axis. Applying reflection operator,

Because | ψ 0 〉 is θ 0 from |β⟩-axis, the angle between | ψ 0 〉 and the reflection line is 2 θ 0 = 2 / N . After the reflection, | ψ 0 〉 will advance 4 θ 0 and we have the following approximation:

| ψ 2 〉 is 5 θ 0 from |β⟩-axis, which gives the next term in the geometric series for rotation angles:

Step 3. Reflection cross line | ψ 2 〉 . Standard reflection operator uses:

This line is 5 θ 0 = 5 / N from |β⟩-axis, and the next state is:

Because | ψ 0 〉 is θ 0 from |β⟩-axis, the angle between | ψ 0 〉 and the reflection line is 4 θ 0 = 4 / N . After the reflection, | ψ 0 〉 will advance 8 θ 0 and we have the following approximation:

| ψ 3 〉 is 9 θ 0 from |β⟩-axis, which gives the next term in the geometric series for rotation angles:

The geometric series for rotation-angle is:

The number of rotation from |β⟩ to |α⟩, R(π/4), is K = log(N). □