Standard Grover’s Algorithm[2]:

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

Exponential Speedup Algorithm[3]:

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-Iteration/Single-Rotation Quantum Search: The author has recently presented the Single-Iteration Quantum Search Algorithm, which achieves amplitude-amplification with exactly one oracle call and one π/4 rotation [1], where the π/4 rotation can be implemented in O(logN) steps using a set of different Reflections. For N = 2ⁿ search spaces, the algorithm achieves success probability exactly 1/2 with one iteration of oracle operator and one π/4 rotation implemented by log(N) double reflections.

This Work:This work presents a simpler and cleaner derivation for π/4 rotation with log(N) diffusion operators, without mentioning reflections and Cartan-Dieudonné theorem [1].

Throughout this paper:

n Number of qubitsN = 2ⁿ Search space sizeK Iteration countk Iteration index|α⟩ the single marked state|β⟩ uniform superposition of N-1 unmarked statesa₁ Amplitude of marked state (assumed to be state |α⟩)a₀ Amplitude of each unmarked stateθ Angle in geometric representation, relative to |β⟩Δθ Rotation angle implemented by U

Section 1 provides motivation and context. Section 2 introduces preliminaries. Section 3 shows that a single oracle call and log(N) different diffusion calls can achieve amplitude-amplification. Section 4 shows that the standard lower bounds for unstructured quantum search still apply. Section 5 makes a few remarks. Section 6 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]. In this paper, the amplitude amplification can be achieved by a single oracle call and a set of different diffusion operators.

The superposition over all 2ⁿ computational states is given in Equation (1):

The oracle operator is:

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

and the uniform superposition of unmarked states in Equation (4)

Mathematical form of the Grover diffusion operator is:

where |ψ⟩ is the uniform superposition in Equation (1). Physical realizability requires that only unitary operators can be implemented in quantum hardware.

Following reference [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) [4][5]. Any state at iteration k can be written in Equation (6):

When k = 0,

Equation (6) can be written in Equation (8):

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

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

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

For large N,

Target angle (k = 1), after only one iteration, is given in Equation (12) [1]:

The required rotation is given in Equation (13):

The oracle operator, O, is given in Equation (2),

So,

From Equations (6) and (7),

The diffusion operator is defined in Equation (5),

Applying both oracle and diffusion,

As an approximation, we can drop O(1/N) term,

The total rotation is from 0 to π/4 for large N. The initial amplitude of |α⟩ in Equation (16) is 1 / N . Equation (20) and (21) indicate that the amplitude is increased approximately by 2 / N . For large N, using identity, sin ( 2 / N ) = 2 / N , the rotation angle, after applying the oracle operator and diffusion operator, is increased roughly by 2 / N .

For the Grover algorithm, the arithmetic series for θ k in Equation (8) is approximately [3]:

As an order of estimate for K, after K iterations, one has:

So Grover’s time complexity is order of

where K is the number of iterations to reach π/4, T_O is the oracle cost and T_A = O(n) is the diffusion operator cost for amplification. Equation (23) is exponential.

The basic ideal is the key identity [10]-[16]:

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

which looks exponential, it can be implemented as follows:

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 Equation (45) 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)

Here the cost means gate count. Finally, if it takes one step for one gate, without gate parallel computation, the time complexity (number of steps without parallel gate execution) is [10]-[18]:

Consider the Diffusion operators:

The Grover diffusion operator is

which can be written as

Using Equation (45),

This operator has a gate count of n, so

The oracle operator has its own time complexity, T_O, which depends on the applications and is not a variable in this paper.

THEOREM 4.1. The time complexity of D₁ is:

Proof. Consider:

By definition (Equation (30)),

So,

Adding all costs in Equation (49) yields Equation (48). □

THEOREM 4.2 The time complexities of the Diffusion Operators are:

where

Proof. By definition,

Similarly,

To compute the time complexity of D_k, simply add each term in Equation (54) to get Equation (50). Equation (51) was proved in the last section. □

THEOREM 4.3 The time complexity of D_k in Equation (50) is exponential,

Proof. From Equation (50),

From Equation (48),

From Equation (59), dropping lower order terms,

T ( D k ) = O ( 2 k ( T O + n ) ) □

THEOREM 4.4 When k = K,

Proof. When k = K, from Equation (42),

From Equation (55),

The paper claims a single oracle call is used in the entire algorithm. However, in Grover’s algorithm, because Grover’s diffusion operator depends on the state produced by the oracle, constructing and applying the Grover’s diffusion operator inherently requires invoking the oracle repeatedly, otherwise, the diffusion operator has nothing to do with the marked state to be amplified. This seems to be inconsistent with one oracle call.

The diffusion operators in this paper are not Grover’s diffusion operators except D₀. This role of invoking the oracle is implemented inexplicitly:

The Grover diffusion operation, D₀ in Equation (25), is indeed dependent on the state produced by the oracle through Equation (30);The next diffusion operator, D₁ in Equation (26), depend on |ψ₁⟩, which depends on oracle through Equation (31) indirectly;The next diffusion operator, D₂ in Equation (27), depend on |ψ₂⟩, which in turn depend on oracle through Equation (32) indirectly;…

The operators (D_k) are not independent of the unknown marked state. They depend on oracle-derived states. The dependence is obtained in Equations (26)-(28) as follows:

while still claiming a single oracle call. Another way to understand this is Equation (55), which indicates that each step depends on oracle operator.

THEOREM 5.1 The geometric progression of rotation angles at iteration, k, is:

Proof. To connect the approximate state updates to the claimed geometric progression of rotation angles, recall in Equation (8):

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

From Equation (33),

Note that the first term above contains ( 1 / N ) | α 〉 , therefore,

Similarly,