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Quantum Fourier Transform and
Eigenvalue Estimation
- October 24, Antonije Mirkovic
- 1 Introduction Contents
- 2 Derivation and Explanation of the Quantum Fourier Transform
- 2.1 Setup, notation, and the target mapping
- 2.2 From the double sum to a product over output-qubit bits
- 2.3 Integer vs. fractional parts of j/ 2 n − m
- 2.4 Closed form of each single-qubit factor
- 2.5 Preparing the factors by gates: from H and controlled rotations to the full QFT
- 2.6 Explicit 3 -qubit example (all steps)
- 2.7 QFT circuit diagrams
- 2.8 Complexity accounting (gate counts)
- 2.9 Inverse QFT (for completeness)
- 3 Eigenvalue Estimation Using the Quantum Fourier Transform
- 3.1 The Eigenvalue Estimation Problem
- 3.2 The Quantum Algorithm for Eigenvalue Estimation
- 3.3 Phase Extraction via the Inverse QFT
- 3.4 Discussion and Implications
1 Introduction
The Fourier Transform is one of the most powerful mathematical tools in science and engineering, providing a way to represent data in an alternative domain. In classical computing, the Discrete Fourier Transform (DFT) converts a set of N complex numbers in the time domain into another set in the frequency domain , revealing periodicities or frequency components hidden within the original data. For example, a sampled sound wave represented as amplitude versus time, can be expressed as a superposition of pure frequency components, such as the musical notes C 4 , E 4 , and G 4. However, the classical computation of the DFT is expensive. The direct method requires O( N^2 ) additions and multiplications, while the Fast Fourier Transform (FFT) reduces this to O( N log N ) operations, which is still exponential in the number of bits n = log 2 N. The Quantum Fourier Transform (QFT) provides a fundamentally different, quantum- mechanical approach to this problem. It is the quantum analogue of the DFT and can be implemented as a unitary operation on an n -qubit quantum register. Mathematically, the QFT performs the same linear transformation as the classical Fourier Transform, but it operates on quantum amplitudes instead of classical data. Because it is realized by a sequence of quantum gates, specifically Hadamard and controlled phase-rotation gates, it can be executed with a complexity of only O( n^2 ) operations, corresponding to O((log N )^2 ) in terms of N. This exponential improvement in scaling is not typically used for direct Fourier analysis but plays a crucial role as a subroutine in several quantum algorithms that exhibit exponential speedups. These include period finding , phase estimation , and notably, order finding in Shor’s factoring algorithm. Hence, the Quantum Fourier Transform serves as one of the most central building blocks in quantum computing.
2 Derivation and Explanation of the Quantum Fourier Trans-
form
2.1 Setup, notation, and the target mapping
Let n ∈ N and N := 2 n. We use the n -qubit computational basis {| j ⟩} N j =0^ −^1 with the binary expansion
j =
n ∑− 1
r =
jr 2 r, jr ∈ { 0 , 1 }. (1)
The N -point Quantum Fourier Transform (QFT) is the unitary QF TN defined on basis states by
QF TN | j ⟩ =
N
N ∑ − 1
k =
exp
( (^2) πi jk N
) | k ⟩ =
2 n
(^2) ∑ n − 1
k =
exp
( (^2) πi jk 2 n
) | k ⟩. (2)
Our goal is to derive step by step a factorized product form of QF T 2 n^ | j ⟩ that makes a direct quantum circuit construction evident.
Binary-point notation. For bits b 1 , b 2 ,... , bm ∈ { 0 , 1 } define
0 .b 1 b 2 · · · bm :=
∑^ m
ℓ =
bℓ 2 ℓ^
2.2 From the double sum to a product over output-qubit bits
Write k in binary as k =
∑ n − 1 m =0 km^^2 m (^) with km ∈ { 0 , 1 }. Then
jk 2 n^
j 2 n
n ∑− 1
m =
km 2 m^ =
n ∑− 1
m =
km j 2 n − m^
2.5 Preparing the factors by gates: from H and controlled rotations to the
full QFT
Each factor in (14) is of the form
1 √ 2
( | 0 ⟩ + eiθ^ | 1 ⟩
) , θ = 2 π · 0_. j_ (^) n − m − 1 · · · j 0_._ (15)
To create (15) from | 0 ⟩:
- Apply a Hadamard gate H : | 0 ⟩ 7 → (| 0 ⟩ + | 1 ⟩) /
- Apply a phase Rz ( θ ) to the | 1 ⟩ component. In circuit language we use controlled phase rotations Rk :=
( 1 0 0 e^2 πi/^2 k
) , CRk = | 0 ⟩⟨ 0 | ⊗ I + | 1 ⟩⟨ 1 | ⊗ Rk. (16)
The binary-fraction phase θ in (15) is built as the product of elementary phases controlled by the higher-significance input bits of j :
eiθ^ =
n ∏− m
ℓ =
exp
( 2 πi j (^) n − m − ℓ 2 ℓ
) , (17)
which is realized by applying CR 1 , CR 2 ,... , CRn − m controlled from (in order) j (^) n − m − 1 , j (^) n − m − 2 ,... , j 0 onto the current target qubit that just received H.
2.6 Explicit 3 -qubit example (all steps)
Let n = 3 and N = 8. Write j = j 2 j 1 j 0. Then, from (14),
QF T 8 | j 2 j 1 j 0 ⟩ =
23 /^2
( | 0 ⟩ + e^2 πi ·^0 .j^2 j^1 j^0 | 1 ⟩
) ⊗
( | 0 ⟩ + e^2 πi ·^0 .j^1 j^0 | 1 ⟩
) ⊗
( | 0 ⟩ + e^2 πi ·^0 .j^0 | 1 ⟩
) . (18)
Gate-by-gate:
- On the top input qubit ( j 2 ): apply H , then controlled rotations from j 1 and j 0 :
H on j 2 → CR 2 (control j 1 ) → CR 3 (control j 0 ).
This prepares √^12 (| 0 ⟩ + e^2 πi ·^0 .j^2 j^1 j^0 | 1 ⟩) on the first output factor.
- On the middle input qubit ( j 1 ): apply H , then CR 2 controlled by j 0 , yielding √^12 (| 0 ⟩ + e^2 πi ·^0 .j^1 j^0 | 1 ⟩).
- On the bottom input qubit ( j 0 ): apply H only, yielding √^12 (| 0 ⟩ + e^2 πi ·^0 .j^0 | 1 ⟩).
Finally, apply SWAP between the top and bottom wires to correct bit order (bit-reversal).
2.7 QFT circuit diagrams
Below are QFT circuits illustrating the characteristic ladder structure of the Quantum Fourier Transform, where each qubit undergoes a Hadamard gate followed by a sequence of descending controlled- Rk rotations. The circuits conclude with output SWAP operations that correct the bit order so that the most significant qubit appears at the top.
Three-qubit QFT.
| j 2 ⟩ (^) H
( | 0 ⟩ + e^2 πi^^0 .j^0 | 1 ⟩
)
| j 1 ⟩ R 2 H
( | 0 ⟩ + e^2 πi^^0 .j^1 j^0 | 1 ⟩
)
| j 0 ⟩ (^) R 3 R 2 H
√^1
( | 0 ⟩ + e^2 πi^^0 .j^2 j^1 j^0 | 1 ⟩
)
Here Rk = diag (1 , e^2 πi/^2 k ). The final SWAP exchanges the first and third wires (bit-reversal).
2.8 Complexity accounting (gate counts)
From the construction above:
- Hadamards: exactly n (one per qubit).
- Controlled rotations: ( n − 1) + ( n − 2) + · · · + 1 = n ( n 2 − 1).
- SWAPs for bit-reversal: n/ 2 (if the output needs MSB on top).
Thus the exact QFT uses a total of
n + n ( n − 1) 2
n ( n + 1) 2
one- and two-qubit gates, plus n/ 2 SWAPs, i.e. overall O ( n^2 ) gate complexity.
2.9 Inverse QFT (for completeness)
The inverse transform QF T (^) 2 † n is implemented by:
- Reversing the order of operations in the QFT circuit.
- Replacing each Rk by R † k = diag(1 , e −^2 πi/^2 k ).
- Reversing the SWAP ladder to undo the bit-reversal in the opposite order.
3 Eigenvalue Estimation Using the Quantum Fourier Transform
3.1 The Eigenvalue Estimation Problem
In many quantum algorithms, we encounter a unitary operator U whose eigenvalues encode quantities of physical or computational interest. Let | u ⟩ be an eigenvector of U with corresponding eigenvalue e^2 πiφ , where φ ∈ [0 , 1) is a real number interpreted as a phase :
U | u ⟩ = e^2 πiφ^ | u ⟩. (20)
Our goal is to estimate φ to m bits of precision using quantum resources. This task is known as the quantum eigenvalue estimation problem (or equivalently, phase estimation problem ).
Motivation. Eigenvalue estimation plays a central role in several quantum algorithms. For example, in quantum chemistry and Hamiltonian simulation, U = e − iHt^ where H is the Hamiltonian of a system; its eigenphases yield the energy spectrum. In Shor’s algorithm, the same procedure is used to estimate periodicities and multiplicative orders. Hence, eigenvalue estimation acts as a fundamental subroutine for extracting spectral information efficiently.
Summary of the Objective Given access to the state (24), our goal in the next sections is to apply the inverse QFT to the control register to extract an m -bit binary estimate of φ :
QF T (^) 2 − m^1
( 1 2 m/^2
(^2) ∑ m − 1
k =
e^2 πikφ^ | k ⟩
) ≈ | φ ˜⟩ ,
where φ ˜ represents the best m -bit approximation to the true phase φ. The success probability approaches unity as m increases, and the estimation precision scales as 1 / 2 m. Thus, the eigenvalue estimation problem reduces to preparing (24) and then decoding φ via the inverse QFT, a procedure that achieves exponentially better scaling than any known classical method.
3.2 The Quantum Algorithm for Eigenvalue Estimation
We now derive step by step the complete quantum circuit and its mathematical action for estimating the eigenvalue phase φ of a unitary operator U , given that U | u ⟩ = e^2 πiφ^ | u ⟩. The algorithm combines two essential ingredients:
- controlled applications of increasing powers of U , which encode the eigenphase φ into the amplitudes of the control register;
- the inverse Quantum Fourier Transform, which extracts the binary digits of φ from those amplitudes.
Quantum Circuit Structure The eigenvalue estimation circuit operates on two registers:
- an m -qubit control register initially in | 0 ⟩⊗ m ,
- a target register containing the eigenstate | u ⟩ of U.
The procedure can be summarized schematically as follows:
...
Control qubits (^) H ⊗ m^ QF T (^) 2 − m^1
Target ( | u ⟩ ) (^) U^20 U^21 U^2 m −^1
The Hadamard layer prepares an equal superposition of all computational basis states in the control register. Then, each controlled- U^2 r operation applies the 2 r -th power of U to the target system, conditional on the r -th control qubit being | 1 ⟩. Finally, the inverse QFT converts the phase-encoded amplitudes into a binary estimate of φ.
Action of the Controlled Powers of U Starting from the superposition state in (22), we apply the controlled unitaries sequentially. Let the binary expansion of k be
k = km − 12 m −^1 + km − 22 m −^2 + · · · + k 121 + k 020 , kr ∈ { 0 , 1 }.
Then the controlled operations act as follows:
| k ⟩ | u ⟩ all controlled- U^
2 r 7 −−−−−−−−−−−−→ U
∑ m − 1 r =0 kr^^2 r | u ⟩ ⊗ | k ⟩ = e^2 πiφ^
∑ m − 1 r =0 kr^^2 r | u ⟩ | k ⟩ = e^2 πikφ^ | u ⟩ | k ⟩. (25)
Therefore, the joint system after all controlled powers is
| ψ 1 ⟩ =
2 m/^2
2 m ∑− 1
k =
e^2 πikφ^ | k ⟩ ⊗ | u ⟩. (26)
The eigenstate | u ⟩ factors out, it is untouched by the algorithm. Hence, we can drop it in what follows and focus solely on the control register:
| ψ ˜ 1 ⟩ =
2 m/^2
(^2) ∑ m − 1
k =
e^2 πikφ^ | k ⟩. (27)
Decomposition in Terms of Binary Fractions
Write the phase φ in binary as
φ = 0 .φ 1 φ 2 · · · φmφm +1 · · · =
∑^ ∞
r =
φr 2 r^
, φr ∈ { 0 , 1 }. (28)
Then e^2 πikφ^ = e^2 πik^
∑∞ r =1 φr^ /^2 r =
∏^ ∞
r =
e^2 πikφr^ /^2
r
. (29)
Although the infinite product form is exact, in practice we can approximate φ up to m bits of precision, since the m -qubit register can only resolve 2 − m^ intervals. The amplitude structure of | ψ ˜ 1 ⟩ thus contains discrete samples of a complex exponential with frequency φ on the integers k = 0 ,... , 2 m^ − 1. This is precisely the input form expected by the inverse Quantum Fourier Transform.
Preparation for the Inverse QFT The key observation is that the QFT performs the transformation
QF T 2 m^ : | k ⟩ 7 →
2 m/^2
(^2) ∑ m − 1
y =
e^2 πiky/^2 m | y ⟩.
Thus, the inverse QFT acts as
QF T (^) 2 − m^1 : | y ⟩ 7 →
2 m/^2
(^2) ∑ m − 1
k =
e −^2 πiky/^2 m | k ⟩.
When QF T (^) 2 − m^1 is applied to | ψ ˜ 1 ⟩ in (27), the amplitude of outcome | y ⟩ becomes
⟨ y | QF T (^) 2 − m^1 | ψ ˜ 1 ⟩ =
2 m
(^2) ∑ m − 1
k =
e^2 πikφe −^2 πiky/^2
m
2 m
(^2) ∑ m − 1
k =
e^2 πik ( φ − y/^2 m )
. (30)
This is a finite geometric series that can be summed exactly.
Applying the Inverse QFT
Recall from the previous section that the control register is in the state
| ψ ˜ 1 ⟩ =
2 m/^2
(^2) ∑ m − 1
k =
e^2 πikφ^ | k ⟩. (35)
Applying QF T (^) 2 − m^1 to this register yields
QF T (^) 2 − m^1 | ψ ˜ 1 ⟩ =
2 m
(^2) ∑ m − 1
y =
(^2) ∑ m − 1
k =
e^2 πik ( φ − y/^2 m ) | y ⟩ (36)
as derived in (30). We again denote
δy := φ −
y 2 m^
so that the amplitude for the basis state | y ⟩ is
ay =
2 m
(^2) ∑ m − 1
k =
e^2 πikδy^. (37)
Summing the Geometric Series
Since e^2 πiδy^ is constant for each term in the sum, we can sum the finite geometric series explicitly:
ay =
2 m
1 − e^2 πi^2 mδy
1 − e^2 πiδy^
Multiplying numerator and denominator by e − πiδy^ and using
1 − eiθ^ = eiθ/^2 (− 2 i ) sin
( θ 2
) ,
we obtain
ay =
2 m^
eπi ( m −1) δy sin( π 2 mδy ) sin( πδy )
Thus, the probability of measuring outcome y is
P ( y ) = | ay |^2 =
22 m
( sin( π 2 mδy ) sin( πδy )
) 2
. (40)
This function is sharply peaked whenever δy is close to zero, i.e. when y ≈ 2 mφ.
Exact and Approximate Recoveries of φ
Case 1: Exact binary representation. If φ can be represented exactly with m binary digits, φ = y 0 2 m^ , y 0 ∈ { 0 , 1 ,... , 2 m^ − 1 } ,
then δy 0 = 0, and using lim x → 0 sin( π^2
mx ) sin( πx ) = 2
m , we find
P ( y 0 ) = 1 , P ( y ̸= y 0 ) = 0_._
Thus, the inverse QFT perfectly maps | ψ ˜ 1 ⟩ to the computational basis state | y 0 ⟩, giving the exact binary expansion of φ : QF T (^) 2 − m^1 | ψ ˜ 1 ⟩ = | y 0 ⟩.
Case 2: Inexact binary representation. If φ cannot be represented exactly with m bits, then δy ̸= 0 for all integers y , and the probability distribution (40) has a main peak centered at y 0 = ⌊ 2 mφ ⌋. The peak width is approximately 1 , so the algorithm still returns an estimate y 0 such that (^) ∣ ∣∣ y 0 2 m^ − φ
∣∣ ∣ <
2 m +^
with probability exceeding 4 /π^2 ≈ 0_._ 405. Hence, measuring the control register yields a high- accuracy approximation of the phase even when it cannot be represented exactly in m bits.
Interpretation of the Measurement Outcome
The measurement result y 0 corresponds to the binary fraction
0_. y_ 0 , 1 y 0 , 2 · · · y 0 ,m = y 0 2 m^
≈ φ.
Thus, the eigenvalue e^2 πiφ^ of U can be reconstructed to m -bit accuracy directly from the measured control register. The quantum circuit therefore implements the mapping
1 2 m/^2
(^2) ∑ m − 1
k =
e^2 πikφ^ | k ⟩ QF T^
− 1 −−−−−→ | φ ˜⟩ ,
where φ ˜ is the best m -bit approximation to φ. In the special case where φ = y 0 / 2 m , the output is deterministic, | y 0 ⟩.
Summary
The inverse QFT acts as the decoder that translates the phase-encoded amplitudes of the control register into the classical binary representation of the eigenphase φ. This conversion from global phase information to a measurable binary output is the key step that allows the eigenvalue estimation algorithm to achieve exponential precision in m using only O( m^2 ) quantum gates.
3.4 Discussion and Implications
The eigenvalue estimation algorithm, as derived above, demonstrates one of the most profound capabilities of quantum computation: the ability to estimate eigenphases of unitary operators with exponentially greater precision than is achievable classically, while using only a polynomial number of gates.
Algorithmic Efficiency
Let us summarize the resource requirements:
- The circuit uses m control qubits and as many controlled- U^2 r gates for r = 0 , 1 ,... , m − 1.
- The cost of implementing the inverse QFT is O( m^2 ) gates, dominated by controlled phase rotations.
- The total number of elementary operations therefore scales polynomially in m , i.e. Total cost = O( m^2 + CU ) ,
where CU is the cost of implementing U and its powers.
If U can be applied efficiently (for instance, via Hamiltonian simulation or modular exponentia- tion), then the entire procedure provides an exponential improvement over classical eigenvalue estimation, which generally scales as O( N^3 ) for an N × N operator.
