**4. QFT permutations**

Universal computation, by its very nature, must involve some set of permutation operators [17–20]. As with other universal gates applied in quantum computation, in this section, we show that the QFT can generate operators that have the properties of a permutation. Consider a successive application of the QFT such as *<sup>Q</sup>*<sup>2</sup> <sup>¼</sup> *QQ* and let us analyze the matrix elements of such an operation:

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

$$\begin{aligned} \left[\left[QQ\right]\_{j,k} &= \frac{1}{N} \sum\_{m=0}^{N-1} \left(e^{\frac{2\pi im}{N}} |j\rangle\langle m|\right) \left(e^{\frac{2\pi mk}{N}} |m\rangle\langle k|\right) \\ &= \frac{1}{N} \sum\_{m=0}^{N-1} e^{\frac{2\pi im}{N}(j+k)} \left\langle m|m\rangle |j\rangle\langle k| \\ &= \begin{cases} 0 & j+k \neq 0 \mod N \\\\ 1 & j+k = 0 \mod N \end{cases} \end{aligned} \tag{46}$$
 
$$\equiv \left[P\_{Q^2}\right]\_{j,k}.$$

along with its square root

*Quantum Fourier Operators and Their Application DOI: http://dx.doi.org/10.5772/intechopen.94902*

it follows that

where *Q* is a QFT matrix.

*PQ*<sup>2</sup>

b. *<sup>Q</sup><sup>k</sup>* <sup>¼</sup> *PQ*<sup>2</sup> if *<sup>k</sup>* <sup>¼</sup> 2 mod 4.

c. *<sup>Q</sup><sup>k</sup>* <sup>¼</sup> *<sup>Q</sup>*�<sup>1</sup> if *<sup>k</sup>* <sup>¼</sup> 3 mod 4.

d. *<sup>Q</sup><sup>k</sup>* <sup>¼</sup> *<sup>I</sup>* if *<sup>k</sup>* <sup>¼</sup> 0 mod 4.

plexity of QFT implementations.

within the context of universal computation.

**5. Conclusions**

**Acknowledgements**

#1560214.

**25**

<sup>p</sup> if *<sup>k</sup>* <sup>¼</sup> 1 mod 4.

of the following outcomes

a. *<sup>Q</sup><sup>k</sup>* <sup>¼</sup> ffiffiffiffiffiffiffi

ffiffiffiffiffi *Sw* <sup>p</sup> <sup>¼</sup>

*PQ*<sup>2</sup> h i 0 1 2

0 1 2

**Theorem 1** *Given the N* � *N inversion permutation matrix defined as*

*Q* ¼

�

In a similar manner, Eq. (46) leads us to the following

In addition, given that *<sup>Q</sup>*<sup>4</sup> <sup>¼</sup> *<sup>I</sup>* we have the following

10 00

00 01

*<sup>j</sup>*,*<sup>k</sup>* <sup>¼</sup> <sup>0</sup> *<sup>j</sup>* <sup>þ</sup> *<sup>k</sup>* 6¼ 0 mod *<sup>N</sup>* 1 *j* þ *k* ¼ 0 mod *N*

> ffiffiffiffiffiffiffi *PQ*<sup>2</sup> q

**Corollary 1** Any algorithm that iteratively applies the QFT can result in only one

These results indicate a deeper connection between universal computation, per-

In this work, we have revisited the quantum Fourier transform which is central

This research is funded by a grant from the National Science Foundation NSF

mutations and the QFT. Furthermore, decomposing the QFT calculation into a product of permutations indicates a potential for reducing the computational com-

to many algorithms applied in the field of quantum computation. As a natural extension of the discrete Fourier transform, the QFT can be implemented using efficient tensor products of quantum operators. Part of the thrust of current research deals with reducing the QFT computational complexity. With this goal in mind, we have phrased the QFT as a permutation operator. Future research will be directed toward quantum circuit implementation using QFT permutation operators

ð Þ 1 þ *i* 0

*:* (50)

, (51)

(52)

ð Þ 1 þ *i* 0

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>i</sup>* <sup>1</sup> 2

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>i</sup>* <sup>1</sup> 2

For an *<sup>n</sup>* qubit system <sup>∣</sup>*qn*�<sup>1</sup>⋯*q*1*q*0i, it should be clear that *PQ*<sup>2</sup> is a permutation operator that leaves the position of ∣*q*0i unchanged and inverts the order of the remaining qubits to form <sup>∣</sup>*q*1⋯*qn*�<sup>1</sup>*q*0i. For example, the CNOT operator in Eq. (22) is equal to *PQ*<sup>2</sup> for *n* ¼ 2

$$\mathbf{CNOT} = \mathbf{Q}^2 = \begin{bmatrix} \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} \\ \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \end{bmatrix} = P\_{Q^2} \tag{47}$$

having properties similar to that of a Sylvester shift matrix (i.e. a generalization of a Pauli matrix). It is sensible that a CNOT operation followed by a CNOT operation should result in the identity operation and, hence, that *PQ*2*PQ*<sup>2</sup> <sup>¼</sup> *<sup>Q</sup>*<sup>4</sup> <sup>¼</sup> *<sup>I</sup>* (i.e. a double inversion recovers the original qubit sequence). These results can be generalized for any *n*. For example, with *n* ¼ 3, Eq. (46) becomes

$$P\_{Q^2} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix} \tag{48}$$

which, after the appropriate sequence of swaps, can be transformed into a Toffoli (CCNOT) gate. Hence, *PQ*<sup>2</sup> can be thought of as a generalization of swap permutation operators and the QFT can be phrased as its square root. For example, it is common to define a two qubit swap operator as

$$\mathbf{S}\_w = \begin{bmatrix} \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} \end{bmatrix} \tag{49}$$

*Quantum Fourier Operators and Their Application DOI: http://dx.doi.org/10.5772/intechopen.94902*

along with its square root

½ � *QQ <sup>j</sup>*,*<sup>k</sup>* <sup>¼</sup> <sup>1</sup>

*N* X *N*�1

¼ 1 *N* X *N*�1

> 8 < :

� *PQ*<sup>2</sup> h i

*CNOT* <sup>¼</sup> *<sup>Q</sup>*<sup>2</sup> <sup>¼</sup>

*j*,*k :*

of a Pauli matrix). It is sensible that a CNOT operation followed by a CNOT operation should result in the identity operation and, hence, that *PQ*2*PQ*<sup>2</sup> <sup>¼</sup> *<sup>Q</sup>*<sup>4</sup> <sup>¼</sup> *<sup>I</sup>* (i.e. a double inversion recovers the original qubit sequence). These results can be

generalized for any *n*. For example, with *n* ¼ 3, Eq. (46) becomes

*PQ*<sup>2</sup> ¼

it is common to define a two qubit swap operator as

**24**

*Sw* ¼

¼

is equal to *PQ*<sup>2</sup> for *n* ¼ 2

*m*¼0 *e i* 2*π <sup>N</sup>jm*∣*j*ih*m*<sup>∣</sup> � �

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

*m*¼0 *e i* 2*π*

*e i* 2*π <sup>N</sup>mk*∣*m*ih*k*<sup>∣</sup> � �

(46)

(48)

(49)

*<sup>N</sup>m j* ð Þ <sup>þ</sup>*<sup>k</sup>* h i *<sup>m</sup>*j*<sup>m</sup>* <sup>∣</sup>*j*ih*k*<sup>∣</sup>

0 *j* þ *k* 6¼ 0 mod*N*

1 *j* þ *k* ¼ 0 mod*N*

For an *<sup>n</sup>* qubit system <sup>∣</sup>*qn*�<sup>1</sup>⋯*q*1*q*0i, it should be clear that *PQ*<sup>2</sup> is a permutation operator that leaves the position of ∣*q*0i unchanged and inverts the order of the remaining qubits to form <sup>∣</sup>*q*1⋯*qn*�<sup>1</sup>*q*0i. For example, the CNOT operator in Eq. (22)

having properties similar to that of a Sylvester shift matrix (i.e. a generalization

which, after the appropriate sequence of swaps, can be transformed into a Toffoli (CCNOT) gate. Hence, *PQ*<sup>2</sup> can be thought of as a generalization of swap permutation operators and the QFT can be phrased as its square root. For example,

¼ *PQ*<sup>2</sup> (47)

$$
\sqrt{\mathbb{S}\_w} = \begin{bmatrix}
\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \frac{1}{2}(\mathbf{1} + i) & \frac{1}{2}(\mathbf{1} + i) & \mathbf{0} \\
\mathbf{0} & \frac{1}{2}(\mathbf{1} + i) & \frac{1}{2}(\mathbf{1} + i) & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1}
\end{bmatrix}.\tag{50}
$$

In a similar manner, Eq. (46) leads us to the following **Theorem 1** *Given the N* � *N inversion permutation matrix defined as*

$$
\begin{bmatrix} P\_{Q^2} \end{bmatrix}\_{j,k} = \begin{cases} \mathbf{0} & j+k \neq \mathbf{0} \mod N \\ \mathbf{1} & j+k = \mathbf{0} \mod N \end{cases} \tag{51}
$$

it follows that

$$Q = \sqrt{P\_{Q^2}}\tag{52}$$

where *Q* is a QFT matrix.

In addition, given that *<sup>Q</sup>*<sup>4</sup> <sup>¼</sup> *<sup>I</sup>* we have the following

**Corollary 1** Any algorithm that iteratively applies the QFT can result in only one of the following outcomes

$$\text{a. } Q^k = \sqrt{P\_Q^\cdot} \text{ if } k = 1 \mod 4.$$

$$\text{b. } Q^k = P\_{Q^2} \text{ if } k = 2 \mod 4.$$

$$\text{c. } Q^k = Q^{-1} \text{ if } k = 3 \mod 4.$$

$$\text{d. } Q^k = I \text{ if } k = 0 \mod 4.$$

These results indicate a deeper connection between universal computation, permutations and the QFT. Furthermore, decomposing the QFT calculation into a product of permutations indicates a potential for reducing the computational complexity of QFT implementations.

## **5. Conclusions**

In this work, we have revisited the quantum Fourier transform which is central to many algorithms applied in the field of quantum computation. As a natural extension of the discrete Fourier transform, the QFT can be implemented using efficient tensor products of quantum operators. Part of the thrust of current research deals with reducing the QFT computational complexity. With this goal in mind, we have phrased the QFT as a permutation operator. Future research will be directed toward quantum circuit implementation using QFT permutation operators within the context of universal computation.

### **Acknowledgements**

This research is funded by a grant from the National Science Foundation NSF #1560214.

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

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