Quantum Fourier transform in computational basis

Abstract

The quantum Fourier transform, with exponential speed-up compared to the classical fast Fourier transform, has played an important role in quantum computation as a vital part of many quantum algorithms (most prominently, Shor’s factoring algorithm). However, situations arise where it is not sufficient to encode the Fourier coefficients within the quantum amplitudes, for example in the implementation of control operations that depend on Fourier coefficients. In this paper, we detail a new quantum scheme to encode Fourier coefficients in the computational basis, with fidelity \(1 - \delta \) and digit accuracy \(\epsilon \) for each Fourier coefficient. Its time complexity depends polynomially on \(\log (N)\), where N is the problem size, and linearly on \(1/\delta \) and \(1/\epsilon \). We also discuss an application of potential practical importance, namely the simulation of circulant Hamiltonians.

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Notes

  1. 1.

    \(\left| y_k - {\tilde{y}}_k \right| < \epsilon \), where \({\tilde{y}}_k\) is the truncated value of \(y_k\) with accuracy \(\epsilon = 2^{-p_0}\).

  2. 2.

    \(\left| \left\langle {\Psi ^{\text {final}}}\right| \left( \frac{1}{\sqrt{N}} \sum _{k=0}^{N-1} \left| {k}\right\rangle \left| {{\tilde{y}}_k}\right\rangle \right) \right| \ge 1 - \delta \), where \(\left| {\Psi ^{\text {final}}}\right\rangle \) is the state obtained through the QFTC algorithm.

  3. 3.

    \(\Vert \hbox {e}^{-iCt} - \widetilde{\hbox {e}^{-iCt}}\Vert \le \delta \), where \(\widetilde{\hbox {e}^{-iCt}}\) represents the operator that is actually performed by this algorithm.

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Acknowledgements

The authors would like to thank Ashley Montanaro for constructive comments and Jeremy O’Brien, Jonathan Matthews, Xiaogang Qiang, Lyle Noakes, Chuheng Zhang and Hanwen Zha for helpful discussions.

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Correspondence to J. B. Wang.

Appendices

Appendix 1: Quantum arithmetic

Addition and multiplication are basic elements of arithmetic in classical computer. There have been several proposals on how to build quantum adders and multipliers [36,37,38,39], constructed predominately using CNOT gates and Toffoli gates. Draper’s addition quantum circuits, however, utilize the quantum Fourier transformation (QFT) [40]. QFT-based multiplication and related quantum arithmetic have also been proposed [41,42,43,44]. In this appendix, for completeness, we outline the construction of the quantum arithmetic gates required for the QFTC algorithm in detail.

We show here, using QFT-based circuits and fixed-point number representation, all elementary quantum arithmetic gates used to construct the QFTC circuit (including adders, multipliers and cosine gates) have \({\mathcal {O}}(\mathrm {poly}(n))\) complexity, where n is the number of qubits (number of digits) representing the number. With accuracy \(\epsilon \), this results in \({\mathcal {O}}(\mathrm {polylog}(1/\epsilon ))\) complexity.

QFT multiply–adder

We begin by describing a quantum multiply–adder for real inputs a and b between 0 and 1. Let \(\left| {a}\right\rangle = \left| {a_1}\right\rangle \left| {a_2}\right\rangle \cdots \left| {a_m}\right\rangle \) represent the fixed-point number \(a = 0.a_1 a_2 \cdots a_m\) (same for b). Using this representation, the quantum multiply–adder (QMA), as shown in Fig. 5a, can realize the following transformation,

$$\begin{aligned} \Pi ^\pm _{m,n}\left| {a}\right\rangle \left| {b}\right\rangle \left| {c}\right\rangle = \left| {a}\right\rangle \left| {b}\right\rangle \left| {c \pm a\times b}\right\rangle , \end{aligned}$$
(33)

where m and n denote the number of digits of a and b, respectively.

In quantum multiply–adders, the outputs, unlike the inputs, can be negative and we use the complemental code \(c^{(C)} = c_0.c_1 c_2 \cdots c_{m+n} \in [0,2)\) to represent the output \(c \in (-1,1)\) and \(c = c^{(C)}\) if c is non-negative and \(c = c^{(C)}-2\) if c is negative. \(\left| {c}\right\rangle \) is composed of \(\left| {c_0}\right\rangle \left| {c_1}\right\rangle \cdots \left| {c_{m+n}}\right\rangle \). Note that this quantum multiply–adder also applies to any fixed-point-represented numbers by cleverly choosing the appropriate positions of the fractional points.

Fig. 5
figure5

Quantum circuit of the multiply–adder, a quantum multiply–adder, b intermediate multiply–adder

The quantum multiply–adder can be decomposed into the following form, as shown in Fig. 5b:

$$\begin{aligned} \Pi ^\pm _{m,n} = ({\mathbb {I}}\otimes {\mathbb {I}}\otimes \mathrm {QFT}^\dagger )\times \mathfrak {\pi }^\pm _{m,n} \times ({\mathbb {I}}\otimes {\mathbb {I}}\otimes \mathrm {QFT}), \end{aligned}$$
(34)

where \(\mathfrak {\pi }^\pm _{m,n}\) represents an intermediate quantum multiply–adder,

$$\begin{aligned} \pi ^\pm _{m,n}\left| {a}\right\rangle \left| {b}\right\rangle \left| {\phi (c)}\right\rangle =\left| {a}\right\rangle \left| {b}\right\rangle \left| {\phi (c\pm a\times b)}\right\rangle \end{aligned}$$
(35)

with \(\left| {\phi (c)}\right\rangle :=\mathrm {QFT}\left| {c}\right\rangle \) and \(\left| {\phi _{k}(c)}\right\rangle = \frac{1}{\sqrt{2}}(\left| {0}\right\rangle +\hbox {e}^{2\pi i c\times 2^{m+n-k}}\left| {1}\right\rangle )\), \(k=1,2,\cdots ,m+n+1\).

Figure 6 shows a detailed quantum circuit construction of \(\mathfrak {\pi }^\pm _{m,n}\), using the QFT adders \(2^{-l} \Sigma ^\pm _{m,n}\), which act as follows:

$$\begin{aligned} \boxed {2^{-l} \Sigma ^\pm _{m,n}}\left| {b}\right\rangle \left| {\phi (c)}\right\rangle = \left| {b}\right\rangle \left| {\phi (c\pm 2^{-l} b)}\right\rangle . \end{aligned}$$
(36)

The QFT adders are constructed via controlled phase operations, as shown in Fig. 6c.

Fig. 6
figure6

Quantum circuit of \(\mathfrak {\pi }^\pm _{m,n}\), (a), \(\mathfrak {\pi }^\pm _{m,n}\) gate (b), QFT adder, (c) detailed quantum circuit construction of the QFT adder \(2^{-l} \Sigma ^\pm _{m,n}\), \(R^\pm _k = \left| {0}\right\rangle \left\langle {0}\right| +\hbox {e}^{\pm 2\pi i/2^{k}}\left| {1}\right\rangle \left\langle {1}\right| \)

After applying the QFT adder \(2^{-m} \Sigma ^\pm _{m,n}\) (controlled by \(\left| {a_m}\right\rangle \)) in Fig. 6, we obtain

$$\begin{aligned} \left| {\phi (c)}\right\rangle \longrightarrow \left| {\phi (c \pm a_m 2^{-m} b)}\right\rangle . \end{aligned}$$
(37)

Proceeding in a similar fashion, it can be seen that the final output state of the intermediate multiply–adder is

$$\begin{aligned} \left| {\phi (c + a_m 2^{-m} b +\cdots + a_1 2^{-1}b)}\right\rangle = \left| {\phi (c \pm a \times b)}\right\rangle . \end{aligned}$$
(38)

To illustrate how the circuit works, take for example the evolution of \(\phi _{m+n-l}(c)\) after \(R^\pm _1,\ldots ,R^\pm _n\):

$$\begin{aligned} \left| {0}\right\rangle + \hbox {e}^{2\pi i c \times 2^{l}}\left| {1}\right\rangle \longrightarrow \left| {0}\right\rangle + \hbox {e}^{2\pi i c \times 2^{l} \pm b }\left| {1}\right\rangle . \end{aligned}$$
(39)

We then have

$$\begin{aligned} \left| {\phi _{k}(c)}\right\rangle \rightarrow \left| {\phi _{k}(c\pm 2^{-l}b)}\right\rangle . \end{aligned}$$

It is clear from Fig. 6c that the QFT adder uses \({\mathcal {O}}\big ((m+n)n\big )\) one- or two-qubit gates. Hence, the total complexity of the intermediate QFT multiply–adders is \({\mathcal {O}}\big ((m+n)mn\big )\). Thus, with QFT scaling \({\mathcal {O}}\big ((m+n)^2\big )\), the total complexity of the quantum multiply–adder \(\Pi ^\pm _{m,n}\) is \(\max \{{\mathcal {O}}(mn^2),{\mathcal {O}}(nm^2)\}\).

Note that if we choose \(l=0\) in \(2^{-l} \Sigma ^\pm _{m,n}\) and perform a QFT and an inverse QFT before and after the application of the QFT adder in Eq. 36, we have a quantum adder

$$\begin{aligned} \left| {b}\right\rangle \left| {c}\right\rangle \rightarrow \left| {b}\right\rangle \left| {c\pm b}\right\rangle . \end{aligned}$$
(40)

We can also add (or subtract) two numbers without having to destroy their original values encoded in the computational basis, i.e.

$$\begin{aligned} \left| {b}\right\rangle \left| {c}\right\rangle \left| {0}\right\rangle \rightarrow \left| {b}\right\rangle \left| {c}\right\rangle \left| {b}\right\rangle \rightarrow \left| {b}\right\rangle \left| {c}\right\rangle \left| {b \pm c}\right\rangle \end{aligned}$$
(41)

by using Eq. 40 twice.

Quantum sine and cosine gate

By implementing the Taylor series using the quantum multiply–adder, we are able to build a quantum sine (and cosine) gate. Suppose \(x = 0. x_1 x_2 \cdots x_n\) and \(x \in [0,1)\). We aim to build a sine gate calculating the value of \(\sin \pi x\), performing \(\left| {x}\right\rangle \left| {0^n}\right\rangle \left| {0^m}\right\rangle \rightarrow \left| {x}\right\rangle \left| {\sin \pi x}\right\rangle \left| {\Psi ^{\text {ancilla}}}\right\rangle \).

Fig. 7
figure7

Quantum circuits of the sine and cosine gates (\(\left| {0}\right\rangle \) represents a number of qubits in above circuits where the numbers are omitted). Pauli-X gates are used to transform \(\left| {0}\right\rangle \) into \(\left| {x}\right\rangle \), and the subscript for all the quantum multiply–adders in above circuits is \((p',p')\), a sine gate, b cosine gate

We now consider the error in the truncated Taylor series. First, the error introduced by imprecision in the n-digit representation of x is \({\mathcal {O}}(2^{-n})\), since the derivative of \(\sin \pi x\) is bounded. The Taylor series of \(\sin \pi x\) at around \(x=0\) is

$$\begin{aligned} \sin \pi x = \pi x - \frac{(\pi x)^3}{3!} + \frac{(\pi x)^5}{5!} - \cdots + (-1)^{t}\frac{(\pi x)^{2t+1}}{(2t+1)!} + \frac{(-1)^{t+1}\cos \pi z}{(2t+3)!}(\pi x)^{(2t+3)}. \end{aligned}$$
(42)

The remainder term for the kth term in the expansion is \(\frac{f^{(k+1)}(z)}{(k+1)!}x^{k+1}\), where \(z\in (0, x)\), according to Taylor’s Theorem [45]. As a result, in Eq. (42), the reminder term (error) is \(\frac{(-1)^{t+1}\cos \pi z}{(2t+3)!}(\pi x)^{(2t+3)}\) and is obviously bounded by \({\mathcal {O}}(2^{-n})\) for \(t=n\).

In the sine gate, the \(t+1\) terms \(\left\{ \pi x,\frac{(\pi x)^3}{3!},\cdots ,(-1)^{t}\frac{(\pi x)^{2t+1}}{(2t+1)!}\right\} \) are first calculated and then added (or subtracted) together. Suppose each of the \(t+1\) terms has an error within \(2^{-p}\). Taking \(p = n + \lceil \log n\rceil = {\mathcal {O}}(n)\), the error introduced by adding and subtracting will be \({\mathcal {O}}(t\times 2^{-p}) = {\mathcal {O}}(2^{-n})\). Suppose all multiply–adders have \(p'\) digits inputs. When errors in \(y_1,y_2\) are within \(2^{-(\ell +1)}\) and \(y_1,y_2\le 1-2^{-(\ell +1)}\), \((y_1+2^{-(\ell +1)})(y_2+2^{-(\ell +1)}) = y_1y_2 + 2^{-\ell }(y_1+y_2)/2 + 2^{-2\ell -2} \le y_1y_2 + 2^{-\ell }.\) It means that by applying the multiply–adders 2t times, the error will be \(2^{2t}\) times larger. Thus, we can choose a \(p'={\mathcal {O}}(p+2t)={\mathcal {O}}(n)\) which guarantees accuracy \(2^{-p}\) in all the powers of x and also all the \(t+1\) terms in the Taylor series.

We conclude that we can choose \(t = {\mathcal {O}}(n)\) and \(p' = {\mathcal {O}}(n)\) so that the total accuracy of the sine gate is bounded by \(2^{-n}\). Figure 7 shows the quantum circuit for the sine and cosine gate. The complexity of the quantum sine gate can be calculated based on the scaling of quantum multiply–adders which equals to \({\mathcal {O}}(p'^3)\). The total complexity of the quantum sine gate is \({\mathcal {O}}(t p'^3) = {\mathcal {O}}(n^4)\) for accuracy \(2^{-n}\). To put it in another way, \({\mathcal {O}}(\text {polylog}(1/\epsilon ))\) one- or two-qubit gates are required to achieve accuracy \(\epsilon \).

Appendix 2: Implementing circulant operators

Fig. 8
figure8

Implementation of circulant matrices. Here \(R\left| {k}\right\rangle \left| {j}\right\rangle = \hbox {e}^{2\pi ikj/N}\left| {k}\right\rangle \left| {j}\right\rangle \)

Consider an arbitrary state \(\left| {s}\right\rangle \). We wish to obtain \(C\left| {s}\right\rangle \), where C is an arbitrary circulant matrix. Below, we present a possible algorithm for implementing a circulant matrix quantum operator (see Fig. 8).

Step 1 :

Perform the inverse QFT on \(\left| {s}\right\rangle \):

$$\begin{aligned} \sum _{k=0}^{N-1} s_k \left| {k}\right\rangle \rightarrow \sum _{k=0}^{N-1} {\mathfrak {s}}_k \left| {k}\right\rangle . \end{aligned}$$
(43)
Step 2 :

Add another register prepared to \(\sum _{j=0}^{N-1} c_j\left| {j}\right\rangle \) using \(O_x\) (\({{\varvec{x}}}={{\varvec{c}}}\) in Eq. 6):

$$\begin{aligned} \sum _{k=0}^{N-1} {\mathfrak {s}}_k \left| {k}\right\rangle \rightarrow \sum _{j,k=0}^{N-1} {\mathfrak {s}}_k c_j\left| {k}\right\rangle \left| {j}\right\rangle . \end{aligned}$$
(44)
Step 3 :

Apply the controlled phase gate so that \(\left| {k}\right\rangle \left| {j}\right\rangle \rightarrow \hbox {e}^{2\pi ikj/N}\left| {k}\right\rangle \left| {j}\right\rangle \):

$$\begin{aligned} \sum _{j,k=0}^{N-1} {\mathfrak {s}}_k c_j\left| {k}\right\rangle \left| {j}\right\rangle \rightarrow \sum _{j,k=0}^{N-1} {\mathfrak {s}}_k c_j \hbox {e}^{2\pi ijk/N}\left| {k}\right\rangle \left| {j}\right\rangle . \end{aligned}$$
(45)
Step 4 :

Apply Hadamard gates to \(\left| {j}\right\rangle \):

$$\begin{aligned} \sum _{j,k=0}^{N-1} {\mathfrak {s}}_k c_j \hbox {e}^{2\pi ijk/N}\left| {k}\right\rangle \left| {j}\right\rangle \rightarrow \sum _{j,k=0}^{N-1} {\mathfrak {s}}_k \left| {k}\right\rangle \left( F_k \left| {0^L}\right\rangle + \sqrt{1-F_k^2}\left| {0^\perp }\right\rangle \right) , \end{aligned}$$
(46)

where \(\left| {0^\perp }\right\rangle \) represents any states perpendicular to \(\left| {0^L}\right\rangle \).

Step 5 :

By post-selecting the ancillary qubit state \(\left| {0^L}\right\rangle \), the quantum state in the first register collapses to

$$\begin{aligned} \frac{1}{\sqrt{\sum _{k}|F_k {\mathfrak {s}}_k|^2}} \sum _{k=0}^{N-1} F_k {\mathfrak {s}}_k \left| {k}\right\rangle . \end{aligned}$$
(47)
Step 6 :

Perform the QFT:

$$\begin{aligned} \text {QFT}\sum _{k=0}^{N-1} {\mathfrak {s}}_k F_k \left| {k}\right\rangle \propto C\left| {s}\right\rangle . \end{aligned}$$
(48)

Note that the post-selection probability of obtaining the correct state in Step 5 is

$$\begin{aligned} p = \sum _{k=0}^{N-1} \left| {\mathfrak {s}}_k F_k \right| ^2, \end{aligned}$$
(49)

and p equals to 1 / N when C is unitary. Therefore, using amplitude amplification [7], \({\mathcal {O}}((\log N)^2/\sqrt{p})\) one- or two-qubit gates, as well as \({\mathcal {O}}(1/\sqrt{p})\) calls of \(O_x\), \(O_s\) and their inverses, are needed to implement a circulant matrix operation C, where \(O_s \left| {0^L}\right\rangle = \sum _{k=0}^{N-1}s_k\left| {k}\right\rangle \).

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Zhou, S.S., Loke, T., Izaac, J.A. et al. Quantum Fourier transform in computational basis. Quantum Inf Process 16, 82 (2017). https://doi.org/10.1007/s11128-017-1515-0

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Keywords

  • Quantum algorithm
  • Quantum Fourier transform
  • Computational basis state
  • Controlled quantum gates