Introduction

Scaling a real matrix O with non-negative entries means finding diagonal matrices \(D_1, D_2\) such that \(B=D_1OD_2\) is bistochastic. Sinkhorn theorem presents a necessary and sufficient condition for existence of the decomposition of a matrix. Moreover, the iterative Sinkhorn–Knopp algorithm finds the bistochastic matrix B [1]. Such decomposition can be used for ranking web pages [2], preconditioning sparse matrices [3] and understanding traffic circulation [4].

Since unitary matrices are complex analogue of orthogonal matrices, it is natural to ask whether there exist a counterpart of Sinkhorn theorem for them. De Vos and De Baerdemacker considered whether it is possible, that for arbitrary unitary matrix \(U\in \mathbf {U}(n)\), there exist two unitary diagonal matrices \(U_1, U_2\) such, that matrix \(U_1UU_2\) has all lines sums equal to 1. Such decomposition exists for arbitrary unitary matrix, and an algorithm for finding it approximately was presented [5]. Matrices called negators were treated as quantum counterpart of bistochastic matrices and form a group \(\mathbf {XU}(n)\) under multiplication. Idel and Wolf propose an application of the quantum scaling in quantum optics [6].

Algorithm converges for arbitrary unitary matrix U [7]. Similar decomposition of unitary matrices \(U\in \mathbf {U}(2m)\) called bZbXbZ decomposition was presented [8]. They show that there always exist matrices \(A, B, C, D\in \mathbf {U}(m)\) such that

$$\begin{aligned} U = \begin{bmatrix} A&\quad 0 \\ 0&\quad B \end{bmatrix} \frac{1}{2}\begin{bmatrix} {I}+C&\quad {I}-C \\ {I}-C&\quad {I}+C \end{bmatrix} \begin{bmatrix} {I}&\quad 0 \\ 0&\quad D \end{bmatrix}, \end{aligned}$$
(1)

where \({I}\) is identity matrix. Matrix in the middle is a block-negator matrix (which is also a negator matrix), while left and right matrices are block diagonal matrices. In [9], an algorithm of finding such decomposition was presented.

Group \(\mathbf {XU}(2^n)\) is isomorphic to \(\mathbf { U}(2^n-1)\) and can be generated by single-qubit negator and controlled-\(\sqrt{\text{ NOT }}\) gates [10]. However, the proof is non-constructive since a decomposition designed for generating random matrices was used [11]. Although it is proved that it exists for any unitary matrix, obtaining such a decomposition is a very complex task. Therefore, another approach is needed for efficient decomposition procedure.

In this article, using similar method presented by de Vos and de Baerdemacker [10], we demonstrate an implementation of arbitrary k-qubit unitary operation using one-qubit ancilla with controlled-\( \sqrt{\text {NOT}}\) and single-qubit negator gates. Since product of these basic negator gates is still a negator matrix, our result can be seen as quantum analogue of scaling matrix. More precisely, we prove that for arbitrary matrix \(U\in \mathbf {U}(2^k)\), which is performed on system \(\mathcal {H}\), there exist a negator \(N\in \mathbf {XU}(2^{k+1})\) such that for arbitrary state \(| \psi \rangle \in \mathcal {H}\), we have

$$\begin{aligned} U| \psi \rangle = \Psi (N \Phi (| \psi \rangle )). \end{aligned}$$
(2)

Here, \(\Phi \) denotes the operation of extending the system with an ancilla register in \(| - \rangle \) state and \(\Psi \) denotes partial trace over the ancilla system. Since after performing operations \(\Phi \) and N the state is of the form \(| - \rangle \otimes U| \psi \rangle \), the partial trace is simply removing the ancilla system giving a pure state \(U| \psi \rangle \). We describe an efficient algorithm that for given U returns explicit and exact form of N with decomposition into a sequence of single-qubit negator and controlled-\(\sqrt{\text {NOT}}\) gates only in contrast to results of de Vos and de Baerdemacker [9, 10].

In Sect. 2, we recall basic facts. In Sect. 3, we show how to perform such transformation efficiently and demonstrate the cost in terms of controlled-\( \sqrt{\text {NOT}}\) gates.

To illustrate the transformation method, a transformation of Grover’s search algorithm is presented step by step in Sect. 4.

Basic facts

Negator gates of dimension 2 were introduced by de Vos and de Baerdemacker [10] as unitary matrices \(N\in \mathbf {U}(2)\) which are also a convex combination of identity matrix and NOT gate. Simple calculation shows that they are of the form

$$\begin{aligned} N(\theta )=\frac{1}{2} \begin{bmatrix} 1 + e^{i\theta }&\quad 1 - e^{i\theta } \\ 1 - e^{i\theta }&\quad 1 + e^{i\theta } \end{bmatrix}, \end{aligned}$$

where \(\theta \in [0,2\pi )\). Negators form a subgroup of single-qubit unitary operations, i.e., \(N(\phi )N(\psi )=N(\phi +\psi )\) for any values of \(\phi \) and \(\psi \). In the following, we will also use a 2-qubit negator operation controlled-\(\sqrt{\text {NOT}}\) gate (which is also controlled-\(N(\frac{\pi }{2})\) gate)

$$\begin{aligned} \begin{bmatrix} 1&\quad 0&\quad 0&\quad 0 \\ 0&\quad 1&\quad 0&\quad 0 \\ 0&\quad 0&\quad \frac{1+i}{2}&\quad \frac{1-i}{2} \\ 0&\quad 0&\quad \frac{1-i}{2}&\quad \frac{1+i}{2} \\ \end{bmatrix}. \end{aligned}$$

As these gates are used as basic operators, we will use a simplified notation in circuit, respectively

figure a

These two kinds of unitary matrices will be called NCN gates (Negators-Controlled-Negator).

In Sect. 3, decomposition of single-qubit unitary gates will be needed. Every unitary matrix \(U\in \mathbf {U}(2)\) can be presented as a product of global phase, two z-rotators and one y-rotator [12]

$$\begin{aligned} U= & {} e^{i\phi _0} \begin{bmatrix} \cos \frac{\phi _1}{2} e^{i\phi _2}&\sin \frac{\phi _1}{2} e^{i\phi _3}\\ -\sin \frac{\phi _1}{2} e^{-i\phi _3}&\cos \frac{\phi _1}{2} e^{-i\phi _2} \end{bmatrix} \nonumber \\= & {} e^{i\phi _0} \begin{bmatrix} e^{i\frac{\phi _2+\phi _3}{2}}&0 \\ 0&e^{-i\frac{\phi _2+\phi _3}{2}} \end{bmatrix} \begin{bmatrix} \cos \frac{\phi _1}{2}&\sin \frac{\phi _1}{2}\\ -\sin \frac{\phi _1}{2}&\cos \frac{\phi _1}{2} \end{bmatrix} \begin{bmatrix} e^{i\frac{\phi _2-\phi _3}{2}}&0 \\ 0&e^{-i\frac{\phi _2-\phi _3}{2}} \end{bmatrix} \nonumber \\= & {} e^{i\phi _0} R_z(-\phi _2-\phi _3)R_y(\phi _1)R_z(\phi _3-\phi _2). \end{aligned}$$
(3)

Since global phase is not measurable, we can simplify this representation without loss of information

$$\begin{aligned} U \cong R_z(\gamma )R_y(\beta )R_z(\alpha ), \end{aligned}$$
(4)

where ‘\(\cong \)’ means equality up to a global phase. The same applies in the case of global phase change on one of the registers of a bigger system

$$\begin{aligned} U_1\otimes e^{i\phi } U_2 \otimes U_3 = e^{i\phi }(U_1\otimes U_2\otimes U_3) \cong U_1\otimes U_2\otimes U_3 . \end{aligned}$$
(5)

Using these two facts, we can say that in any situation, we can ignore global phase change on any register.

While it may lead to a conclusion that our transformation is mainly applied to group \(\mathbf {SU}(n)\), we decided to stay with the unitary matrices formalism, since negator gates are not special unitary matrices. The result may be written using the special matrices; however, then the negators gates column and row sums will equal \(e^{i\theta }\) in general.

Circuit transformation method

In this section, we provide complete description of the transformation method. We recall a sketch of a proof of universality theorem between quantum gates and negator gates from the work of de Vos and de Baerdemacker [10]. Next, we present transformation method of arbitrary single-qubit gate into NCN product. Then, we provide a method of decomposition for arbitrary k-qubit circuit, based on the single-qubit case. Finally, we analyze the cost of presented transformation.

Universality theorem

De Vos and de Baerdemacker proved a universality theorem: Group \(\mathbf {XU}(2^k)\) generated by negators and controlled-\( \sqrt{\text {NOT}}\) is isomorphic to \(\mathbf {U}(2^k-1)\) [10]. The proof consists of several steps:

  1. 1.

    Every matrix \(U\in \mathbf {U}(2^k-1)\) can be decomposed into a product of m gates \(U_1U_2\cdots U_m\), where matrices \(U_i\in \mathbf {U}(2^k-1)\) are of some special forms [11].

  2. 2.

    Group \(\mathbf {U}(2^k-1)\) is isomorphic to group

    $$\begin{aligned} \mathbf {^{1}U}(2^k) = \left\{ \begin{bmatrix} 1&\quad \mathbf 0 ^T\\ \mathbf 0&\quad U\end{bmatrix}: U\in \mathbf {U}(2^k-1) \right\} , \end{aligned}$$
    (6)

    because of the isomorphism \(h: \mathbf {U}(2^k-1) \rightarrow \mathbf {^{1}U}(2^k) \)

    $$\begin{aligned} h(U) = \begin{bmatrix} 1&\quad \mathbf 0 \\ \mathbf 0&\quad U \end{bmatrix}. \end{aligned}$$
    (7)
  3. 3.

    Function \(f : \mathbf {^{1}U}(2^k) \rightarrow \mathbf {XU}(2^k)\) of the form \(f(U)=(H\otimes {I}_{2^k})U(H\otimes {I}_{2^k})\) is an isomorphism.

  4. 4.

    Decomposition of every \(f(h(U_i))\) into a product of NCN gates is possible, where \(U_i\) comes from point 1.

The proof used the decomposition presented in the work of Poźniak et al. [11], because it is proven that the decomposition exists for any unitary matrix. However, obtaining such decomposition is a very complex task. Therefore, we need to choose a different decomposition in order to find an efficient decomposition procedure.

Obviously, group \(\mathbf {U}(2^k)\) is isomorphic to some subgroup of \(\mathbf {XU}(2^{k+1})\). In other words, with ancilla (one additional qubit), every unitary matrix can be replaced with a sequence of NCN gates. For our purpose, we choose function \(g:\mathbf {U}(2^k) \rightarrow \mathbf {XU}(2^{k+1})\)

$$\begin{aligned} g(U) = \frac{1}{2}H\otimes {I}(| 0 \rangle \langle 0 |\otimes {I}+ | 1 \rangle \langle 1 |\otimes U)H\otimes {I}= \frac{1}{2} \begin{bmatrix} {I}&\quad {I}\\ {I}&\quad -{I}\end{bmatrix} \begin{bmatrix} {I}&\quad \mathbf 0\\ \mathbf 0&\quad U \end{bmatrix}\begin{bmatrix} {I}&\quad {I}\\ {I}&\quad -{I}\end{bmatrix}. \end{aligned}$$
(8)

Using the function g, every gate U changes into controlled-U. Using circuit notation, we can present this fact as

figure b

Note that if we assume that the first qubit is set to \(| - \rangle \), the control qubit does not influence the result (the condition is always ‘true’).

Single-qubit gate transformation

Now, we aim at decomposition of arbitrary single-qubit gate into NCN gates. With Eq. (4) for any \((U \in {\mathbf {U}}(2))\), there exist real parameters \(\alpha ,\beta ,\gamma \) such that

$$\begin{aligned} U \cong R_z(\gamma )R_y(\beta )R_z(\alpha ). \end{aligned}$$
(9)

Therefore, after applying function g, we have

figure c

We change the rotators with neighboring Hadamard gates into NCN gates as shown in Fig. 1

figure d

Let us note that the symbols of controlled-NOT, controlled-\(\sqrt{\text {NOT}}^\dagger \) and controlled-negator used in the decomposed circuit do not mean that these gates cannot be transformed. We use these symbols as a simplified notation for its decomposition with use of controlled-\(\sqrt{\text {NOT}}\) gates as shown in Fig. 2.

Fig. 1
figure 1

Decomposition of controlled-y-rotator, controlled-z-rotator and Toffoli gate. Decompositions use the simplified notation from Fig. 2

Fig. 2
figure 2

Decomposition of controlled-NOT, controlled-\( \sqrt{\text {NOT}}^\dagger \) gates and controlled-negator [10]

General transformation method

Now, we consider transformation of arbitrary k-qubit circuit. Let us assume that we have a circuit which consists of unitary operation \( (U \in {\mathbf {U}}(2^k))\), generalized measurement \(\mathbf M = \{M_a\in {L}(\mathbb {C}^{2^k}):a\in \Sigma \}\), where \(\Sigma \) is a set of classical outputs of measurement, and starting state \(| \phi _0 \rangle \)

figure e

In order to construct a decomposition of unitary U into a sequence of negator gates, we begin with obtaining a decomposition of U into controlled-NOT and single-qubit gates

figure f

here denoted by a sequence of gates \(U=V_m\cdots V_1\). Contrary to the decomposition presented in the work of Poźniak, Życzkowski and Kuś, there exist efficient methods for constructing such circuit [13]. Next, we need to add an additional qubit, transform \(V_i\) gates into controlled-\(V_i\) gates and add Hadamard gates as below (since \(HH={I}\))

figure g

Let us note that product \(H\cdot \text {controlled-}V_j\cdot H\) is an image of homomorphism presented in Eq. (8) on \(V_j\). Next, we replace the product with the sequence of NCN gates (here denoted by \(\mathbf N_j\)) as in previous subsection (if \(V_j\) is controlled-NOT, then we choose Toffoli gate transformation from Fig. 1)

figure h

For the sake of simplicity, we may change the starting state and resulting state on the first wire

figure i

Now, we have an equivalent circuit which consists of negators and controlled-\( \sqrt{\text {NOT}}\) gates only.

Transformation cost

Now, we consider upper bound of cost of decomposition into negator circuit. Two kinds will be discussed: memory complexity and number of single- and two-qubit gates. In the first case for arbitrary k-qubit circuit transformation requires one additional qubit.

Let \(c_{\text {CNOT}}(k) \) and \(c_s(k)\) denote upper bound of the number of, respectively, controlled-NOT and single-qubit gates needed for the implementation of an arbitrary k-qubit circuit. Using the operation presented above, we need \(17c_{\text {CNOT}}(k) + 64c_s(k)\) controlled-\( \sqrt{\text {NOT}}\) gates and \(11c_{\text {CNOT}}(k)+34c_s(k)\) negators to implement an equivalent circuit (up to global phase).

Any circuit which consists of controlled-NOT and single-qubit gates can be simplified in such a way that \(c_s(k)\le 2c_{\text {CNOT}}(k)+k\). This estimation is based on the worst case, when there are two single-qubit gates between every controlled-NOT gate. Taking this into account, we can express the previous result in terms of \(c_{\text {CNOT}}\) only, because only \(17c_{\text {CNOT}}(k)+ 64c_s(k) \le 145 c_{\text {CNOT}}(k)+64k\) controlled-\( \sqrt{\text {NOT}}\) gates are needed. In fact, if \(c_{\text {CNOT}} = O(4^k)\), then so is the number of controlled-\( \sqrt{\text {NOT}}\) gates.

Step-by-step transformation example

To illustrate the introduced decomposition, we will present Grover’s algorithm for \(k=2\) qubits as NCN circuit. The original circuit for this algorithm is presented in Fig. 3, where \(\omega \) denotes the searched state.

Fig. 3
figure 3

Original Grover’s search algorithm circuit in case \(k=2\). G is Grover diffusion operator, \(U_\omega \) is quantum black box and we perform measurement M. Algorithm comes from [14]

Fig. 4
figure 4

Grover’s search algorithm decomposition. Unnecessary Hadamard gates have already been removed (a) \(| 1 \rangle \) changes into \(| - \rangle \); (b) measurement \(\mathbf {M}\) does not change, on first wire we end with \(| - \rangle \) state; (c) subcircuit simplification; (d) subcircuit transforming into NCN gates. Any other transformation left can be done similarly, except the \(U_\omega \) case

As in the previous section, we will add one qubit, change every H and G gate into controlled-H and controlled-G, respectively, and add Hadamard gates on the ancilla register. Former steps of the decomposition are explicitly presented in Fig. 4. The following facts were used

  • the decomposition of Hadamard gate is \(H\cong R_z(\pi )R_y(\frac{\pi }{2})R_z(0)=R_z(\pi )R_y(\frac{\pi }{2})\),

  • the decomposition of NOT gate is \(\text{ NOT }\cong R_z(\pi )R_y(\pi )R_z(0)=R_z(\pi )R_y(\pi )\),

  • for any \((U, V \in {\mathbf {U}}(2))\), we have

    figure j
  • Grover’s diffusion operator can be decomposed in the following way

    figure k

Decomposition of \(U_\omega \) depends strictly on the value of \(\omega \); therefore, it is not presented in the example. The full decomposition is presented in Fig. 4.

Concluding remarks

In the presented work, we provide a constructive method of scaling arbitrary unitary matrices \(U\in \mathbf {U}(2^k)\). More precisely, we proved that for arbitrary unitary matrix \(U\in \mathbf {U}(2^k)\), there exists unitary negator matrix \(N\in \mathbf {XU}(2^{k+1})\) such that for arbitrary state \(| \psi \rangle \), we have

$$\begin{aligned} U| \psi \rangle = \Psi (N \Phi (| \psi \rangle )). \end{aligned}$$
(10)

Here, \(\Phi \) denotes the operation of extending the system with an ancilla register in \(| - \rangle \) state and \(\Psi \) denotes partial trace over the ancilla system. We described efficient algorithm of decomposing N into product of single-qubit negator and controlled-\(\sqrt{\text{ NOT }}\) gates. Our decomposition consists of \(O(4^k)\) entangling gates which is proved to be optimal and needs one-qubit ancilla.

Our result can be seen as complex analogue of Sinkhorn–Knopp algorithm, which is known to have wide applications. The result is in contrast to the previous results [10], which could be only used to prove the existence of such decomposition. Moreover, our transformation is exact and can be found constructively. In contrast to [9], our transformation consists only of negator gates. The main difference is that transformation needs one-qubit ancilla.