A universal quantum circuit scheme for finding complex eigenvalues

Abstract

We present a general quantum circuit design for finding eigenvalues of non-unitary matrices on quantum computers using the iterative phase estimation algorithm. In addition, we show how the method can be used for the simulation of resonance states for quantum systems.

Appendices

Appendix 1: Verification of the circuit in Fig. 2

The circuit is divided in three blocks: Formation, Combination, and Input Modification as shown in the figure. In the Input Modification block, the initial input is modified to $|\varphi \rangle$. Then, the matrix $V$ is constructed in the Formation and Combination blocks.

1.1 Formation block

The second block in Fig. 2 is called Formation Block which consists of uniformly controlled rotation gates for each element of $U$. For the matrix element $u_{ij}$, assumed to be real, the rotation gate $R_{ij}(\theta _{ij})$ is defined as follows:

$$\begin{aligned} R_{ij}(\theta _{ij})= \left( \begin{array}{l@{\quad }l@{}} \cos \left( \frac{\theta _{ij}}{2}\right) &{}\sin \left( \frac{\theta _{ij}}{2}\right) \\ -\sin \left( \frac{\theta _{ij}}{2}\right) &{}\cos \left( \frac{\theta _{ij}}{2}\right) \end{array}\right) \end{aligned}$$

(27)

where $\theta _{ij}$ is determined from the value of the element in accordance with the equality $cos(\frac{\theta _{ij}}{2})=u_{ij}$ to get the following:

$$\begin{aligned} R_{ij}(\theta _{ij})= \left( \begin{array}{c@{\quad }c@{}} u_{ij} &{}\sqrt{1-u_{ij}}\\ -\sqrt{1-u_{ij}}&{}u_{ij} \end{array}\right) \end{aligned}$$

(28)

Based on the linear indices of the elements (row-wise), $R_{ij}$s are controlled by the different states of $(2n)$ qubits: e.g., $u_{12}$ is the second element in the matrix and has a linear index of $2$. Hence, the control of $R_{12}$ is set accordingly so that this gate operates when the first $(2n)$ qubits in $\left| 0\dots 01\right\rangle$ state. Combination of these controlled rotation gates forms a network as the second block in Fig. 2. This block has the following matrix representation in the computational basis:

$$\begin{aligned} F=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{}} R_{11}&{} &{} &{}\\ &{}R_{12}&{} &{}\\ &{} &{}\ddots &{}\\ &{} &{} &{}R_{NN} \end{array}\right) ,\ R_{ij}= \left( \begin{array}{c@{\quad }c@{}} u_{ij} &{}\sqrt{1-u_{ij}}\\ -\sqrt{1-u_{ij}}&{}u_{ij} \end{array}\right) . \end{aligned}$$

(29)

Here, we have used rotations around the y-axis since the matrix elements are assumed to be real. However, for the complex elements of $U$, a product of $R_z$ and $R_y$ gates is needed for each $R_{ij}$. For instance, if $u_{ij} = e^{i\phi }cos(\theta ),\,e^{i\phi }$ is created by $R_z$, and $cos(\theta )$ is by $R_y$.

1.2 Combination block

For a system of $(2n+1)$ qubits, the third block in Fig. 2 is defined as the application of the Hadamard gates to $(n+1)\hbox {th }, (n+2)\hbox {th }, \ldots , (2n-1)\hbox {th}$, and $(2n)$th qubits from the top of the circuit: i.e. $(I^{\otimes n}\otimes H^{\otimes n}\otimes I)$, where $H$ and $I$ are the Hadamard and identity matrices, respectively. The matrix representation of this block is as follows:

$$\begin{aligned} C=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{}} C_{block}&{} &{} &{}\\ &{}C_{block} &{} &{}\\ &{} &{}\ddots &{}\\ &{} &{} &{}C_{block}\\ \end{array}\right) ,\quad \hbox {where } C_{block}= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{}} k&{}0&{}\dots &{} k &{}0\\ 0&{}k&{}\dots &{}0&{}k\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ k&{}0&{}\dots &{}k&{}0\\ 0&{}k&{}\dots &{}0&{}k\\ \end{array}\right) _{2N\times 2N}.\nonumber \\ \end{aligned}$$

(30)

Note that $C_{block}$ also has negative elements, however, they are not located in the first row. Therefore, they shall not affect the predetermined states. The application of the above matrix $C$ to the matrix $F$ defined in Eq. (29) forms the matrix $V$ defined in Eq. (3) where the same row elements of $U$ are located on the leading rows of each $V_i$ ($i$ represents the row index of $U$):

$$\begin{aligned} V\!&= \!CF\nonumber \\ \!&= \! \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{}} C_{block}&{} &{} &{}\\ &{}C_{block} &{} &{}\\ &{} &{}\ddots &{} \\ &{} &{} &{}C_{block}\\ \end{array}\right) \left( \begin{array}{c@{\quad }c@{\quad }c@{}} \begin{array}{c@{\quad }c@{}} u_{11}&{}\sqrt{1-u_{11}}\\ -\sqrt{1-u_{11}}&{}u_{11} \end{array}&{}\\ &{}\ddots &{}\\ &{}&{}\begin{array}{c@{\quad }c@{}} u_{NN}&{}\sqrt{1-u_{NN}}\\ -\sqrt{1-u_{NN}}&{}u_{NN} \end{array} \end{array}\right) \nonumber \\ \!&= \! \left( \begin{array}{c@{\quad }c@{\quad }c@{}} \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{}} ku_{11}&{}\bullet &{}ku_{12}&{}\dots &{}\bullet &{} ku_{1N}&{}\bullet \\ \vdots &{}\vdots &{}\vdots &{}&{}\vdots &{}\vdots &{}\vdots \end{array} &{} &{} \\ &{} \ddots &{}\\ &{} &{} \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{}} ku_{N1}&{}\bullet &{}ku_{N2}&{}\dots &{}\bullet &{} ku_{NN}&{}\bullet \\ \vdots &{}\vdots &{}\vdots &{}&{}\vdots &{}\vdots &{}\vdots \end{array} \end{array}\right) . \end{aligned}$$

(31)

Since the negative elements are not in the first row of $C_{block}$, the resulting leading rows are not affected by these elements. The elements represented by the symbol “$\bullet$” are disregarded by modifying the input to the circuit in the modification block explained below.

1.3 Input modification block

The initial input to the circuit in Fig. 2 is defined as $\tilde{\left| \alpha \right\rangle }=\left| 0..0\right\rangle \otimes \left| \alpha \right\rangle$ where $\left| 0\ldots 0\right\rangle$ is the input to the ancilla qubits. The first block in Fig. 2 consists of the Hadamard gates on the first $n$ qubits and sequential swap operations between the $(n+1)$th and the remaining last $n$ qubits (We apply swap operations between the ancilla $(n+1)$th qubit and the main qubits: First we swap the $(n+1)$th qubit with the $(2n+1)$th, then the $(n+1)$th with the $(2n)$th, then the $(n+1)$th with the $(2n-1)$th, and so on. Finally, we swap the $(n+1)$th with the $(n+2)$th). This block modifies the input $\tilde{\left| \alpha \right\rangle }$ in a way that in the application of $V=CF$ to the input, the output is not affected by the elements represented by “$\bullet$” between $ku_{ij}$ and $ku_{i(j+1)}$. Hence, the application of this block to the initial input transforms $\tilde{\left| \alpha \right\rangle }$ to $\left| \varphi \right\rangle$ in Eq. (3):

$$\begin{aligned} \tilde{\left| \alpha \right\rangle } \!&\rightarrow \! \left| \varphi \right\rangle \nonumber \\ {[}\alpha _1\, \alpha _2\,\dots \alpha _N\,0 \dots 0{]}^T \!&\rightarrow \! {[}\gamma \alpha _1\,0\,\gamma \alpha _2\dots 0\,\gamma \alpha _N\dots \,\gamma \alpha _1\,0\, \gamma \alpha _2\, \dots \, 0\, \gamma \alpha _N\, 0{]}^T,\qquad \quad \end{aligned}$$

(32)

where $\gamma$ is a normalization constant.

Consequently, the circuit in Fig. 2 describes the operation $V\left| \varphi \right\rangle =CF\left| \varphi \right\rangle$ which simulates any matrix $U$, having elements less than or equal to 1, on the following normalized set of $N$ states: $\{\left| 0\dots 000\right\rangle _a\left| 0\dots 0\right\rangle ,\,\dots ,\,\left| 0\dots 110\right\rangle _a\left| 0\dots 0\right\rangle \}$, where $\left| \dots \right\rangle _a$ represents the ancilla qubits. This is shown in Eq. (3).

For illustration purposes, we also present the full forms of the operators for the formation, combination and input modification blocks, and the output vector for the simulation of the following $2 \times 2$ arbitrary matrix [31]:

$$\begin{aligned} U= \left( \begin{array}{cc} u_{11}&{}\quad u_{12}\\ u_{21}&{}\quad u_{22}\\ \end{array}\right) \end{aligned}$$

(33)

The full form of the matrix for the formation block is as follows:

$$\begin{aligned} F\!=\! \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{}} u_{11} &{} \sqrt{1-u_{11}^2} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -\sqrt{1-u_{11}^2} &{} u_{11} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} u_{12} &{} \sqrt{1-u_{12}^2} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} -\sqrt{1-u_{12}^2} &{} u_{12} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} u_{21} &{} \sqrt{1-u_{21}^2} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -\sqrt{1-u_{21}^2} &{} u_{21} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} u_{22} &{} \sqrt{1-u_{22}^2} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -\sqrt{1-u_{22}^2} &{} u_{22} \end{array} \right) \nonumber \\ \end{aligned}$$

(34)

The combination matrix $C$ and the matrix for the input modification $M$ are defined as:

$$\begin{aligned}&C = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{}} \frac{1}{\sqrt{2}} &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 &{} 0 \\ \frac{1}{\sqrt{2}} &{} 0 &{} -\frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} -\frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} \frac{1}{\sqrt{2}} \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} -\frac{1}{\sqrt{2}} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} -\frac{1}{\sqrt{2}} \end{array} \right) ,\nonumber \\&M = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{}} \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 \\ 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 &{} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 &{} -\frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 &{} -\frac{1}{\sqrt{2}} &{} 0 \\ 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 &{} -\frac{1}{\sqrt{2}} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 &{} -\frac{1}{\sqrt{2}} \end{array} \right) \end{aligned}$$

(35)

For the initial input $\left| \mathbf{0}\right\rangle \left| \alpha \right\rangle =\hat{\left| \alpha \right\rangle }$, we find the following final state:

$$\begin{aligned} CFM\hat{\left| \alpha \right\rangle }=V\left| \varphi \right\rangle =\frac{1}{2} \left( \begin{array}{c} \alpha _{1} u_{11}+\alpha _{2} u_{12} \\ - \alpha _{1} \sqrt{1-u_{11}^2}- \alpha _{2} \sqrt{1-u_{12}^2} \\ \alpha _{1} u_{11}-\alpha _{2} u_{12} \\ -\alpha _{1} \sqrt{1-u_{11}^2}+ \alpha _{2} \sqrt{1-u_{12}^2} \\ \alpha _{1} u_{21}+\alpha _{2} u_{22}\\ - \alpha _{1} \sqrt{1-u_{21}^2}- \alpha _{2} \sqrt{1-u_{22}^2} \\ \alpha _{1} u_{21}-\alpha _{2} u_{22} \\ - \alpha _{1} \sqrt{1-u_{21}^2}+ \alpha _{2} \sqrt{1-u_{22}^2} \end{array}\right) \end{aligned}$$

(36)

Clearly, the normalized states $|000\rangle$ and $|100\rangle$ simulate the original given system.

Appendix 2: Simulation details

1.1 The decomposition of a multi-controlled network

The circuit in Fig. 2 includes a network of rotation gates in the formation block which dominates the complexity of the circuit. The uniformly controlled networks such as the one in Fig. 5a controlled by $k$ qubits can be decomposed in terms of $2^k$ CNOT gates and $2^k$ single rotation gates [38]. For instance, the circuit as illustrated for $k=2$ in Fig. 5a can be decomposed as in Fig. 5b.

Fig. 5

(a) A uniformly controlled multi-qubit network. (b) The decomposition of the network in (a) into CNOT and single quantum gates. The change of the bits in the gray code representations determines the control qubit for the CNOT gates. The network in the phase estimation algorithm includes two consecutive circuits such as in (b) with $R_z$ and $R_y$ single gates

The angle values in the decomposed circuit are solutions of the system of the linear equation $M^{k}{\varvec{\theta }}={\varvec{\phi }}$:

$$\begin{aligned} M^{k}\left( \begin{array}{c} \theta _1\\ \theta _2\\ \vdots \\ \theta _{2^{k}} \end{array}\right) =\left( \begin{array}{c} \phi _1\\ \phi _2\\ \vdots \\ \phi _{2^{k}} \end{array}\right) , \end{aligned}$$

(37)

where $k$ is the number of control qubits in the network, and the entries of $M$ are defined as:

$$\begin{aligned} M_{ij}=(-1)^{b_{i-1}.g_{j-1}}, \end{aligned}$$

(38)

in which the power term is found by taking the dot product of the standard binary code of the index $i-1,\,b_{i-1}$ and the binary representation of $j-1$th Gray-coded integer, $g_{j-1}$. Since $M^k$ is a column permuted version of the Hadamard matrix, we see that $M$ is unitary. Thus, $(M^k)^{-1}=2^{-k}(M^k)^T$ and the new angle values in the decomposed circuit are results of the matrix vector multiplication [38]:

$$\begin{aligned} {\varvec{\theta }}=2^{-k}(M^k)^{T}{\varvec{\phi }}. \end{aligned}$$

(39)

1.2 Simulation details for the example system

In Fig. 4, we have a multi-controlled network composed of 4 gates. This network basically comes from the Formation step of the circuit design method, which has been represented in matrix form as $F$ in Eq. (29). Since the elements of $U$ are complex, we need to have rotations around the $z-$axis and the $y-$axis. Hence, the above matrix is the product of two matrices $F_z$ for rotation around the z-axis and $F_y$ for rotations around the y-axis: $F=F_zF_y$:

$$\begin{aligned} F=\left( \begin{array}{c@{\quad }c@{\quad }c@{}} R_{11}^z&{} &{}\\ &{}\ddots &{}\\ &{}&{}R_{NN}^z \end{array}\right) \left( \begin{array}{c@{\quad }c@{\quad }c@{}} R_{11}^y&{} &{}\\ &{}\ddots &{}\\ &{} &{}R_{NN}^y \end{array}\right) \end{aligned}$$

(40)

To complete this network to a whole uniformly controlled network, we assume that the initial four other gates are identity. Hence, the final decomposition for $2 \times 2$ matrix, the decomposed circuit includes: 4 Hadamard gates, 16 CNOT and 16 single gates (8 CNOTs and 8 $R_y$ gates for the $F_y$; and 8 CNOTs and 8 $R_z$ gates for the $F_z$), and 2 swaps.

The parameters for the last iteration (The operator is $U^{2^0}$.) of the phase estimation algorithm is shown below, where after scaling the elements of $U$ by the matrix norm $||U||_1$ (maximum of the absolute sums of the columns), we find the angle values for $R_y$ and $R_z$ gates:

Matrix Elements	Scaled Elements	Angles for $R_y$s	Angles for $R_z$s
0.2588 $+$ 1.1214i	0.1469 $+$ 0.6364i	1.0521	$-$0.9634
$-$0.4569 $-$ 0.1109i	$-$0.2593 $-$ 0.0629i	0.0278	0.1837
$-$0.4569 $-$ 0.1109i	$-$0.2593 $-$ 0.0629i	$-$0.2486	1.9401
1.0594 $+$ 0.7394i	0.6012 $+$ 0.4196i	0.0278	0.1837
		$-$0.0278	$-$0.1837
		0.2486	$-$1.9401
		$-$0.0278	$-$0.1837
		$-$1.0521	0.9634

The angle value for the $R_z$ gate on the first qubit is $-$2.51572849. The output of the phase estimation algorithm on the chosen states and the normalized probability of the phase qubit are shown below.

States	Probabilities on the Chosen States	Normalized Probabilities of the phase qubit
0	0.0002	0.0058
1	0.001	0.9942
2	0.0393
3	0.1765

The bit values of the phase is found as $(11100110100)$ which corresponds to 0.9004. The absolute value of the eigenvalue is found from the ratio of the probability values of the phase qubit:

$$\begin{aligned} \frac{(1+\frac{|\lambda |}{||U||_1})^2}{(1-\frac{|\lambda |}{||U||_1})^2}=\frac{0.9942}{0.0058} \end{aligned}$$

(41)

From the above, we find $|\lambda |=0.8560\times ||U||_1=1.5084$. The eigenvalue of $U$ is found as $e^{i2\pi 0.9004}\times 1.5084=1.22255 + 0.88355i$. The eigenvalue of the Hamitlonian matrix is found from $(log(1.22255 + 0.88355i)/i)$ which is $0.62581 - 0.41105i$.