Unilateral protection scheme for N-qubit GHZ states against decoherence: a resource-efficient approach

Abstract

In this paper, we propose a novel protection scheme for N-qubit Greenberger–Horne–Zeilinger (GHZ) states against amplitude damping noise, using unilateral operations. The key innovation lies in the implementation of local operations on a single qubit, irrespective of the total number of qubits, thereby significantly reducing the operational complexity and resource requirements compared to existing methods. The scheme involves a unilateral rotation to transform the GHZ state into a less vulnerable configuration before exposing it to noise, followed by a weak measurement and another unilateral rotation to recover the initial state. We provide a comprehensive mathematical analysis of the proposed unilateral protection scheme, deriving expressions for both fidelity and success probability. The practical feasibility of our scheme is demonstrated through simulations using the Qiskit SDK, in which we provide the corresponding quantum circuit for performing the partial weak measurement used in our scheme. Comparative analysis with pioneer protection schemes reveals a substantial enhancement in success probability and fidelity for both GHZ and generalized GHZ states, emphasizing the superiority of the proposed approach.

Appendices

Appendix A

The initial N-qubit general GHZ state, as described in Eq. (1), can be represented by its corresponding density matrix, given by

$$\begin{aligned} \rho _{\rm in}^N=\left[\begin{array}{ccccc}\alpha ^2&{}\dots &{}\alpha \beta ^\dagger \\ &{}\ddots &{} \\ \alpha \beta &{}\dots &{}\beta ^2 \end{array} \right]_{2^N \times 2^N}. \end{aligned}$$

(A1)

In the first step, a unilateral rotation is applied to the first qubit of the GHZ state. The N-qubit representation of the unilateral rotation is expressed as:

$$\begin{aligned} R_{x}^N=&R_x \otimes \left[ \begin{array}{cccc} 1&{}\quad 0\\ 0&{} \quad 1 \end{array} \right]^{\bigotimes N-1}. \end{aligned}$$

(A2)

Consequently, the state of the N-qubit GHZ state, following the application of the unilateral rotation, is calculated as:

$$\begin{aligned} \rho _{x}^N= R_{x}^N\rho _{\rm in}^N{R_{x}^N}^T=\left[\begin{array}{ccccc}\ddots &{} &{} &{}.^{.^{.}}\\ &{}\left| \beta \right| ^2&{}\alpha \beta &{} \\ &{}\alpha \beta ^\dagger &{}|\alpha |^2&{}\\ .^{.^{.}}&{} &{} &{}\ddots \end{array} \right]_{2^N \times 2^N}. \end{aligned}$$

(A3)

The notation $A^T$ denotes the transpose of matrix A, this notation is used here, as all elements within the operation matrices are real.

After the unilateral rotation, the GHZ state transfers through ADC. Considering that each qubit of the N-qubit GHZ state undergoes ADC, the corresponding N-qubit Kraus operators of ADC are expressed as:

$$\begin{aligned} E_0=&\underbrace{e_0\otimes e_0\otimes \dots \otimes e_0\otimes e_0}_{N}\\ E_1=&\underbrace{e_0\otimes e_0\otimes \dots \otimes e_0\otimes e_1}_{N}\\&\vdots \\ E_{2^N-1}=&\underbrace{e_1\otimes e_1\otimes \dots \otimes e_1\otimes e_1}_{N}. \end{aligned}$$

(A4)

Hence, the state of N-qubit GHZ state after passing through ADC is given by

$$\begin{aligned} \rho _d^N=\sum _{i=1}^{2^N-1}E_i\rho _{x}^NE_i^T= \left[ \begin{array}{cccc} E_d&{} &{} &{}.^{.^{.}}\\ &{}B_d&{}C_d&{} \\ &{}D_d&{}A_d&{}\\ .^{.^{.}}&{} &{} &{}\ddots \end{array} \right]_{2^N \times 2^N}, \end{aligned}$$

(A5)

where $A_{d}=\left| \alpha \right| ^2(1-r),B_{d}=|\beta |^2 (1-r)^{N-1},C_{d}=D_{d}^{\dagger }=\alpha \beta \left( 1-r\right) ^ \frac{N}{2} \text { and } E_d=|\alpha |^2r{\left| 0 \right\rangle } {\left\langle 0 \right| } ^{\otimes N-1} +|\beta |^2(r{\left| 0 \right\rangle } {\left\langle 0 \right| } + (1-r){\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N-1}-\beta ^2 (1-r)^{N-1}({\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N-1}$ After the ADC, the weak measurement operator in Eq. (3) applies to the first qubit. Therefore, the weak measurement operator for the N-qubit GHZ state is expressed as

$$\begin{aligned} M^N&=M\otimes \underbrace{\left[ \begin{array}{cccc} 1&{}0\\ 0&{}1 \end{array} \right]\otimes \dots \otimes \left[ \begin{array}{cccc} 1&{}0\\ 0&{}1 \end{array} \right]}_{N-1}. \end{aligned}$$

(A6)

Hence, the state of N-qubit GHZ state after employing the weak measurement is derived as

$$\begin{aligned} \rho _m^N=M^N\rho _d^N{M^N}^T=\left[ \begin{array}{cccc} E_m&{} &{} &{}.^{.^{.}}\\ &{}B_m&{}C_m&{} \\ &{}D_m&{}A_m&{}\\ .^{.^{.}}&{} &{} &{}\ddots \end{array} \right]_{2^N \times 2^N}, \end{aligned}$$

(A7)

where $A_{m}=|\alpha |^2(1-r),B_{m}=|\beta |^2 (1-q)(1-r)^{N-1},C_{m}=D_{m}^{\dagger }=\alpha \beta (1-q)^\frac{1}{2}\left( 1-r\right) ^ \frac{N}{2} \text { and } E_m=(1-q )(|\alpha |^2r{\left| 0 \right\rangle } {\left\langle 0 \right| } ^{\otimes N-1} +|\beta |^2(r{\left| 0 \right\rangle } {\left\langle 0 \right| } + (1-r){\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N-1}-\beta ^2 (1-r)^{N-1}({\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N-1})$.

Finally, in the last step, we apply the same unilateral operation as the one before the ADC to restore the state to its initial structure. Hence, the final state after the whole process of protection is represented as

$$\begin{aligned} \rho _f^N=R_{x}^N\rho _{m}^N{R_{x}^N}^T=\left[ \begin{array}{cccc} A_f &{} &{} &{} C_f \\ &{} \ddots &{}.^{.^{.}} &{} \\ &{}.^{.^{.}} &{} E_f &{}\\ D_f &{} &{} &{} B_f \end{array} \right]_{2^N \times 2^N}, \end{aligned}$$

(A8)

where $A_{f}=|\alpha |^2(1-r),B_{f}=|\beta |^2 (1-q)(1-r)^{N-1},C_{f}=D_{f}^{\dagger }=\alpha \beta (1-q)^\frac{1}{2}\left( 1-r\right) ^ \frac{N}{2} \text { and } E_f=(1-q )(|\alpha |^2r{\left| 0 \right\rangle } {\left\langle 0 \right| } ^{\otimes N-1} +|\beta |^2(r{\left| 0 \right\rangle } {\left\langle 0 \right| } + (1-r){\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N-1}-\beta ^2 (1-r)^{N-1}({\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N-1})$.

Appendix B

In WMRPS, before sending qubits through the ADC, weak measurements are employed on all qubits, followed by a subsequent weak measurement reversal after passing through the ADC. Assuming identical parameters (such as measurement strengths and the decay rate of the ADC) for all qubits, the operators for weak measurement, denoted as $M_{\text {WM}}$, and weak measurement reversal, denoted as $M_{\text {WMR}}$, are defined as

$$\begin{aligned} M_{\text {WM}}^N = \begin{pmatrix} 1 &{} 0 \\ 0 &{} \sqrt{1 - p_1} \end{pmatrix}^{\otimes N}, \ M_{\text {WMR}}^N = \begin{pmatrix} \sqrt{1 - p_2} &{} 0 \\ 0 &{} 1 \end{pmatrix}^{\otimes N}, \end{aligned}$$

(B9)

where $p_1$ and $p_2$ represent the weak measurement and weak measurement reversal strength, respectively. Considering the initial N-qubit GHZ state in Eq. (A1), the density matrix resulting from the application of weak measurement $M_{\text {WM}}$ on all qubits is represented as

$$\begin{aligned} {\rho _{\text {WM}}^{\text {WMRPS}}}^N=M_{\text {WM}}^N\rho _{\rm in}^N{M_{\text {WM}}^N}^T=\left[ \begin{array}{cccc} F&{} &{} &{} G\\ &{}\ddots &{}.^{.^.}&{} \\ &{}.^{.^.}&{}\ddots &{} \\ H&{} &{} &{} K \end{array} \right]_{2^N \times 2^N}, \end{aligned}$$

(B10)

where $F_{\text {WM}}=|\alpha |^2, G_{\text {WM}}=H^{\dagger }=\alpha \beta (1-p_1)^\frac{N}{2}\ \text {and} \ K_{\text {WM}}=|\beta |^2 (1-p_1)^{N}$, all other elements of the density matrix at this step are zero.

Subsequently, all qubits will be sent through ADCs, resulting in the transformation of the elements of the density matrix to:

$$\begin{aligned} {\rho _{\text {ADC}}^{\text {WMRPS}}}^N=\sum _{i=1}^{2^N-1}E_i{\rho _{\text {WM}}^{\text {WMRPS}}}^NE_i^T=\left[ \begin{array}{cccc} F_{\text {ADC}}&{} &{}G_{\text {ADC}}\\ &{} L_{\text {ADC}} &{} \\ H_{\text {ADC}}&{} &{} K_{\text {ADC}} \end{array} \right]_{2^N \times 2^N} \end{aligned}$$

(B11)

where $F_{\text {ADC}}=|\alpha |^2+|\beta |^2 (1-p_1)^{N}r^{N},G_{\text {ADC}}=H^{\dagger }_{\text {ADC}}=\alpha \beta (1-p_1)^\frac{N}{2}r^\frac{N}{2}, K_{\text {ADC}}=|\beta |^2 (1-p_1)^{N}(1-r )^{N}\text {and}$ $L_{\text {ADC}}=|\beta |^2(1-p_1)^{N}[(r{\left| 0 \right\rangle } {\left\langle 0 \right| } + (1-r){\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N}-r^{N}{\left| 0 \right\rangle } {\left\langle 0 \right| }^{\otimes N}-(1-r)^{ N}{\left| 1 \right\rangle } {\left\langle 1 \right| } ^{\otimes N}]$.

Finally, by employing the weak measurement reversal $M_{\text {WMR}}$ on all qubits, the final density matrix after the whole WMRPS is calculated as

$$\begin{aligned} \rho _f^{\text {WMRPS}}=M_{\text {WMR}}^N {\rho _{\text {ADC}}^{\text {WMRPS}}}^N{M_{\text {WMR}}^N}^T=\left[ \begin{array}{cccc} F_f &{} &{} G_f \\ &{} L_f &{}\\ H_f &{} &{} K_f \end{array} \right]_{2^N \times 2^N}, \end{aligned}$$

(B12)

where $F_f=|\alpha |^2(1-p_2)^{N}+|\beta |^2 (1-p_1)^{N}r^{N}(1-p_2)^{N},G_f=H_f^\dagger =\alpha \beta (1-p_1)^\frac{N}{2}(1-p_2)^\frac{N}{2}r^\frac{N}{2}, \text { and }K_f=|\beta |^2 (1-p_1)^{N}(1-r )^{N}\text { and } L_f=|\beta |^2(1-p_1)^{N}((r(1-p_2){\left| 0 \right\rangle } {\left\langle 0 \right| } + (1-r){\left| 1 \right\rangle } {\left\langle 1 \right| } )^{\otimes N}-(r(1-p_2))^{\otimes N}{\left| 0 \right\rangle } {\left\langle 0 \right| }^{\otimes N}-(1-r)^{N}{\left| 1 \right\rangle } {\left\langle 1 \right| } ^{N})$. Here, we note that the matrix $\rho _f^{\text {WMRPS}}$ is a $2^N \times 2^N$ matrix where the elements $F_f$, $G_f$, $H_f$, and $K_f$ are scalar values located at the corners of the matrix. The central part of the matrix, denoted by $L_f$, is a $2^N \times 2^N$ matrix as well whose corner elements are all 0.

Hence, by considering the final state following the entire WMRPS process in Eq. (B12), the success probability of WMRPS is obtained as:

$$\begin{aligned} {P}_\text {WMRPS}&=\text {trace}(\rho _f^{\text {WMRPS}})=F_f+K_f+\text {trace}(L_f)\\ &=| \alpha |^2 (1 - p_{2})^N + | \beta |^2 (1 - p_{1})^N (r(1 - p_{2}) + s)^N. \end{aligned}$$

(B13)

Also, the fidelity of WMRPS is calculated as

$$\begin{aligned}&\text {Fid}_{\text {WMRPS}}=\langle \psi _{\rm in}|\rho _f^\text {WMRPS}|\psi _{\rm in}\rangle = \frac{1}{{P}_{\text {WMRPS}}}\left[ |\alpha |^4(1 - p_2)^N \right. \\ & \quad +|\alpha \beta |^2 (1 - p_1)^Nr^N(1 - p_2)^N \quad + \left. 2|\alpha \beta |^2(1 - p_1)^{\frac{N}{2}}s^{\frac{N}{2}}(1 - p_{r})^{\frac{N}{2}}+|\beta |^4(1 - p_1)^Ns^N\right] . \end{aligned}$$

(B14)