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XpookyNet: advancement in quantum system analysis through convolutional neural networks for detection of entanglement

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Abstract

Quantum system attributes, notably entanglement, are indispensable in manipulating quantum information tasks. Ergo, machine learning applications to help harness the sophistication of quantum information theory have surged. However, relying on unsuited prototypes and not bridling quantum information data for usual processors has often resulted in sub-optimal efficaciousness and inconvenience. We develop a custom deep convolutional neural network, XpookyNet, which is streamlined with respect to the interrelationships of density matrices of two-qubit systems to underpin three-qubit systems, breaking new ground to more qubit systems by their subsystems. Comparative implementation of XpookyNet provides instantaneous and meticulous results with an accuracy of 98.53% in merely a few epochs. XpookyNet effectively handles the inherent complexity of quantum information, equipping deeper insights into many-body systems as a bedrock. The study also investigates quantum features and their relation to the purity of a density matrix directly related to noise and system decoherence in NISQ-era quantum computation. Preparing the density matrix in a compact and compatible format with customary convolutional neural networks plays a determining role in dissecting quantum features. The procedure renders a convex criterion that detects entanglement and is a yardstick for quantum system coherence.

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Data Availability

The data associated with this project is available on GitHub. You can access the repository, which includes all relevant datasets, code, and documentation, at the following link: https://github.com/AKookani/XpookyNet

Code Availability

The codes that support the plots within the paper are publicly available on https://github.com/AKookani/XpookyNet

References

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Author information

Authors and Affiliations

  1. Faculty of Engineering, College of Farabi, University of Tehran, Tehran, Iran

    Ali Kookani & Kazim Fouladi

  2. School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran

    Yousef Mafi & Payman Kazemikhah

  3. Department of Engineering, Loyola University Maryland, Baltimore, MD, USA

    Hossein Aghababa

  4. Swanson School of Engineering, Electrical Engineering, University of Pittsburgh, Pittsburgh, PA, USA

    Masoud Barati

Authors
  1. Ali Kookani
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  2. Yousef Mafi
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  3. Payman Kazemikhah
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  4. Hossein Aghababa
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  5. Kazim Fouladi
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  6. Masoud Barati
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Contributions

AK: conceptualization, methodology, investigation, programming, literature survey, writing. YM: data adjustment, visualization, literature survey, writing. PK: literature survey, revising manuscript. HA: revising manuscript, project administration, supervision. KF: revising manuscript, supervision. MB: revising manuscript, supervision.

Corresponding author

Correspondence to Hossein Aghababa.

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Supplementary file 1 (mp4 21310 KB)

Appendices

Appendix A: Two-qubit density matrix

Two-qubit separable mixed states are expressed:

$$\begin{aligned} \rho _{sep}=\sum _{i=1}^{m}{\lambda _i\rho _i^A\otimes \rho _i^B}, \end{aligned}$$
(A1)

where \(\sum _{i}\lambda _i=1\) and \(0\le \lambda _i\le 1\), summing over \(i=1\) to m. \(\rho _i^A\) and \(\rho _i^B\) are arbitrary density matrices of A and B qubits, respectively. Two-qubit entangled mixed states are expressed as

$$\begin{aligned} \rho _{ent}=\sum _{i=1}^{m}{\lambda _i\rho _{i}^{AB}}, \end{aligned}$$
(A2)

\(\rho _{i}^{AB}\) is the arbitrary density matrix of the two-qubit entangled state.

Appendix B: Generation of three-qubit density matrix

In the three qubits case, two-qubit entangled states are required to build partial entangled states.

1.1 B.1 Three-qubit separable density matrix

Separable mixed states are expressed:

$$\begin{aligned} \rho _{sep}=\sum _{i=1}^{m}{\lambda _i\rho _i^A\otimes \rho _i^B\otimes \rho _i^C}, \end{aligned}$$
(B3)

where \(\sum _{i}\lambda _i=1\) and \(0\le \lambda _i\le 1\), with m iterating from 1 to an arbitrary number. The greater the value of m, the less pure the state.

1.2 B.2 Partial entangled density matrix

Partial entangled mixed states of a three-qubit state are produced in three cases.

Case 1: Partial entangled pairs \(|\psi _{A}\rangle \) and \(|\psi _{B}\rangle \) are generated by

$$\begin{aligned} \rho _{AB|C}=\sum _{i=1}^{m}{\lambda _i\rho _i^{AB}\otimes \rho _i^C}, \end{aligned}$$
(B4)

where \(\sum _{i}\lambda _i=1\) and \(0\le \lambda _i\le 1\), with m iterating from 1 to an arbitrary number. \(\rho ^{AB}\) is a density matrix of a two-qubit entangled state.

Case 2: Partial entangled pairs \(|\psi _{B}\rangle \) and \(|\psi _{C}\rangle \) are generated by

$$\begin{aligned} \rho _{A|BC}=\sum _{i=1}^{m}{\lambda _i\rho _i^A\otimes \rho _i^{BC}}, \end{aligned}$$
(B5)

where \(\sum _{i}\lambda _i=1\) and \(0\le \lambda _i\le 1\), \(\rho _i^{BC}\) is an arbitrary density matrix of a two-qubit entangled state.

Case 3: Partial entangled pairs \(|\psi _{A}\rangle \) and \(|\psi _{C}\rangle \) are generated by state vectors of the entangled state \(|\psi _{AC}\rangle \) and separated state \(|\psi _B\rangle \) considered as

$$\begin{aligned} |\psi _{AC}\rangle =[a_0\ \ a_1\ \ a_2\ \ a_3]^T \end{aligned}$$
(B6)

and

$$\begin{aligned} |\psi _{B}\rangle =[b_0\ \ b_1]^T. \end{aligned}$$
(B7)

Therefore, the partial entangled states are considered as

$$\begin{aligned} |\psi _{AC|B}\rangle =[a_0b_0\ \ a_1b_0\ \ a_0b_1\ \ a_1b_1\ \ a_2b_0\ \ a_3b_0\ \ a_2b_1\ \ a_3b_1]^T, \end{aligned}$$
(B8)

where a and b are corespondent amplitudes of states (B6) and (B7). Ultimately, the partial entangled states are generated by

$$\begin{aligned} \rho _{AC|B}=\sum _{i=1}^{m}{\lambda _i(|\psi _{B|AC}\rangle _i\langle \psi _{B|AC}|_i)}, \end{aligned}$$
(B9)

where \(\sum _{i}\lambda _i=1\) and \(0\le \lambda _i\le 1\) and \(|\psi _{B|AC}\rangle \) is the generated partial entangled state.

1.3 B.3 Three-qubit GHZ state

The state vector of a random three-qubit state that GHZ is applied to is described as

$$\begin{aligned} |\Psi _{GHZ}\rangle =\frac{1}{\sqrt{N_{GHZ}}}\left( \cos {(\epsilon )}|000\rangle +\sin {(\epsilon )}e^{i\phi }|\Phi _{ABC}\rangle \right) \end{aligned}$$
(B10)

with initial states:

$$\begin{aligned}&|\Phi _{ABC}\rangle = |\varphi _A\rangle |\varphi _B\rangle |\varphi _C\rangle \end{aligned}$$
(B11)
$$\begin{aligned}&|\varphi _A\rangle =\cos {(\theta _A)}|0\rangle +e^{i\phi _A}\sin {(\theta _A)}|1\rangle ,\end{aligned}$$
(B12)
$$\begin{aligned}&|\varphi _B\rangle =\cos {(\theta _B)}|0\rangle +e^{i\phi _B}\sin {(\theta _B)}|1\rangle ,\end{aligned}$$
(B13)
$$\begin{aligned}&|\varphi _C\rangle =\cos {(\theta _C)}|0\rangle +e^{i\phi _C}\sin {(\theta _C)}|1\rangle , \end{aligned}$$
(B14)

where

$$N_{GHZ}=1/(1+\cos {(\delta )}\sin {(\delta )}\cos {(\alpha )}\cos {(\beta )}\cos {(\phi )}).$$

The angles belong to the intervals \(\delta \in (0,\ \pi /4]\), \((\alpha ,\ \beta ,\ \gamma )\in (0,\ \pi /2]\), and \(\phi \in [0,\ 2\pi )\).

1.4 B.4 Three-qubit W-state

A random state vector that undergoes the W-state operation can be written as

$$\begin{aligned} \left| \Psi _{\text {W}}\right\rangle =&\frac{1}{\sqrt{N_{\text {W}}}} (a\ |001\rangle + b\ |010\rangle + c |100\rangle - d\ |\phi \rangle ), \end{aligned}$$
(B15)

where

$$N_W=1/\sqrt{|a|^2+|b|^2+|c|^2+|d|^2}$$

is the normalization factor and \(|\phi \rangle \) is a superposition of remaining states with W-state.

1.5 B.5 Three-qubit graph state

The graph state can be considered as

$$\begin{aligned} \left| \Psi _{\text {Graph}}\right\rangle= & \frac{1}{\sqrt{N_{\text {Graph}}}} (\alpha _0\ |000\rangle + \alpha _1\ |001\rangle + \alpha _2\ |010\rangle \nonumber \\ & - \alpha _3\ |011\rangle + \alpha _4\ |100\rangle + \alpha _5\ |101\rangle - \alpha _6\ |110\rangle \nonumber \\ & + \alpha _7\ |111\rangle ) \end{aligned}$$
(B16)

where normalization factor is

$$\begin{aligned} N_{Graph}=1/\sqrt{|\alpha _0|^2+\ldots +|\alpha _7|^2}. \end{aligned}$$

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Kookani, A., Mafi, Y., Kazemikhah, P. et al. XpookyNet: advancement in quantum system analysis through convolutional neural networks for detection of entanglement. Quantum Mach. Intell. 6, 50 (2024). https://doi.org/10.1007/s42484-024-00183-y

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