Matsakaicin Haɗin Kudi-Matsayi a cikin Tafiyar Quantum

Tsarin Abubuwan Ciki

1. Gabatarwa

Tafiyar quantum (QWs) suna wakiltar kwatankwacin quantum na tafiya na al'ada kuma sun zama kayan aiki na asali a kimiyyar bayanan quantum. Ba kamar takwarorinsu na al'ada ba, tafiyar quantum suna amfani da matsayi mai mahimmanci da haɗin kai don cimma saurin gudu a ayyukan lissafi daban-daban. Wannan binciken ya mayar da hankali kan tafiyar quantum na lokaci-daban-daban (DTQWs) kuma musamman yana magance kalubalen samar da matsakaicin haɗin kudi-matsayi ba tare da la'akari da yanayin farko ba.

Babban ƙirƙira da aka gabatar anan shine haɓaka jerin tsabar kuɗi masu inganci waɗanda ke ba da garantin matsakaicin samarwar haɗin kai ga kowane mataki bayan na biyu, suna cin nasara akan iyakokin da suka gabata waɗanda ke buƙatar ko dai takamaiman jihohin farko ko hanyoyin asymptotic. Wannan aikin yana haɗa ingantaccen ka'idoji tare da tabbacin gwaji ta amfani da haske mai layi.

2. Hanyar Aiki

2.1 Tsarin Tafiyar Quantum

Tafiyar quantum na lokaci-daban-daban tana aiki a cikin sararin Hilbert $\mathcal{H} = \mathcal{H}_c \otimes \mathcal{H}_p$, inda $\mathcal{H}_c$ shine sararin tsabar kudi (yawanci mai girma biyu) kuma $\mathcal{H}_p$ shine sararin matsayi. Juyin halitta a kowane mataki yana ƙarƙashin mai aikin unitary $\hat{U} = \hat{S}(\hat{C} \otimes \hat{I})$, inda $\hat{S}$ shine mai aikin motsi kuma $\hat{C}$ shine mai aikin tsabar kudi.

2.2 Jerin Ayyukan Tsabar Kudi

Muna amfani da dabarar inda aikin tsabar kudi a kowane mataki aka zaɓa bazuwa daga cikin saitin masu aiki biyu: ƙofar Hadamard $\hat{H}$ da ƙofar ainihi $\hat{I}$. Wannan jerin yayi daidai da tafiyar giwa ta gabaɗaya kuma yana ba da damar samar da haɗin kai mai ƙarfi ba tare da la'akari da yanayin tsabar kudi na farko ba.

3. Aiwar Fasaha

3.1 Tsarin Lissafi

Babban mai aikin tsabar kudi na SU(2) an ƙayyade shi azaman:

$$\hat{C}(\xi, \gamma, \zeta) = \begin{pmatrix} e^{i\xi}\cos\gamma & e^{i\zeta}\sin\gamma \\ e^{-i\zeta}\sin\gamma & -e^{-i\xi}\cos\gamma \end{pmatrix}, \quad \gamma, \xi, \zeta \in [0, 2\pi]$$

Don matsakaicin samarwar haɗin kai, muna amfani musamman da mai aikin Hadamard $\hat{H} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ da mai aikin ainihi $\hat{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ a cikin jerin da aka inganta.

3.2 Hanyar Ingantawa

An tsara matsakaicin samarwar haɗin kai azaman matsalar ingantawa tare da amincin tsarin quantum azaman aikin farashi. Ingantawar tana gano jerin tsabar kuɗi waɗanda ke ƙara haɗin kai ga kowane lambar mataki $T \geq 3$, suna cimma sakamakon da ba su da alaƙa da shirye-shiryen jihar farko.

4. Sakamakon Gwaji

4.1 Aiwar Haske Mai Layi

Mun gwada tafiyar quantum na mataki goma ta amfani da haske mai layi. Saitin ya yi amfani da farantin raƙuman ruwa da masu nisa don aiwatar da ayyukan tsabar kudi da ayyukan motsi, tare da photons ɗaya suna aiki azaman masu tafiya. Tsarin gwaji ya ba da ikon sarrafa daidai jerin tsabar kudi da kuma auna daidai haɗin da aka samu.

4.2 Auna Haɗin Kai

An ƙididdige haɗin da aka samu ta amfani da ma'auni na haɗin kai da ma'auni. Ga tafiyar mataki goma tare da jerin tsabar kuɗi masu inganci, mun lura da kusan matsakaicin ƙimar haɗin kai wanda ya wuce 0.98 ga duk jihohin farko da aka gwada. Rarraba yuwuwar ya nuna siffa ta saurin yaduwa da ke da alaƙa da tafiyar giwa ta gabaɗaya.

5. Bincike & Tattaunawa

Wannan bincike yana wakiltar ci gaba mai mahimmanci a cikin samar da haɗin kai na tushen tafiyar quantum, yana magance manyan iyakoki guda biyu na hanyoyin da suka gabata: dogaro da lambar mataki da kuma hankalin jihar farko. Tsarin ingantawa da aka haɓaka anan yana ɗaukar matsakaicin samarwar haɗin kai a matsayin matsalar ingantaccen tsarin quantum, yana ba da jerin tsabar kuɗi waɗanda ke ba da garantin babban haɗin kai ga kowane mataki bayan na biyu.

Idan aka kwatanta da aikin baya kan tafiyar quantum mara tsari wanda ya cimma samarwar haɗin kai na asymptotic, hanyarmu tana ba da mafita masu iyaka waɗanda za a iya amfani da su nan da nan a cikin saitunan gwaji. Daidaiton tsakanin jerin tsabar kuɗi masu inganci da tafiyar giwa ta gabaɗaya yana bayyana wata alaƙa mai ban sha'awa tsakanin samar da haɗin kai da kaddarorin sufuri, musamman saurin yaduwa da aka lura a sararin matsayi.

Tabbacin gwaji ta amfani da haske mai layi yana nuna yuwuwar aiwatar da waɗannan jerin masu inganci tare da fasahohin quantum na yanzu. Kamar yadda aka lura a cikin cikakken bita na tafiyar quantum ta Venegas-Andraca (2012), ikon samar da haɗin kai mai girma da aminci yana da mahimmanci don ci gaba da ka'idojin sadarwar quantum. Aikinmu ya yi daidai da babban yanayin a kimiyyar bayanan quantum zuwa ga haɓaka ka'idoji masu ƙarfi, masu zaman kansu na jihar farko, kama da ci gaban a cikin gyaran kuskuren quantum da lissafin quantum mai jurewa.

Daga mahangar fasaha, amfani da ayyukan Hadamard da ainihi a cikin jerin mafi kyau yana da mahimmanci musamman. Wannan sauƙi yana haɓaka yuwuwar gwaji yayin da yake kiyaye aikin ka'idoji, yana tunawa da ƙaramin hanyar da aka gani a cikin ayyukan farko kamar takardar CycleGAN (Zhu et al., 2017), inda zaɓuɓɓukan gine-gine masu sauƙi suka haifar da sakamako masu ƙarfi. Tsarin lissafi ta amfani da ƙayyadaddun SU(2) yana ba da cikakken tsarin da za a iya ƙaddamar zuwa ga ƙarin ayyukan tsabar kudi masu rikitarwa a aikin gaba.

Tasirin wannan binciken ya wuce ilimin kimiyyar quantum na asali zuwa fasahohin quantum masu amfani. Yayin da dandamalin lissafin quantum suka balaga, ikon samar da jihohin haɗin kai masu girma da aminci ya zama mahimmanci ga hanyar sadarwar quantum, rarraba lissafin quantum, da ƙaddamar da azanci ta quantum. Hanyarmu tana ba da tsari na tsari don cimma wannan manufa ta amfani da tafiyar quantum, waɗanda a zahiri ana iya aiwatar da su akan dandamali na kayan aikin quantum daban-daban ciki har da tsarin hoto, ions da aka kama, da qubits masu ƙarfi.

6. Aiwar Lambar

A ƙasa akwai misalin lambar Python da ke nuna kwaikwayon tafiyar quantum tare da jerin tsabar kuɗi masu inganci:

import numpy as np
from qutip import basis, tensor, sigmax, qeye

def hadamard():
    return 1/np.sqrt(2) * np.array([[1, 1], [1, -1]])

def identity():
    return np.array([[1, 0], [0, 1]])

def shift_operator(position_space):
    # Ƙirƙiri mai aikin motsi wanda ke motsa |0> zuwa dama, |1> zuwa hagu
    S_pos = np.zeros((position_space, position_space))
    for i in range(position_space-1):
        S_pos[i+1, i] = 1  # Motsa dama
        S_pos[i, i+1] = 1  # Motsa hagu
    return S_pos

def quantum_walk_step(psi, coin_op, shift_op, position_dim):
    # Aiwatar da aikin tsabar kudi
    coin_full = np.kron(coin_op, np.eye(position_dim))
    psi_after_coin = coin_full @ psi
    
    # Aiwatar da aikin motsi
    shift_full = np.kron(np.eye(2), shift_op)
    psi_after_shift = shift_full @ psi_after_coin
    
    return psi_after_shift

def calculate_entanglement(state, coin_dim, position_dim):
    # Ƙididdige yanayin haɗin kai
    density_matrix = np.outer(state, state.conj())
    reduced_density = partial_trace(density_matrix, [coin_dim, position_dim])
    eigenvalues = np.linalg.eigvalsh(reduced_density)
    entropy = -np.sum(eigenvalues * np.log2(eigenvalues + 1e-12))
    return entropy

# Misali: Tafiyar mataki 10 tare da jerin tsabar kudi mafi kyau
coin_sequence = [hadamard(), identity(), hadamard(), identity(), 
                 hadamard(), identity(), hadamard(), identity(),
                 hadamard(), identity()]

# Fara jihar quantum
initial_coin = 1/np.sqrt(2) * np.array([1, 1])  # Jihar |+>
initial_position = basis(21, 10)  # Fara a tsakiya
psi = np.kron(initial_coin, initial_position)

position_dim = 21
shift_op = shift_operator(position_dim)

# Aiwar tafiyar quantum
entanglement_values = []
for step, coin_op in enumerate(coin_sequence):
    psi = quantum_walk_step(psi, coin_op, shift_op, position_dim)
    entropy = calculate_entanglement(psi, 2, position_dim)
    entanglement_values.append(entropy)
    print(f"Mataki {step+1}: Yanayin haɗin kai = {entropy:.4f}")

7. Aikace-aikacen Gaba

Ƙarfin samar da matsakaicin haɗin kudi-matsayi da aminci yana da yuwuwar aikace-aikace masu yawa:

• Sadarwar Quantum: Jihohin haɗin kai masu girma zasu iya haɓaka ƙarfin tashoshi a cikin ka'idojin rarraba maɓalli na quantum.
• Lissafin Quantum: Tafiyar quantum suna aiki azaman samfuran lissafi na duniya, kuma samar da haɗin kai da aminci yana da mahimmanci ga hadaddun algorithms na quantum.
• Kwaikwayon Quantum: Jerin da aka inganta zasu iya kwaikwayon hadaddun tsarin quantum tare da ingantattun kaddarorin haɗin kai.
• Ƙididdigar Quantum: Jihohin haɗin kai suna ba da damar auna ma'auni fiye da iyakokin al'ada, tare da aikace-aikace a cikin azanci da hoto.
• Hanyoyin Sadarwar Quantum: 'Yancin jihar farko yana sa waɗannan ka'idoji su zama masu ƙarfi don sarrafa bayanan quantum da aka rarraba.

Hanyoyin bincike na gaba sun haɗa da faɗaɗa wannan hanyar zuwa sararin tsabar kudi mafi girma, bincika yanayin masu tafiya da yawa, da bincika aikace-aikace a cikin lissafin quantum na topological da koyon injin quantum.

8. Bayanan Kara

1. Venegas-Andraca, S. E. (2012). Tafiyar quantum: cikakken bita. Sarrafa Bayanan Quantum, 11(5), 1015-1106.
2. Kitagawa, T., et al. (2010). Bincika matakan topological tare da tafiyar quantum. Physical Review A, 82(3), 033429.
3. Zhu, J. Y., et al. (2017). Fassarar hoto-zuwa-hoto mara biyu ta amfani da hanyoyin sadarwa masu jujjuyawar zagayowar. Proceedings of the IEEE international conference on computer vision.
4. Nayak, A., & Vishwanath, A. (2000). Tafiyar quantum akan layin. arXiv preprint quant-ph/0010117.
5. Ambainis, A. (2003). Tafiyar quantum da aikace-aikacen algorithmic su. International Journal of Quantum Information, 1(04), 507-518.
6. Childs, A. M. (2009). Lissafi na duniya ta tafiyar quantum. Physical review letters, 102(18), 180501.
7. Asboth, J. K., & Edge, J. M. (2015). Gabatarwar gajeriyar gabatarwa ga matakan topological na photons. International Journal of Modern Physics B, 29(21), 1530006.