"Ina tsammanin zan iya faɗi cikin aminci cewa babu wanda ya fahimci injiniyoyin ƙididdiga." - Richard Feynman
Batun lissafin ƙididdiga ya kasance yana burge marubutan fasaha da ƴan jarida koyaushe. Ƙarfin lissafinsa da sarƙaƙƙiyarsa sun ba shi wani ƙaƙƙarfan aura. Sau da yawa, abubuwan da ke da alaƙa da bayanan bayanai suna bayyana dalla-dalla dalla-dalla iri-iri iri-iri na wannan masana'antar, yayin da kawai ke taɓa aikace-aikacen sa: wannan na iya ɓatar da mai karatu mai hankali.
Shahararrun labaran kimiyya sun watsar da bayanin tsarin ƙididdiga kuma suna yin maganganu kamar:
Bitar na yau da kullun na iya zama 1 ko 0, amma qubit na iya zama 1 da 0 a lokaci guda.
Idan kun yi sa'a (wanda ban tabbata ba), za a gaya muku cewa:
Qubit yana cikin matsayi mai girma tsakanin "1" da "0".
Babu ɗaya daga cikin waɗannan bayanan da ke da alama, tun da muna ƙoƙarin ƙirƙira ƙirar ƙira ta hanyar amfani da harshe da aka haɓaka a cikin duniyar gargajiya. Don bayyana ƙa'idodin ƙididdiga na ƙididdiga, wajibi ne a yi amfani da wani harshe - lissafi.
A cikin wannan koyawa, zan rufe kayan aikin lissafin da ake buƙata don ƙira da fahimtar tsarin ƙididdiga na ƙididdigewa, da kuma yadda ake kwatantawa da amfani da dabaru na ƙididdigar ƙididdiga. Bugu da ƙari, zan ba da misali na ƙididdiga algorithm kuma in gaya muku menene fa'idarsa akan kwamfutar gargajiya.
Zan yi iya ƙoƙarina don bayyana duk wannan cikin bayyanannen harshe, amma har yanzu ina fatan masu karatun wannan labarin suna da fahimtar algebra madaidaiciya da dabaru na dijital (an rufe algebra madaidaiciya. , game da dabaru na dijital - ).
Da farko, bari mu wuce ƙa'idodin dabaru na dijital. Ya dogara ne akan amfani da na'urorin lantarki don aiwatar da lissafi. Don sanya bayanin mu ya zama mai ma'ana, bari mu sauƙaƙa yanayin wayar lantarki zuwa "1" ko "0", wanda zai dace da jihohi "a kunne" ko "kashe". Ta hanyar tsara transistor a cikin wani jeri, za mu ƙirƙiri abubuwan da ake kira abubuwan dabaru waɗanda ke ɗaukar ƙimar siginar shigarwa ɗaya ko fiye kuma mu canza su zuwa siginar fitarwa dangane da wasu ƙa'idodi na dabaru na Boolean.
Ƙofofin dabaru na gama-gari da teburan jihohinsu
Dangane da sarƙoƙin irin waɗannan abubuwa na asali, ana iya ƙirƙirar abubuwa masu rikitarwa, kuma bisa ga sarƙoƙi na abubuwan da suka fi rikitarwa, zamu iya ƙarshe, tare da babban matakin abstraction, muna tsammanin samun analogue na na'ura ta tsakiya.
Kamar yadda na ambata a baya, muna buƙatar wata hanya don wakiltar dabaru na dijital ta hanyar lissafi. Da farko, bari mu gabatar da dabaru na gargajiya na lissafi. Yin amfani da algebra na linzamin kwamfuta, za a iya wakilta raƙuman raƙuman da ke da kimar "1" da "0" a matsayin ginshiƙai guda biyu:
inda lambobin hagu suke vector. Ta hanyar wakiltar raƙuman mu ta wannan hanya, za mu iya ƙirƙira ayyuka masu ma'ana akan raƙuman ruwa ta amfani da canjin vector. Lura: ko da yake yin amfani da ragowa biyu a cikin ƙofofin dabaru na iya yin ayyuka da yawa (DA, BA, XOR, da sauransu), lokacin amfani da bit ɗaya, ana iya aiwatar da ayyukan guda huɗu kawai: canjin ainihi, ƙididdigewa, ƙididdige ƙimar “0” akai-akai da lissafin "1" akai-akai. Tare da tuba na ainihi, bit ɗin ya kasance baya canzawa, tare da ƙin yarda, ƙimar bit ɗin ta canza zuwa akasin haka (daga "0" zuwa "1" ko daga "1" zuwa "0"), da lissafin madaidaicin "1" ko "0" yana saita bit zuwa "1" ko "0" ba tare da la'akari da ƙimarsa ta baya ba.
| Identity | Canji na ainihi |
| da Negation | Rashin damuwa |
| Tsayawa-0 | Lissafi na akai-akai "0" |
| Tsayawa-1 | Lissafi na akai-akai "1" |
Dangane da sabon wakilcin da muka gabatar na ɗan kaɗan, yana da sauƙin aiwatar da ayyuka akan abin da ya dace ta amfani da canjin vector:
Kafin mu ci gaba, bari mu dubi manufar , wanda kawai ke nuna cewa don tabbatar da jujjuyawar aiki ko ma'auni, ya zama dole a ƙayyade jerin ƙimar siginar shigarwa dangane da siginar fitarwa da sunayen ayyukan da aka yi amfani da su. Don haka, zamu iya yanke shawarar cewa canji na ainihi da ƙin yarda suna canzawa, amma ayyukan ƙididdige ƙididdiga "1" da "0" ba su kasance ba. Godiya ga Makanikan ƙididdiga, kwamfutoci masu ƙididdigewa suna amfani da ayyuka masu juyawa na musamman, don haka abin da za mu mai da hankali a kai ke nan. Bayan haka, muna canza abubuwan da ba za su iya jurewa ba zuwa abubuwan da za su iya juyar da su don ba da damar yin amfani da su ta hanyar kwamfutoci masu yawa.
Tare da taimakon za a iya wakilta rago ɗaya da ragi da yawa:
Yanzu da muke da kusan dukkanin mahimman ra'ayoyin ilimin lissafi, bari mu matsa zuwa ƙofar mu ta ƙididdigewa ta farko. Wannan shi ne ma'aikacin , ko Ba a sarrafa shi ba (BA), wanda yake da mahimmanci a cikin juzu'i da ƙididdiga masu yawa. Abun CNOT ya shafi rago biyu kuma ya dawo da rago biyu. Na farko bit an sanya shi a matsayin "control" bit, kuma na biyu a matsayin "control" bit. Idan an saita bit ɗin sarrafawa zuwa "1", bit ɗin sarrafawa yana canza ƙimarsa; Idan an saita bit ɗin sarrafawa zuwa "0", ba a canza bit ɗin sarrafawa ba.
Ana iya wakilta wannan ma'aikacin azaman vector canji mai zuwa:
Don nuna duk abin da muka rufe zuwa yanzu, zan nuna muku yadda ake amfani da kashi na CNOT akan ragi da yawa:
Don taƙaita abin da aka riga aka faɗa: a cikin misali na farko mun bazu | 10⟩ cikin sassa na samfuran tensor ɗin sa kuma amfani da matrix CNOT don samun sabon yanayin yanayin samfurin; sai mu sanya shi zuwa |11⟩ bisa ga tebur na ƙimar CNOT da aka bayar a baya.
Don haka, mun tuna da duk ƙa'idodin lissafin da za su taimaka mana mu fahimci lissafin gargajiya da na yau da kullun, kuma a ƙarshe za mu iya ci gaba zuwa ƙididdige ƙididdiga na zamani da qubits.
Idan kun karanta wannan zuwa yanzu, to ina da albishir a gare ku: ana iya bayyana qubits cikin sauƙi ta hanyar lissafi. Gabaɗaya, idan za'a iya saita bit (cbit) na gargajiya zuwa |1⟩ ko |0⟩, qubit ɗin yana cikin babban matsayi kuma yana iya zama duka | 0⟩ da |1⟩ kafin aunawa. Bayan an auna, yana faduwa zuwa |0⟩ ko |1⟩. A wasu kalmomi, ana iya wakilta qubit a matsayin haɗin kai tsaye na |0⟩ da |1⟩ bisa ga dabarar da ke ƙasa:
inda a₀ и a₁ wakiltar, bi da bi, amplitudes | 0⟩ da | 1⟩. Ana iya la'akari da waɗannan a matsayin "ƙirar quantum", wanda ke wakiltar yiwuwar faɗuwar qubit zuwa ɗaya daga cikin jihohi bayan an auna shi, tun da yake a cikin injiniyoyin ƙididdiga wani abu a cikin babban matsayi ya rushe zuwa ɗaya daga cikin jihohi bayan an gyara shi. Bari mu fadada wannan magana mu sami abubuwa masu zuwa:
Don sauƙaƙe bayanina, wannan shine wakilcin da zan yi amfani da shi a cikin wannan labarin.
Don wannan qubit, damar rugujewa zuwa ƙimar a₀ bayan ma'auni daidai yake da |a₀|², da damar rugujewa zuwa ƙima a₁ daidai yake da |a₁|². Misali, don qubit mai zuwa:
damar rugujewa zuwa “1” daidai yake da |1/√2|², ko ½, wato, 50/50.
Tun da a cikin tsarin gargajiya dole ne a haɗa dukkan yuwuwar har zuwa ɗaya (don cikakkiyar rarrabawar yiwuwar), zamu iya yanke shawarar cewa murabba'ai na cikakkiyar ƙimar amplitudes | 0⟩ da | 1⟩ dole ne su ƙara zuwa ɗaya. Dangane da wannan bayanin za mu iya tsara ma'auni mai zuwa:
Idan kun saba da trigonometry, zaku lura cewa wannan ma'auni yayi daidai da ka'idar Pythagorean (a²+b²=c²), wato, zamu iya wakiltan yuwuwar jihohin qubit a matsayin maki akan da'irar naúrar, wato:
Ana amfani da ma'aikata masu ma'ana da abubuwa zuwa qubits kamar yadda yake a cikin halin da ake ciki tare da ragi na gargajiya - dangane da canjin matrix. Duk ma'aikatan matrix masu jujjuyawar da muka tuna zuwa yanzu, musamman CNOT, ana iya amfani da su don aiki tare da qubits. Irin waɗannan masu sarrafa matrix suna ba ku damar amfani da kowane girman qubit ba tare da aunawa da rushe shi ba. Bari in ba ku misali na amfani da ma'aikacin negation akan qubit:
Kafin mu ci gaba, bari in tunatar da ku cewa girman darajar a₀ kuma a₁ a zahiri , don haka ana iya tsara yanayin qubit daidai a kan yanki mai girma uku, wanda kuma aka sani da :
Koyaya, don sauƙaƙe bayanin, zamu iyakance kanmu anan zuwa lambobi na gaske.
Yana da lokaci don tattauna wasu abubuwa masu ma'ana waɗanda ke da ma'ana kawai a cikin mahallin lissafin ƙididdiga.
Ɗaya daga cikin mafi mahimmancin masu aiki shine "Hadamard element": yana ɗaukar ɗan lokaci a cikin "0" ko "1" jihar kuma yana sanya shi a cikin matsayi mai dacewa tare da damar 50% na rushewa zuwa "1" ko "0" bayan aunawa.
Lura cewa akwai lambar mara kyau a gefen dama na ma'aikacin Hadamard. Wannan shi ne saboda sakamakon yin amfani da mai aiki ya dogara da ƙimar siginar shigarwa: - |1⟩ ko |0⟩, don haka lissafin yana canzawa.
Wani muhimmin batu game da sinadarin Hadamard shine jujjuyawar sa, ma'ana yana iya daukar qubit a cikin yanayin da ya dace ya canza shi zuwa |0⟩ ko |1⟩.
Wannan yana da mahimmanci sosai domin yana ba mu ikon canzawa daga yanayin ƙididdigewa ba tare da ƙayyade yanayin qubit ba - kuma, bisa ga haka, ba tare da rushe shi ba. Don haka, zamu iya tsara ƙididdigar ƙididdiga bisa ƙayyadaddun ƙayyadaddun ƙayyadaddun ƙayyadaddun ƙa'ida maimakon ƙa'ida mai yiwuwa.
Ma'aikatan ƙididdiga masu ƙunshe da lambobi na gaske kawai akasin nasu ne, don haka za mu iya wakiltar sakamakon amfani da ma'aikacin zuwa qubit a matsayin canji a cikin da'irar naúrar ta hanyar injin jiha:
Don haka, qubit, yanayin da aka gabatar a cikin zanen da ke sama, bayan yin amfani da aikin Hadamard, an canza shi zuwa yanayin da aka nuna ta kibiya mai dacewa. Hakazalika, za mu iya gina wani na'ura na jiha wanda zai kwatanta canjin qubit ta amfani da ma'aikacin negation kamar yadda aka nuna a sama (wanda kuma aka sani da Pauli negation operator, ko bit inversion), kamar yadda aka nuna a kasa:
Don yin ƙarin hadaddun ayyuka akan qubit ɗin mu, za mu iya yin sarƙoƙi masu aiki da yawa ko kuma amfani da abubuwa sau da yawa. Misalin canji na serial bisa kama da wannan:
Wato idan muka fara da bit |0⟩ sai a shafa kadan daga nan sai a yi aikin Hadamard, sai kuma a sake yin wani aiki na Hadamard, sai a binne na karshe, sai mu karasa da vector din da aka bayar ta hanyar on. gefen dama na sarkar. Ta hanyar shimfiɗa injinan jihohi daban-daban a saman juna, za mu iya farawa a |0⟩ kuma mu bibiyi kibiyoyi masu launi daidai da kowane canji don fahimtar yadda duk suke aiki.
Tun da mun zo wannan nisa, lokaci ya yi da za a yi la'akari da ɗaya daga cikin nau'o'in ƙididdigar ƙididdiga, wato - , kuma ya nuna fa'idarsa akan kwamfuta ta gargajiya. Ya kamata a lura da cewa Deutsch-Jozsa algorithm ne gaba daya deterministic, wato, ya mayar da daidai amsar 100% na lokaci (sabanin sauran adadi algorithms dangane da yiwuwar ma'anar qubits).
Bari mu yi tunanin cewa kana da akwatin baƙar fata wanda ya ƙunshi aiki / mai aiki a kan guda ɗaya (tuna - tare da bit guda ɗaya, ayyuka hudu kawai za a iya yi: fassarar ainihi, rashin fahimta, kimantawa na "0" akai-akai da kimantawa na akai-akai "1". "). Menene ainihin aikin da aka yi a cikin akwatin? Ba ku san wanne ba, amma kuna iya bi ta yawancin bambance-bambancen ƙimar shigarwa kamar yadda kuke so kuma ku kimanta sakamakon fitarwa.
Nawa abubuwan shigarwa da fitarwa nawa za ku yi amfani da su ta cikin akwatin baƙar fata don gano wane aiki ake amfani da shi? Ka yi tunanin wannan na daƙiƙa guda.
A cikin yanayin kwamfutar gargajiya, kuna buƙatar yin tambayoyi 2 don tantance aikin da za ku yi amfani da su. Misali, idan shigar da “1” ta samar da fitowar “0”, zai bayyana a fili cewa ko dai aikin lissafin “0” akai-akai ko kuma ana amfani da aikin negation, bayan haka dole ne a canza darajar siginar shigarwa. zuwa "0" kuma duba abin da ke faruwa a wurin fita.
Dangane da kwamfuta mai ƙididdigewa, kuma za a buƙaci tambayoyi guda biyu, tunda har yanzu kuna buƙatar ƙimar fitarwa daban-daban guda biyu don ayyana daidai aikin don amfani da ƙimar shigarwar. Koyaya, idan kun sake fasalin tambayar kaɗan, ya zama cewa kwamfutoci masu ƙididdigewa har yanzu suna da fa'ida sosai: idan kuna son sanin ko aikin da ake amfani da shi na dindindin ne ko kuma mai canzawa, kwamfutoci masu yawa zasu sami fa'ida.
Ayyukan da aka yi amfani da su a cikin akwatin yana da canji idan dabi'u daban-daban na siginar shigarwar sun haifar da sakamako daban-daban a wurin fitarwa (misali, canjin ainihi da bit inversion), kuma idan ƙimar fitarwa ba ta canzawa ba tare da la'akari da ƙimar shigarwa ba, to aiki akai akai (misali, ƙididdige "1" akai-akai ko ƙididdige "0").
Yin amfani da ƙididdiga algorithm, za ku iya ƙayyade ko aiki a cikin akwatin baƙar fata yana dawwama ko canzawa bisa tambaya ɗaya kawai. Amma kafin mu kalli yadda ake yin hakan dalla-dalla, muna buƙatar nemo hanyar da za mu tsara kowane ɗayan waɗannan ayyuka akan kwamfuta ta ƙididdigewa. Tun da kowane ma'aikacin adadi dole ne ya zama mai jujjuyawa, nan da nan za mu fuskanci matsala: ayyuka don ƙididdige adadin "1" da "0" ba su kasance ba.
Magani na gama gari da ake amfani da shi a cikin ƙididdiga na ƙididdigewa shine ƙara ƙarin abin fitarwa wanda ke dawo da duk ƙimar shigarwar da aikin ya karɓa.
| Zuwa ga: | Bayan: |
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Ta wannan hanyar, za mu iya ƙayyade ƙimar shigarwar kawai bisa ƙimar fitarwa, kuma aikin ya zama mai jujjuyawa. Tsarin da'irori na ƙididdigewa yana haifar da buƙatar ƙarin abin shigarwa. Don haɓaka masu aiki masu dacewa, za mu ɗauka cewa ƙarin shigar qubit an saita zuwa | 0⟩.
Yin amfani da wakilcin da'ira iri ɗaya wanda muka yi amfani da shi a baya, bari mu ga yadda kowane ɗayan abubuwa guda huɗu (canji na ainihi, rashin daidaituwa, kimantawa na "0" akai-akai da kimantawa na dindindin "1") za'a iya aiwatar da su ta amfani da ma'aikatan ƙididdiga.
Misali, wannan shine yadda zaku iya aiwatar da aikin don ƙididdige “0” akai-akai:
Lissafi na akai-akai "0":
Anan ba ma buƙatar masu aiki kwata-kwata. Qubit na farko (wanda muka ɗauka shine |0⟩) yana dawowa da ƙima ɗaya, kuma ƙimar shigarwa ta biyu ta dawo kanta - kamar yadda aka saba.
Tare da aikin ƙididdige "1" akai-akai yanayin ya ɗan bambanta:
Lissafi na akai-akai "1":
Tunda mun zaci cewa qubit na farko koyaushe yana saita zuwa | 0⟩, sakamakon amfani da na'urar inversion ta atomatik shine koyaushe yana samar da guda ɗaya a wurin fitarwa. Kuma kamar yadda aka saba, qubit na biyu yana ba da ƙimarsa a wurin fitarwa.
Lokacin yin taswirar ma'aikacin canjin ainihi, aikin zai fara zama mai rikitarwa. Ga yadda za a yi:
Sauyi iri ɗaya:
Alamar da aka yi amfani da ita a nan tana nuna nau'in CNOT: layi na sama yana nuna maɓallin sarrafawa, kuma layin ƙasa yana nuna maɓallin sarrafawa. Bari in tunatar da ku cewa lokacin amfani da afaretan CNOT, ƙimar sarrafawar bit ɗin yana canzawa idan bit ɗin sarrafawa daidai yake da | 1⟩, amma ya kasance baya canzawa idan bit ɗin sarrafawa yayi daidai da | 0⟩. Tunda mun zaci cewa darajar saman layi koyaushe daidai yake da |0⟩, ana sanya darajarsa koyaushe zuwa layin ƙasa.
Muna ci gaba ta hanya irin wannan tare da ma'aikacin negation:
Karya:
Mu kawai mu juya da bit a karshen fitarwa line.
Yanzu da muka sami wannan fahimtar ta farko, bari mu kalli takamaiman fa'idodin da ke tattare da kwamfuta ta ƙididdiga akan kwamfutar gargajiya idan aka zo ga tantance dawwama ko bambancin aikin da ke ɓoye a cikin akwatin baƙi ta amfani da tambaya ɗaya kawai.
Don magance wannan matsala ta amfani da ƙididdigar ƙididdiga a cikin buƙatu ɗaya, dole ne a sanya qubits ɗin shigarwa a cikin babban matsayi kafin a wuce su zuwa aikin, kamar yadda aka nuna a ƙasa:
Ana sake yin amfani da kashi na Hadamard zuwa sakamakon aikin don karya qubits daga babban matsayi kuma sanya algorithm mai kayyadewa. Mun fara tsarin a jihar | 00⟩ kuma, saboda dalilai zan yi bayani ba da jimawa ba, samun sakamakon | 11⟩ idan aikin da aka yi amfani da shi ya kasance akai-akai. Idan aikin da ke cikin akwatin baƙar fata yana canzawa, to bayan an auna tsarin yana dawo da sakamakon | 01⟩.
Don fahimtar sauran labarin, bari mu dubi misalin da na nuna a baya:
Ta amfani da na'urar inversion bit sannan kuma amfani da sinadarin Hadamard zuwa ga ƙimar shigarwar biyu daidai da | 0⟩, muna tabbatar da cewa an fassara su zuwa matsayi iri ɗaya na | 0⟩ da | 1⟩, kamar haka:
Yin amfani da misalin ƙaddamar da wannan ƙimar zuwa aikin akwatin baƙar fata, yana da sauƙi a nuna cewa duka ƙimar ƙimar suna fitowa |11⟩.
Lissafi na akai-akai "0":
Hakazalika, mun ga cewa aikin lissafin “1” akai-akai shima yana samar da |11⟩ a matsayin fitarwa, wato:
Lissafi na akai-akai "1":
Lura cewa fitarwa zai kasance | 1⟩, tun -1² = 1.
Ta wannan ka'ida, zamu iya tabbatar da cewa lokacin amfani da ayyuka masu canzawa biyu, koyaushe za mu sami | 01⟩ a wurin fitarwa (idan muka yi amfani da hanya ɗaya), kodayake komai ya ɗan fi rikitarwa.
Sauyi iri ɗaya:
Tunda CNOT ma'aikacin qubit ne guda biyu, ba za'a iya wakilta shi azaman na'ura mai sauƙi na jiha ba, saboda haka yana da mahimmanci don ayyana siginar fitarwa guda biyu dangane da samfurin tensor na qubits ɗin shigarwa da ninkawa ta matrix CNOT kamar yadda aka bayyana a baya:
Tare da wannan hanyar za mu iya tabbatar da cewa ƙimar fitarwa | 01⟩ ana karɓar idan an ɓoye aikin ɓoye a cikin akwatin baƙar fata:
Karya:
Don haka, yanzu mun nuna yanayin da kwamfuta ta ƙididdigewa ta fi dacewa fiye da kwamfutar da aka saba.
Menene na gaba?
Ina ba da shawarar mu ƙare a nan. Mun riga mun yi babban aiki. Idan kun fahimci duk abin da na rufe, Ina tsammanin yanzu kuna da kyakkyawar fahimtar tushen ƙididdigar ƙididdiga da ƙididdigar ƙididdiga, kuma me yasa ƙididdigar ƙididdiga na iya zama mafi inganci fiye da lissafin gargajiya a wasu yanayi.
Ba za a iya kiran bayanina cikakken jagora ga ƙididdigar ƙididdiga da algorithms ba - a maimakon haka, taƙaitacciyar gabatarwa ce ga ilimin lissafi da rubutu, wanda aka tsara don kawar da ra'ayoyin masu karatu game da batun da mashahuran majiyoyin kimiyya suka ƙulla (hakika, da gaske da yawa ba za su iya fahimta ba. hali!). Ban sami lokacin da zan tabo batutuwa masu mahimmanci da yawa, kamar , rikitaccen ƙimar girman girman | 0⟩ da | 1⟩ da aiki na nau'ikan dabaru daban-daban yayin canji ta hanyar Bloch Sphere.
Idan kuna son tsarawa da tsara ilimin ku game da kwamfutoci masu yawa, cikin gaggawa Ina ba ku shawara ku karanta Emma Strubel: duk da ɗimbin darussan lissafi, wannan littafi yana magana ne akan algorithms na ƙididdigewa dalla-dalla.
source: www.habr.com