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Quantum Computing - Emerging Frontiers | PPTX
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Quantum Computing - Emerging Frontiers

The document discusses various aspects of quantum computing, focusing on its theoretical foundations, such as quantum gates, entangled states, and universal gate sets. It also examines the implications of quantum complexity theory, especially the distinctions between classical and quantum computations. Additionally, the text touches upon holonomic computing and the geometric properties that influence quantum systems and their capabilities.

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Q U A N T U M C O M P U T I N G E M E R G I N G F R O N T I E R S A N D F U T U R E P E R S P E C T I V E S
Q U A N T U M R E S I S T A N T L E D G E R
Q U A N T U M B L O C K C H A I N B Y R Q C Q U I D I T C O M P U T I N G M O D E L
Q U A N T U M B L O C K C H A I N A R C H I T E C T U R E
T O P O L O G I C A L Q U A N T U M C O M P U T I N G
T O P O L O G I C A L C O M P U T A T I O N
T O P O L O G I C A L D Y N A M I C S
H O L O N O M I C C O M P U T I N G
H O L O N O M I C E V O L U T I O N
H O L O N O M I C L O O P S
C O M P U T A T I O N I S A C O M B I N A T I O N O F A N A L G O R I T H M ( A M A T H E M A T I C A L O B J E C T ) A N D A C O M P U T I N G D E V I C E ( P H Y S I C A L O B J E C T ) . W H A T I S T H E B A S I C P R E M I S E O F C O M P U T A T I O N
I F A N A B S T R A C T C O M P U T I N G D E V I C E I S C O N S I D E R E D F O R C L A S S I C A L C O M P U T A T I O N ( T U R I N G M A C H I N E A B S T R A C T I O N ) , A P H Y S I C A L S Y S T E M O B E Y I N G L A W S O F Q U A N T U M P H Y S I C S I S C O N S I D E R E D A S C O M P U T I N G D E V I C E F O R Q U A N T U M C O M P U T I N G C L A S S I C A L V / S Q U A N T U M C O M P U T A T I O N
T H E C E N T R A L P R O B L E M I N C O M P U T I N G S C I E N C E S I S T H E C O N J E C T U R E T H A T T W O C O M P L E X I T Y C L A S S E S , P ( P O L Y N O M I A L T I M E ) A N D N P ( N O N D E T E R M I N I S T I C P O L Y N O M I A L T I M E ) A R E D I S T I N C T I N T H E S T A N D A R D T U R I N G M O D E L O F C O M P U T A T I O N . P V / S N P P R O B L E M
L A W S O F Q U A N T U M C O M P U T I N G E X H I B I T T H E M A T H E M A T I C A L F E A T U R E S N E C E S S A R Y T O S U P P O R T A M O D E L C A P A B L E O F S O L V I N G A L L P P R O B L E M S , A C O M P U T A T I O N A L L Y I N T R A C T A B L E C L A S S I N P O L Y N O M I A L T I M E .
A S E T O F Q U A N T U M G A T E S G A C T I N G O N A Q U B I T C I R C U I T M O D E L W I T H N Q U B I T S I S D E F I N E D T O B E A N - Q U I B I T U N I V E R S A L G A T E S E T I F F O R A N Y E > 0 , A S E Q U E N C E O F G A T E S F R O M T H I S S E T G C A N B E U S E D T O A P P R O X I M A T E A N Y U N I T A R Y E V O L U T I O N O N A L L N - Q U B I T S T O A C C U R A C Y E . D E F I N I T I O N O F A U N I V E R S A L G A T E S E T
T H E C O S T C A N B E P L A C E D O N T H E B R E D T H O F T H E C I R C U I T , T H E D E P T H O F T H E C I R C U I T , T H E T O T A L N U M B E R O F G A T E S U S E D E T C . C O S T F O R I M P L E M E N T I N G A U N I V E R S A L G A T E S E T
I T S E E K S T O U N D E R S T A N D H O W T H E C O S T F O R B U I L D I N G Q U A N T U M G A T E S G R O W F O R V A R I O U S Q U A N T U M A L G O R I T H M S Q U A N T U M C O M P U T A T I O N A L C O M P L E X I T Y
T H E D E P T H C O R R E S P O N D S T O T O T A L R U N N I N G T I M E O F T H E C I R C U I T . D E P T H O F Q U A N T U M C O M P U T I N G C I R C U I T
T H E O R E M T E L L S U S T H E D E P T H D I F F E R E N C E B E T W E E N C I R C U I T S C O N S T R U C T E D W I T H D I F F E R E N T U N I V E R S A L S E T S O F G A T E S T O A N A C C U R A C Y O F O ( L O G - C ( 1 / E ) ) S O L O V A Y - K I T A E V T H E O R E M
Q U A N T U M C O M P U T I N G I S P E R F O R M E D W I T H T H E A I D O F A Q U A N T U M S Y S T E M W H O S E E V O L U T I O N I S D E S C R I B E D B Y T H E U N I T A R Y O P E R A T O R U ( T ) = E X P - I H T W H E R E H I S T H E H A M I L T O N I A N O F T H E S Y S T E M . S Y S T E M R E Q U I R E M E N T S F O R T H E Q U A N T U M C O M P U T I N G
T H E S T A T E O F A C O M P O S I T E S Y S T E M T H A T C A N ’ T B E W R I T T E N A S T H E P R O D U C T O F S T A T E S O F I T S C O M P O N E N T S Y S T E M I S C A L L E D E N T A N G L E D S T A T E . E N T A N G L E D S T A T E O F A Q U A N T U M C O M P U T E R
B E L L B A S I S I S A O R T H O N O R M A L B A S I S F O R T H E T W O Q U B I T S T A T E S P A C E B E L L S T A T E S
A T W O Q U B I T G A T E A I S A U N I V E R S A L F O R T H E Q U A N T U M C O M P U T A T I O N I F A T O G E T H E R W I T H L O C A L U N I T A R Y T R A N S F O R M A T I O N S G E N E R A T E S A L L U N I T A R Y T R A N S F O R M A T I O N S O F T H E C O M P L E X V E C T O R S P A C E O F D I M E N S I O N 2 - E X P - N T O I T S E L F . U N I V E R S A L Q U A N T U M G A T E
A G A T E A I S S A I D T O B E E N T A N G L I N G I F T H E R E I S A V E C T O R | A B > = | A > X | B > S U C H T H A T A | A B > I S N O T D E C O M P O S A B L E A S A T E N S O R P R O D U C T O F T W O Q U B I T S . P R O O F O F E N T A N G L E M E N T
T W O Q U B I T G A T E A I S U N I V E R S A L I F A N D O N L Y I F I T I S E N T A N G L I N G . U N I V E R S A L I T Y A N D E N T A N G L E M E N T
A P H A S E I S N O T A S T A T E O F M A T T E R B U T A C O M P L E X N U M B E R O F U N I T M O D U L E S . T H E P H A S E S I S O B T A I N E D W I T H T H E C Y C L I C E V O L U T I O N O F A P H Y S I C A L S Y S T E M . G E O M E T R I C C H A R A C T E R O F Q U A N T U M C O M P U T I N G
C Y C L I C V A R I A T I O N O F E X T E R N A L P A R A M E T E R S O F T E N L E A D T O A N E T E V O L U T I O N I N V O L V I N G A P H A S E D E P E N D I N G O N L Y O N T H E G E O M E T R Y O F T H E P A T H T R A V E R S E D I N T H E P A R A M E T E R S P A C E . E V O L U T I O N O F P H A S E P R O P E R T Y
P R E S E N C E O F G E O M E T R I C P H A S E I S Q U A N T U M S Y S T E M S A N D I T S P R O P E R T I E S H A S B E E N U S E D T O C O N S T R U C T A N E W M O D E L F O R Q U A N T U M C O M P U T I N G . H O L O N O M Y I S A P H A S E T H A T D E P E N D S O N T H E G E O M E T R Y O F A L O O P A N D I S I N D E P E N D E N T O F A N Y C O O R D I N A T E C H O I C E . H O L O N O M I C Q U A N T U M C O M P U T I N G
C L A S S I C A L G A M E S C A N N O T B E Q U A N T I Z E D I N A U N I Q U E W A Y B E C A U S E T H E Y A R E O N L Y A S Y M P T O T I C A L S H A D O W S O F A W I D E S P E C T R A O F Q U A N T U M M O D E L S . F O L L O W I N G T W O S T A G E S A R E I M P O R T A N T : 1 . P R E - Q U A N T I Z A T I O N 2 . Q U A N T I Z A T I O N Q U A N T U M G A M E T H E O R Y
Q F T D O E S N O T E X P L I C I T L Y P R O D U C E A N Y O F T H E F O U R I E R C O E F F I C I E N T S . Q F T C A N B E R E G A R D E D A S C O M P U T I N G A L L T H E P R O B A B I L I T I E S O F A P R O B A B I L I T Y D I S T R I B U T I O N O R S A M P L I N G A D I S T R I B U T I O N Q U A N T U M F O U R I E R T R A N S F O R M

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Quantum Computing - Emerging Frontiers

  • 1.
    Q U AN T U M C O M P U T I N G E M E R G I N G F R O N T I E R S A N D F U T U R E P E R S P E C T I V E S
  • 2.
    Q U AN T U M R E S I S T A N T L E D G E R
  • 3.
    Q U AN T U M B L O C K C H A I N B Y R Q C Q U I D I T C O M P U T I N G M O D E L
  • 4.
    Q U AN T U M B L O C K C H A I N A R C H I T E C T U R E
  • 6.
    T O PO L O G I C A L Q U A N T U M C O M P U T I N G
  • 7.
    T O PO L O G I C A L C O M P U T A T I O N
  • 8.
    T O PO L O G I C A L D Y N A M I C S
  • 9.
    H O LO N O M I C C O M P U T I N G
  • 10.
    H O LO N O M I C E V O L U T I O N
  • 11.
    H O LO N O M I C L O O P S
  • 12.
    C O MP U T A T I O N I S A C O M B I N A T I O N O F A N A L G O R I T H M ( A M A T H E M A T I C A L O B J E C T ) A N D A C O M P U T I N G D E V I C E ( P H Y S I C A L O B J E C T ) . W H A T I S T H E B A S I C P R E M I S E O F C O M P U T A T I O N
  • 13.
    I F AN A B S T R A C T C O M P U T I N G D E V I C E I S C O N S I D E R E D F O R C L A S S I C A L C O M P U T A T I O N ( T U R I N G M A C H I N E A B S T R A C T I O N ) , A P H Y S I C A L S Y S T E M O B E Y I N G L A W S O F Q U A N T U M P H Y S I C S I S C O N S I D E R E D A S C O M P U T I N G D E V I C E F O R Q U A N T U M C O M P U T I N G C L A S S I C A L V / S Q U A N T U M C O M P U T A T I O N
  • 14.
    T H EC E N T R A L P R O B L E M I N C O M P U T I N G S C I E N C E S I S T H E C O N J E C T U R E T H A T T W O C O M P L E X I T Y C L A S S E S , P ( P O L Y N O M I A L T I M E ) A N D N P ( N O N D E T E R M I N I S T I C P O L Y N O M I A L T I M E ) A R E D I S T I N C T I N T H E S T A N D A R D T U R I N G M O D E L O F C O M P U T A T I O N . P V / S N P P R O B L E M
  • 15.
    L A WS O F Q U A N T U M C O M P U T I N G E X H I B I T T H E M A T H E M A T I C A L F E A T U R E S N E C E S S A R Y T O S U P P O R T A M O D E L C A P A B L E O F S O L V I N G A L L P P R O B L E M S , A C O M P U T A T I O N A L L Y I N T R A C T A B L E C L A S S I N P O L Y N O M I A L T I M E .
  • 16.
    A S ET O F Q U A N T U M G A T E S G A C T I N G O N A Q U B I T C I R C U I T M O D E L W I T H N Q U B I T S I S D E F I N E D T O B E A N - Q U I B I T U N I V E R S A L G A T E S E T I F F O R A N Y E > 0 , A S E Q U E N C E O F G A T E S F R O M T H I S S E T G C A N B E U S E D T O A P P R O X I M A T E A N Y U N I T A R Y E V O L U T I O N O N A L L N - Q U B I T S T O A C C U R A C Y E . D E F I N I T I O N O F A U N I V E R S A L G A T E S E T
  • 17.
    T H EC O S T C A N B E P L A C E D O N T H E B R E D T H O F T H E C I R C U I T , T H E D E P T H O F T H E C I R C U I T , T H E T O T A L N U M B E R O F G A T E S U S E D E T C . C O S T F O R I M P L E M E N T I N G A U N I V E R S A L G A T E S E T
  • 18.
    I T SE E K S T O U N D E R S T A N D H O W T H E C O S T F O R B U I L D I N G Q U A N T U M G A T E S G R O W F O R V A R I O U S Q U A N T U M A L G O R I T H M S Q U A N T U M C O M P U T A T I O N A L C O M P L E X I T Y
  • 19.
    T H ED E P T H C O R R E S P O N D S T O T O T A L R U N N I N G T I M E O F T H E C I R C U I T . D E P T H O F Q U A N T U M C O M P U T I N G C I R C U I T
  • 20.
    T H EO R E M T E L L S U S T H E D E P T H D I F F E R E N C E B E T W E E N C I R C U I T S C O N S T R U C T E D W I T H D I F F E R E N T U N I V E R S A L S E T S O F G A T E S T O A N A C C U R A C Y O F O ( L O G - C ( 1 / E ) ) S O L O V A Y - K I T A E V T H E O R E M
  • 21.
    Q U AN T U M C O M P U T I N G I S P E R F O R M E D W I T H T H E A I D O F A Q U A N T U M S Y S T E M W H O S E E V O L U T I O N I S D E S C R I B E D B Y T H E U N I T A R Y O P E R A T O R U ( T ) = E X P - I H T W H E R E H I S T H E H A M I L T O N I A N O F T H E S Y S T E M . S Y S T E M R E Q U I R E M E N T S F O R T H E Q U A N T U M C O M P U T I N G
  • 22.
    T H ES T A T E O F A C O M P O S I T E S Y S T E M T H A T C A N ’ T B E W R I T T E N A S T H E P R O D U C T O F S T A T E S O F I T S C O M P O N E N T S Y S T E M I S C A L L E D E N T A N G L E D S T A T E . E N T A N G L E D S T A T E O F A Q U A N T U M C O M P U T E R
  • 23.
    B E LL B A S I S I S A O R T H O N O R M A L B A S I S F O R T H E T W O Q U B I T S T A T E S P A C E B E L L S T A T E S
  • 24.
    A T WO Q U B I T G A T E A I S A U N I V E R S A L F O R T H E Q U A N T U M C O M P U T A T I O N I F A T O G E T H E R W I T H L O C A L U N I T A R Y T R A N S F O R M A T I O N S G E N E R A T E S A L L U N I T A R Y T R A N S F O R M A T I O N S O F T H E C O M P L E X V E C T O R S P A C E O F D I M E N S I O N 2 - E X P - N T O I T S E L F . U N I V E R S A L Q U A N T U M G A T E
  • 25.
    A G AT E A I S S A I D T O B E E N T A N G L I N G I F T H E R E I S A V E C T O R | A B > = | A > X | B > S U C H T H A T A | A B > I S N O T D E C O M P O S A B L E A S A T E N S O R P R O D U C T O F T W O Q U B I T S . P R O O F O F E N T A N G L E M E N T
  • 26.
    T W OQ U B I T G A T E A I S U N I V E R S A L I F A N D O N L Y I F I T I S E N T A N G L I N G . U N I V E R S A L I T Y A N D E N T A N G L E M E N T
  • 27.
    A P HA S E I S N O T A S T A T E O F M A T T E R B U T A C O M P L E X N U M B E R O F U N I T M O D U L E S . T H E P H A S E S I S O B T A I N E D W I T H T H E C Y C L I C E V O L U T I O N O F A P H Y S I C A L S Y S T E M . G E O M E T R I C C H A R A C T E R O F Q U A N T U M C O M P U T I N G
  • 28.
    C Y CL I C V A R I A T I O N O F E X T E R N A L P A R A M E T E R S O F T E N L E A D T O A N E T E V O L U T I O N I N V O L V I N G A P H A S E D E P E N D I N G O N L Y O N T H E G E O M E T R Y O F T H E P A T H T R A V E R S E D I N T H E P A R A M E T E R S P A C E . E V O L U T I O N O F P H A S E P R O P E R T Y
  • 29.
    P R ES E N C E O F G E O M E T R I C P H A S E I S Q U A N T U M S Y S T E M S A N D I T S P R O P E R T I E S H A S B E E N U S E D T O C O N S T R U C T A N E W M O D E L F O R Q U A N T U M C O M P U T I N G . H O L O N O M Y I S A P H A S E T H A T D E P E N D S O N T H E G E O M E T R Y O F A L O O P A N D I S I N D E P E N D E N T O F A N Y C O O R D I N A T E C H O I C E . H O L O N O M I C Q U A N T U M C O M P U T I N G
  • 30.
    C L AS S I C A L G A M E S C A N N O T B E Q U A N T I Z E D I N A U N I Q U E W A Y B E C A U S E T H E Y A R E O N L Y A S Y M P T O T I C A L S H A D O W S O F A W I D E S P E C T R A O F Q U A N T U M M O D E L S . F O L L O W I N G T W O S T A G E S A R E I M P O R T A N T : 1 . P R E - Q U A N T I Z A T I O N 2 . Q U A N T I Z A T I O N Q U A N T U M G A M E T H E O R Y
  • 31.
    Q F TD O E S N O T E X P L I C I T L Y P R O D U C E A N Y O F T H E F O U R I E R C O E F F I C I E N T S . Q F T C A N B E R E G A R D E D A S C O M P U T I N G A L L T H E P R O B A B I L I T I E S O F A P R O B A B I L I T Y D I S T R I B U T I O N O R S A M P L I N G A D I S T R I B U T I O N Q U A N T U M F O U R I E R T R A N S F O R M