A Quantum Test Algorithm quantph0501108

1
A Quantum Test Algorithm (quant-ph/0501108)

• Jacob D. Biamonte and Marek A. Perkowski
• biamonte_at_ieee.org

2
Goal Avoid expanding a full set of basis vectors
to characterize the system

• Instead of a complete set of the space of the
  system, consider a complete set for the function
  of the gate instead.
• Force each gate to first act on all classical
  basis states
• Force each gate to act on all possible quantum
  states and next remove entanglement
• Each qubit needs to be initialized and measured
  in both basis states

3
k-CN Gate with classical inputs needs 4 tests

• Set all the controls to ?1? toggle the target,
  set the controls to ?0? toggle the target. Now
  the EXOR gate just acted exhaustively on all
  classical basis vectors.
• This is a total of 4 finite operations on a
  complete basis

?x1?
?x1?
?x2?
?x2?
?xk-1?
?xk-1?
?xk??(?x1? ? ?x2? ? ? ?xk-1?)
?xk?
4
k-CN Gate with Quantum inputs takes 4 tests

• Set all the controls to ???k (activating and
  non-activating states)
• ?? at the target (nothing should happen)
• ? - ? phase goes backwards
• Set all the controls to ? - ??k (activating and
  non-activating states)
• ?? at the target (nothing should happen)
• ? - ? phase goes backwards
• Now the EXOR gate just acted exhaustively on all
  quantum basis vectors
• The gate is capable of sending phase backward
  with - ?a? activating state, not sending phase
  backwards with a - ?a?
• The gate is capable of NOT sending phase backward
  with - ?n? activating state, not sending phase
  backwards with a - ?n?
• This is a grand total of eight finite quantum
  operations on a complete basis

?x1?
Non-Local Description
?x2?
??k?
?xk-1?
?xk?
5
How to make oracles

• There is something known as a PPRM in classical
  circuit design. This means you can realize any
  oracle without inverters on the top bits (this
  method works for general ESOPs expansions but
  this example is much cleaner since the
  entanglement gets even worse).

f(x1,x2,x3,x4) 1?x1?x2?x3?x3x4?x1x3x4?x1x2x3x4
?x2x3x4
?x1?
?x1?
?x2?
?x2?
?x3?
?x3?
?x4?
?x4?
?1?
f(x1,x2,x3,x4)
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T1 (?0??k ?1??k)??0? --- How to concurrently
test multiple classical degrees of freedom for
the oracle

• Known as a PPRM in classical circuit design.

??ab? (?0000? ?1111?)? ?0? ?0000???0?
?1111???0?
?0?
?0?
H
?0?
?0?
?0?
?0?
?0?
?0?
H
?0?
?1?
?1?
?0?
?0?
?1?
?0?
?0?
?0?
?1111???0?
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T2 (?0??k - ?1??k)??1? --- concurrently testing
multiple classical degrees of freedom for the
oracle II
Each gate that acted on ?1? in the last test now
acts on ?0? in this test.
??ab? (?0000? - ?1111?)? ?0? ?0000???0? -
?1111???0?
?1?
?1?
H
?1?
?1?
?1?
?1?
?1?
?1?
H
?1?
?1?
?0?
?0?
?0?
?1?
?1?
?1?
?1?
?1111???0?
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First what Happens inside of the oracle?
??? ?0000? ?0001? - ?0010? - ?0100? -
?1000? ?1100? ?1010? ?1001? ?0110?
-?0101? ?0011? ?1110? ?1011? - ?1101?
?0111? - ?1111?
?0?
H
Each term in the superposition will get a phase
flip for each activated gate
?0?
H
?0?
H
?0?
H
?1?
H
- ?0010? (thanks to this gate this term gets
tagged
The parity of the number of gates that each term
in the superposition turns on is encoded in the
phase.
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The Quantum Test Sequence ???k? - ?
??? ?0000? ?0001? - ?0010? - ?0100? -
?1000? ?1100? ?1010? ?1001? ?0110?
-?0101? ?0011? ?1110? ?1011? - ?1101?
?0111? - ?1111?
???k? - ?
Remove Entanglement
H
H
?0?
?0?
H
?0?
H
?0?
?0?
?0?
H
H
?0?
?1?
H
H
? - ?
? - ?
? - ?
? - ?
? - ?
? - ?
? - ?
? - ?
? - ?
?1?
?1?
H
H
? - ? (up to global phase)
10
The Quantum Test Sequence ? - ??k? - ?
??? - ?0000? - ?0001? ?0010? ?0100?
?1000? - ?1100? - ?1010? - ?1001? - ?0110?
?0101? - ?0011? - ?1110? - ?1011? ?1101? -
?0111? ?1111?
The Dual of T3
? - ??k? - ?
Remove Entanglement
H
H
?1?
?1?
H
?1?
H
?1?
?1?
?1?
H
H
?1?
?0?
H
H
? - ?
? - ?
? - ?
? - ?
? - ?
? - ?
? - ?
? - ?
? - ?
?1?
?1?
H
H
? - ? (up to global phase)
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T5 and T6 --- The Quantum Test Sequences
???k?? and ?-??k??
??? ?0000? ?0001? ?0010? ?0100?
?1000? ?1100? ?1010? ?1001? ?0110?
?0101? ?0011? ?1110? ?1011? ?1101?
?0111? ?1111?
?0?
H
H
?0?
???k??
?0?
?0?
H
H
Nothing should happen in these tests.
?0?
?0?
H
H
?0?
?0?
H
H
?0?
?0?
H
H
??? ?0000? - ?0001? - ?0010? - ?0100? -
?1000? - ?1100? - ?1010? - ?1001? - ?0110? -
?0101? - ?0011? - ?1110? - ?1011? - ?1101? -
?0111? - ?1111?
?1?
H
H
?1?
?1?
H
?1?
H
?1?
?1?
H
H
? - ??k??
?1?
?1?
H
H
?0?
?0?
H
H