Modeling Quantum Computing in Haskell

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Modeling Quantum Computing in Haskell

• Amr Sabry
• ???? ????
• Indiana University

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Our Abstractions are Broken!

• CS prides itself on the fact that it is
  insensitive to the changes in the hardware and
  its technology
• Turing Machines, complexity classes, lambda
  calculus, etc, were supposed to be perfect
  abstractions
• Changes in the model of physics (classical vs.
  quantum) broke through our abstractions

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Turing 36 Deutsch 97

• Turing hoped that his abstracted-paper-tape
  model was so simple, so transparent and well
  defined, that it would not depend on any
  assumptions about physics that could conceivably
  be falsified, and therefore that it could become
  the basis of an abstract theory of computation
  that was independent of the underlying physics.
• 'He thought,' as Feynman once put it,
  'that he understood paper.' But he was mistaken.
  Real, quantum-mechanical paper is wildly
  different from the abstract stuff that the Turing
  machine uses. The Turing machine is entirely
  classical, and does not allow for the possibility
  the paper might have different symbols written on
  it in different universes, and that those might
  interfere with one another.

• Computing is normally done by writing
  certain symbols on paper. We may suppose this
  paper is divided into squares like a childs
  arithmetic book.
• If we regard a symbol as literally printed
  on a square we may suppose that the square is 0 lt
  x lt 1, 0 lt y lt1. The symbol is defined as a set
  of points in this square, viz. the set occupied
  by printers ink.

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Challenge to PL Research

• Quantum physics is affecting hardware design,
  complexity theory, and propagates all the way to
  our high level programming languages
• Develop a new appropriate programming
  paradigm

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Outline

• Quantum data
• Operations/Functions
• I/O or Measurement
• Example --- Wave/Particle Duality
• Conclusions and Related Work

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Quantum Data
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Bits and Qubits
Examples of qubits
qFT
or
False
True
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Generalize
Given a type a with constructors C1, C2, ,
Cn Quantum values of type QV a
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Examples
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Pairs

• Pairs of type (QV a, QV b) No surprises
• Pairs of type QV (a, b) can be assembled from two
  quantum values using a tensor product

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Non Compositionality and Entanglement

• There is no way to describe the state of the
  pair p2 in terms of the state of its two
  components. The values are ENTANGLED.

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Functions / Operations
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Two Simple Operations
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Controlled-Not (cnot)
cnot(c,v) (x,y) where x is always the same as
c if c is False, y v if c is True, y not v
cnotM
What is cnot (qFT,qF) ?
cnot(qFT,qF) the pair p2
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Our First Circuit
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Measurement
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Probabilities

• observe qFT should produce False with 50
  probability and produce True with 50 probability
• What happens if we observe a quantum value qFT
  more than once?

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Collapse

• MUST return (True,True,True) or
  (False,False,False)
• Not allowed to produce (True,False,True) or any
  other mixed values

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Observing Pairs

• Given a pair of type QV (a,b) we can make three
  observations
• Observe the state of the pair itself
• Observe the left component only
• Observe the right component only

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Observing Pairs
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The EPR Paradox
observeLeft must affect the right particle
Faster than light communication? Multiple
universes? Signals back from the future? Hidden
local state?
Spooky action at a distance. HOW?
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Spooky Action at a Distance ) Side-effects
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Wave/Particle Duality
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Lets Implement a Simple Example

• The individual operations are easy H is
  hadamard, V and VT are phase-shifting operations
• But given a generally entangled triple QV
  (a,b,c), how can we conveniently apply an
  operation on the first component, or the second
  and third, or the first and third, etc given that
  we cannot decompose the tuple into its three
  components.

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Virtual Values Adaptors
Pointer to the real value
Adaptor
(c,a)
(a,b,c)
b
A quantum value with entangled subvalues
A virtual value to operate on the third and first
components
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Apply cnot to our virtual value
cnotM

• Promote cnotM to work on the full quantum value
  as follows
• Extract the required components using the
  adaptor, apply cnot
• Other components are left unchanged

Promoted cnotM
((a,b,c),(x,b,z)) cnotM((c,a),(z,x))
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Example
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Conclusions and Related Work
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Summary

• Quantum values are maps from classical values to
  probability amplitudes (complex numbers)
• Functions are matrices
• Observation causes collapse (modeled by
  side-effects)
• Programming without destructors

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A Lambda Calculus for QC by André van Tonder

• ?-calculus with (quantum) constants
• Uses sequences of terms (history) to make
  reductions reversible
• State a superposition of sequences
• To avoid having history terms entangled with
  computational terms, functions are linear (cannot
  discard superpositions)
• No observation
• No datatypes still talks about qubits and gates
• Can use pattern-matching on entangled values!

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And of course

• Jerzy Karczmarczuk