Modeling Quantum Computing in Haskell

Amr Sabry

صبرى عمروIndiana University

August 28, 2003 Haskell Workshop2 of 31

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

August 28, 2003 Haskell Workshop3 of 31

Turing ‘36 Deutsch ‘97

Computing is normally

done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child’s arithmetic book….

If we regard a symbol as literally printed on a square we may suppose that the square is 0 < x < 1, 0 < y <1. The symbol is defined as a set of points in this square, viz. the set occupied by printers ink.

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.

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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

Quantum Data

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Bits and Qubits

orTrue

FalseqFT = True

False

11

Examples of qubits

True

False

01qF =

True

False

10qT =

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Generalize

Given a type a with constructors C1, C2, …, Cn

Quantum values of type QV a

C1 1

… …

C2

Cn

2

n

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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:

q2 = True

False 1

2True

Falseq1 = 1

2

q1 &* q2 = (False,False) 1 * 1

(False,True) 1 * 2

(True,False) 2 * 1

(True,True) 2 * 2

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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.ENTANGLED.

Functions / Operations

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Two Simple Operations

What is hadamard qF ?

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Controlled-Not (cnot)

c x

notv y

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

What is cnot (qFT,qF) ?

cnot(qFT,qF) = the pair p2

((False,False),(False,False)) 1

((False, True),(False, True)) 1

((True, False),(True, True)) 1

((True, True),(True, False)) 1

cnotM =

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Our First Circuit

not

hadamardqF

qF

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:

1. Observe the state of the pair itself

2. Observe the left component only

3. Observe the right component only

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Observing Pairs

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The EPR Paradox

Entangled particles

Light-years apart

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

Wave/Particle Duality

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Let’s 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.

August 28, 2003 Haskell Workshop25 of 31

Virtual Values & Adaptors

(False,False,False)

(True,True,True) 1

1

A quantum value with entangled subvalues

A virtual value to operate on the third and first components

Pointer to the real value

Adaptor

(a,b,c)(c,a)

b

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Apply cnot to our virtual value

cnotM = ((False,False),(False,False)) 1

((False, True),(False, True)) 1

((True, False),(True, True)) 1

((True, True),(True, False)) 1

((a,b,c),(x,b,z)) cnotM((c,a),(z,x))

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 =

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Example

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

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BBC World on SAS flight

It is a bizarre place … … normal laws of physics break … … the fastest machine today will seem like

an abacus …

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Computer Science

Physics

Hardware

Assembly, C, etc

Functional Programming, lambda calculus, etc

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Matrix Representation

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Conventions in Physics

Quantum values represented by a vector of probability amplitudes (in some implicit order)

Quantum operations represented by matrices giving for each input constructor its contribution to each output constructor

Applying an operation to a value is multiplication

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Potential Rewards

Nature’s programming paradigm might be better than what we have.

Information flow in QM is a mystery (EPR paradox). An executable semantics for QC might help.

There are some appealing connections between quantum computing and functional programming (operations must be pure and reversible, evaluation is different from observation, quantum values cannot be cloned i.e. are linear, etc)

comp.lang.functional discussion about free will and predestination

August 28, 2003 Haskell Workshop37 of 31

Bits and Qubits

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Generalize

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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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Virtual Values & Adaptors

Quantum value with entangled subvalues

Adaptor

Virtual Value

1

2

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Applying Operations to Virtual Values

Input

Function

Output

Promote the function

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Examples

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Observing Pairs (I)

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Accessing Substructures

In conventional programming: composable references, façade pattern, …

In quantum computing: virtual registers, shuffle wires, or our proposal of virtual values and adaptors