[PPT] - Coalgebraic Walks in Quantum and Turing Computation Bart Jacobs PowerPoint Presentation

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Coalgebraic Walks in Quantum and Turing Computation

Bart Jacobs

Institute for Computing and Information Sciences – Digital Security Radboud University Nijmegen

30/10/10, Oxford

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 1 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Outline

Introduction Walks illustrating computation types Non-deterministic walks Probabilistic walks Quantum walks Quantum walks, coalgebraically Matrix representation Reversible computation Turing Machines Conclusions

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 2 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Outline

Introduction Walks illustrating computation types Non-deterministic walks Probabilistic walks Quantum walks Quantum walks, coalgebraically Matrix representation Reversible computation Turing Machines Conclusions

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 3 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Aims of this presentation

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 4 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Aims of this presentation

Explore quantum computation in relation to well-known

computation paradigms:

deterministic / non-deterministic / probabilistic / . . .
Seek connections quantum-coalgebra connection, within

paradigm for state-based computation.

Byproduct: coalgebraic modeling of Turing machines

(long-standing open problem)

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 4 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Aims of this presentation

Explore quantum computation in relation to well-known

computation paradigms:

deterministic / non-deterministic / probabilistic / . . .
Seek connections quantum-coalgebra connection, within

paradigm for state-based computation.

Byproduct: coalgebraic modeling of Turing machines

(long-standing open problem) ☛ ✡ ✟ ✠

Walks on a line form the leading example

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 4 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Why coalgebras?

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 5 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Why coalgebras?

Coalgebras have emerged as a generic formalism for

state-based computation, including e.g.

its own “coalgebraic modal logic”
bisimilarity as observational indistinguishability

(with generic ‘bisimilarity-is-congruence’ proofs)

canonical (final) models, with sound & complete languages

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 5 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Why coalgebras?

Coalgebras have emerged as a generic formalism for

state-based computation, including e.g.

its own “coalgebraic modal logic”
bisimilarity as observational indistinguishability

(with generic ‘bisimilarity-is-congruence’ proofs)

canonical (final) models, with sound & complete languages
The language of quantum mechanics is very much related:
states and observations
observations disturbe the state (have side-effects)
unfinished debates about determinism & probability

(Einstein: God does not play dice; Bohr: internal state is unkown)

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 5 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Coalgebras

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 6 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Coalgebras

Mathematical models for state-based computation:
state space, say X
transition map X → F(X)
F is a functor that fixes the type of computation

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 6 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Coalgebras

Mathematical models for state-based computation:
state space, say X
transition map X → F(X)
F is a functor that fixes the type of computation
A deterministic automaton involves a pair of maps:

X

step,final?

X A × 2

where 2 = {0, 1}.

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 6 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Coalgebras

Mathematical models for state-based computation:
state space, say X
transition map X → F(X)
F is a functor that fixes the type of computation
A deterministic automaton involves a pair of maps:

X

step,final?

X A × 2

where 2 = {0, 1}.

A non-deterministic automaton, or transition system, with

input actions A is a coalgebra: X

succs

P(X)A

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 6 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Coalgebras

Mathematical models for state-based computation:
state space, say X
transition map X → F(X)
F is a functor that fixes the type of computation
A deterministic automaton involves a pair of maps:

X

step,final?

X A × 2

where 2 = {0, 1}.

A non-deterministic automaton, or transition system, with

input actions A is a coalgebra: X

succs

P(X)A

By replacing powerset P by distribution functor D one obtains

probabilistic automata (aka. Markov chain)

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 6 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Coalgebras of monads

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 7 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Coalgebras of monads

In this context we use coalgebras X → T(X) where T is a

monad

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 7 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Coalgebras of monads

In this context we use coalgebras X → T(X) where T is a

monad

As a result, coalgebras can be composed
More formally: the set of coalgebras
TX

X forms a monoid

Even more formally: a monoid in the category of T-algebras

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 7 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Coalgebras of monads

In this context we use coalgebras X → T(X) where T is a

monad

As a result, coalgebras can be composed
More formally: the set of coalgebras
TX

X forms a monoid

Even more formally: a monoid in the category of T-algebras
Main monad examples:
Powerset P
Distribution D
Multiset M

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Distribution monad D

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Distribution monad D

For a set X, define D(X) = {ϕ: X → [0, 1]

support(ϕ) is finite, and

x ϕ(x) = 1}

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Distribution monad D

For a set X, define D(X) = {ϕ: X → [0, 1]

support(ϕ) is finite, and

x ϕ(x) = 1}

Such ϕ ∈ D(X) is a formal convex combination: r1x1 + · · · + rnxn where        support(ϕ) = {x1, . . . , xn} ri = ϕ(xi) > 0 r1 + · · · + rn = 1

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 8 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Distribution monad D

For a set X, define D(X) = {ϕ: X → [0, 1]

support(ϕ) is finite, and

x ϕ(x) = 1}

Such ϕ ∈ D(X) is a formal convex combination: r1x1 + · · · + rnxn where        support(ϕ) = {x1, . . . , xn} ri = ϕ(xi) > 0 r1 + · · · + rn = 1 Coalgebras X → D(X) are Markov chains, giving probabilistic transitions: x

ri

− → xi with

i ri = 1.

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 8 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Multiset monad M

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 9 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Multiset monad M

The mulitset monad M over complex numbers is similar to, but simpler than, the distribution monad D. For a set X, now define

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 9 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Multiset monad M

The mulitset monad M over complex numbers is similar to, but simpler than, the distribution monad D. For a set X, now define M(X) = {ϕ: X → C

support(ϕ) is finite }

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 9 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Multiset monad M

The mulitset monad M over complex numbers is similar to, but simpler than, the distribution monad D. For a set X, now define M(X) = {ϕ: X → C

support(ϕ) is finite }

Such ϕ ∈ M(X) is a formal linear combination: z1x1 + · · · + znxn where

support(ϕ) = {x1, . . . , xn}

zi = ϕ(xi) ∈ C Such formal linear combinations form the free vector space on X.

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 9 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Multiset monad M

The mulitset monad M over complex numbers is similar to, but simpler than, the distribution monad D. For a set X, now define M(X) = {ϕ: X → C

support(ϕ) is finite }

Such ϕ ∈ M(X) is a formal linear combination: z1x1 + · · · + znxn where

support(ϕ) = {x1, . . . , xn}

zi = ϕ(xi) ∈ C Such formal linear combinations form the free vector space on X. Coalgebras X → M(X) are like weighted automata, adding weights/resources in C as labels to transitions.

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 9 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Outline

Introduction Walks illustrating computation types Non-deterministic walks Probabilistic walks Quantum walks Quantum walks, coalgebraically Matrix representation Reversible computation Turing Machines Conclusions

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 10 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Walk the walk

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 11 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Walk the walk

Consider a line of integer points . . . , −2, −1, 0, 1, 2, . . . ∈ Z. Different styles of walks, say of a drunkard, on this line will be described next:

non-deterministic
probabilistic
quantum

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 11 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Walk the walk

Consider a line of integer points . . . , −2, −1, 0, 1, 2, . . . ∈ Z. Different styles of walks, say of a drunkard, on this line will be described next:

non-deterministic
probabilistic
quantum

All three computational styles can & will be represented coalgebraically.

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 11 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Non-deterministic walks: definition

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 12 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Non-deterministic walks: definition

Coalgebraic represenation, of possible left-or-right stepping: Z

s

P(Z)

k

{k − 1, k + 1}

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 12 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Non-deterministic walks: definition

Coalgebraic represenation, of possible left-or-right stepping: Z

s

P(Z)

k

{k − 1, k + 1}

Iteration, starting in 0 ∈ Z, yields: 0 → {−1, 1}

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 12 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Non-deterministic walks: definition

Coalgebraic represenation, of possible left-or-right stepping: Z

s

P(Z)

k

{k − 1, k + 1}

Iteration, starting in 0 ∈ Z, yields: 0 → {−1, 1} → {−2, 0, 2}

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 12 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Non-deterministic walks: definition

Coalgebraic represenation, of possible left-or-right stepping: Z

s

P(Z)

k

{k − 1, k + 1}

Iteration, starting in 0 ∈ Z, yields: 0 → {−1, 1} → {−2, 0, 2} → {−3, −1, 1, 3}

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 12 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Non-deterministic walks: definition

Coalgebraic represenation, of possible left-or-right stepping: Z

s

P(Z)

k

{k − 1, k + 1}

Iteration, starting in 0 ∈ Z, yields: 0 → {−1, 1} → {−2, 0, 2} → {−3, −1, 1, 3} · · · {−n, −n + 2, · · · n − 2, n}

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 12 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Non-deterministic walks: definition

Coalgebraic represenation, of possible left-or-right stepping: Z

s

P(Z)

k

{k − 1, k + 1}

Iteration, starting in 0 ∈ Z, yields: 0 → {−1, 1} → {−2, 0, 2} → {−3, −1, 1, 3} · · · {−n, −n + 2, · · · n − 2, n}

· · ·

3
2
1

1 2 3 · · ·

etc

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Non-deterministic walks: iteration

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 13 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Non-deterministic walks: iteration

Formally, iteration is done via the Kleisli extension endomap: P(Z)

s#

P(Z)

U

k∈U{k − 1, k + 1}

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 13 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Non-deterministic walks: iteration

Formally, iteration is done via the Kleisli extension endomap: P(Z)

s#

P(Z)

U

k∈U{k − 1, k + 1}

The subset of successors of 0 ∈ Z, after n steps, is obtained as the n-th iterate:

s#n

({0}) = {−n, −n + 2, · · · n − 2, n}.

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 13 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Non-deterministic walks: iteration

Formally, iteration is done via the Kleisli extension endomap: P(Z)

s#

P(Z)

U

k∈U{k − 1, k + 1}

The subset of successors of 0 ∈ Z, after n steps, is obtained as the n-th iterate:

s#n

({0}) = {−n, −n + 2, · · · n − 2, n}. Aside: categorically, this can be described directly as iteration in the Kleisli category of the powerset monad P.

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 13 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Probabilistic walks: definition

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 14 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Probabilistic walks: definition

Probabilistic left-or-right stepping, each with chance 1

2 is expressed

via a formal convex sum / distribution, as: Z

d

D(Z)

k

1 2(k − 1) + 1 2(k + 1)

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 14 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Probabilistic walks: definition

Probabilistic left-or-right stepping, each with chance 1

2 is expressed

via a formal convex sum / distribution, as: Z

d

D(Z)

k

1 2(k − 1) + 1 2(k + 1)

Iteration, starting in 0 ∈ Z, now yields: 0 → 1

2(-1) + 1 2(1)

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 14 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Probabilistic walks: definition

Probabilistic left-or-right stepping, each with chance 1

2 is expressed

via a formal convex sum / distribution, as: Z

d

D(Z)

k

1 2(k − 1) + 1 2(k + 1)

Iteration, starting in 0 ∈ Z, now yields: 0 → 1

2(-1) + 1 2(1)

→ 1

4(-2) + 1 2(0) + 1 4(2)

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 14 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Probabilistic walks: definition

Probabilistic left-or-right stepping, each with chance 1

2 is expressed

via a formal convex sum / distribution, as: Z

d

D(Z)

k

1 2(k − 1) + 1 2(k + 1)

Iteration, starting in 0 ∈ Z, now yields: 0 → 1

2(-1) + 1 2(1)

→ 1

4(-2) + 1 2(0) + 1 4(2)

→ 1

8(-3) + 3 8(-1) + 3 8(1) + 1 8(3)

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 14 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

Radboud University Nijmegen

Probabilistic walks: definition

Probabilistic left-or-right stepping, each with chance 1

2 is expressed

via a formal convex sum / distribution, as: Z

d

D(Z)

k

1 2(k − 1) + 1 2(k + 1)

Iteration, starting in 0 ∈ Z, now yields: 0 → 1

2(-1) + 1 2(1)

→ 1

4(-2) + 1 2(0) + 1 4(2)

→ 1

8(-3) + 3 8(-1) + 3 8(1) + 1 8(3)

· · ·

3
2
1

1 2 3 · · · 1

1

2

1

2

1

4

1

2

1

4

1

8

3

8

3

8

1

8

1

16 1 4 5 8 1 4 1 16

etc

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Probabilistic walks: the general formula

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Probabilistic walks: the general formula

This tree of probabilities involves Pascal’s triangle. Starting in k ∈ Z, after n iterations one obtains the formal convex sum: n

2n (k−n)+

n

1

2n (k−n+2)+

n

2

2n (k−n+4)+. . .+

n

n−1

2n

(k+n−2)+ n

n

2n (k+n)

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Probabilistic walks: the general formula

This tree of probabilities involves Pascal’s triangle. Starting in k ∈ Z, after n iterations one obtains the formal convex sum: n

2n (k−n)+

n

1

2n (k−n+2)+

n

2

2n (k−n+4)+. . .+

n

n−1

2n

(k+n−2)+ n

n

2n (k+n)

Using the sum formula for binomial coefficients: n

+

n

1

+

n

2

+ · · · +

n

n−1

+

n

=

2n.

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Probabilistic walks: iteration

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Probabilistic walks: iteration

Again there is a Kleisli extension endomap d# : D(Z) → D(Z)

r1k1 + · · · + rnkn
−

→

1

2r1(k1 − 1) + 1 2r1(k1 + 1) + · · · + 1 2rn(kn − 1) + 1 2rn(kn + 1)

Bart Jacobs

30/10/10, Oxford Coalgebraic Walks 16 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Probabilistic walks: iteration

Again there is a Kleisli extension endomap d# : D(Z) → D(Z)

r1k1 + · · · + rnkn
−

→

1

2r1(k1 − 1) + 1 2r1(k1 + 1) + · · · + 1 2rn(kn − 1) + 1 2rn(kn + 1)

The subset of successors of 0 ∈ Z, after n steps, is obtained as the

n-th iterate:

d#n

(1 0) = n

2n (−n)+

n

1

2n (−n+2)+. . .+

n

n−1

2n

(n−2)+ n

n

2n (n)

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 16 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Probabilistic walks: iteration

Again there is a Kleisli extension endomap d# : D(Z) → D(Z)

r1k1 + · · · + rnkn
−

→

1

2r1(k1 − 1) + 1 2r1(k1 + 1) + · · · + 1 2rn(kn − 1) + 1 2rn(kn + 1)

The subset of successors of 0 ∈ Z, after n steps, is obtained as the

n-th iterate:

d#n

(1 0) = n

2n (−n)+

n

1

2n (−n+2)+. . .+

n

n−1

2n

(n−2)+ n

n

2n (n)

It is iteration in the Kleisli category of the distribution monad D.

Bart Jacobs 30/10/10, Oxford Coalgebraic Walks 16 / 42

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Quantum walks

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Quantum walks

Situation / plan

Quantum walks are standardly described as endomap S → S

(see eg. work of Julia Kempe)

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Quantum walks

Situation / plan

Quantum walks are standardly described as endomap S → S

(see eg. work of Julia Kempe)

Here it will be shown that there is also an (equivalent)

coalgebraic / monadic description.

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Quantum walks: endomap definition I

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Quantum walks: endomap definition I

Two vector spaces

M(Z), the free vector space on Z (over C), with base vectors

written as: | k ∈ M(Z), for k ∈ Z

C2, the one qubit space, with base vectors | ↑ and | ↓

(think of the direction of the walk)

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Quantum walks: endomap definition I

Two vector spaces

M(Z), the free vector space on Z (over C), with base vectors

written as: | k ∈ M(Z), for k ∈ Z

C2, the one qubit space, with base vectors | ↑ and | ↓

(think of the direction of the walk)

The state space is C2 ⊗ M(Z), with basis

| ↑ ⊗ | k

| ↓ ⊗ | k

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Quantum walks: endomap definition II

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Quantum walks: endomap definition II

The relevant endomap is: C2 ⊗ M(Z)

q

C2 ⊗ M(Z)

| ↑ ⊗ | k

1

√ 2| ↑ ⊗ | k − 1 + 1 √ 2| ↓ ⊗ | k + 1

| ↓ ⊗ | k

1

√ 2| ↑ ⊗ | k − 1 – 1 √ 2| ↓ ⊗ | k + 1

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Quantum walks: endomap definition II

The relevant endomap is: C2 ⊗ M(Z)

q

C2 ⊗ M(Z)

| ↑ ⊗ | k

1

√ 2| ↑ ⊗ | k − 1 + 1 √ 2| ↓ ⊗ | k + 1

| ↓ ⊗ | k

1

√ 2| ↑ ⊗ | k − 1 – 1 √ 2| ↓ ⊗ | k + 1

Implicitly, the Hadamard operator H : C2 → C2 is used: H = 1 √ 2

  1

1 1 −1

 

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Quantum walks: iteration and probabilities I

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Quantum walks: iteration and probabilities I

Obtaining probabilities in quantum walks

We are interested in the probability of reaching | k after n
steps. Starting from 0 — more precisely, from | ↑ ⊗ | 0 .

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Quantum walks: iteration and probabilities I

Obtaining probabilities in quantum walks

We are interested in the probability of reaching | k after n
steps. Starting from 0 — more precisely, from | ↑ ⊗ | 0 .
We do so by n-times iterating the endomap q, and then

measuring in the basis | k .

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Quantum walks: iteration and probabilities I

Obtaining probabilities in quantum walks

We are interested in the probability of reaching | k after n
steps. Starting from 0 — more precisely, from | ↑ ⊗ | 0 .
We do so by n-times iterating the endomap q, and then

measuring in the basis | k .

The sum of norms |αi|2 of amplitudes αi for | k then gives

the probability.

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Quantum walks: iteration and probabilities II

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Quantum walks: iteration and probabilities II

| ↑ ⊗ | 0 →

1 √ 2| ↑ ⊗ | -1 + 1 √ 2| ↓ ⊗ | 1

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Quantum walks: iteration and probabilities II

| ↑ ⊗ | 0 →

1 √ 2| ↑ ⊗ | -1 + 1 √ 2| ↓ ⊗ | 1

probabilities    | -1 | 1

√ 2|2 = 1 2

| 1 | 1

√ 2|2 = 1 2

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Quantum walks: iteration and probabilities II

| ↑ ⊗ | 0 →

1 √ 2| ↑ ⊗ | -1 + 1 √ 2| ↓ ⊗ | 1

probabilities    | -1 | 1

√ 2|2 = 1 2

| 1 | 1

√ 2|2 = 1 2

→

1 2| ↑ ⊗ | -2 + 1 2| ↓ ⊗ | 0

+

1 2| ↑ ⊗ | 0 − 1 2| ↓ ⊗ | 2

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Quantum walks: iteration and probabilities II

| ↑ ⊗ | 0 →

1 √ 2| ↑ ⊗ | -1 + 1 √ 2| ↓ ⊗ | 1

probabilities    | -1 | 1

√ 2|2 = 1 2

| 1 | 1

√ 2|2 = 1 2

→

1 2| ↑ ⊗ | -2 + 1 2| ↓ ⊗ | 0

+

1 2| ↑ ⊗ | 0 − 1 2| ↓ ⊗ | 2

probabilities        | -2 | 1

2|2 = 1 4

| 0 | 1

2|2 + | 1 2|2 = 1 2

| 2 | − 1

2|2 = 1 4

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Quantum walks: iteration and probabilities II

| ↑ ⊗ | 0 →

1 √ 2| ↑ ⊗ | -1 + 1 √ 2| ↓ ⊗ | 1

probabilities    | -1 | 1

√ 2|2 = 1 2

| 1 | 1

√ 2|2 = 1 2

→

1 2| ↑ ⊗ | -2 + 1 2| ↓ ⊗ | 0

+

1 2| ↑ ⊗ | 0 − 1 2| ↓ ⊗ | 2

probabilities        | -2 | 1

2|2 = 1 4

| 0 | 1

2|2 + | 1 2|2 = 1 2

| 2 | − 1

2|2 = 1 4

→ · · · (next page)

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Quantum walks: iteration and probabilities II

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Quantum walks: iteration and probabilities II

· · · →

1 2 √ 2| ↑ ⊗ | -3 + 1 2 √ 2| ↓ ⊗ | -1 + 1 2 √ 2| ↑ ⊗ | -1

−

1 2 √ 2| ↓ ⊗ | 1 + 1 2 √ 2| ↑ ⊗ | -1 + 1 2 √ 2| ↓ ⊗ | 1

−

1 2 √ 2| ↑ ⊗ | 1 + 1 2 √ 2| ↓ ⊗ | 3

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Quantum walks: iteration and probabilities II

· · · →

1 2 √ 2| ↑ ⊗ | -3 + 1 2 √ 2| ↓ ⊗ | -1 + 1 2 √ 2| ↑ ⊗ | -1

−

1 2 √ 2| ↓ ⊗ | 1 + 1 2 √ 2| ↑ ⊗ | -1 + 1 2 √ 2| ↓ ⊗ | 1

−

1 2 √ 2| ↑ ⊗ | 1 + 1 2 √ 2| ↓ ⊗ | 3

=

1 2 √ 2| ↑ ⊗ | -3 + 1 √ 2| ↑ ⊗ | -1 + 1 2 √ 2| ↓ ⊗ | -1

−

1 2 √ 2| ↑ ⊗ | 1 + 1 2 √ 2| ↓ ⊗ | 3

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Quantum walks: iteration and probabilities II

· · · →

1 2 √ 2| ↑ ⊗ | -3 + 1 2 √ 2| ↓ ⊗ | -1 + 1 2 √ 2| ↑ ⊗ | -1

−

1 2 √ 2| ↓ ⊗ | 1 + 1 2 √ 2| ↑ ⊗ | -1 + 1 2 √ 2| ↓ ⊗ | 1

−

1 2 √ 2| ↑ ⊗ | 1 + 1 2 √ 2| ↓ ⊗ | 3

=

1 2 √ 2| ↑ ⊗ | -3 + 1 √ 2| ↑ ⊗ | -1 + 1 2 √ 2| ↓ ⊗ | -1

−

1 2 √ 2| ↑ ⊗ | 1 + 1 2 √ 2| ↓ ⊗ | 3

probabilities              | -3 |

1 2 √ 2|2 = 1 8

| -1 | 1

√ 2|2 + | 1 2 √ 2|2 = 5 8

| 1 | −

1 2 √ 2|2 = 1 8

| 3 |

1 2 √ 2|2 = 1 8

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Quantum walks: iteration and probabilities II

· · · →

1 2 √ 2| ↑ ⊗ | -3 + 1 2 √ 2| ↓ ⊗ | -1 + 1 2 √ 2| ↑ ⊗ | -1

−

1 2 √ 2| ↓ ⊗ | 1 + 1 2 √ 2| ↑ ⊗ | -1 + 1 2 √ 2| ↓ ⊗ | 1

−

1 2 √ 2| ↑ ⊗ | 1 + 1 2 √ 2| ↓ ⊗ | 3

=

1 2 √ 2| ↑ ⊗ | -3 + 1 √ 2| ↑ ⊗ | -1 + 1 2 √ 2| ↓ ⊗ | -1

−

1 2 √ 2| ↑ ⊗ | 1 + 1 2 √ 2| ↓ ⊗ | 3

probabilities              | -3 |

1 2 √ 2|2 = 1 8

| -1 | 1

√ 2|2 + | 1 2 √ 2|2 = 5 8

| 1 | −

1 2 √ 2|2 = 1 8

| 3 |

1 2 √ 2|2 = 1 8

There is “drift to the left” due to interference.

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Quantum walks: iteration and probabilities III

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Quantum walks: iteration and probabilities III

The resulting tree of quantum walk probabilities starts as: · · ·

3
2
1

1 2 3 · · · 1

1

2

1

2

1

4

1

2

1

4

1

8

5

8

1

8

1

8

1

16 5 8 1 8 1 8 1 16

The matrix involved — Hadamard’s H in this case — determines the drifting, and thus how the tree is traversed. This may yield

ptimisations in data processing.

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Basic isomorphism

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Basic isomorphism

LEMMA C2 ⊗ M(X) ∼ = M(X + X) where + is disjoint union

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Basic isomorphism

LEMMA C2 ⊗ M(X) ∼ = M(X + X) where + is disjoint union Proof By the following chain of isomorphisms: C2 ⊗ M(X) = (C ⊕ C) ⊗ M(X)

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Basic isomorphism

LEMMA C2 ⊗ M(X) ∼ = M(X + X) where + is disjoint union Proof By the following chain of isomorphisms: C2 ⊗ M(X) = (C ⊕ C) ⊗ M(X) ∼ = C ⊗ M(X) ⊕ C ⊗ M(X) by distributivity

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Basic isomorphism

LEMMA C2 ⊗ M(X) ∼ = M(X + X) where + is disjoint union Proof By the following chain of isomorphisms: C2 ⊗ M(X) = (C ⊕ C) ⊗ M(X) ∼ = C ⊗ M(X) ⊕ C ⊗ M(X) by distributivity ∼ = M(X) ⊕ M(X) since C is tensor unit

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Basic isomorphism

LEMMA C2 ⊗ M(X) ∼ = M(X + X) where + is disjoint union Proof By the following chain of isomorphisms: C2 ⊗ M(X) = (C ⊕ C) ⊗ M(X) ∼ = C ⊗ M(X) ⊕ C ⊗ M(X) by distributivity ∼ = M(X) ⊕ M(X) since C is tensor unit ∼ = M(X + X) ⊕ is also coproduct of spaces, and M is a free functor.

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Quantum endomorphism as coalgebra

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Quantum endomorphism as coalgebra

THEOREM Linear maps (in VectC) C2 ⊗ M(Z) − → C2 ⊗ M(Z) correspond bijectively to functions (in Sets) Z − → M(Z + Z)2 . that is, to coalgebras with state space Z of the functor M(2 · −)2.

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Quantum endomorphism as coalgebra

THEOREM Linear maps (in VectC) C2 ⊗ M(Z) − → C2 ⊗ M(Z) correspond bijectively to functions (in Sets) Z − → M(Z + Z)2 . that is, to coalgebras with state space Z of the functor M(2 · −)2. Proof C2 ⊗ M(Z) − → C2 ⊗ M(Z) linear = = = = = = = = = = = = = = = = = = = = = = M(Z + Z) − → M(Z + Z) linear

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Quantum endomorphism as coalgebra

THEOREM Linear maps (in VectC) C2 ⊗ M(Z) − → C2 ⊗ M(Z) correspond bijectively to functions (in Sets) Z − → M(Z + Z)2 . that is, to coalgebras with state space Z of the functor M(2 · −)2. Proof C2 ⊗ M(Z) − → C2 ⊗ M(Z) linear = = = = = = = = = = = = = = = = = = = = = = M(Z + Z) − → M(Z + Z) linear = = = = = = = = = = = = = = = = = = = = Z + Z − → M(Z + Z)

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Quantum endomorphism as coalgebra

THEOREM Linear maps (in VectC) C2 ⊗ M(Z) − → C2 ⊗ M(Z) correspond bijectively to functions (in Sets) Z − → M(Z + Z)2 . that is, to coalgebras with state space Z of the functor M(2 · −)2. Proof C2 ⊗ M(Z) − → C2 ⊗ M(Z) linear = = = = = = = = = = = = = = = = = = = = = = M(Z + Z) − → M(Z + Z) linear = = = = = = = = = = = = = = = = = = = = Z + Z − → M(Z + Z) = = = = = = = = = = = = = = = = = Z − → M(Z + Z)2

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Qubit action is monadic!

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Qubit action is monadic!

In the background, there is a new monad transformer result.

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Qubit action is monadic!

In the background, there is a new monad transformer result. LEMMA If T is a monad, then so is X → T(n · X)n, for each n ∈ N

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Qubit action is monadic!

In the background, there is a new monad transformer result. LEMMA If T is a monad, then so is X → T(n · X)n, for each n ∈ N For quantum walks we use T = M and n = 2.

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Outline

Introduction Walks illustrating computation types Non-deterministic walks Probabilistic walks Quantum walks Quantum walks, coalgebraically Matrix representation Reversible computation Turing Machines Conclusions

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Three representations of quantum walks

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Three representations of quantum walks

1 Endomaps C2 ⊗ M(Z) −

→ C2 ⊗ M(Z)

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Three representations of quantum walks

1 Endomaps C2 ⊗ M(Z) −

→ C2 ⊗ M(Z) | ↑ ⊗ | k

1

√ 2| ↑ ⊗ | k − 1 + 1 √ 2| ↓ ⊗ | k + 1

| ↓ ⊗ | k

1

√ 2| ↑ ⊗ | k − 1 − 1 √ 2| ↓ ⊗ | k + 1

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Three representations of quantum walks

1 Endomaps C2 ⊗ M(Z) −

→ C2 ⊗ M(Z) | ↑ ⊗ | k

1

√ 2| ↑ ⊗ | k − 1 + 1 √ 2| ↓ ⊗ | k + 1

| ↓ ⊗ | k

1

√ 2| ↑ ⊗ | k − 1 − 1 √ 2| ↓ ⊗ | k + 1 2 Coalgebras Z −

→ M(Z + Z)2

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Three representations of quantum walks

1 Endomaps C2 ⊗ M(Z) −

→ C2 ⊗ M(Z) | ↑ ⊗ | k

1

√ 2| ↑ ⊗ | k − 1 + 1 √ 2| ↓ ⊗ | k + 1

| ↓ ⊗ | k

1

√ 2| ↑ ⊗ | k − 1 − 1 √ 2| ↓ ⊗ | k + 1 2 Coalgebras Z −

→ M(Z + Z)2 Z

M(Z + Z)2

m

1

√ 2κ1| m − 1 + 1 √ 2κ2| m + 1 , 1 √ 2κ1| m − 1 − 1 √ 2κ2| m + 1

Bart Jacobs

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Three representations of quantum walks

1 Endomaps C2 ⊗ M(Z) −

→ C2 ⊗ M(Z) | ↑ ⊗ | k

1

√ 2| ↑ ⊗ | k − 1 + 1 √ 2| ↓ ⊗ | k + 1

| ↓ ⊗ | k

1

√ 2| ↑ ⊗ | k − 1 − 1 √ 2| ↓ ⊗ | k + 1 2 Coalgebras Z −

→ M(Z + Z)2 Z

M(Z + Z)2

m

1

√ 2κ1| m − 1 + 1 √ 2κ2| m + 1 , 1 √ 2κ1| m − 1 − 1 √ 2κ2| m + 1

3 2 × 2 matrices of coalgebras Z → M(Z) — see below

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Where do these matrices come from?

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Where do these matrices come from?

Multiset M is an additive monad

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Where do these matrices come from?

Multiset M is an additive monad

It sends finite coproducts to products
M(0) ∼

= 1 and M(X + Y ) ∼ = M(X) × M(Y ), canonically

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Where do these matrices come from?

Multiset M is an additive monad

It sends finite coproducts to products
M(0) ∼

= 1 and M(X + Y ) ∼ = M(X) × M(Y ), canonically Hence: Z − → M(Z + Z)2 ∼ =

M(Z) × M(Z)

2 ∼ = M(Z)4

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Where do these matrices come from?

Multiset M is an additive monad

It sends finite coproducts to products
M(0) ∼

= 1 and M(X + Y ) ∼ = M(X) × M(Y ), canonically Hence: Z − → M(Z + Z)2 ∼ =

M(Z) × M(Z)

2 ∼ = M(Z)4 These four coalgebras Z → M(Z) will be used as entries in a 2 × 2 matrix

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Quantum walks as matrix

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Quantum walks as matrix

Matrix entries give 4 possible transitions between | ↑ and | ↓ , in:   λk.

1 √ 2| k − 1

λk.

1 √ 2| k − 1

λk.

1 √ 2| k + 1

λk. −

1 √ 2| k + 1

 

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Quantum walks as matrix

Matrix entries give 4 possible transitions between | ↑ and | ↓ , in:   λk.

1 √ 2| k − 1

λk.

1 √ 2| k − 1

λk.

1 √ 2| k + 1

λk. −

1 √ 2| k + 1

  Moreover: matrix composition corresponds to

endomap composition, for C2 ⊗ M(Z) −

→ C2 ⊗ M(Z)

Kleisli composition, for Z −

→ M(Z + Z)2

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Quantum walks as matrix

Matrix entries give 4 possible transitions between | ↑ and | ↓ , in:   λk.

1 √ 2| k − 1

λk.

1 √ 2| k − 1

λk.

1 √ 2| k + 1

λk. −

1 √ 2| k + 1

  Moreover: matrix composition corresponds to

endomap composition, for C2 ⊗ M(Z) −

→ C2 ⊗ M(Z)

Kleisli composition, for Z −

→ M(Z + Z)2 Question: can one calculate fixed points as eigenvectors?

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Underlying structure

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Underlying structure

LEMMA The set of coalgebras M(X)X is a ring with:

sums inherited pointwise from vector space M(X)
Kleisli composition as multiplication

(Hence we have an algebra: a monoid in VectC) We do matrix algebra over this (semi)ring.

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Outline

Introduction Walks illustrating computation types Non-deterministic walks Probabilistic walks Quantum walks Quantum walks, coalgebraically Matrix representation Reversible computation Turing Machines Conclusions

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Quantum computation is reversible

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Quantum computation is reversible

Reversibility captured by requirement that endomap C2 ⊗ M(Z)

q

− → C2 ⊗ M(Z) is unitary: q is an isomorphism with q−1 = q† (adjoint transpose)

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Quantum computation is reversible

Reversibility captured by requirement that endomap C2 ⊗ M(Z)

q

− → C2 ⊗ M(Z) is unitary: q is an isomorphism with q−1 = q† (adjoint transpose) What does this mean for (matrices of) coalgebras?

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Swappable coaglebras

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Swappable coaglebras

Recall that a coalgebra c : X → M(X) is a function c : X → (X → C) such that:

each c(x): X → C has finite support

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Swappable coaglebras

Recall that a coalgebra c : X → M(X) is a function c : X → (X → C) such that:

each c(x): X → C has finite support

DEFINITION Call c : X → M(X) swappable if:

Each c(−)(y): X → C also has finite support

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Swappable coaglebras

Recall that a coalgebra c : X → M(X) is a function c : X → (X → C) such that:

each c(x): X → C has finite support

DEFINITION Call c : X → M(X) swappable if:

Each c(−)(y): X → C also has finite support

Define an involution on swappable coalgebras via swapping and conjugation (on C): c = λy. λx. c(x)(y) : X − → M(X)

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Introduction Walks illustrating computation types Matrix representation Reversible computation Turing Machines Conclusions

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Swappable coaglebras

Recall that a coalgebra c : X → M(X) is a function c : X → (X → C) such that:

each c(x): X → C has finite support

DEFINITION Call c : X → M(X) swappable if:

Each c(−)(y): X → C also has finite support

Define an involution on swappable coalgebras via swapping and conjugation (on C): c = λy. λx. c(x)(y) : X − → M(X) LEMMA The subset of swappable coalgebras M(X)X is an involutive semiring

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Swappability and unitarity

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Swappability and unitarity

Call a matrix of swappable coalgebras unitary if its adjoint transpose is its inverse.

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Swappability and unitarity

Call a matrix of swappable coalgebras unitary if its adjoint transpose is its inverse. THEOREM A unitary n × n matrix over M(X)X yields a unitary endomap Cn ⊗ M(X) → Cn ⊗ M(X).

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Swappability and unitarity

Call a matrix of swappable coalgebras unitary if its adjoint transpose is its inverse. THEOREM A unitary n × n matrix over M(X)X yields a unitary endomap Cn ⊗ M(X) → Cn ⊗ M(X). It is not clear if the converse also holds. Possibly the notion of swappability is too strong.

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Outline

Introduction Walks illustrating computation types Non-deterministic walks Probabilistic walks Quantum walks Quantum walks, coalgebraically Matrix representation Reversible computation Turing Machines Conclusions

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What we have so far

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What we have so far

Quantum walks on Z use 2 auxiliary states: | ↑ and | ↓

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What we have so far

Quantum walks on Z use 2 auxiliary states: | ↑ and | ↓
This number of states (two) reappears:
in the monad M(2 · −)2
in the size (2 × 2) of the matrix involved

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What we have so far

Quantum walks on Z use 2 auxiliary states: | ↑ and | ↓
This number of states (two) reappears:
in the monad M(2 · −)2
in the size (2 × 2) of the matrix involved
Next idea:
see Turing machines as “walks on a tape”
tape forms state space X
states of the Turing machine captured by n in X → T(n · X)n

(where n · X = X + · · · + X)

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Illustration, for non-deterministic Turing Machine

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Illustration, for non-deterministic Turing Machine

Three-state Turing machine: 1

00R

00R
11R
2

00R

3

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Illustration, for non-deterministic Turing Machine

Three-state Turing machine: 1

00R

00R
11R
2

00R

3

Use: “stream + position” as 2-dimensional tape: T = 2Z × Z

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Illustration, for non-deterministic Turing Machine

Three-state Turing machine: 1

00R

00R
11R
2

00R

3

Use: “stream + position” as 2-dimensional tape: T = 2Z × Z Three equivalent descriptions, using (finite) powerset monad Pfin

1 Endomap 23 ⊗ Pfin(T) −

→ 23 ⊗ Pfin(T) in the category of join semi-lattices (algebras of Pfin)

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Illustration, for non-deterministic Turing Machine

Three-state Turing machine: 1

00R

00R
11R
2

00R

3

Use: “stream + position” as 2-dimensional tape: T = 2Z × Z Three equivalent descriptions, using (finite) powerset monad Pfin

1 Endomap 23 ⊗ Pfin(T) −

→ 23 ⊗ Pfin(T) in the category of join semi-lattices (algebras of Pfin)

2 Coalgebra T −

→ Pfin

T + T + T

3

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Illustration, for non-deterministic Turing Machine

Three-state Turing machine: 1

00R

00R
11R
2

00R

3

Use: “stream + position” as 2-dimensional tape: T = 2Z × Z Three equivalent descriptions, using (finite) powerset monad Pfin

1 Endomap 23 ⊗ Pfin(T) −

→ 23 ⊗ Pfin(T) in the category of join semi-lattices (algebras of Pfin)

2 Coalgebra T −

→ Pfin

T + T + T

3

3 3 × 3 matrix

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Turing machine coalgebra, in detail

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Turing machine coalgebra, in detail

T

Pfin(T + T + T)3

(t, p)

{κ1(t, p + 1)} ∪ {κ2(t, p + 1) | t(p) = 0},

{κ3(t, p + 1) | t(p) = 0}, ∅

Coprojection κj at projection i captures transition from state i to

state j.

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Outline

Introduction Walks illustrating computation types Non-deterministic walks Probabilistic walks Quantum walks Quantum walks, coalgebraically Matrix representation Reversible computation Turing Machines Conclusions

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Concluding remarks

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Concluding remarks

Qbit-action, like in quantum walks, fits in coalgebraic setting,

via monad construction T(n · −)n

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Concluding remarks

Qbit-action, like in quantum walks, fits in coalgebraic setting,

via monad construction T(n · −)n

This provides more uniformity in state-based computation

(and makes quantum computation “more discrete”)

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Concluding remarks

Qbit-action, like in quantum walks, fits in coalgebraic setting,

via monad construction T(n · −)n

This provides more uniformity in state-based computation

(and makes quantum computation “more discrete”)

Also Turing computation is coalgebraic

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Concluding remarks

Qbit-action, like in quantum walks, fits in coalgebraic setting,

via monad construction T(n · −)n

This provides more uniformity in state-based computation

(and makes quantum computation “more discrete”)

Also Turing computation is coalgebraic
Coalgebraic Turing Machine description provides:
genericity in type of computation (eg. probabilistic)
transitions via composition (Kleisli or matrix)
linear algebraic toolbox

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Future work

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Future work

spectral decomposition for getting fixed (stable) points

(both for quantum walks and for Turing machines)

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Future work

spectral decomposition for getting fixed (stable) points

(both for quantum walks and for Turing machines)

Elaborate more examples, esp. for understanding the

swappability-unitarity question

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Future work

spectral decomposition for getting fixed (stable) points

(both for quantum walks and for Turing machines)

Elaborate more examples, esp. for understanding the

swappability-unitarity question

include measurements in this same coalgebraic framework.

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