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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey Diagrammatic Reasoning and Quantum Computation Aleks Kissinger ACA, Kalamata November 4, 2015 Q UANTUM G ROUP Introduction Quantum circuits Spiders ZX-calculus MBQC

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Diagrammatic Reasoning and Quantum Computation

Aleks Kissinger

ACA, Kalamata

November 4, 2015

QUANTUM GROUP

⇒

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Picturing Quantum Processes

A first course in quantum theory and diagrammatic reasoning Bob Coecke & Aleks Kissinger CUP 2015

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebra and rewriting

  • An algebraic theory consists of a set of operations and constants, satisfying

certain equations

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebra and rewriting

  • An algebraic theory consists of a set of operations and constants, satisfying

certain equations

  • e.g. a monoid consists of a binary operation and constant e such that:

(a · b) · c = a · (b · c) and a · e = a = e · a

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebra and rewriting

  • An algebraic theory consists of a set of operations and constants, satisfying

certain equations

  • e.g. a monoid consists of a binary operation and constant e such that:

(a · b) · c = a · (b · c) and a · e = a = e · a

  • We can apply an equation as a term rewrite rule
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebra and rewriting

  • An algebraic theory consists of a set of operations and constants, satisfying

certain equations

  • e.g. a monoid consists of a binary operation and constant e such that:

(a · b) · c = a · (b · c) and a · e = a = e · a

  • We can apply an equation as a term rewrite rule
  • Instantiate free variables:

(a · b) · c = a · (b · c)

    a := x b := (y · e) c := z

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebra and rewriting

  • An algebraic theory consists of a set of operations and constants, satisfying

certain equations

  • e.g. a monoid consists of a binary operation and constant e such that:

(a · b) · c = a · (b · c) and a · e = a = e · a

  • We can apply an equation as a term rewrite rule
  • Instantiate free variables:

(a · b) · c = a · (b · c)

    a := x b := (y · e) c := z then replace a sub-term: w · ((x · (y · e)) · z)

  • w · (x · ((y · e) · z))
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebra and rewriting

  • Alternatively, we could write these equations as trees:

= a b c b c a = = a a a

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebra and rewriting

  • Alternatively, we could write these equations as trees:

= a b c b c a = = a a a

  • In which case:

w · ((x · (y · e)) · z)

  • w · (x · ((y · e) · z))
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebra and rewriting

  • Alternatively, we could write these equations as trees:

= a b c b c a = = a a a

  • In which case:

w · ((x · (y · e)) · z)

  • w · (x · ((y · e) · z))

becomes: w x y z x z w y

  • z

w x y

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Diagram substitution

  • Note we can drop the free variables:

= a b c b c a

  • =
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Diagram substitution

  • Note we can drop the free variables:

= a b c b c a

  • =
  • The role of variables is replaced by the fact that the LHS and RHS have a

shared boundary:

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Diagram substitution

  • Note we can drop the free variables:

= a b c b c a

  • =
  • The role of variables is replaced by the fact that the LHS and RHS have a

shared boundary:

  • This treats inputs and outputs symmetrically
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebra and coalgebra

  • We can consider structures with many outputs as well as inputs.
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebra and coalgebra

  • We can consider structures with many outputs as well as inputs.
  • Coalgebraic structures: algebraic structures “upside-down”
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebra and coalgebra

  • We can consider structures with many outputs as well as inputs.
  • Coalgebraic structures: algebraic structures “upside-down”
  • e.g. a comonoid satisfies:

= = =

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebra and coalgebra

  • We can consider structures with many outputs as well as inputs.
  • Coalgebraic structures: algebraic structures “upside-down”
  • e.g. a comonoid satisfies:

= = =

  • The most interesting structures consist of algebras interacting with

coalgebras: = = =

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Equational reasoning with diagram substitution

  • Again, we use equations to perform substitutions, but on graphs rather than

just trees =

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Equational reasoning with diagram substitution

  • Again, we use equations to perform substitutions, but on graphs rather than

just trees =

  • For example:
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Example: Quantum circuit rewriting

= ⊕

H H

⊕

H H

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Example: Quantum circuit rewriting

= ⊕

H H

⊕

H H H

  • ⊕

Zα

⊕ ⊕

Zα

⊕ ⊕

  • ⊕

Zα H

⊕

H H H H H

⊕ ⊕ ⊕ ⊕

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Example: Quantum circuit rewriting

= ⊕

H H

⊕

H H H

  • ⊕

Zα

⊕ ⊕

Zα

⊕ ⊕

  • ⊕

Zα H

⊕

H H H H H

⊕ ⊕ ⊕ ⊕ So, we can define an equational theory for quantum circuits, using rewriting.

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Why an equational theory for quantum circuits?

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Why an equational theory for quantum circuits?

  • circuit optimization:

H

⊕

Zα

⊕ ⊕ = ⊕

Zα H

⊕

H H H H

⊕ ⊕ ⊕ ⊕

H H

=

H

⊕

Zα

=

H

⊕

H Zα

⊕

H H H

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Why an equational theory for quantum circuits?

  • circuit optimization:

H

⊕

Zα

⊕ ⊕ = ⊕

Zα H

⊕

H H H H

⊕ ⊕ ⊕ ⊕

H H

=

H

⊕

Zα

=

H

⊕

H Zα

⊕

H H H

  • verify equivalence (e.g. when adding error-correction)

encode decode

Z Z H H H H H H H

=

Z H

  • (automated) translation to other gate sets and paradigms
  • exploit algebraic invariants to prove properties about computations
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

A complete set of gate identities

  • These equations are complete for Clifford circuits:

(Selinger 2013)

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

As an equational theory

  • The good:
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

As an equational theory

  • The good:
  • complete for Clifford circuits:

C1 = C2 = ⇒ C1 =E C2

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

As an equational theory

  • The good:
  • complete for Clifford circuits:

C1 = C2 = ⇒ C1 =E C2

  • unique normal forms
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

As an equational theory

  • The good:
  • complete for Clifford circuits:

C1 = C2 = ⇒ C1 =E C2

  • unique normal forms
  • relatively compact (3 generators, 15 rules)
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

As an equational theory

  • The good:
  • complete for Clifford circuits:

C1 = C2 = ⇒ C1 =E C2

  • unique normal forms
  • relatively compact (3 generators, 15 rules)
  • The bad:
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

As an equational theory

  • The good:
  • complete for Clifford circuits:

C1 = C2 = ⇒ C1 =E C2

  • unique normal forms
  • relatively compact (3 generators, 15 rules)
  • The bad:
  • rules are large, and don’t carry any intuition or algebraic structure
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

As an equational theory

  • The good:
  • complete for Clifford circuits:

C1 = C2 = ⇒ C1 =E C2

  • unique normal forms
  • relatively compact (3 generators, 15 rules)
  • The bad:
  • rules are large, and don’t carry any intuition or algebraic structure
  • rewrite strategy is complicated (17 derived gates, 100 derived rules)
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

As an equational theory

  • The good:
  • complete for Clifford circuits:

C1 = C2 = ⇒ C1 =E C2

  • unique normal forms
  • relatively compact (3 generators, 15 rules)
  • The bad:
  • rules are large, and don’t carry any intuition or algebraic structure
  • rewrite strategy is complicated (17 derived gates, 100 derived rules)
  • The ugly:
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

As an equational theory

  • The good:
  • complete for Clifford circuits:

C1 = C2 = ⇒ C1 =E C2

  • unique normal forms
  • relatively compact (3 generators, 15 rules)
  • The bad:
  • rules are large, and don’t carry any intuition or algebraic structure
  • rewrite strategy is complicated (17 derived gates, 100 derived rules)
  • The ugly:
  • proof of completeness is extremely complicated (> 100 pages long! though

mostly machine-generated)

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Can we do better?

  • Yes!
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Can we do better?

  • Yes!
  • We can capture underlying algebraic structure by decomposing gates into

smaller pieces

H H H

⊕ ⊕ ⊕ ⊕

H Zα

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Can we do better?

  • Yes!
  • We can capture underlying algebraic structure by decomposing gates into

smaller pieces

H H H

⊕ ⊕ ⊕ ⊕

H Zα

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Decomposing CNOT

⊕

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Decomposing CNOT

⊕

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Decomposing CNOT

⊕

|i |j

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Decomposing CNOT

⊕

copy |i |i |i |j

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Decomposing CNOT

⊕

copy xor |i |i |j |i ⊕ j

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

‘Copy’ maps

  • |0 → |00

|1 → |11 ⊕            |00 → |0 |01 → |1 |10 → |1 |11 → |0

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

‘Copy’ maps

  • |0 → |00

|1 → |11 ⊕

  • |++ → |+

|−− → |−

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

‘Copy’ maps

  • |0 → |00

|1 → |11 ⊕

  • |+ → |++

|− → |−−

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

‘Copy’ maps

  • |0 → |00

|1 → |11

  • |+ → |++

|− → |−−

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

‘Copy’ maps

  • |0 → |00

|1 → |11

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

‘Copy’ maps

  • |0 → |00

|1 → |11

  • |0 → 1

|1 → 1

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

‘Copy’ maps

  • |0 → |00

|1 → |11

  • 0| + 1|
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

‘Copy’ maps

  • |0 → |00

|1 → |11

  • 0| + 1|
  • |00 → |0

|11 → |1

  • |0 + |1
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Algebraic identities...

These satisfy 8 identities: = = = = = = = ...making them a commutative Frobenius algebra.

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

But luckily...

...you don’t need to remember all that! The only thing to remember is, for: ... ... :=

  • |0..0 → |0...0

|1..1 → |1...1

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

But luckily...

...you don’t need to remember all that! The only thing to remember is, for: ... ... :=

  • |0..0 → |0...0

|1..1 → |1...1 we have: ... ... ... ...

...

= ... ...

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

But luckily...

...you don’t need to remember all that! The only thing to remember is, for: ... ... :=

  • |0..0 → |0...0

|1..1 → |1...1 we have: ... ... ... ...

...

= ... ...

  • r equivalently:

=

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

But luckily...

...you don’t need to remember all that! The only thing to remember is, for: ... ... :=

  • |+..+ → |+...+

|−..− → |−...− we have: ... ... ... ...

...

= ... ...

  • r equivalently:

=

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

What about 2-colour diagrams?

Direction of edges doesn’t matter: = =:

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

What about 2-colour diagrams?

Direction of edges doesn’t matter: = =: ...in fact, only topology matters: = =

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Interaction: Hopf algebra

Red + green spiders also satisfy: = = =

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Interaction: Hopf algebra

Red + green spiders also satisfy: = = = ...from which we can derive: = make the overall structure into a Hopf algebra

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Circuit calculation

= ⊕ ⊕ ⊕

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Circuit calculation

=

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Circuit calculation

=

  • ⇐
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Making spiders universal

... ... :=

  • |0..0 → |0...0

|1..1 → |1...1 ... ... :=

  • |+..+ → |+...+

|−..− → |−...−

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Making spiders universal

α

... ... :=

  • |0..0 → |0...0

|1..1 → eiα |1...1

α

... ... :=

  • |+..+ → |+...+

|−..− → eiα |−...−

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Making spiders universal

α

... ... :=

  • |0..0 → |0...0

|1..1 → eiα |1...1 ...

α + β

... ... ...

α β ...

= ... ...

α

... ... :=

  • |+..+ → |+...+

|−..− → eiα |−...− ...

α + β

... ... ...

α β ...

= ... ...

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Making spiders universal

Theorem

Phased spiders are universal for qubit quantum computation.

Proof.

Let: ⊕ :=

U

:=

γ β α

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

The ZX-calculus

The ZX-calculus consists of the two spider-fusion rules:

...

α + β

... ... ...

α β ...

= ... ... ...

α + β

... ... ...

α β ...

= ... ...

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

The ZX-calculus

The ZX-calculus consists of the two spider-fusion rules:

...

α + β

... ... ...

α β ...

= ... ... ...

α + β

... ... ...

α β ...

= ... ...

four Interaction rules:

= = =

π π π π α π

=

  • α
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

The ZX-calculus

The ZX-calculus consists of the two spider-fusion rules:

...

α + β

... ... ...

α β ...

= ... ... ...

α + β

... ... ...

α β ...

= ... ...

four Interaction rules:

= = =

π π π π α π

=

  • α

and the Colour Change rule:

α

· · ·

α

· · · · · · · · = · · · · ·

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

The ZX-calculus

The ZX-calculus consists of the two spider-fusion rules:

...

α + β

... ... ...

α β ...

= ... ... ...

α + β

... ... ...

α β ...

= ... ...

four Interaction rules:

= = =

π π π π α π

=

  • α

and the Colour Change rule:

α

· · ·

α

· · · · · · · · = · · · · ·

where

π 2 π 2 π 2

:=

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Completeness

Theorem (Backens 2013)

The ZX-calculus is complete for Clifford ZX-diagrams: D1 = D2 = ⇒ D1 =zx D2 D1 :=

. . . . . .

· · · · · ·

π 2

· · · · · ·

π 2

D2 :=

. . . . . .

· · · · · ·

π 2

· · · · · ·

π 2

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

  • Measurement-based quantum computing is an alternative (and

equivalent) paradigm to the circuit model

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

  • Measurement-based quantum computing is an alternative (and

equivalent) paradigm to the circuit model

  • Rather than repeatedly applying operations to a small number of systems,

start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

  • Measurement-based quantum computing is an alternative (and

equivalent) paradigm to the circuit model

  • Rather than repeatedly applying operations to a small number of systems,

start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

  • Measurement-based quantum computing is an alternative (and

equivalent) paradigm to the circuit model

  • Rather than repeatedly applying operations to a small number of systems,

start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

  • Measurement-based quantum computing is an alternative (and

equivalent) paradigm to the circuit model

  • Rather than repeatedly applying operations to a small number of systems,

start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

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SLIDE 79

Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

  • Measurement-based quantum computing is an alternative (and

equivalent) paradigm to the circuit model

  • Rather than repeatedly applying operations to a small number of systems,

start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

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SLIDE 80

Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

  • Measurement-based quantum computing is an alternative (and

equivalent) paradigm to the circuit model

  • Rather than repeatedly applying operations to a small number of systems,

start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

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SLIDE 81

Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

  • Measurement-based quantum computing is an alternative (and

equivalent) paradigm to the circuit model

  • Rather than repeatedly applying operations to a small number of systems,

start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

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SLIDE 82

Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

  • Measurement-based quantum computing is an alternative (and

equivalent) paradigm to the circuit model

  • Rather than repeatedly applying operations to a small number of systems,

start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

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SLIDE 83

Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

  • Measurement-based quantum computing is an alternative (and

equivalent) paradigm to the circuit model

  • Rather than repeatedly applying operations to a small number of systems,

start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

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SLIDE 84

Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

  • Measurement-based quantum computing is an alternative (and

equivalent) paradigm to the circuit model

  • Rather than repeatedly applying operations to a small number of systems,

start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

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SLIDE 85

Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurement-based quantum computing

  • Measurement-based quantum computing is an alternative (and

equivalent) paradigm to the circuit model

  • Rather than repeatedly applying operations to a small number of systems,

start with a big entangled state called a graph state and do many local measurements in different bases: ... ... ... ... ... ...

  • But crucially, the choices of measurements can depend on past

measurement outcomes. This is called feed-forward, and it’s where all the magic happens.

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Graph states and cluster states

  • Graph states are prepared by starting with many qubits in the |+ state and

creating entanglement with controlled-Z operations:

=

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Graph states and cluster states

  • Graph states are prepared by starting with many qubits in the |+ state and

creating entanglement with controlled-Z operations:

=

  • Since controlled-Z’s commute, the only relevant part is the graph:

... ... ... ... ... ... ← →

... ... ... ... ... ... ... ... ...

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurements and feed-forward

  • Compute with single qubit ONB measurements of this form:

π

,

  • α

α + π

,

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurements and feed-forward

  • Compute with single qubit ONB measurements of this form:

π

,

  • α

α + π

,

  • We want to get the first outcome and treat the second outcome as an error:

... α π error

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurements and feed-forward

  • We can propagate the error out using the ZX-rules:

... α π = ... π α ... π α = π π ... α π = = π α ... π

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurements and feed-forward

  • We can propagate the error out using the ZX-rules:

... α π = ... π α ... π α = π π ... α π = = π α ... π

  • If we know an error occurred, we can modify our later measurement choices

to account for it: π α π ... −β γ + π = ... α β γ

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Measurements and feed-forward

  • We can propagate the error out using the ZX-rules:

... α π = ... π α ... π α = π π ... α π = = π α ... π

  • If we know an error occurred, we can modify our later measurement choices

to account for it: π α π ... −β γ + π = ... α β γ

  • ⇐
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Notable results

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Notable results: MBQC

  • Duncan & Perdrix used the ZX-calculus to offer a new technique for

transforming MBQC patterns to circuits, which has some advantages over

  • ther known methods, e.g. not requiring ancillas.1
  • For more details, Ducan has written a self-contained introduction to MBQC

from the diagrammatic/ZX point of view, which is available on the arXiv.2

1Rewriting measurement-based quantum computations with generalised flow. R. Duncan,

  • S. Perdrix, ICALP 2010.

personal.strath.ac.uk/ross.duncan/papers/gflow.pdf

2A graphical approach to measurement-based quantum computing. R. Duncan.

arXiv:1203.6242

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Notable results: quantum algorithms

  • Vicary gave graphical characterisations of standard quantum algorithms3
  • ...a framework since used by Vicary & Zeng to develop new algorithms as

generalisations4

3The Topology of Quantum Algorithms. LICS 2013, J. Vicary. arXiv:1209.3917 4Abstract structure of unitary oracles for quantum algorithms. J.Vicary, W. Zeng.

arXiv:1406.1278

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Notable results: quantum protocols

  • Coecke, along with 3 Wangs and a Zhang give graphical proof of QKD5
  • Hillebrand gave rewriting proofs of many (∼ 25) quantum protocols.6
  • Zamdzhiev used ZX-calculus to verify 3 kinds of quantum secret sharing.7

5Graphical Calculus for Quantum Key Distribution. B. Coecke, Q. Wang, B. Wang,

  • Y. Wang, and Q. Zhang. QPL 2011.

6Quantum Protocols involving Multiparticle Entanglement and their Representations in the

zx-calculus. A. Hillebrand. Masters thesis, Oxford 2011. www.cs.ox.ac.uk/people/bob.coecke/Anne.pdf

7An Abstract Approach towards Quantum Secret Sharing. Masters thesis, Oxford 2012.

www.cs.ox.ac.uk/people/bob.coecke/VladimirZamdzhievThesis.pdf

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Notable results: quantum non-locality

  • AK, Coecke, Duncan, and Wang gave diagrammatic presentation of

GHZ/Mermin non-locality argument8

αi

=

αi αi

=

α3 α2 α1

=

  • ...which has since been generalised to arbitrary dimensions and

quantum-like theories9

8Strong Complementarity and Non-locality in Categorical Quantum Mechanics. B. Coecke,

  • R. Duncan, A. Kissinger, Q. Wang. LICS 2012.

9Mermin Non-Locality in Abstract Process Theories. QPL 2015

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Where do we go from here?

  • Completeness (Clifford + T, full)
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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Where do we go from here?

  • Completeness (Clifford + T, full)
  • Automation: implementation of Clifford decision procedure, theory synthesis
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SLIDE 100

Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Where do we go from here?

  • Completeness (Clifford + T, full)
  • Automation: implementation of Clifford decision procedure, theory synthesis
  • Bigger algorithms, more sophisticated protocols, and generally more

expressiveness of the diagrammatic language

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Introduction Quantum circuits Spiders ZX-calculus MBQC Survey

Thanks!

  • Quantomatic is joint work with Lucas Dixon, Alex Merry, Ross Duncan,

Vladimir Zamdzhiev, and David Quick

  • See: quantomatic.github.io