the zx-calculuspersonal.strath.ac.uk/ross.duncan/talks/2015/vaxjo.pdf · ross duncan quantum...

Ross Duncan ●Quantum Foundations ● Växjo 2015

The ZX-calculus

diagrams for quantum computing

Ross Duncan

University of Strathclyde

Pessimistic Diagram Convention

PAST / HEAVEN

FUTURE / HELL

|�Í Input register

Quantum Circuits

|�ÕÍ Output register

U : C2n - C2n

“Time”

Universality

We say that a model of quantum computation is universal if it can represent all unitary maps. The circuit model requires a small set of gates to be universal:

Universality

We say that a model of quantum computation is universal if it can represent all unitary maps. The circuit model requires a small set of gates to be universal:

Quantum CircuitsExample 3.9 (Bell state). A Bell state, |00Í + |11Í, is prepared by the circuitbelow, on the left.

|0Í |0Í

ddddb= H úæ úæ úæ

The corresponding zx-calculus derivation is a proof of the correctness of thiscircuit.

To simplify the next example, we introduce some shorthand for some usefulcircuit elements.

|+Í:=

Example 3.10 (1D-cluster state). A 1-dimensional cluster state consists of acollection qubits 1, . . . , n initialised in the state |+Í = |0Í + |1Í, which are thenentangled by applying a ·Z operation to qubits i and i ≠ 1, and to i and i + 1.The corresponding circuit is shown below:

|+Í |+Í |+Í |+Í

... ...

ddddb= H H H

H H...

The zx-calculus translation can be radically simplified by repeated use of thespider rule:

H H...

úæ H H H H H... ...

The 1D-cluster is a special case of a more general class of states: graph states.These states are the basis of measurement-based quantum computation, and willplay an important role later in the paper.

3.2 Circuit-like diagrams

While the preceding has demonstrated the ease with quantum circuits can betranslated into diagrams, there are many diagrams which do not correspond to

Preparing a Bell state

Example 3.9 (Bell state). A Bell state, |00Í + |11Í, is prepared by the circuitbelow, on the left.

|0Í |0Í

|+Í:=

|+Í |+Í |+Í |+Í

... ...

ddddb= H H H

H H...

úæ H H H H H... ...

Controlled-Z gate

Quantum Circuits

Good Points: + Universal + Clear operational interpretation Bad points: - Inflexible - Mathematically ad hoc - Few nice algebraic properties

What is the ZX-calculus?

• Abstract diagrammatic theory (GPT)

• Includes a lot of qubit quantum theory

• Based on the properties of the Pauli Z and X observables

ZX-calculusGood Points: + Universal + Derived from the basic algebra of complementarity + Powerful algebraic theory + Can represent almost anything Bad point: - Need to impose operational meaning post-hoc Meh Point: - Not complete (see Simon Perdrix’s talk later!)

WARNINGIn this talk:

• ignoring global scalar factors • mostly talking about pure states and processes

Linear maps Linear maps

A new process theory from an old one...

I The theory of pure quantum maps has types:

I and processes:

for all processes f from linear maps.

Aleks Kissinger (QTFT, Vaxjo 2015) Picturing Quantum Processes June 9, 2015 10 / 33

WARNINGIn this talk:

• ignoring global scalar factors • mostly talking about pure states and processes

Linear maps Linear maps

A new process theory from an old one...

I The theory of pure quantum maps has types:

I and processes:

for all processes f from linear maps.

|0i 7! |00i|1i 7! |11i

h0|+ ei↵ h1|

|0i 7! |00i|1i 7! |11i

h0|+ ei↵ h1|

|0i 7! |00i|1i 7! |11i

h0|+ ei↵ h1|

This is unitary

|0i 7! |00i|1i 7! |11i

h0|+ ei↵ h1|

This is unitary

✓1 00 ei↵

|0i 7! |00i|1i 7! |11i

h0|+ ei↵ h1|

This is unitary

✓1 00 ei↵

“Phase”

–5–3

double

B= ≠– –

double

B= k k k œ {0, fi}

double

a k + fi

b = k ≠ fi

2 k + fi

2 k œ {0, fi}

= k k œ {0, fi}

2 ≠ fi

k ≠ fi

2 k + fi

= k k œ {0, fi}

2 ≠ fi

–i–1 –

· · · · · · =q

œ [0, 2fi)

✓1 00 ei↵

✓1 00 ei�

✓1 00 ei(↵+�)

– + —

Lemma: phases form an abelian group

Generalised SpiderThm: Any connected monochrome diagram with phases is determined completely by its arity and the sum of its phases.

– + —

–5–3

i –i

ZX-calculus syntax

Defn: A diagram is an undirected open graph generated by the above vertices.

–...

Figure 2: Interior vertices of diagrams

• X vertices with m inputs and n outputs, labelled by an angle – œ [0, 2fi);these are these are denoted Xm

n (–), and shown graphically as (dark) redcircles,

• H (or Hadamard) vertices, restricted to degree 2; shown as squares.

If a X or Z vertex has – = 0 then the label is entirely omitted. The allowedvertices are shown in Figure ??.

Since the inputs and outputs of of a diagram are totally ordered, we canidentify them with natural numbers and speak of the kth input, etc.

Remark 3.4. When a vertex occurs inside the graph, the distinction betweeninputs and outputs is purely conventional: one can view them simply as verticesof degree n + m; however, this distinction allows the semantics to be stated moredirectly, see below.

The collection of diagrams forms a compact category in the obvious way: theobjects are natural numbers and the arrows m æ n are those diagrams withm inputs and n outputs; composition g ¶ f is formed by identifying the inputsof g with the outputs of f and erasing the corresponding vertices; f ¢ g is thediagram formed by the disjoint union of f and g with If ordered before Ig, andsimilarly for the outputs. This is basically the free (self-dual) compact categorygenerated by the arrows shown in Figure ??.

We can make this category †-compact by specifying that f† is the samediagram as f , but with the inputs and outputs exchanged, and all the anglesnegated.

This construction yields a category that does not incorporate the algebraicstructure of strongly complementary observables. To obtain the desired categorywe must quotient by the equations shown in Figure ??. We denote the categoryso-obtained by D.

Remark 3.5. The equations shown in Figure ?? are not exactly those describedin Sections ?? and ??, however they are equivalent to them. We shall therefore,on occasion, use properties discussed earlier as derived rules in computations.

Since D is a monoidal category we can assign an interpretation to any diagramby providing a monoidal functor from D to any other monoidal category. Sincewe interested in quantum mechanics, the obvious target category is fdHilb.

Definition 3.6. Let J·K : D æ fdHilb be a symmetric monoidal functor definedon objects by

J1K = C2

–...

J1K = C2

↵ 2 [0, 2⇡)

ZX-calculus semantics

–...

J1K = C2

–...

J1K = C2

|+i⌦n 7! |+i⌦m

|�i⌦n 7! ei↵ |�i⌦m

|0i⌦n 7! |0i⌦m

|1i⌦n 7! ei↵ |1i⌦m

Representing Qubits

Representing Phase shifts

Representing Paulis

Representing CNot

The ZX-calculus is universal

Theorem: Let U be a unitary map on n qubits; then there exists a ZX-calculus term D such that:

JDK = U

The ZX-calculus is universal

Theorem: Let U be a unitary map on n qubits; then there exists a ZX-calculus term D such that:

JDK = U

Equations

EquationsGeneralised Spider

Equations

Equations“Strong Complementarity”

Equations

· · ·

↵+ n�2

· · ·

�⇡2

�⇡2 �⇡

�⇡2

(colour change)

Equations

· · ·

↵+ n�2

· · ·

�⇡2

�⇡2 �⇡

�⇡2

(colour change)

A weird one specific to ZX

Example: CNOTS

– + —

–5–3

i –i

– + —

–5–3

i –i

Example: CNOTS

= = = =

– + —

–5–3

i –i

Representing Hadamard

H = 1Ô2

31 11 ≠1

4= J K

More on the Hadamard

· · ·

↵+ n�2

· · ·

↵+ n�2

· · ·

�⇡2 �⇡

· · ·

↵+ n�2

· · ·

�⇡2 �⇡

· · ·

Corollary: total symmetry between red and green

Example: preparing a Bell stateExample 3.9 (Bell state). A Bell state, |00Í + |11Í, is prepared by the circuitbelow, on the left.

|0Í |0Í

|+Í:=

|+Í |+Í |+Í |+Í

... ...

ddddb= H H H

H H...

úæ H H H H H... ...

Proof. It su�ces to show that there are zx-calculus terms for the matrices Z–,H and ·X. We have

J H K = H, J – K = Z– and J K = ·X

which can be verified by direct calculation. Note that

J K = J K

so the presentation of ·X is unambiguous.

Example 3.12 (The ·Z-gate). The ·Z-gate can be obtained by using aHadamard (H) gate to transform the second qubit of a ·X gate. We obtain asimpler representation using the colour-change rule

From the presentation of ·Z in the zx-calculus, we can immediately read o�that it is symmetric in its inputs. Furthermore, we can prove one of the basicproperties of the ·Z gate, namely that is self-inverse.

Example 3.13 (Bell state). The following is a zx-calculus term representing aquantum circuit which produces a Bell state, |00Í + |11Í. We can verify this factby the equations of the calculus.

H = = =

The zx-calculus can represent many things which do not correspond toquantum circuits. We now present a criterion to recognise which diagrams docorrespond to quantum circuits.

Example: preparing a Bell state

J H K = H, J – K = Z– and J K = ·X

J K = J K

H = = =

Example: Controlled-Z

J H K = H, J – K = Z– and J K = ·X

J K = J K

H = = =

|0Í |0Í

|+Í:=

|+Í |+Í |+Í |+Í

... ...

ddddb= H H H

H H...

úæ H H H H H... ...

Example: Controlled-Z

J H K = H, J – K = Z– and J K = ·X

J K = J K

H = = =

Which diagrams are circuits?

Defn: a diagram is called circuit-like if:

• All of its , , and boundary vertices can be covered by set of disjoint paths P, each of which ends in an output.

• The subgraph determined by each p in P is acyclic

• Every cycle in the diagram which overlaps with 2 paths in P traverses an edge in the opposite direction to P.

• It is 2-coloured (or 3-coloured, if including H)

H HNO :(

Which diagrams are circuits?Thm: if a diagram is circuit-like then is a unitary embedding.

Being circuit-like is a stronger than being a unitary embedding: * particular gate set * particular representation * minimality w.r.t. to some of the rules. * &c …

Graph StatesLet G = (V,E) be a simple, undirected graph. Then define: !!!Viewed as circuit we get this:

|Gi =O

(v,u)2E

|0Í |0Í

|+Í:=

|+Í |+Í |+Í |+Í

... ...

ddddb= H H H

H H...

úæ H H H H H... ...

Graph StatesLet G = (V,E) be a simple, undirected graph. Then define: !!!Viewed as circuit we get this:

|Gi =O

(v,u)2E

|0Í |0Í

|+Í:=

|+Í |+Í |+Í |+Í

... ...

ddddb= H H H

H H...

úæ H H H H H... ...

Graph StatesLet G = (V,E) be a simple, undirected graph. Then define: !!!But we can also view it as a PEPS:

|Gi =O

(v,u)2E

· · · · · ·

X XYX YX YXY XYY

fi fi fi

”= fi

H H H H H H H

Graph StatesLet G = (V,E) be a simple, undirected graph. Then define: !!!!

|Gi =O

(v,u)2E

|0Í |0Í

|+Í:=

|+Í |+Í |+Í |+Í

... ...

ddddb= H H H

H H...

úæ H H H H H... ...

Graph StatesLet G = (V,E) be a simple, undirected graph. Then define: !!!Or in 2D:

|Gi =O

(v,u)2E

· · · · · ·

X XYX YX YXY XYY

fi fi fi

”= fi

Graph StatesLet G = (V,E) be a simple, undirected graph. Then define: !!!Or in 2D:

|Gi =O

(v,u)2E

· · · · · ·

X XYX YX YXY XYY

fi fi fi

”= fi

Hopf Algebra Equivalence

Thm: any Hopf algebra expression can be put into normal form:

Application 1: MBQC

1WQC is a quantum computer design based on single qubit projective measurements on a graph state. !• We can use the ZX-calculus to translate from the 1WQC to

circuit model • Relies on the Hopf algebra normal form • Produces circuits with minimal space complexity

GFlow Strategy

Flow Strategy

Application 2: QECCZX-calculus can demonstrate the correctness Quantum Error Correcting Codes:

Application 3: Mermin-GHZThe Mermin-GHZ argument is a no-go result for LHV theories which does not rely on probabilities. !We can show:

• The argument can be carried out in the ZX-calculus • Any pair of observables satisfying the assumptions of

the argument are strongly complementary.

Mermin-GHZ|GHZi = |000i+ |111i

X X X +1

X Y Y -1

Y X Y -1

Y Y X -1

X X X +1

X Y Y -1

Y X Y -1

Y Y X -1

+1 +1 +1 +1 ≠ -1

Ingredients:

• Doubling (like in Aleks’s talk) • Measurements of X and Y • Parity computation on the outcomes • GHZ state • LHV state

DoublingQuantum systems —> Double wires Classical systems —> Single wires

Linear maps

Quantum spiders

I A quantum spider is a classical spider, doubled:

...:= double

I An example is the GHZ state:

GHZ := = double

I They also fuse:...

–5–3

i –i

double

B= ≠– –

DoublingEigenstates of X and Y

–5–3

i –i

double

B= ≠– –

double

B= k k k œ {0, fi}

double

a k + fi2

b = k ≠ fi2 k + fi

2 k œ {0, fi}

≠k k

= k k œ {0, fi}

–5–3

i –i

double

B= ≠– –

double

B= k k k œ {0, fi}

double

a k + fi2

b = k ≠ fi2 k + fi

2 k œ {0, fi}

≠k k

= k k œ {0, fi}

–5–3

i –i

double

B= ≠– –

double

B= k k k œ {0, fi}

double

a k + fi2

b = k ≠ fi2 k + fi

2 k œ {0, fi}

≠k k

= k k œ {0, fi}

MeasurementsX Measurement:

–5–3

i –i

Sanity check:

–5–3

i –i

double

B= ≠– –

double

B= k k k œ {0, fi}

double

a k + fi2

b = k ≠ fi2 k + fi

2 k œ {0, fi}

= k k œ {0, fi}

MeasurementsY Measurement:

Sanity check:

–5–3

i –i

double

B= ≠– –

double

B= k k k œ {0, fi}

double

a k + fi2

b = k ≠ fi2 k + fi

2 k œ {0, fi}

= k k œ {0, fi}

+ fi2 ≠ fi

–5–3

i –i

double

B= ≠– –

double

B= k k k œ {0, fi}

double

a k + fi2

b = k ≠ fi2 k + fi

2 k œ {0, fi}

= k k œ {0, fi}

+ fi2 ≠ fi

k ≠ fi2 k + fi

= k k œ {0, fi}

+ fi2 ≠ fi

Parity Computation

–5–3

i –i

double

B= ≠– –

double

B= k k k œ {0, fi}

double

a k + fi2

b = k ≠ fi2 k + fi

2 k œ {0, fi}

= k k œ {0, fi}

+ fi2 ≠ fi

k ≠ fi2 k + fi

= k k œ {0, fi}

+ fi2 ≠ fi

–i–1 –n· · · · · · =

qi –i

–i œ [0, 2fi)

+ fi2 ≠ fi

kik1 kn· · · · · · =

ki œ {0, fi}

Parity Computation

–5–3

i –i

double

B= ≠– –

double

B= k k k œ {0, fi}

double

a k + fi2

b = k ≠ fi2 k + fi

2 k œ {0, fi}

= k k œ {0, fi}

+ fi2 ≠ fi

k ≠ fi2 k + fi

= k k œ {0, fi}

+ fi2 ≠ fi

–i–1 –n· · · · · · =

qi –i

–i œ [0, 2fi)

+ fi2 ≠ fi

kik1 kn· · · · · · =

ki œ {0, fi}

Parity Computation

· · · · · ·

GHZ State· · · · · ·

double

Measuring the GHZ state· · · · · ·

double

≠–1+–1 +–2 ≠–2 +–3 ≠–3

i –iq

i –i

i –iq

i –i

Measuring the GHZ state

2 ≠ fi

kik1 k

· · · · · · =q

œ {0, fi}

· · · · · ·

≠–1+–1 +–2 ≠–2 +–3 ≠–3

· · · · · ·

double

≠–1+–1 +–2 ≠–2 +–3 ≠–3

i –iq

i –i

i –iq

i –i

Measuring the GHZ state: XXX

· · · · · ·

double

≠–1+–1 +–2 ≠–2 +–3 ≠–3

i –iq

i –i

i –iq

i –i

Measuring the GHZ state: XXX

· · · · · ·

double

≠–1+–1 +–2 ≠–2 +–3 ≠–3

i –iq

i –i

i –iq

i –i

· · · · · ·

double

≠–1+–1 +–2 ≠–2 +–3 ≠–3

i –iq

i –i

i –iq

i –i

Measuring the GHZ state: XYY

· · · · · ·

double

≠–1+–1 +–2 ≠–2 +–3 ≠–3

i –iq

i –i

i –iq

i –i

Measuring the GHZ state: XYY

· · · · · ·

double

≠–1+–1 +–2 ≠–2 +–3 ≠–3

i –iq

i –i

i –iq

i –i

Local Hidden State⇤ = (kx1 , k

1 , kx

2 , ky

2 , kx

3 , ky

· · · · · ·

double

≠–1+–1 +–2 ≠–2 +–3 ≠–3

X XYX YX YXY XYY

Local Hidden State⇤ = (kx1 , k

1 , kx

2 , ky

2 , kx

3 , ky

Parity of LHV outcomeskx

X XYX YX YXY XYY

Parity of QM outcomes· · · · · ·

X XYX YX YXY XYY

fi fi fi

Parity of QM outcomes

· · · · · ·

X XYX YX YXY XYY

fi fi fi

Parity of QM outcomes

· · · · · ·

X XYX YX YXY XYY

fi fi fi

Contradiction!

· · · · · ·

X XYX YX YXY XYY

fi fi fi

”= fi

Comments

• The proof relied on the strong complementarity of X and Z • This can be generalised to higher dimensions, with some

other abelian group replacing parity • We have the following converse: The assumptions used by Mermin imply the observables in the proof must be strongly complementary

Thanks!XZ-calculus: B. Coecke and RD, Interacting Quantum Observables: Categorical Algebra and Diagrammatics; NJP (2011), arXiv:0906.4725 MBQC: RD and S. Perdrix, Rewriting measurement-based quantum computations with generalised flow, ICALP (2010), doi:10.1007/978-3-642-14162-1\_24 RD, A graphical approach to measurement-based quantum computing, Quantum Physics and Linguistics, OUP 2011 arxiv:1203.6242 Steane Code RD and Maxime Lucas, Verifying the Steane code with Quantomatic, QPL 2013, doi:10.4204/EPTCS.171.4 GHZ-Mermin: B. Coecke, RD, A. Kissinger, Q. Wang, Strong Complementarity and Non-locality in Categorical Quantum Mechanics, LiCS 2012, doi: 10.1109/LICS.2012.35 B. Coecke, RD, A. Kissinger, Q. Wang, Generalised Compositional Theories and Diagrammatic Reasoning; forthcoming in Giulio and Rob’s book; arxiv soon!

the zx-calculuspersonal.strath.ac.uk/ross.duncan/talks/2015/vaxjo.pdf · ross duncan quantum...

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