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Ross Duncan ●Quantum Foundations ● Växjo 2015
The ZX-calculus
diagrams for quantum computing
Ross Duncan
University of Strathclyde
Ross Duncan ●Quantum Foundations ● Växjo 2015
Pessimistic Diagram Convention
PAST / HEAVEN
FUTURE / HELL
Ross Duncan ●Quantum Foundations ● Växjo 2015
|�Í Input register
Quantum Circuits
|�ÕÍ Output register
Unita
ry g
ates
U : C2n - C2n
“Time”
Ross Duncan ●Quantum Foundations ● Växjo 2015
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:
Ross Duncan ●Quantum Foundations ● Växjo 2015
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:
Ross Duncan ●Quantum Foundations ● Växjo 2015
Quantum CircuitsExample 3.9 (Bell state). A Bell state, |00Í + |11Í, is prepared by the circuitbelow, on the left.
T
Q
cccca
|0Í |0Í
H
R
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.
Z :=H
H
|+Í:=
|0Í
H
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:
T
Q
cccca
|+Í |+Í |+Í |+Í
Z Z Z
Z Z
... ...
... ...
R
ddddb= H H H
H H...
...
...
...
The zx-calculus translation can be radically simplified by repeated use of thespider rule:
H H H
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
16
Preparing a Bell state
Example 3.9 (Bell state). A Bell state, |00Í + |11Í, is prepared by the circuitbelow, on the left.
T
Q
cccca
|0Í |0Í
H
R
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.
Z :=H
H
|+Í:=
|0Í
H
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:
T
Q
cccca
|+Í |+Í |+Í |+Í
Z Z Z
Z Z
... ...
... ...
R
ddddb= H H H
H H...
...
...
...
The zx-calculus translation can be radically simplified by repeated use of thespider rule:
H H H
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
16
Controlled-Z gate
Ross Duncan ●Quantum Foundations ● Växjo 2015
Quantum Circuits
Good Points: + Universal + Clear operational interpretation Bad points: - Inflexible - Mathematically ad hoc - Few nice algebraic properties
Ross Duncan ●Quantum Foundations ● Växjo 2015
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
Ross Duncan ●Quantum Foundations ● Växjo 2015
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!)
Ross Duncan ●Quantum Foundations ● Växjo 2015
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:
:=
bA
I and processes:
=bf
ff
for all processes f from linear maps.
Aleks Kissinger (QTFT, Vaxjo 2015) Picturing Quantum Processes June 9, 2015 10 / 33
Ross Duncan ●Quantum Foundations ● Växjo 2015
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:
:=
bA
I and processes:
=bf
ff
for all processes f from linear maps.
Aleks Kissinger (QTFT, Vaxjo 2015) Picturing Quantum Processes June 9, 2015 10 / 33
Ross Duncan ●Quantum Foundations ● Växjo 2015
|0i 7! |00i|1i 7! |11i
= = =
= = =
=
–
–
1
Ross Duncan ●Quantum Foundations ● Växjo 2015
|0i 7! |00i|1i 7! |11i
= = =
= = =
=
–
1
h0|+ ei↵ h1|
= = =
= = =
=
–
–
1
Ross Duncan ●Quantum Foundations ● Växjo 2015
|0i 7! |00i|1i 7! |11i
h0|+ ei↵ h1|
= = =
= = =
=
–
–
1
Ross Duncan ●Quantum Foundations ● Växjo 2015
|0i 7! |00i|1i 7! |11i
h0|+ ei↵ h1|
= = =
= = =
=
–
–
1
This is unitary
Ross Duncan ●Quantum Foundations ● Växjo 2015
|0i 7! |00i|1i 7! |11i
h0|+ ei↵ h1|
= = =
= = =
=
–
–
1
This is unitary
✓1 00 ei↵
◆
Ross Duncan ●Quantum Foundations ● Växjo 2015
|0i 7! |00i|1i 7! |11i
h0|+ ei↵ h1|
= = =
= = =
=
–
–
1
This is unitary
✓1 00 ei↵
◆
“Phase”
Ross Duncan ●Quantum Foundations ● Växjo 2015
–
–
—
=–
—
–1
–5–3
–2
–4
=q
i
–i
double
A–
B= ≠– –
double
Ak
B= k k k œ {0, fi}
double
Q
a k + fi
2
R
b = k ≠ fi
2 k + fi
2 k œ {0, fi}
k k
= k k œ {0, fi}
+ fi
2 ≠ fi
2
k ≠ fi
2 k + fi
2
=k k
= k k œ {0, fi}
+ fi
2 ≠ fi
2
–i–1 –
n
· · · · · · =q
i
–i
–i
œ [0, 2fi)
1
✓1 00 ei↵
◆
✓1 00 ei�
◆
Ross Duncan ●Quantum Foundations ● Växjo 2015
✓1 00 ei(↵+�)
◆
= = =
= = =
=
–
–
–
—
1
Ross Duncan ●Quantum Foundations ● Växjo 2015
= = =
= = =
=
–
– + —
=–
—
1
Lemma: phases form an abelian group
Ross Duncan ●Quantum Foundations ● Växjo 2015
Generalised SpiderThm: Any connected monochrome diagram with phases is determined completely by its arity and the sum of its phases.
= = =
= = =
=
–
– + —
=–
—
–1
–5–3
–2
–4
=q
i –i
1
Ross Duncan ●Quantum Foundations ● Växjo 2015
ZX-calculus syntax
Defn: A diagram is an undirected open graph generated by the above vertices.
...
...
–...
...
– H
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
18
...
...
–...
...
– H
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
18
↵ 2 [0, 2⇡)
Ross Duncan ●Quantum Foundations ● Växjo 2015
ZX-calculus semantics
...
...
–...
...
– H
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
18
...
...
–...
...
– H
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
18
|+i⌦n 7! |+i⌦m
|�i⌦n 7! ei↵ |�i⌦m
|0i⌦n 7! |0i⌦m
|1i⌦n 7! ei↵ |1i⌦m
Ross Duncan ●Quantum Foundations ● Växjo 2015
Representing Qubits
Ross Duncan ●Quantum Foundations ● Växjo 2015
Representing Phase shifts
Ross Duncan ●Quantum Foundations ● Växjo 2015
Representing Paulis
Ross Duncan ●Quantum Foundations ● Växjo 2015
Representing CNot
Ross Duncan ●Quantum Foundations ● Växjo 2015
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
Ross Duncan ●Quantum Foundations ● Växjo 2015
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
7!
7!
7!
Ross Duncan ●Quantum Foundations ● Växjo 2015
Equations
Ross Duncan ●Quantum Foundations ● Växjo 2015
EquationsGeneralised Spider
Ross Duncan ●Quantum Foundations ● Växjo 2015
Equations
Ross Duncan ●Quantum Foundations ● Växjo 2015
Equations“Strong Complementarity”
Ross Duncan ●Quantum Foundations ● Växjo 2015
Equations
· · ·
· · ·
�2
↵+ n�2
�2
�2
�2
· · ·
· · ·
�⇡2
↵
�⇡2 �⇡
2
�⇡2
=
(colour change)
Ross Duncan ●Quantum Foundations ● Växjo 2015
Equations
· · ·
· · ·
�2
↵+ n�2
�2
�2
�2
· · ·
· · ·
�⇡2
↵
�⇡2 �⇡
2
�⇡2
=
(colour change)
A weird one specific to ZX
Ross Duncan ●Quantum Foundations ● Växjo 2015
Example: CNOTS
= = =
= = =
=
–
– + —
=–
—
–1
–5–3
–2
–4
=q
i –i
1
= = =
= = =
=
–
– + —
=–
—
–1
–5–3
–2
–4
=q
i –i
1
?
Ross Duncan ●Quantum Foundations ● Växjo 2015
Example: CNOTS
= = =
= = = =
–
– + —
=–
—
–1
–5–3
–2
–4
=q
i –i
1
Ross Duncan ●Quantum Foundations ● Växjo 2015
Representing Hadamard
fi2
fi2
fi2
H = 1Ô2
31 11 ≠1
4= J K
Ross Duncan ●Quantum Foundations ● Växjo 2015
More on the Hadamard
fi2
fi2
fi2
H :=
Ross Duncan ●Quantum Foundations ● Växjo 2015
More on the Hadamard
H H
H H
· · ·
· · ·
↵
Ross Duncan ●Quantum Foundations ● Växjo 2015
More on the Hadamard
H H
H H
· · ·
· · ·
↵
· · ·
· · ·
↵+ n�2
⇡2
�2
⇡2
�2
⇡2
�2
⇡2
�2
=
Ross Duncan ●Quantum Foundations ● Växjo 2015
More on the Hadamard
H H
H H
· · ·
· · ·
↵
· · ·
· · ·
↵+ n�2
⇡2
�2
⇡2
�2
⇡2
�2
⇡2
�2
=
· · ·
· · ·
↵
⇡2
⇡2
⇡2
⇡2
�⇡2 �⇡
2
�⇡2 �⇡
2
=
Ross Duncan ●Quantum Foundations ● Växjo 2015
More on the Hadamard
H H
H H
· · ·
· · ·
↵
· · ·
· · ·
↵+ n�2
⇡2
�2
⇡2
�2
⇡2
�2
⇡2
�2
=
· · ·
· · ·
↵
⇡2
⇡2
⇡2
⇡2
�⇡2 �⇡
2
�⇡2 �⇡
2
=
· · ·
· · ·
↵=
Corollary: total symmetry between red and green
Ross Duncan ●Quantum Foundations ● Växjo 2015
Example: preparing a Bell stateExample 3.9 (Bell state). A Bell state, |00Í + |11Í, is prepared by the circuitbelow, on the left.
T
Q
cccca
|0Í |0Í
H
R
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.
Z :=H
H
|+Í:=
|0Í
H
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:
T
Q
cccca
|+Í |+Í |+Í |+Í
Z Z Z
Z Z
... ...
... ...
R
ddddb= H H H
H H...
...
...
...
The zx-calculus translation can be radically simplified by repeated use of thespider rule:
H H H
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
16
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
H
H
= H
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.
H
H
=H
H
=H
H
=H
H
=H
H
=
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 corresponding zx-calculus derivation is a proof of the correctness of thiscircuit.
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.
21
7!
Ross Duncan ●Quantum Foundations ● Växjo 2015
Example: preparing a Bell state
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
H
H
= H
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.
H
H
=H
H
=H
H
=H
H
=H
H
=
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 corresponding zx-calculus derivation is a proof of the correctness of thiscircuit.
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.
21
Ross Duncan ●Quantum Foundations ● Växjo 2015
Example: Controlled-Z
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
H
H
= H
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.
H
H
=H
H
=H
H
=H
H
=H
H
=
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 corresponding zx-calculus derivation is a proof of the correctness of thiscircuit.
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.
21
Example 3.9 (Bell state). A Bell state, |00Í + |11Í, is prepared by the circuitbelow, on the left.
T
Q
cccca
|0Í |0Í
H
R
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.
Z :=H
H
|+Í:=
|0Í
H
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:
T
Q
cccca
|+Í |+Í |+Í |+Í
Z Z Z
Z Z
... ...
... ...
R
ddddb= H H H
H H...
...
...
...
The zx-calculus translation can be radically simplified by repeated use of thespider rule:
H H H
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
16
7!
Ross Duncan ●Quantum Foundations ● Växjo 2015
Example: Controlled-Z
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
H
H
= H
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.
H
H
=H
H
=H
H
=H
H
=H
H
=
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 corresponding zx-calculus derivation is a proof of the correctness of thiscircuit.
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.
21
Ross Duncan ●Quantum Foundations ● Växjo 2015
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)
Ross Duncan ●Quantum Foundations ● Växjo 2015
Which diagrams are circuits?
H
Ross Duncan ●Quantum Foundations ● Växjo 2015
Which diagrams are circuits?
HYES!
Ross Duncan ●Quantum Foundations ● Växjo 2015
Which diagrams are circuits?
H
H
=
Ross Duncan ●Quantum Foundations ● Växjo 2015
Which diagrams are circuits?
Ross Duncan ●Quantum Foundations ● Växjo 2015
Which diagrams are circuits?
NO :(
Ross Duncan ●Quantum Foundations ● Växjo 2015
Which diagrams are circuits?
H H
Ross Duncan ●Quantum Foundations ● Växjo 2015
Which diagrams are circuits?
H HNO :(
Ross Duncan ●Quantum Foundations ● Växjo 2015
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 …
Ross Duncan ●Quantum Foundations ● Växjo 2015
Graph StatesLet G = (V,E) be a simple, undirected graph. Then define: !!!Viewed as circuit we get this:
|Gi =O
(v,u)2E
CZvu
Ov2V
|+i
Example 3.9 (Bell state). A Bell state, |00Í + |11Í, is prepared by the circuitbelow, on the left.
T
Q
cccca
|0Í |0Í
H
R
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.
Z :=H
H
|+Í:=
|0Í
H
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:
T
Q
cccca
|+Í |+Í |+Í |+Í
Z Z Z
Z Z
... ...
... ...
R
ddddb= H H H
H H...
...
...
...
The zx-calculus translation can be radically simplified by repeated use of thespider rule:
H H H
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
16
Ross Duncan ●Quantum Foundations ● Växjo 2015
Graph StatesLet G = (V,E) be a simple, undirected graph. Then define: !!!Viewed as circuit we get this:
|Gi =O
(v,u)2E
CZvu
Ov2V
|+i
Example 3.9 (Bell state). A Bell state, |00Í + |11Í, is prepared by the circuitbelow, on the left.
T
Q
cccca
|0Í |0Í
H
R
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.
Z :=H
H
|+Í:=
|0Í
H
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:
T
Q
cccca
|+Í |+Í |+Í |+Í
Z Z Z
Z Z
... ...
... ...
R
ddddb= H H H
H H...
...
...
...
The zx-calculus translation can be radically simplified by repeated use of thespider rule:
H H H
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
16
Ross Duncan ●Quantum Foundations ● Växjo 2015
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
CZvu
Ov2V
|+i
· · · · · ·
kx
1 ky
1 ky
2kx
2 ky
3kx
3
fi
X XYX YX YXY XYY
fi fi
fi fi fi
fi
”= fi
H H H H H H H
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Graph StatesLet G = (V,E) be a simple, undirected graph. Then define: !!!!
|Gi =O
(v,u)2E
CZvu
Ov2V
|+i
Example 3.9 (Bell state). A Bell state, |00Í + |11Í, is prepared by the circuitbelow, on the left.
T
Q
cccca
|0Í |0Í
H
R
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.
Z :=H
H
|+Í:=
|0Í
H
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:
T
Q
cccca
|+Í |+Í |+Í |+Í
Z Z Z
Z Z
... ...
... ...
R
ddddb= H H H
H H...
...
...
...
The zx-calculus translation can be radically simplified by repeated use of thespider rule:
H H H
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
16
Ross Duncan ●Quantum Foundations ● Växjo 2015
Graph StatesLet G = (V,E) be a simple, undirected graph. Then define: !!!Or in 2D:
|Gi =O
(v,u)2E
CZvu
Ov2V
|+i
· · · · · ·
kx
1 ky
1 ky
2kx
2 ky
3kx
3
fi
X XYX YX YXY XYY
fi fi
fi fi fi
fi
”= fi
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Graph StatesLet G = (V,E) be a simple, undirected graph. Then define: !!!Or in 2D:
|Gi =O
(v,u)2E
CZvu
Ov2V
|+i
· · · · · ·
kx
1 ky
1 ky
2kx
2 ky
3kx
3
fi
X XYX YX YXY XYY
fi fi
fi fi fi
fi
”= fi
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Hopf Algebra Equivalence
Thm: any Hopf algebra expression can be put into normal form:
Ross Duncan ●Quantum Foundations ● Växjo 2015
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
Ross Duncan ●Quantum Foundations ● Växjo 2015
GFlow Strategy
Ross Duncan ●Quantum Foundations ● Växjo 2015
Flow Strategy
Ross Duncan ●Quantum Foundations ● Växjo 2015
Application 2: QECCZX-calculus can demonstrate the correctness Quantum Error Correcting Codes:
Ross Duncan ●Quantum Foundations ● Växjo 2015
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.
Ross Duncan ●Quantum Foundations ● Växjo 2015
Mermin-GHZ|GHZi = |000i+ |111i
X X X
X Y Y
Y X Y
Y Y X
Ross Duncan ●Quantum Foundations ● Växjo 2015
Mermin-GHZ|GHZi = |000i+ |111i
X X X +1
X Y Y -1
Y X Y -1
Y Y X -1
-1
QM
Ross Duncan ●Quantum Foundations ● Växjo 2015
Mermin-GHZ|GHZi = |000i+ |111i
X X X +1
X Y Y -1
Y X Y -1
Y Y X -1
+1 +1 +1 +1 ≠ -1
LHV
Ross Duncan ●Quantum Foundations ● Växjo 2015
Ingredients:
• Doubling (like in Aleks’s talk) • Measurements of X and Y • Parity computation on the outcomes • GHZ state • LHV state
Ross Duncan ●Quantum Foundations ● Växjo 2015
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
Pi
i i i
!
I They also fuse:...
...
...
...
... =
...
...
Aleks Kissinger (QTFT, Vaxjo 2015) Picturing Quantum Processes June 10, 2015 21 / 26
–1
–5–3
–2
–4
=q
i –i
double
A–
B= ≠– –
1
Ross Duncan ●Quantum Foundations ● Växjo 2015
DoublingEigenstates of X and Y
–1
–5–3
–2
–4
=q
i –i
double
A–
B= ≠– –
double
Ak
B= k k k œ {0, fi}
double
Q
a k + fi2
R
b = k ≠ fi2 k + fi
2 k œ {0, fi}
≠k k
= k k œ {0, fi}
1
–1
–5–3
–2
–4
=q
i –i
double
A–
B= ≠– –
double
Ak
B= k k k œ {0, fi}
double
Q
a k + fi2
R
b = k ≠ fi2 k + fi
2 k œ {0, fi}
≠k k
= k k œ {0, fi}
1
–1
–5–3
–2
–4
=q
i –i
double
A–
B= ≠– –
double
Ak
B= k k k œ {0, fi}
double
Q
a k + fi2
R
b = k ≠ fi2 k + fi
2 k œ {0, fi}
≠k k
= k k œ {0, fi}
1
Ross Duncan ●Quantum Foundations ● Växjo 2015
MeasurementsX Measurement:
–1
–5–3
–2
–4
=q
i –i
1
Sanity check:
–1
–5–3
–2
–4
=q
i –i
double
A–
B= ≠– –
double
Ak
B= k k k œ {0, fi}
double
Q
a k + fi2
R
b = k ≠ fi2 k + fi
2 k œ {0, fi}
k k
= k k œ {0, fi}
1
Ross Duncan ●Quantum Foundations ● Växjo 2015
MeasurementsY Measurement:
Sanity check:
–1
–5–3
–2
–4
=q
i –i
double
A–
B= ≠– –
double
Ak
B= k k k œ {0, fi}
double
Q
a k + fi2
R
b = k ≠ fi2 k + fi
2 k œ {0, fi}
k k
= k k œ {0, fi}
+ fi2 ≠ fi
2
1
–1
–5–3
–2
–4
=q
i –i
double
A–
B= ≠– –
double
Ak
B= k k k œ {0, fi}
double
Q
a k + fi2
R
b = k ≠ fi2 k + fi
2 k œ {0, fi}
k k
= k k œ {0, fi}
+ fi2 ≠ fi
2
k ≠ fi2 k + fi
2
=k k
= k k œ {0, fi}
+ fi2 ≠ fi
2
1
Ross Duncan ●Quantum Foundations ● Växjo 2015
Parity Computation
–1
–5–3
–2
–4
=q
i –i
double
A–
B= ≠– –
double
Ak
B= k k k œ {0, fi}
double
Q
a k + fi2
R
b = k ≠ fi2 k + fi
2 k œ {0, fi}
k k
= k k œ {0, fi}
+ fi2 ≠ fi
2
k ≠ fi2 k + fi
2
=k k
= k k œ {0, fi}
+ fi2 ≠ fi
2
–i–1 –n· · · · · · =
qi –i
–i œ [0, 2fi)
+ fi2 ≠ fi
2
kik1 kn· · · · · · =
qi ki
ki œ {0, fi}
1
Ross Duncan ●Quantum Foundations ● Växjo 2015
Parity Computation
–1
–5–3
–2
–4
=q
i –i
double
A–
B= ≠– –
double
Ak
B= k k k œ {0, fi}
double
Q
a k + fi2
R
b = k ≠ fi2 k + fi
2 k œ {0, fi}
k k
= k k œ {0, fi}
+ fi2 ≠ fi
2
k ≠ fi2 k + fi
2
=k k
= k k œ {0, fi}
+ fi2 ≠ fi
2
–i–1 –n· · · · · · =
qi –i
–i œ [0, 2fi)
+ fi2 ≠ fi
2
kik1 kn· · · · · · =
qi ki
ki œ {0, fi}
1
Ross Duncan ●Quantum Foundations ● Växjo 2015
Parity Computation
· · · · · ·
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
GHZ State· · · · · ·
double
3 4=
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Measuring the GHZ state· · · · · ·
double
3 4=
≠–1+–1 +–2 ≠–2 +–3 ≠–3
≠q
i –iq
i –i
≠q
i –iq
i –i
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Measuring the GHZ state
+ fi
2 ≠ fi
2
kik1 k
n
· · · · · · =q
i
ki
ki
œ {0, fi}
· · · · · ·
≠–1+–1 +–2 ≠–2 +–3 ≠–3
≠q
i
–i
qi
–i
≠q
i
–i
qi
–i
kx
1 ky
1 ky
2kx
2 ky
3kx
3
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Measuring the GHZ state
· · · · · ·
double
3 4=
≠–1+–1 +–2 ≠–2 +–3 ≠–3
≠q
i –iq
i –i
≠q
i –iq
i –i
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Measuring the GHZ state: XXX
· · · · · ·
double
3 4=
≠–1+–1 +–2 ≠–2 +–3 ≠–3
≠q
i –iq
i –i
≠q
i –iq
i –i
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Measuring the GHZ state: XXX
· · · · · ·
double
3 4=
≠–1+–1 +–2 ≠–2 +–3 ≠–3
≠q
i –iq
i –i
≠q
i –iq
i –i
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Measuring the GHZ state
· · · · · ·
double
3 4=
≠–1+–1 +–2 ≠–2 +–3 ≠–3
≠q
i –iq
i –i
≠q
i –iq
i –i
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Measuring the GHZ state: XYY
· · · · · ·
double
3 4=
≠–1+–1 +–2 ≠–2 +–3 ≠–3
≠q
i –iq
i –i
≠q
i –iq
i –i
fi fi
fi
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Measuring the GHZ state: XYY
· · · · · ·
double
3 4=
≠–1+–1 +–2 ≠–2 +–3 ≠–3
≠q
i –iq
i –i
≠q
i –iq
i –i
fi fi
fi
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Local Hidden State⇤ = (kx1 , k
y
1 , kx
2 , ky
2 , kx
3 , ky
3)
· · · · · ·
double
3 4=
≠–1+–1 +–2 ≠–2 +–3 ≠–3
≠q
i
–i
qi
–i
≠q
i
–i
qi
–i
fi fi
fi
kx
1 ky
1 ky
2kx
2 ky
3kx
3
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
kx
1 ky
1 ky
2kx
2 ky
3kx
3
X XYX YX YXY XYY
3
Local Hidden State⇤ = (kx1 , k
y
1 , kx
2 , ky
2 , kx
3 , ky
3)
Ross Duncan ●Quantum Foundations ● Växjo 2015
Parity of LHV outcomeskx
1 ky
1 ky
2kx
2 ky
3kx
3
X XYX YX YXY XYY
kx
1 ky
1 ky
2kx
2 ky
3kx
3
3
Ross Duncan ●Quantum Foundations ● Växjo 2015
Parity of LHV outcomeskx
1 ky
1 ky
2kx
2 ky
3kx
3
3
Ross Duncan ●Quantum Foundations ● Växjo 2015
Parity of LHV outcomeskx
1 ky
1 ky
2kx
2 ky
3kx
3
kx
1 ky
1 ky
2kx
2 ky
3kx
3
3
Ross Duncan ●Quantum Foundations ● Växjo 2015
Parity of QM outcomes· · · · · ·
fi
X XYX YX YXY XYY
fi fi
fi fi fi
fi
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Parity of QM outcomes
· · · · · ·
fi
X XYX YX YXY XYY
fi fi
fi fi fi
fi
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Parity of QM outcomes
· · · · · ·
fi
X XYX YX YXY XYY
fi fi
fi fi fi
fi
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
Contradiction!
· · · · · ·
kx
1 ky
1 ky
2kx
2 ky
3kx
3
fi
X XYX YX YXY XYY
fi fi
fi fi fi
fi
”= fi
2
Ross Duncan ●Quantum Foundations ● Växjo 2015
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
Ross Duncan ●Quantum Foundations ● Växjo 2015
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!