Picturing Quantum Processes
Aleks Kissinger and Bob Coecke
Radboud University and Oxford University
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www.cambridge.org/pqp
20% discount @ CUP with code: COECKE2017
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Chapters 11-13:
Quantum Foundations, Computation, and Resources
Mermin once summarized a popular attitude towards quantum theory as
“Shut up and calculate”. We suggest a different slogan: “Shut up and
contemplate”!
— Lucien Hardy and Rob Spekkens, 2010.
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Spiders
n
n
...
:=
...
i
i
...
i
i
i
...
i
P
i
m
m
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Spiders
n
n
...
:=
i
...
i
i
...
i
i
i
...
i
P
m
m
...
...
...
...
Kissinger & Coecke
...
=
...
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...
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(Classical) spiders are linear maps:
...
...
...so we can double them to get quantum spiders:
...
...
... !
:= double
=
...
...
...
...which also fuse together:
...
...
...
...
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...
=
...
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Measuring is a spider:
i
=
P
i
=
i
i
P
i
i
i
...and so is encoding:
:=
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They connect classical to quantum:
...
...
...
...
...giving something new:
...
...
...
...
=
...
...
...
...
Bastard spiders!
...
...
...
...
:=
...
...
...
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Bastard spiders fuse together:
...
...
...
...
...
=
...
...
They also absorb other kinds of spider:
...
...
...
...
...
...
...
=
...
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...
...
...
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...
=
...
...
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Useful bastards
Bastard spiders arise naturally in the interaction between classical and
quantum data, e.g.
• Discarding:
=
=
=
• Decoherence:
=
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Useful bastards (cont’d)
• Non-demolition measurement:
=
=
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=
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Multicoloured spiders
...
i
=
i
...
...
...
i
...
...
...
...
i
P
...
=
...
...
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...
...
Picturing Quantum Processes
...
i
i
...
i
...
...
=
...
i
...
i
i
P
...
=
...
...
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Complementary bases
0
1
0
1
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Complementarity
Theorem. Two bases are complementary (a.k.a. mutually unbiased):
∀i, j
j
:
=
i
1
D
if and only if:
=
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1
D
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Complementarity
Theorem. Two bases are complementary (a.k.a. mutually unbiased):
∀i, j
j
:
=
i
1
D
if and only if:
=
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1
D
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Complementarity
=
(encode in
Kissinger & Coecke
1
D
) THEN (measure in
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) = (no data flow)
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Stern-Gerlach
N
S
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Stern-Gerlach
.
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&
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Stern-Gerlach
:=
Kissinger & Coecke
:=
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Stern-Gerlach
:=
:=
Z -measurement
N
S
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Stern-Gerlach
:=
:=
Z -measurement
X -measurement
N
S
N
S
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Stern-Gerlach
S
N
N
S
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Stern-Gerlach
S
N
N
S
X -measurement
Z -measurement
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=
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Stern-Gerlach
N
S
N
S
N
S
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blocked!
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Stern-Gerlach
N
S
S
N
N
blocked!
S
0
0
2nd Z -measurement
X -measurement
=
1st Z -measurement
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Teleportation with spiders
Aleks
Bob
ρ
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Teleportation with spiders
=
Kissinger & Coecke
=
=
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=
=
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Power of the Graphical Language
We now have a fairly powerful language, it is natural to ask:
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Power of the Graphical Language
We now have a fairly powerful language, it is natural to ask:
Q: Is it universal?
Can we express any map in terms of spiders?
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Power of the Graphical Language
We now have a fairly powerful language, it is natural to ask:
Q: Is it universal?
Can we express any map in terms of spiders?
Q: Is it complete?
Can we prove every equation between maps
using some set of spider-equations?
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Universality
Q: Can we write any linear map as a diagram of:
...
...
?
and
...
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Universality
Q: Can we write any linear map as a diagram of:
...
...
?
and
...
A: Clearly not! e.g.
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...
1 0
0 i
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Universality
Q: Can we write any linear map as a diagram of:
...
...
?
and
...
A: Clearly not! e.g.
...
1 0
0 i
Q2: How much more do we need?
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Phases
0
1
0
1
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Phases
0
α
1
D
α
1
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Phase spiders
= α+β
α
Kissinger & Coecke
β
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Phase spiders
= α+β
α
β
...
...
α
...
Kissinger & Coecke
:=
α
...
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Phase spiders
= α+β
α
β
...
...
:=
α
...
α
...
α
...
...
...
...
=
β
...
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α+β
...
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Universality
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Universality
Theorem. Any linear map with 2D inputs and outputs can be expressed
as a diagram of:
...
...
and
α
...
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α
...
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Universality
Theorem. Any linear map with 2D inputs and outputs can be expressed
as a diagram of:
...
...
and
α
α
...
...
Corollary. Any quantum map from qubits to qubits can be expressed as
a diagram of:
...
α
...
Kissinger & Coecke
...
,
α
,
and
...
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Universality
Proof. (idea)
γ
Any single-qubit unitary:
Ub
=
+
β
CNOT:
α
...gives any unitary, which gives any state:
...
ψb
...
=
Ub
0
... 0
...which gives any map by process-state duality.
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Completeness
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Completeness
Q. Can we find a complete set of equations to describe
the behaviour of:
...
...
α
α
and
?
...
...
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Completeness
Q. Can we find a complete set of equations to describe
the behaviour of:
...
...
α
α
and
?
...
...
A. Yes! The ZX-calculus.
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ZX-calculus
...
α
...
...
...
...
=
α+β
...
...
...
β
...
α
...
...
...
....
Kissinger & Coecke
α+β
...
....
...
π
2
π
2
=
...
=
β
- π2
=
...
- π2
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- π2
- π2
π
2
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ZX-calculus
Theorem. The ZX-calculus is complete for Clifford diagrams, i.e.
diagrams where α ∈ {0, π, ± π2 }.
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ZX-calculus
Theorem. The ZX-calculus is complete for Clifford diagrams, i.e.
diagrams where α ∈ {0, π, ± π2 }.
Theorem. (Hot off the press!
complete for all diagrams.
1
2
1,2 )
A bigger version of the ZX-calculus is
Jeandel, Perdrix, Vilmart. 31 May, 2017. arXiv:1705.11151
Wang & Ng. 29 June, 2017. arXiv:1706.09877
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Application: Quantum Computing
The quantum circuit model:
measure some outputs
- π4
π
4
quantum gates
π
4
- π2
π
4
- π4
prepare a state
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Quantum algorithms
Quantum process
Classical computation
bit
⇒
f
Ubf
:=
fb
N bits
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Quantum algorithms
Quantum process
Classical computation
bit
⇒
f
Ubf
:=
fb
N bits
⇒ Deutsch-Jozsa, Bernstein-Vazirani, quantum seach, and hidden
subgroup (e.g. factoring) algorithms.
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Automation
Quantomatic:
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Spekkens’ toy model
:: ∗ 7→ {(0, 0)}
:: ∗ 7→ {(0, 1)}
:: ∗ 7→ {(1, 0)}
:: ∗ 7→ {(1, 1)}
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Spekkens’ toy model
Z :: (a, b) 7→ (a, b ⊕ 1)
Z :: (a, b) 7→ (a, a ⊕ b)
H :: (a, b) 7→ (b, a)
CZ :: ((a, b), (c, d)) 7→ ((a, b ⊕ c), (c, a ⊕ d))
MEASURE(a, b) := reveal a, randomize b
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Spekkens’ toy model
...
...
...
...
a,b
...
=
...
...
...
...
a+c , b+d
c ,d
...
...
a,b
....
Kissinger & Coecke
...
...
a, a
a, a
=
...
b, b
=
...
a+c , b+d
c ,d
...
....
=
a, a
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b, b
b, b
b, b
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GHZ/Mermin non-locality
quantum theory
0
0
π
2
π
2
π
2
0
π
2
π
2
π
2
0
ziA
yiA
ziB
Kissinger & Coecke
π
π
ziA
yiA
ziB
=
π
=
0
yiB
ziC
yiC
yiB
ziC
yiC
=
0
=
0
any local theory
π
0
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Thanks!
...
Coecke, Abramsky, Backens, Duncan, Edwards, Gogioso, Hadzihasanovic, Heunen, Lal,
Dixon, Merry, Pavlovic, Perdrix, Quick, Selinger, Zamdzhiev, ...
Special thanks ⇒
Jamie Vicary and David Reutter - qubit.zone simulators
www.cambridge.org/pqp
Kissinger & Coecke
quantomatic.github.io
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