Upload
others
View
13
Download
0
Embed Size (px)
Citation preview
Algebraic (and Diagrammatic) Structures inQuantum Theory
Aleks Kissinger
Institute for Computing and Information SciencesRadboud University Nijmegen
Quantum Software in NL 2017
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 1 / 26
Software = components + composition
• Effective software comes from understanding relevantcomponents and how they compose
• Effective quantum software will be exactly the same
• But now the “components” could mean:
devices physical processes mathematicalprocesses
ρ 7→∑
i BiρB†i
• ...each with its own (related) notion of composition
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 2 / 26
Process theories
• The moral: take processes as primitive
• A process is just a box with inputs and outputs:
quicksort
lists
lists
cooking
bacon
breakfast
eggs food
baby
love
poonoise
• Composition means forming diagrams of processes:
CA
D
B
g
A
f h
A
• A collection of processes that make sense to ‘plug together’ iscalled a process theory
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 3 / 26
Quantum Picturalism := quantum process theory
The idea: Describe quantum theory entirely in terms of:
f
BA
C D
processes connectivity
B
A C
g
A
f
D
h
A
interaction
=
Not in terms of:
• Hilbert space
• self-adjoint operators, unitary transformations
• calculations with matrices/complex numbers
• ....
(though some may be emergent notions)
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 4 / 26
PICTURING QUANTUM PROCESSESA First Course in Quantum Theory and Diagrammatic Reasoning
BOB COECKE ALEKS KISSINGER Cambridge University Press
March 2017 (pre-order)
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 5 / 26
Why?
• Simpler!
(1⊗σ⊗k)◦(σ⊗1⊗1⊗1)◦(f ⊗ g ⊗ 1⊗ 1) ◦ (h⊗ 1) =(g ⊗ f ) ◦ (1⊗ k) ◦ (h ⊗ 1)
vs. k
k
h
f
=f
g
h
g
• New perspective = new insights• Deriving QT from first principles ⇐ ‘diagrammatic backbone’
+ a bit of information theory
e.g. Pavia 2010 and Hardy 2011Hardy (2010): “we join the quantum picturalism revolution”
• A ‘theory playground’
e.g. QT vs. real/boolean-valued/modal QT,local QT (indefinite causal structure),
Spekken’s toy theory, ...
• New calculational tools, applications in quantuminfo/computation
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 6 / 26
Just diagrams
Q: What kinds of behaviour can we study using just diagrams, andnothing else?
A: (Non-)separability
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 7 / 26
Separability for states
• States are processes with no input:
ψ ψ ψ...
Interpret as: ‘preparing a system (or some systems)’
• Separable states:
ψ = ψ1 ψ2
• ...are boring!
• Non-separable := ‘no such ψ1, ψ2 exist’, but this isn’t veryhelpful
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 8 / 26
Cup-states
‘Non-separable’ isn’t very helpful, but ‘really non-separable’ is:
Definition
A process ψ is called cup-state if there exists a process φ, called acap-effect, such that:
φ
ψ
=
ψ
φ=
ψ looks like a state, but it acts like a wire.
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 9 / 26
Cup-states
• By introducing some clever notation:
:= ψ := φ
• Then these equations:
φ
ψ
=
ψ
φ=
• ...look like this:
= =
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 10 / 26
Yank the wire!
= =
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 11 / 26
Quantum teleportation
BobAleks
ψquantum state
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 12 / 26
Quantum teleportation
BobAleks
ψ quantum state
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 12 / 26
Quantum teleportation
BobAleks
ψquantum state
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 12 / 26
Quantum teleportation
BobAleks
ψentangled state
quantum state
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 12 / 26
Quantum teleportation
BobAleks
ψentangled state
quantum measurement
quantum state
Ui
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 12 / 26
Quantum teleportation
Bob
U†i
Aleks
ψentangled state
Bob’s fix
quantum measurement
quantum state
Ui
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 12 / 26
...and it works
Bob
U†i
Aleks
ψ
Ui
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 13 / 26
...and it works
Bob
U†i
Aleks
ψ
Ui
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 13 / 26
...and it works
Bob
U†i
Aleks
ψ
Ui
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 13 / 26
...and it works
BobAleks
ψ
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 13 / 26
If we change the process theory...
Bob
U†i
Aleks
ψentangled state
Bob’s fix
quantum measurement
quantum state
Ui
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 14 / 26
If we change the process theory...
Bob
U†i
Aleks
bshared random bit
do nothing OR flip bit
compare bits
secret bit
Ui
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 14 / 26
If we change the process theory...
Bob
U†i
Aleks
bshared random bit
do nothing OR flip bit
compare bits
secret bit
Ui
...‘classical teleportation’ is one-time-pad crypto
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 14 / 26
...and it works
Bob
U†i
Aleks
b
Ui
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 15 / 26
...and it works
U†i
Ui
Aleks Bob
b
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 15 / 26
...and it works
b
U†i
Bob
Ui
Aleks
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 15 / 26
...and it works
Aleks
b
Bob
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 15 / 26
One and the same
OTP cryptography1882 Quantum teleportation
111 years laterImage: Hanson lab@TUDelft
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 16 / 26
Quantum algorithms
f
N bits
bit
Classical computation
⇒ f:=Uf
‘copy’
xor
Quantum process
⇒ simple (diagrammatic) derivations of Deutsch-Jozsa,Bernstein-Vazirani, quantum seach, and hidden subgroupalgorithms.
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 17 / 26
Graphical calculus
Quantum circuits + algebraic structure = ZX-calculus
⊕:=
α:=Z (α)
:= αX (α)
α
...
...
...α
...
spiders
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 18 / 26
Graphical calculus
ZX-calculus has 4 equations:
= α+β
...
.....
. α ...
... β
...
... ..
.
α+β
...
...
...
...
...
...
=α
β
π2
-π2
-π2
-π2
-π2=
π2
π2
=
...
...
...
...
...which can prove any equality between Clifford circuits (and a bitmore).
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 19 / 26
Quantum circuit simplification
π4
-π4
-π4
π4
-π2
π4
=
=π
π4
π
-π4
=
⊕
⊕⊕
0 X(π4
)0 S†X
(π3
)
⊕
⊕
0
⊕
⊕
0
X(π3
)
X(π4
)†T †
⊕
⊕
:=
:=1
0
X π
π
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 20 / 26
Measurement-based quantum computation
Z Z
Z Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z Z
α
...
π
=
...
π
=
α
...
...
...
...
απ
π
α
...
......
...
...
...
...
...
...
...
......
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 21 / 26
GHZ/Mermin non-locality
quantum theory any local theory
π2
π20
π2
π20 0
π2
π2 00 0
π0 π π
π
==
yAi yC
izAi yBi zCizBi
yAi yB
i yCizCizBizAi
0
==
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 22 / 26
Causal structures and indeterminism
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 23 / 26
Quantum error correction
Stabiliser code
Zd1Zd2Zb1Zp2Zp4
Zd1Zd3Zb3Zp1Zp4
Zd2Zd3Zb2Zp3Zp4
Zb4Zp1Zp2Zp3Zp4
Xd1Xd2Xb2Xb4Xp1
Xd1Xd3Xb1Xb4Xp3
Xd2Xd3Xb3Xb4Xp2
Xb1Xb2Xb3Xb4Xp4
↔
Quantum paritycheck graph
↔
Encoding circuit
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 24 / 26
Automation
Quantomatic:
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 25 / 26
Thanks!
• Picturing Quantum Processes. CUP (March 2017)
• Categorical Quantum Mechanics I: Causal Quantum Processes. Coecke and Kissinger. arXiv:1510.05468
• Categorical Quantum Mechanics II: Classical-Quantum Interaction. Coecke and Kissinger.arXiv:1605.08617
• Coherent Parity Check Construction for Quantum Error Correction. Chancellor, Kissinger, Zohren,Horsman. (arXiv on Monday!)
http://quantomatic.github.io
Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 26 / 26