Algebraic (and Diagrammatic) Structures in Quantum Theory Aleks - PowerPoint PPT Presentation

Algebraic (and Diagrammatic) Structures in Quantum Theory Aleks Kissinger Institute for Computing and Information Sciences Radboud 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 relevant components and how they compose • Effective quantum software will be exactly the same • But now the “components” could mean: devices physical processes mathematical processes i B i ρ B † ρ �→ � 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: lists breakfast noise poo cooking baby quicksort lists eggs bacon food love • Composition means forming diagrams of processes: A B C g D A f h A • A collection of processes that make sense to ‘plug together’ is called 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: processes connectivity interaction A A C C D g f = D B A B f h A 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 PROCESSES A First Course in Quantum Theory and Diagrammatic Reasoning BOB COECKE Cambridge University Press ALEKS KISSINGER March 2017 (pre-order) Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 5 / 26

Why? • Simpler! k g f (1 ⊗ σ ⊗ k ) ◦ ( σ ⊗ 1 ⊗ 1 ⊗ 1) ◦ vs. ( f ⊗ g ⊗ 1 ⊗ 1) ◦ ( h ⊗ 1) = = f g k ( g ⊗ f ) ◦ (1 ⊗ k ) ◦ ( h ⊗ 1) h h • New perspective = new insights • Deriving QT from first principles ⇐ ‘diagrammatic backbone’ + a bit of information theory e.g. Pavia 2010 and Hardy 2011 Hardy (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 quantum info/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, and nothing 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 1 ψ 2 ψ • ...are boring! • Non-separable := ‘no such ψ 1 , ψ 2 exist’, but this isn’t very helpful 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 a cap-effect , such that: φ φ = = ψ ψ 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: φ := := ψ • Then these equations: φ φ = = ψ ψ • ...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 Aleks Bob quantum state ψ Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 12 / 26

Quantum teleportation Aleks Bob quantum state ψ Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 12 / 26

Quantum teleportation Aleks Bob quantum state ψ Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 12 / 26

Quantum teleportation Aleks Bob quantum state ψ entangled state Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 12 / 26

Quantum teleportation Aleks Bob quantum measurement U i quantum state ψ entangled state Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 12 / 26

Quantum teleportation Aleks Bob U † Bob’s fix i quantum measurement U i quantum state ψ entangled state Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 12 / 26

...and it works Aleks Bob U † i U i ψ Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 13 / 26

...and it works Aleks Bob U † i U i ψ Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 13 / 26

...and it works Aleks Bob U † i U i ψ Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 13 / 26

...and it works Aleks Bob ψ Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 13 / 26

If we change the process theory... Aleks Bob U † Bob’s fix i quantum measurement U i quantum state ψ entangled state Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 14 / 26

If we change the process theory... Aleks Bob U † do nothing OR flip bit i compare bits U i secret bit b shared random bit Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 14 / 26

If we change the process theory... Aleks Bob U † do nothing OR flip bit i compare bits U i secret bit b shared random bit ...‘classical teleportation’ is one-time-pad crypto Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 14 / 26

...and it works Aleks Bob U † i U i b Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 15 / 26

...and it works Aleks Bob U † i U i b Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 15 / 26

...and it works Aleks Bob U † i U i b Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 15 / 26

...and it works Aleks Bob b Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 15 / 26

One and the same OTP cryptography 1882 Quantum teleportation 111 years later Image: Hanson lab@TUDelft Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 16 / 26

Quantum algorithms Classical computation Quantum process xor bit ⇒ := f U f f N bits ‘copy’ ⇒ simple (diagrammatic) derivations of Deutsch-Jozsa , Bernstein-Vazirani , quantum seach , and hidden subgroup algorithms. Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 17 / 26

Graphical calculus Quantum circuits + algebraic structure = ZX-calculus := Z ( α ) α := := X ( α ) α ⊕ spiders α α ... ... ... ... Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 18 / 26

Graphical calculus ZX-calculus has 4 equations: ... ... α α ... ... = = α + β α + β ... ... ... ... ... ... β β ... ... ... ... π - π 2 2 = = - π π - π ... ... ... ... 2 2 2 π - π 2 2 ...which can prove any equality between Clifford circuits (and a bit more). Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 19 / 26

Quantum circuit simplification � π � π � S † � X X π - π π 0 4 2 4 3 3 T † - π 0 ⊕ ⊕ ⊕ 4 := � π � π � † � X ⊕ X ⊕ - π π 0 4 4 4 4 X π 0 ⊕ ⊕ ⊕ ⊕ = - π π 4 4 π = = π π 0 := 1 Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 20 / 26

Measurement-based quantum computation Z Z Z ... ... ... Z π π = = Z Z Z π π α α α Z Z ... ... ... ... Z Z Z ... ... ... ... Z ... ... ... ... Z Z Z α ... ... ... Aleks Kissinger Quantum Software in NL 2017 Diagrammatic Structures in Quantum Theory 21 / 26

GHZ/Mermin non-locality quantum theory any local theory π π π π π π 0 0 0 0 0 0 2 2 2 2 2 2 z A y A z B y B z C y C i i i i i i = = 0 z A y A z B y B z C y C π π π i i i i i i = = 0 π 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