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Feb 22, 2024 •207 likes •563 views

How to compute using quantum walks Viv Kendon PhD students: James Morley (UCL CountingLabs) Adam Callison (Imperial) Jemma Bennett (Durham) Collaborators: JQC & QLM Stephanie Foulds (Durham) Physics Dept Dom Horsman + many UG

  1. How to compute using quantum walks Viv Kendon PhD students: James Morley (UCL → CountingLabs) Adam Callison (Imperial) Jemma Bennett (Durham) Collaborators: JQC & QLM Stephanie Foulds (Durham) Physics Dept Dom Horsman + many UG project students Durham University (Grenoble) viv.kendon@durham.ac.uk also at Durham: Susan Stepney (York) Nick Chancellor (Durham) (UKRI Innovation Fellow) Jie Chen (Durham) Laur Nita (Durham) Quantum Simulation and Quantum Walks CIRM 20th January 2020

  2. QSQW - CIRM 20.1.2020 January 19, 2020 Overview • modeling vs simulation? • abstraction/representation framework • solving classical problems with quantum walks • searching and spin glasses • universal quantum walk computing • summary and outlook ���� ���� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ����� ����� ����� ����� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ����� ����� ����� ����� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ����� ����� ����� ����� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ����� ����� ����� ����� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ���� ���� ����� ����� ����� ����� ����� ����� ����� ����� 0 −9 −8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 2/34 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

  3. QSQW - CIRM 20.1.2020 January 19, 2020 numerical simulation... ⋆ we simulate mathematical models, not physical systems: Mathematical Model Experiments Numerical simulation Analytical calculations COMPARE Revise Model computational physics tests our models when can’t calculate analytically... 3/34 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

  4. QSQW - CIRM 20.1.2020 January 19, 2020 representation relation in physics: ψ : i � ∂ψ ∂ t = H ψ abstract R T physical e − spaces of abstract and physical objects (here, an electron and a wave-function) with a representation relation (modelling) R T mediating between the spaces R is theory dependent, so write R T for theory T ⋆ could represent electron as point charge if doing electrostatics . . . 4/34 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

  5. QSQW - CIRM 20.1.2020 January 19, 2020 science C T ( m p ) m ′ m p ε p m p ′ abstract R T R T physical time goes by p ′ p H ( p ) – physical system p evolves under H ( p ) to p ′ – theory m p calculated C T ( m p ) to obtain m ′ p – “good” theory agrees with observation to within ε : | m ′ p − m p ′ | < ε 5/34 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

  6. QSQW - CIRM 20.1.2020 January 19, 2020 technology technology is making things we designed, here making a p ′ C T ( m p ) m ′ m p p ≈ m p ′ abstract R T � physical R T raw engineer finished p ′ p material product H ( p ) p , T and H such that we can engineer a physical system to our p – effectively inverting R T −→ � specifications m ′ R T – an instantiation representation relation 6/34 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

  7. QSQW - CIRM 20.1.2020 January 19, 2020 computing among many things, we engineer computers c ′ c encode decode m ′ m p p ≈ m p ′ � abstract R T R T physical program runs p ′ p H ( p ) computing : use a physical computer p to calculate abstract problem c encode c into model m p , instantiate � R T into p , run, decode output 7/34 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

  8. QSQW - CIRM 20.1.2020 January 19, 2020 requirements for computing c ′ c encode decode m ′ m p p ≈ m p ′ computing is a high level process... � abstract R T R T physical program runs p ′ p H ( p ) • computations have outputs (else can replace computer with brick...) • representational entity (“owns” the computation) [Stepney/VK “The role of the representational entity...” 219–231 UCNC 2019 & Nat. Comp. 2020] abstract is instantiated in the representational entity (does not need to be human – Horsman/VK/Stepney/Young Abstraction and representation in living organisms: when does a biological system compute? in: Representation and reality: Humans, Animals and Machines. Gordana Dodig-Crnkovic and Raffaela Giovagnoli, Editors. Springer 2016) 8/34 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

  9. QSQW - CIRM 20.1.2020 January 19, 2020 GOAL: increase computing power . . . ⋆ current computers already very powerful – two barriers to more computing power: 1. silicon chip technology reaching limits 2. energy consumption far from optimal: – resource limits; global warming [lots of room to improve on energy consumption – see, e.g., SpiNNaker project for other ways to use Si] note these are related: can’t cool Si chips any faster 9/34 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

  10. QSQW - CIRM 20.1.2020 January 19, 2020 beyond silicon . . . quantum: IBM 5 qubit BZ reaction chemical reservoir computer rat neuron on silicon encoding for DNA computer ⋆ future computing is diversifying ⋆ ⇒ need to co-design algorithms with hardware ⇐ 10/34 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

  11. QSQW - CIRM 20.1.2020 January 19, 2020 hybrid computers . . . practice: co-processors: unconventional: control + substrate: conventional: • quantum • NMR • graphics cards • reservoir • ASIC application-specific integrated circuit • slime mould • FPGA field-programmable gate array ⋆ hybrid computational systems are the norm ⋆ theory: single paradigm: • classical – T uring Machine • analog – Shannon’s GPAC • quantum – gate model, QTM, CV, MBQC, QW, AQC, . . . • linear optics (Bosons) [Aaronson/Arkhipov STOC 2011 ECCC TRI-10 170] 11/34 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

  12. QSQW - CIRM 20.1.2020 January 19, 2020 quantum computing input −→ encode −→ | ψ in � −→ ˆ U −→ | ψ out � −→ decode −→ result ˆ U is unitary evolution (or more generally, open system/environment) – can be gate sequence , or engineer Hamiltonian ˆ H ( t ) such that ∫ dt ˆ | ψ out � = T exp {− i / � H ( t )} | ψ in � ⋆ covers most of quantum information processing . . . . . . including communications, where aim is result=input encode – arbitrary choices: using spin-down |↓� ≡ 0 instead of spin-up |↑� ≡ 0 makes no difference → provided encode and decode done consistently 12/34 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

  13. QSQW - CIRM 20.1.2020 January 19, 2020 quantum information processing quantum information is built on the idea that: quantum logic allows greater EFFICIENCY than classical logic classical quantum bits, 0 or 1 qubits, α | 0 � + β | 1 � yes or no, binary decisions yes and no, superpositions HEADS or TAILS, random numbers random measurement outcomes ⇒ quantum gives different computation from classical: how different? • computability – what can be computed? • complexity – how efficiently can it be computed? ⇒ quantum computability is the same as classical complexity differs: some problems can be computed more EFFICIENTL Y 13/34 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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