Non-local games and verifiable delegation of quantum computation Alex Bredariol Grilo joint work with Andrea Coladangelo, Stacey Jeffery and Thomas Vidick
Why verifiably delegate quantum computation? Non-local games and verifiable delegation of quantum computation 2 / 23
Why verifiably delegate quantum computation? Superiorita Non-local games and verifiable delegation of quantum computation 2 / 23
Why verifiably delegate quantum computation? Superiorita But they are expensive Non-local games and verifiable delegation of quantum computation 2 / 23
Why verifiably delegate quantum computation? Superiorita But they are expensive Online service Non-local games and verifiable delegation of quantum computation 2 / 23
Why verifiably delegate quantum computation? Superiorita But they are expensive Online service Can a client be sure that she is experiencing a quantum speedup? Non-local games and verifiable delegation of quantum computation 2 / 23
Ideal world Goal: Interacrive proof system for BQP where Non-local games and verifiable delegation of quantum computation 3 / 23
Ideal world Goal: Interacrive proof system for BQP where ◮ the verifier runs poly-time prob. computation V x Non-local games and verifiable delegation of quantum computation 3 / 23
Ideal world P Goal: Interacrive proof system for BQP where ◮ the verifier runs poly-time prob. computation ... ◮ an honest prover runs poly-time quantum computation V x Non-local games and verifiable delegation of quantum computation 3 / 23
Ideal world P Goal: Interacrive proof system for BQP where ◮ the verifier runs poly-time prob. computation ... ◮ an honest prover runs poly-time quantum computation ◮ the protocol is sound against any malicious prover V x Non-local games and verifiable delegation of quantum computation 3 / 23
Ideal world P Goal: Interacrive proof system for BQP where ◮ the verifier runs poly-time prob. computation ... ◮ an honest prover runs poly-time quantum computation ◮ the protocol is sound against any malicious prover V ◮ additional property: the prover does not learn the input x Non-local games and verifiable delegation of quantum computation 3 / 23
Relaxed models Exponential-size provers Comput. soundness Almost-classical clients x x P P ... ... ... V V x x V x Non-local games and verifiable delegation of quantum computation 4 / 23
Multiple provers V Non-local games and verifiable delegation of quantum computation 5 / 23
Multiple provers Q Q P 1 | EPR � P 2 Multiple entangled non-communicating P ... V x , Q Non-local games and verifiable delegation of quantum computation 5 / 23
Multiple provers Q Q P 1 | EPR � P 2 Multiple entangled non-communicating P ... Sound against any malicious strategy V x , Q Non-local games and verifiable delegation of quantum computation 5 / 23
Multiple provers Q Q P 1 | EPR � P 2 Multiple entangled non-communicating P ... Sound against any malicious strategy Servers have to keep entangled V x , Q Non-local games and verifiable delegation of quantum computation 5 / 23
Multiple provers Q Q P 1 | EPR � P 2 Multiple entangled non-communicating P ... Sound against any malicious strategy Servers have to keep entangled V “Plug-and-play” x , Q Non-local games and verifiable delegation of quantum computation 5 / 23
Previous works Provers Rounds Total Resources Blind RUV 2012 2 poly( n ) poly( n ) yes Non-local games and verifiable delegation of quantum computation 6 / 23
Previous works Provers Rounds Total Resources Blind ≥ g 8192 RUV 2012 2 poly( n ) yes Non-local games and verifiable delegation of quantum computation 6 / 23
Previous works Provers Rounds Total Resources Blind ≥ g 8192 RUV 2012 2 poly( n ) yes ≥ 2 153 g 22 McKague 2013 poly ( n ) poly( n ) yes ≥ g 2048 GKW 2015 2 poly( n ) yes Θ( g 4 log g ) HDF 2015 poly( n ) poly( n ) yes > g 3 FH 2015 5 poly( n ) no > g 3 NV 2017 7 2 no Non-local games and verifiable delegation of quantum computation 6 / 23
The results Delegate circuit Q on n qubits, with g gates and depth d , 2 provers: Non-local games and verifiable delegation of quantum computation 7 / 23
The results Delegate circuit Q on n qubits, with g gates and depth d , 2 provers: Verifier-on-a-leash protocol: O ( d ) rounds, O ( g log g ) EPR pairs, blind Non-local games and verifiable delegation of quantum computation 7 / 23
The results Delegate circuit Q on n qubits, with g gates and depth d , 2 provers: Verifier-on-a-leash protocol: O ( d ) rounds, O ( g log g ) EPR pairs, blind Dogwalker protocol: 2 rounds, O ( g log g ) EPR pairs Non-local games and verifiable delegation of quantum computation 7 / 23
Comparing to previous works Provers Rounds Total Resources Blind ≥ g 8192 RUV 2012 2 poly( n ) yes ≥ 2 153 g 22 McKague 2013 poly ( n ) poly( n ) yes ≥ g 2048 GKW 2015 2 poly( n ) yes Θ( g 4 log g ) HDF 2015 poly( n ) poly( n ) yes > g 3 FH 2015 5 poly( n ) no > g 3 NV 2017 7 2 no VoL 2 O (depth) Θ( g log g ) yes DW 2 2 Θ( g log g ) no g 3 Relativistic 2 1 no Non-local games and verifiable delegation of quantum computation 8 / 23
Basics on quantum computation 1 General idea 2 Our protocols 3 Open problems 4 Non-local games and verifiable delegation of quantum computation 9 / 23
Very quick introduction to quantum computation 1 qubit ◮ Unit vector in C 2 � 1 � 0 ◮ Basis: | 0 � = � � and | 1 � = 0 1 ◮ | ψ 1 � = α | 0 � + β | 1 � , α, β ∈ C and | α | 2 + | β | 2 = 1 Non-local games and verifiable delegation of quantum computation 10 / 23
Very quick introduction to quantum computation 1 qubit ◮ Unit vector in C 2 � 1 � 0 ◮ Basis: | 0 � = � � and | 1 � = 0 1 ◮ | ψ 1 � = α | 0 � + β | 1 � , α, β ∈ C and | α | 2 + | β | 2 = 1 n qubits ◮ Unit vector in ( C 2 ) ⊗ n ◮ Basis: | i � , i ∈ { 0 , 1 } n i ∈{ 0 , 1 } n α i | i � , α i ∈ C and � | α i | 2 = 1 ◮ | ψ 2 � = � Non-local games and verifiable delegation of quantum computation 10 / 23
Very quick introduction to quantum computation 1 qubit ◮ Unit vector in C 2 � 1 � 0 ◮ Basis: | 0 � = � � and | 1 � = 0 1 ◮ | ψ 1 � = α | 0 � + β | 1 � , α, β ∈ C and | α | 2 + | β | 2 = 1 n qubits ◮ Unit vector in ( C 2 ) ⊗ n ◮ Basis: | i � , i ∈ { 0 , 1 } n i ∈{ 0 , 1 } n α i | i � , α i ∈ C and � | α i | 2 = 1 ◮ | ψ 2 � = � 1 | EPR � = 2 ( | 00 � + | 11 � ) √ ◮ It cannot be written as a product state ◮ Source of quantum “spooky actions” 1 ◮ For every orthonomal basis {| v � , | v ⊥ �} , | EPR � = � | vv � + | v ⊥ v ⊥ � � √ 2 Non-local games and verifiable delegation of quantum computation 10 / 23
Very quick introduction to quantum computation Evolution of quantum states ◮ Unitary operators ◮ Composed by gates picked from a (universal) gate-set Non-local games and verifiable delegation of quantum computation 11 / 23
Very quick introduction to quantum computation Evolution of quantum states ◮ Unitary operators ◮ Composed by gates picked from a (universal) gate-set Projective measurements on | ψ � ◮ Set of projectors { P i } , s.t. � i P i = I ◮ Output i with probability � P i | ψ �� 2 P i | ψ � ◮ After the measurement, the states collapses to � P i | ψ �� Non-local games and verifiable delegation of quantum computation 11 / 23
Very quick introduction to quantum computation Evolution of quantum states ◮ Unitary operators ◮ Composed by gates picked from a (universal) gate-set Projective measurements on | ψ � ◮ Set of projectors { P i } , s.t. � i P i = I ◮ Output i with probability � P i | ψ �� 2 P i | ψ � ◮ After the measurement, the states collapses to � P i | ψ �� 1 | EPR � = 2 ( | 00 � + | 11 � ) √ ◮ If measure the first half, the second half is completely defined (independent of the chosen basis) Non-local games and verifiable delegation of quantum computation 11 / 23
From quantum delegation to classical delegation Q P V x , Q Non-local games and verifiable delegation of quantum computation 12 / 23
From quantum delegation to classical delegation Q V and P share EPR pairs P | EPR � ⊗ t V x , Q Non-local games and verifiable delegation of quantum computation 12 / 23
From quantum delegation to classical delegation Q V and P share EPR pairs V sends z i ∈ R { 0 , 1 } P | EPR � ⊗ t z V x , Q Non-local games and verifiable delegation of quantum computation 12 / 23
From quantum delegation to classical delegation Q V and P share EPR pairs V sends z i ∈ R { 0 , 1 } P P sends back c i ∈ { 0 , 1 } | EPR � ⊗ t z c V x , Q Non-local games and verifiable delegation of quantum computation 12 / 23
From quantum delegation to classical delegation Q V and P share EPR pairs V sends z i ∈ R { 0 , 1 } P P sends back c i ∈ { 0 , 1 } | EPR � ⊗ t z c V measures half of EPR pairs with Clifford observables V x , Q Non-local games and verifiable delegation of quantum computation 12 / 23
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