Alex Bredariol Grilo joint work with Andrea Coladangelo, Stacey - - PowerPoint PPT Presentation

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Alex Bredariol Grilo joint work with Andrea Coladangelo, Stacey - - PowerPoint PPT Presentation

Dec 09, 2023 25 likes •836 views

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

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Non-local games and verifiable delegation of quantum computation

Alex Bredariol Grilo

joint work with Andrea Coladangelo, Stacey Jeffery and Thomas Vidick

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Why verifiably delegate quantum computation?

Non-local games and verifiable delegation of quantum computation 2 / 23

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Why verifiably delegate quantum computation?

Superiorita

Non-local games and verifiable delegation of quantum computation 2 / 23

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Why verifiably delegate quantum computation?

Superiorita But they are expensive

Non-local games and verifiable delegation of quantum computation 2 / 23

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Why verifiably delegate quantum computation?

Superiorita But they are expensive Online service

Non-local games and verifiable delegation of quantum computation 2 / 23

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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

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Ideal world

Goal: Interacrive proof system for BQP where

Non-local games and verifiable delegation of quantum computation 3 / 23

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Ideal world

V x Goal: Interacrive proof system for BQP where

◮ the verifier runs poly-time prob. computation Non-local games and verifiable delegation of quantum computation 3 / 23

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Ideal world

V x P ... Goal: Interacrive proof system for BQP where

◮ the verifier runs poly-time prob. computation ◮ an honest prover runs poly-time quantum computation Non-local games and verifiable delegation of quantum computation 3 / 23

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Ideal world

V x 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 Non-local games and verifiable delegation of quantum computation 3 / 23

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Ideal world

V x 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 ◮ additional property: the prover does not learn the input Non-local games and verifiable delegation of quantum computation 3 / 23

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Relaxed models

Exponential-size provers

V P x x ...

Almost-classical clients

V P x ...

  • Comput. soundness

V x ... x

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Multiple provers

V

Non-local games and verifiable delegation of quantum computation 5 / 23

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Multiple provers

V P1 P2 Q Q x, Q

|EPR

... Multiple entangled non-communicating P

Non-local games and verifiable delegation of quantum computation 5 / 23

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Multiple provers

V P1 P2 Q Q x, Q

|EPR

... Multiple entangled non-communicating P Sound against any malicious strategy

Non-local games and verifiable delegation of quantum computation 5 / 23

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Multiple provers

V P1 P2 Q Q x, Q

|EPR

... Multiple entangled non-communicating P Sound against any malicious strategy Servers have to keep entangled

Non-local games and verifiable delegation of quantum computation 5 / 23

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Multiple provers

V P1 P2 Q Q x, Q

|EPR

... Multiple entangled non-communicating P Sound against any malicious strategy Servers have to keep entangled “Plug-and-play”

Non-local games and verifiable delegation of quantum computation 5 / 23

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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

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Previous works

Provers Rounds Total Resources Blind RUV 2012 2 poly(n) ≥ g8192 yes

Non-local games and verifiable delegation of quantum computation 6 / 23

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Previous works

Provers Rounds Total Resources Blind RUV 2012 2 poly(n) ≥ g8192 yes McKague 2013 poly(n) poly(n) ≥ 2153g22 yes GKW 2015 2 poly(n) ≥ g2048 yes HDF 2015 poly(n) poly(n) Θ(g4 log g) yes FH 2015 5 poly(n) > g3 no NV 2017 7 2 > g3 no

Non-local games and verifiable delegation of quantum computation 6 / 23

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The results

Delegate circuit Q on n qubits, with g gates and depth d, 2 provers:

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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

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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

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Comparing to previous works

Provers Rounds Total Resources Blind RUV 2012 2 poly(n) ≥ g8192 yes McKague 2013 poly(n) poly(n) ≥ 2153g22 yes GKW 2015 2 poly(n) ≥ g2048 yes HDF 2015 poly(n) poly(n) Θ(g4 log g) yes FH 2015 5 poly(n) > g3 no NV 2017 7 2 > g3 no VoL 2 O(depth) Θ(g log g) yes DW 2 2 Θ(g log g) no Relativistic 2 1 g3 no

Non-local games and verifiable delegation of quantum computation 8 / 23

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1

Basics on quantum computation

2

General idea

3

Our protocols

4

Open problems

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Very quick introduction to quantum computation

1 qubit

◮ Unit vector in C2 ◮ Basis: |0 =

1

  • and |1 =

1

  • ◮ |ψ1 = α |0 + β |1 , α, β ∈ C and |α|2 + |β|2 = 1

Non-local games and verifiable delegation of quantum computation 10 / 23

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Very quick introduction to quantum computation

1 qubit

◮ Unit vector in C2 ◮ Basis: |0 =

1

  • and |1 =

1

  • ◮ |ψ1 = α |0 + β |1 , α, β ∈ C and |α|2 + |β|2 = 1

n qubits

◮ Unit vector in (C2)⊗n ◮ Basis: |i , i ∈ {0, 1}n ◮ |ψ2 =

i∈{0,1}n αi |i , αi ∈ C and |αi|2 = 1

Non-local games and verifiable delegation of quantum computation 10 / 23

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Very quick introduction to quantum computation

1 qubit

◮ Unit vector in C2 ◮ Basis: |0 =

1

  • and |1 =

1

  • ◮ |ψ1 = α |0 + β |1 , α, β ∈ C and |α|2 + |β|2 = 1

n qubits

◮ Unit vector in (C2)⊗n ◮ Basis: |i , i ∈ {0, 1}n ◮ |ψ2 =

i∈{0,1}n αi |i , αi ∈ C and |αi|2 = 1

|EPR =

1 √ 2 (|00 + |11)

◮ It cannot be written as a product state ◮ Source of quantum “spooky actions” ◮ For every orthonomal basis {|v , |v ⊥}, |EPR =

1 √ 2

  • |vv + |v ⊥v ⊥
  • Non-local games and verifiable delegation of quantum computation

10 / 23

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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

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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 {Pi}, s.t.

i Pi = I

◮ Output i with probability Pi |ψ2 ◮ After the measurement, the states collapses to

Pi|ψ Pi|ψ

Non-local games and verifiable delegation of quantum computation 11 / 23

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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 {Pi}, s.t.

i Pi = I

◮ Output i with probability Pi |ψ2 ◮ After the measurement, the states collapses to

Pi|ψ Pi|ψ

|EPR =

1 √ 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

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From quantum delegation to classical delegation

V x, Q P Q

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From quantum delegation to classical delegation

V x, Q P Q |EPR⊗t V and P share EPR pairs

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From quantum delegation to classical delegation

V x, Q P Q |EPR⊗t z V and P share EPR pairs V sends zi ∈R {0, 1}

Non-local games and verifiable delegation of quantum computation 12 / 23

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From quantum delegation to classical delegation

V x, Q P Q |EPR⊗t z c V and P share EPR pairs V sends zi ∈R {0, 1} P sends back ci ∈ {0, 1}

Non-local games and verifiable delegation of quantum computation 12 / 23

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From quantum delegation to classical delegation

V x, Q P Q |EPR⊗t z c V and P share EPR pairs V sends zi ∈R {0, 1} P sends back ci ∈ {0, 1} V measures half of EPR pairs with Clifford

  • bservables

Non-local games and verifiable delegation of quantum computation 12 / 23

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From quantum delegation to classical delegation

V x, Q P Q |EPR⊗t z c V and P share EPR pairs V sends zi ∈R {0, 1} P sends back ci ∈ {0, 1} V measures half of EPR pairs with Clifford

  • bservables

V performs checks

Non-local games and verifiable delegation of quantum computation 12 / 23

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From quantum delegation to classical delegation

V x, Q P Q |EPR⊗t z c V and P share EPR pairs V sends zi ∈R {0, 1} P sends back ci ∈ {0, 1} V measures half of EPR pairs with Clifford

  • bservables

V performs checks If P passes tests, then no “harmful” errors

Non-local games and verifiable delegation of quantum computation 12 / 23

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From quantum delegation to classical delegation

V x, Q P Q z c

Non-local games and verifiable delegation of quantum computation 13 / 23

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From quantum delegation to classical delegation

V x, Q P Q z c V ′ x, Q PP Q

Non-local games and verifiable delegation of quantum computation 13 / 23

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From quantum delegation to classical delegation

V x, Q P Q z c V ′ x, Q PP Q PV Q

|EPR

Idea: Delegate V to a prover

Non-local games and verifiable delegation of quantum computation 13 / 23

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From quantum delegation to classical delegation

V x, Q P Q z c V ′ x, Q PP Q PV Q

|EPR

Idea: Delegate V to a prover

Non-local games and verifiable delegation of quantum computation 13 / 23

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From quantum delegation to classical delegation

V x, Q P Q z c V ′ x, Q PP Q PV Q

|EPR

Idea: Delegate V to a prover If PV is honest, we are done

Non-local games and verifiable delegation of quantum computation 13 / 23

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From quantum delegation to classical delegation

V x, Q P Q z c V ′ x, Q PP Q PV Q

|EPR

Idea: Delegate V to a prover If PV is honest, we are done How to test PV?

Non-local games and verifiable delegation of quantum computation 13 / 23

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Non-local games

V P1 P2

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Non-local games

V P1 P2 P1 and P2 share a strategy before the game start and then they do not communicate

Non-local games and verifiable delegation of quantum computation 14 / 23

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Non-local games

V P1 P2 x, y ∼ D P1 and P2 share a strategy before the game start and then they do not communicate V picks x, y from distribution D

Non-local games and verifiable delegation of quantum computation 14 / 23

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Non-local games

V P1 P2 x, y ∼ D x y P1 and P2 share a strategy before the game start and then they do not communicate V picks x, y from distribution D V sends x to P1 and y to P2

Non-local games and verifiable delegation of quantum computation 14 / 23

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Non-local games

V P1 P2 x, y ∼ D x y a b P1 and P2 share a strategy before the game start and then they do not communicate V picks x, y from distribution D V sends x to P1 and y to P2 P1 answers with a and P2 answers with b

Non-local games and verifiable delegation of quantum computation 14 / 23

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Non-local games

V P1 P2 x, y ∼ D x y a b V (a, b|x, y) ∈ {0, 1} P1 and P2 share a strategy before the game start and then they do not communicate V picks x, y from distribution D V sends x to P1 and y to P2 P1 answers with a and P2 answers with b V accepts iff V (a, b|x, y) = 1

Non-local games and verifiable delegation of quantum computation 14 / 23

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Non-local games

V P1 P2 x, y ∼ D x y a b V (a, b|x, y) ∈ {0, 1} P1 and P2 share a strategy before the game start and then they do not communicate V picks x, y from distribution D V sends x to P1 and y to P2 P1 answers with a and P2 answers with b V accepts iff V (a, b|x, y) = 1 Classical value ω(G) and quantum value ω∗(G)

Non-local games and verifiable delegation of quantum computation 14 / 23

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Non-local games

V P1 P2 x, y ∼ D x y a b V (a, b|x, y) ∈ {0, 1} P1 and P2 share a strategy before the game start and then they do not communicate V picks x, y from distribution D V sends x to P1 and y to P2 P1 answers with a and P2 answers with b V accepts iff V (a, b|x, y) = 1 Classical value ω(G) and quantum value ω∗(G) ω∗(G) > ω(G)

Non-local games and verifiable delegation of quantum computation 14 / 23

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Bell inequalities and rigidity theorems - Example CHSH

Non-local games and verifiable delegation of quantum computation 15 / 23

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Bell inequalities and rigidity theorems - Example CHSH

V P1 P2 x, y ∈R {0, 1} x · y = a ⊕ b x a y b

Non-local games and verifiable delegation of quantum computation 15 / 23

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Bell inequalities and rigidity theorems - Example CHSH

V P1 P2 x, y ∈R {0, 1} x · y = a ⊕ b x a y b Classical value ω(CHSH) = 3

4

Quantum value ω∗(CHSH) = cos2( π

8 )

Non-local games and verifiable delegation of quantum computation 15 / 23

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Bell inequalities and rigidity theorems - Example CHSH

V P1 P2 x, y ∈R {0, 1} x · y = a ⊕ b x a y b Classical value ω(CHSH) = 3

4

Quantum value ω∗(CHSH) = cos2( π

8 )

Provers share |EPR and measure 1 P1 X Z P2

X+Z √ 2 Z−X √ 2

Non-local games and verifiable delegation of quantum computation 15 / 23

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Bell inequalities and rigidity theorems - Example CHSH

V P1 P2 x, y ∈R {0, 1} x · y = a ⊕ b x a y b Classical value ω(CHSH) = 3

4

Quantum value ω∗(CHSH) = cos2( π

8 )

Provers share |EPR and measure 1 P1 X Z P2

X+Z √ 2 Z−X √ 2

Rigidity: if acceptance prob. is ω∗(CHSH) − ε, then strategy is O(√ε) close to the previous one

Non-local games and verifiable delegation of quantum computation 15 / 23

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Our rigidity results

Our game

Non-local games and verifiable delegation of quantum computation 16 / 23

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Our rigidity results

Our game

G is a set of one-qubit Clifford observables Game where a constant fraction of the questions are in a random Gm Based on the Pauli Braiding Test

Non-local games and verifiable delegation of quantum computation 16 / 23

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Our rigidity results

Our game

G is a set of one-qubit Clifford observables Game where a constant fraction of the questions are in a random Gm Based on the Pauli Braiding Test

Honest strategy

Share m EPR pairs and on question of the form W ∈ Gm the prover measures the “correct” observable W .

Non-local games and verifiable delegation of quantum computation 16 / 23

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Our rigidity results

Theorem

The honest strategy succeeds with prob. 1 − e−Ω(m) in the game.

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Our rigidity results

Theorem

The honest strategy succeeds with prob. 1 − e−Ω(m) in the game.

Theorem

For any ε > 0, any strategy for the provers that succeeds with prob. 1 − ε must be O(√ε)-close to the honest strategy.

Non-local games and verifiable delegation of quantum computation 17 / 23

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From quantum delegation to classical delegation

V ′ x, Q PV Q PP Q

|EPR

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From quantum delegation to classical delegation

V ′ x, Q PV Q PP Q

|EPR

Protocol

Non-local games and verifiable delegation of quantum computation 18 / 23

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From quantum delegation to classical delegation

V ′ x, Q PV Q PP Q

|EPR

Protocol

◮ With prob. p, play non-local game Non-local games and verifiable delegation of quantum computation 18 / 23

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From quantum delegation to classical delegation

V ′ x, Q PV Q PP Q

|EPR

Protocol

◮ With prob. p, play non-local game ◮ With prob. 1 − p, execute original protocol Non-local games and verifiable delegation of quantum computation 18 / 23

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From quantum delegation to classical delegation

V ′ x, Q PV Q PP Q

|EPR

Protocol

◮ With prob. p, play non-local game ◮ With prob. 1 − p, execute original protocol

Two tests are indistinguishable for PV

Non-local games and verifiable delegation of quantum computation 18 / 23

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From quantum delegation to classical delegation

V ′ x, Q PV Q PP Q

|EPR

Protocol

◮ With prob. p, play non-local game ◮ With prob. 1 − p, execute original protocol

Two tests are indistinguishable for PV PV is tested with the game

Non-local games and verifiable delegation of quantum computation 18 / 23

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From quantum delegation to classical delegation

V ′ x, Q PV Q PP Q

|EPR

Protocol

◮ With prob. p, play non-local game ◮ With prob. 1 − p, execute original protocol

Two tests are indistinguishable for PV PV is tested with the game PP is tested in the original protocol

Non-local games and verifiable delegation of quantum computation 18 / 23

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From quantum delegation to classical delegation

V ′ x, Q PV Q PP Q

|EPR

Protocol

◮ With prob. p, play non-local game ◮ With prob. 1 − p, execute original protocol

Two tests are indistinguishable for PV PV is tested with the game PP is tested in the original protocol If both pass the tests, they perform the computation

Non-local games and verifiable delegation of quantum computation 18 / 23

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Verifier-on-a-leash protocol

1 1 2 2 3 3 Rigidity Test Original protocol Rigidity-Clifford Test rounds Computation rounds PV PP

Non-local games and verifiable delegation of quantum computation 19 / 23

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DogWalker protocol

In Verifier-on-a-leash protocol

Non-local games and verifiable delegation of quantum computation 20 / 23

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DogWalker protocol

In Verifier-on-a-leash protocol

◮ Rounds of communication for blindness Non-local games and verifiable delegation of quantum computation 20 / 23

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DogWalker protocol

In Verifier-on-a-leash protocol

◮ Rounds of communication for blindness

In DogWalker protocol

Non-local games and verifiable delegation of quantum computation 20 / 23

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DogWalker protocol

In Verifier-on-a-leash protocol

◮ Rounds of communication for blindness

In DogWalker protocol

◮ Reveal x to PV Non-local games and verifiable delegation of quantum computation 20 / 23

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DogWalker protocol

In Verifier-on-a-leash protocol

◮ Rounds of communication for blindness

In DogWalker protocol

◮ Reveal x to PV ◮ Extra tests to check if PV is honest Non-local games and verifiable delegation of quantum computation 20 / 23

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DogWalker protocol

1 1 2 2 3 3 4 4 Rigidity Test Original protocol U n i f

  • r

m i t y

  • f

{ ci }i Tomography Test Rigidity-Clifford Test rounds Computation round Rigidity-Tomography PV PP

Non-local games and verifiable delegation of quantum computation 21 / 23

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Open problems

More efficient 1-round schemes ( ˜ O(g) resources)

Non-local games and verifiable delegation of quantum computation 22 / 23

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Open problems

More efficient 1-round schemes ( ˜ O(g) resources) Blind O(1)-round protocols

Non-local games and verifiable delegation of quantum computation 22 / 23

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Open problems

More efficient 1-round schemes ( ˜ O(g) resources) Blind O(1)-round protocols Delegation protocol with non-entangled provers

Non-local games and verifiable delegation of quantum computation 22 / 23

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Thank you for your attention!

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