Determinism and Computational Power of Real Measurement-based - - PowerPoint PPT Presentation

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Determinism and Computational Power of Real Measurement-based - - PowerPoint PPT Presentation

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Determinism and Computational Power of Real Measurement-based Quantum Computation Simon Perdrix, Luc Sanselme CNRS, Inria Project team CARTE, LORIA simon.perdrix@loria.fr FCT17 Bordeaux Introduction - Context MBQC: Measurement-based

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Determinism and Computational Power of Real Measurement-based Quantum Computation

Simon Perdrix, Luc Sanselme

CNRS, Inria Project team CARTE, LORIA simon.perdrix@loria.fr

FCT’17 – Bordeaux

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2/22

Introduction - Context

MBQC: Measurement-based Quantum Computation [Briegel, Raussendorf’01]

  • Universal model of quantum computation.
  • Conceptual and technological breakthrough
  • Decreasing depth of quantum computations [Browne, Kashefi,

Perdrix’10]

  • Blind QC [Broadbent, Fitzsimon, Kashefi’09]
  • Interactive Proofs [McKague]
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3/22

Resource: Graph state

|Φ>

  • Property. Given a graph G, and u ∈ V (G) \ I,

XuZN(u) |G = |G where X = |0 → |1 |1 → |0 and Z = |0 → |0 |1 → − |1

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3/22

Resource: Graph state

X Z Z Z

  • Property. Given a graph G, and u ∈ V (G) \ I,

XuZN(u) |G = |G where X = |0 → |1 |1 → |0 and Z = |0 → |0 |1 → − |1

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The tool: measurements

|Φ>

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The tool: measurements

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The tool: measurements

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The tool: measurements

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The tool: measurements

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The tool: measurements

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The tool: measurements

U|Φ>

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The tool: measurements

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Circuits vs MBQC

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Circuits vs MBQC

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The tool: measurements (ctd)

1-qubit measurements are parametrised by a point of the Bloch sphere.

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The tool: measurements (ctd)

1-qubit measurements are parametrised by a point of the Bloch sphere. In MBQC, measurements are in the:

  • (X,Y)-plane
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The tool: measurements (ctd)

1-qubit measurements are parametrised by a point of the Bloch sphere. In MBQC, measurements are in the:

  • (X,Y)-plane
  • (X,Z)-plane
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The tool: measurements (ctd)

1-qubit measurements are parametrised by a point of the Bloch sphere. In MBQC, measurements are in the:

  • (X,Y)-plane
  • (X,Z)-plane
  • (Z,Y)-plane
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6/22

The tool: measurements (ctd)

1-qubit measurements are parametrised by a point of the Bloch sphere. In MBQC, measurements are in the:

  • (X,Y)-plane
  • (X,Z)-plane
  • (Z,Y)-plane
  • according to X
  • according to Y
  • according to Z

Each measurement is parameterised by λ ∈ {X, Y, Z, (X, Y ), (X, Z), (Z, Y )} and an angle θ when λ is a plane.

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Irreversibility and Non determinism

Each measurement M λ,θ (plane λ, angle θ) is characterised by two projections {P λ,θ(0), P λ,θ(1)}. P λ,θ(0) |ϕ with prob. ||P λ,θ(0) |ϕ ||2 |ϕ

c l a s s i c a l

  • u

t c

  • m

e

✲

classical outcome 1

✲ P λ,θ(1) |ϕ with prob. ||P λ,θ(1) |ϕ ||2

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Towards Reversibility

|ϕ → EG |ϕ

P (0) P (1) P (0) P (1) P (0) P (1)

|ψ00 |ψ01 |ψ10 |ψ11

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Towards Reversibility

|ϕ → EG |ϕ

P (0) P (1) P (0) P (1) P (0) P (1)

|ψ00 |ψ01 |ψ10 |ψ11 = U |ϕ = U |ϕ = U |ϕ = U |ϕ Robust determinism:

  • Deterministic: all branches produce the same quantum state.
  • Uniformity: independent of the angles.
  • Strongness: every branch has a non zero probability.
  • Stepwise.
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Towards Reversibility

P θ(s) P θ(0)EG |ϕ EG |ϕ P

θ

( ) ✲ P θ(1) ✲ P θ(1)EG |ϕ When measurement in the (X, Y )-plane, P θ(1)Z = P θ(0)

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Towards Reversibility

P θ(s) ◦ Zs P θ(0)EG |ϕ EG |ϕ P

θ

( ) ✲ P θ(1)Z = P θ(0) ✲ P θ(0)EG |ϕ When measurement in the (X, Y )-plane, P θ(1)Z = P θ(0) Deterministic ....

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Towards Reversibility

P θ(s) ◦ Zs P θ(0)EG |ϕ EG |ϕ P

θ

( ) ✲ P θ(1)Z = P θ(0) ✲ P θ(0)EG |ϕ When measurement in the (X, Y )-plane, P θ(1)Z = P θ(0) Deterministic ....but acausal

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already meas. to be corrected not yet measured

(X, Y )

u Z

Z X (X, Z)

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already meas. to be corrected not yet measured

(X, Y )

u Z2=I

Z X (X, Z) Z X Z

  • p(u) > u
  • u ∈ N(p(u))
  • ∀v ∈ N(p(u)) \ {u}, v > u.

Causal Flow [DK’04]

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already meas. to be corrected not yet measured

(X, Y )

u \ Z

Z X Z (X, Z) X

  • p(u) > u
  • u ∈ N(p(u))
  • ∀v ∈ N(p(u)) \ {u}, v > u.

Causal Flow [DK’04]

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already meas. to be corrected not yet measured

(X, Y )

u \ Z

Z X

\ Z

(X, Z) X X

  • ∀v ∈ p(u), v > u
  • u ∈ Odd(p(u)), where Odd(D) := {v∈V : |N(v)∩D| = 1 mod 2}.
  • ∀v ∈ Odd(p(u)) \ {u}, v > u.

GFlow [BKMP’07]

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already meas. to be corrected not yet measured

(X, Y )

u \ Z

Z

Z

X (X, Z) X

  • ∀v ∈ p(u), v > u
  • u ∈ Odd(p(u)), where Odd(D) := {v∈V : |N(v)∩D| = 1 mod 2}.
  • ∀v ∈ Odd(p(u)) \ {u}, v > u.

GFlow [BKMP’07]

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already meas. to be corrected not yet measured

(X, Y )

u \ Z

Z

\ Z

X (X, Z) X

  • ∀v ∈ p(u), v > u
  • u ∈ Odd(p(u)), where Odd(D) := {v∈V : |N(v)∩D| = 1 mod 2}.
  • ∀v ∈ Odd(p(u)) \ {u}, v > u or λv = {Z}.

Pauli Flow [BKMP’07]

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already meas. to be corrected not yet measured

(X, Y )

u \ Z

Z X

X

(X, Z) Z Z

  • ∀v ∈ p(u), v > u or λv = {X}.
  • u ∈ Odd(p(u)), where Odd(D) := {v∈V : |N(v)∩D| = 1 mod 2}.
  • ∀v ∈ Odd(p(u)) \ {u}, v > u or λv = {Z}.

Pauli Flow [BKMP’07]

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Pauli Flow

  • Definition. An open graph state (G, I, O, λ) has Pauli flow if there exist a

map p : Oc → 2Ic and a strict partial order < over Oc such that ∀u, v ∈ Oc, —(P1) if v ∈ p(u), u = v, and λv / ∈ {{X}, {Y }} then u < v, —(P2) if v ≤ u, u = v, and λv / ∈ {{Y }, {Z}} then v / ∈ Oddp(u), —(P3) if v ≤ u, u = v, and λv = {Y } then v ∈ p(u) ⇔ v ∈ Oddp(u), —(P4) if λu = {X, Y } then u / ∈ p(u) and u ∈ Oddp(u), —(P5) if λu = {X, Z} then u ∈ p(u) and u ∈ Oddp(u), —(P6) if λu = {Y, Z} then u ∈ p(u) and u / ∈ Oddp(u), —(P7) if λu = {X} then u ∈ Oddp(u), —(P8) if λu = {Z} then u ∈ p(u), —(P9) if λu = {Y } then either: u / ∈ p(u) & u ∈ Oddp(u)

  • r

u ∈ p(u) & u / ∈ Oddp(u). Theorem [Browne, Kashefi, Mhalla, Perdrix 07] Pauli flow is a sufficient condition for robust determinism. Theorem [Browne, Kashefi, Mhalla, Perdrix 07] Pauli Fow is necessary for robust determinism when ∀u, |λu| = 2. Is Pauli Flow necessary for robust determinism ?

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Counter Example

(X, Z), α 1 3 Y 2 No Pauli flow but robust determinism.

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Restrictions

  • {X, Y }-MBQC: ∀u, λu ∈ {{X}, {Y }, {X, Y }}.
  • Universal model of quantum computation

Robustly deterministic but no Pauli flow: (X, Y ), α 1 3 X 2

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Restrictions

  • {X, Y }-MBQC: ∀u, λu ∈ {{X}, {Y }, {X, Y }}.
  • Universal model of quantum computation

Robustly deterministic but no Pauli flow: (X, Y ), α 1 3 X 2

  • {Y, Z}-MBQC: ∀u, λu ∈ {{Y }, {Z}, {Y, Z}}.
  • Universal model of quantum computation

Robustly deterministic but no Pauli flow: (Y, Z), α 1 3 Z 2

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Restrictions

  • {X, Y }-MBQC: ∀u, λu ∈ {{X}, {Y }, {X, Y }}.
  • Universal model of quantum computation

Robustly deterministic but no Pauli flow: (X, Y ), α 1 3 X 2

  • {Y, Z}-MBQC: ∀u, λu ∈ {{Y }, {Z}, {Y, Z}}.
  • Universal model of quantum computation

Robustly deterministic but no Pauli flow: (Y, Z), α 1 3 Z 2

  • {X, Z}-MBQC: ∀u, λu ∈ {{X}, {Z}, {X, Z}}.
  • Real MBQC
  • Universal model of quantum computation.
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Restrictions

  • {X, Y }-MBQC: ∀u, λu ∈ {{X}, {Y }, {X, Y }}.
  • Universal model of quantum computation

Robustly deterministic but no Pauli flow: (X, Y ), α 1 3 X 2

  • {Y, Z}-MBQC: ∀u, λu ∈ {{Y }, {Z}, {Y, Z}}.
  • Universal model of quantum computation

Robustly deterministic but no Pauli flow: (Y, Z), α 1 3 Z 2

  • {X, Z}-MBQC: ∀u, λu ∈ {{X}, {Z}, {X, Z}}.
  • Real MBQC
  • Universal model of quantum computation.
  • Theorem. Pauli flow is necessary and sufficient for real (i.e. {X,Z}-MBQC)

robust determinism.

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Application: Interactive Proofs

McKague’s interactive proofs protocol based on real MBQC:

  • A classical verifier
  • Polynomial number of quantum provers.
  • real MBQC is crucial: no way for the verifier to distinguish a state and its

conjugate. McKague’s open question: Protocol with 2 provers only.

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Application: Interactive Proofs

McKague’s interactive proofs protocol based on real MBQC:

  • A classical verifier
  • Polynomial number of quantum provers.
  • real MBQC is crucial: no way for the verifier to distinguish a state and its

conjugate. McKague’s open question: Protocol with 2 provers only. ⇔ Universal real MBQC on bipartite graphs.

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Application: Interactive Proofs

McKague’s interactive proofs protocol based on real MBQC:

  • A classical verifier
  • Polynomial number of quantum provers.
  • real MBQC is crucial: no way for the verifier to distinguish a state and its

conjugate. McKague’s open question: Protocol with 2 provers only. ⇔ Universal real MBQC on bipartite graphs.

  • Theorem. All measurements of a robustly deterministic real bipartite MBQC

can be performed in parallel.

  • Corollary. McKague’s protocol does not work with 2 provers.
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Conclusion and future work

  • Pauli flow characterises robust determinism for real-MBQC.

→ Efficient algorithm for Pauli flow? (efficient algorithm exists for GFlow [Mhalla,Perdrix’08])

  • Counter examples for {X, Y } and {Y, Z} planes.

→ Extension of Pauli flow?

  • Weak power of real bipartite MBQC.

→ Equivalent to commuting quantum circuits?