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
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] 2/22
Resource: Graph state |Φ> Property. Given a graph G , and u ∈ V ( G ) \ I , X u Z N ( u ) | G � = | G � where X = | 0 � �→ | 1 � and Z = | 0 � �→ | 0 � | 1 � �→ | 0 � | 1 � �→ − | 1 � 3/22
Resource: Graph state Z Z X Z Property. Given a graph G , and u ∈ V ( G ) \ I , X u Z N ( u ) | G � = | G � where X = | 0 � �→ | 1 � and Z = | 0 � �→ | 0 � | 1 � �→ | 0 � | 1 � �→ − | 1 � 3/22
The tool: measurements |Φ> 4/22
The tool: measurements 4/22
The tool: measurements 4/22
The tool: measurements 4/22
The tool: measurements 4/22
The tool: measurements 4/22
The tool: measurements U |Φ> 4/22
The tool: measurements 4/22
Circuits vs MBQC 5/22
Circuits vs MBQC 5/22
The tool: measurements (ctd) 1-qubit measurements are parametrised by a point of the Bloch sphere. 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 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 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 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. 6/22
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) } . 0 P λ,θ (0) | ϕ � with prob. || P λ,θ (0) | ϕ � || 2 e m o c u t o ✲ a l c s i a s l c | ϕ � classical outcome 1 P λ,θ (1) | ϕ � with prob. || P λ,θ (1) | ϕ � || 2 ✲ 7/22
Towards Reversibility | ψ 00 � P (0) P (0) P (1) | ψ 01 � | ϕ � �→ E G | ϕ � | ψ 10 � P (0) P (1) P (1) | ψ 11 � 8/22
Towards Reversibility | ψ 00 � = U | ϕ � P (0) P (0) P (1) | ψ 01 � = U | ϕ � | ϕ � �→ E G | ϕ � | ψ 10 � = U | ϕ � P (0) P (1) P (1) | ψ 11 � = 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. 8/22
Towards Reversibility P θ ( s ) ) P θ (0) E G | ϕ � 0 ( θ P ✲ E G | ϕ � P θ (1) P θ (1) E G | ϕ � ✲ When measurement in the ( X, Y ) -plane, P θ (1) Z = P θ (0) 9/22
Towards Reversibility P θ ( s ) ◦ Z s ) P θ (0) E G | ϕ � 0 ( θ P ✲ E G | ϕ � P θ (1) Z = P θ (0) P θ (0) E G | ϕ � ✲ When measurement in the ( X, Y ) -plane, P θ (1) Z = P θ (0) Deterministic .... 9/22
Towards Reversibility P θ ( s ) ◦ Z s ) P θ (0) E G | ϕ � 0 ( θ P ✲ E G | ϕ � P θ (1) Z = P θ (0) P θ (0) E G | ϕ � ✲ When measurement in the ( X, Y ) -plane, P θ (1) Z = P θ (0) Deterministic ....but acausal 9/22
already meas. not yet measured to be corrected Z ( X, Y ) ( X, Z ) u Z X 10/22
already meas. not yet measured to be corrected Z Z ( X, Y ) Z X ( X, Z ) u Z 2 = I X • p ( u ) > u • u ∈ N ( p ( u )) • ∀ v ∈ N ( p ( u )) \ { u } , v > u . Causal Flow [DK’04] 11/22
already meas. not yet measured to be corrected Z ( X, Y ) ( X, Z ) u X \ Z Z X • p ( u ) > u • u ∈ N ( p ( u )) • ∀ v ∈ N ( p ( u )) \ { u } , v > u . Causal Flow [DK’04] 12/22
already meas. not yet measured to be corrected Z ( X, Y ) ( X, Z ) u X \ Z \ 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] 13/22
already meas. not yet measured to be corrected Z X Z ( X, Y ) ( X, Z ) u \ 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] 14/22
already meas. not yet measured to be corrected Z \ X Z ( X, Y ) ( X, Z ) u \ 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 λ v = { Z } . Pauli Flow [BKMP’07] 15/22
already meas. not yet measured to be corrected Z ( X, Y ) ( X, Z ) u Z \ Z X Z X • ∀ v ∈ p ( u ) , v > u or λ v = { X } . • u ∈ Odd ( p ( u )) , where Odd ( D ) := { v ∈ V : | N ( v ) ∩ D | = 1 mod 2 } . • ∀ v ∈ Odd ( p ( u )) \ { u } , v > u or λ v = { Z } . Pauli Flow [BKMP’07] 16/22
Pauli Flow Definition. An open graph state ( G, I, O, λ ) has Pauli flow if there exist a map p : O c → 2 I c and a strict partial order < over O c such that ∀ u, v ∈ O c , —(P1) if v ∈ p ( u ) , u � = v , and λ v / ∈ {{ X } , { Y }} then u < v , —(P2) if v ≤ u , u � = v , and λ v / ∈ {{ Y } , { Z }} then v / ∈ Odd p ( u ) , —(P3) if v ≤ u , u � = v , and λ v = { Y } then v ∈ p ( u ) ⇔ v ∈ Odd p ( u ) , —(P4) if λ u = { X, Y } then u / ∈ p ( u ) and u ∈ Odd p ( u ) , —(P5) if λ u = { X, Z } then u ∈ p ( u ) and u ∈ Odd p ( u ) , —(P6) if λ u = { Y, Z } then u ∈ p ( u ) and u / ∈ Odd p ( u ) , —(P7) if λ u = { X } then u ∈ Odd p ( u ) , —(P8) if λ u = { Z } then u ∈ p ( u ) , —(P9) if λ u = { Y } then either: u / ∈ p ( u ) & u ∈ Odd p ( u ) or u ∈ p ( u ) & u / ∈ Odd p ( 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, | λ u | = 2 . Is Pauli Flow necessary for robust determinism ? 17/22
Counter Example Y ( X, Z ) , α 2 1 3 No Pauli flow but robust determinism. 18/22
Restrictions • { X, Y } -MBQC: ∀ u, λ u ∈ {{ X } , { Y } , { X, Y }} . • Universal model of quantum computation ( X, Y ) , α X Robustly deterministic 1 2 but no Pauli flow: 3 19/22
Restrictions • { X, Y } -MBQC: ∀ u, λ u ∈ {{ X } , { Y } , { X, Y }} . • Universal model of quantum computation ( X, Y ) , α X Robustly deterministic 1 2 but no Pauli flow: 3 • { Y, Z } -MBQC: ∀ u, λ u ∈ {{ Y } , { Z } , { Y, Z }} . • Universal model of quantum computation ( Y, Z ) , α Z Robustly deterministic but no Pauli flow: 1 2 3 19/22
Restrictions • { X, Y } -MBQC: ∀ u, λ u ∈ {{ X } , { Y } , { X, Y }} . • Universal model of quantum computation ( X, Y ) , α X Robustly deterministic 1 2 but no Pauli flow: 3 • { Y, Z } -MBQC: ∀ u, λ u ∈ {{ Y } , { Z } , { Y, Z }} . • Universal model of quantum computation ( Y, Z ) , α Z Robustly deterministic but no Pauli flow: 1 2 3 • { X, Z } -MBQC: ∀ u, λ u ∈ {{ X } , { Z } , { X, Z }} . • Real MBQC • Universal model of quantum computation. 19/22
Restrictions • { X, Y } -MBQC: ∀ u, λ u ∈ {{ X } , { Y } , { X, Y }} . • Universal model of quantum computation ( X, Y ) , α X Robustly deterministic 1 2 but no Pauli flow: 3 • { Y, Z } -MBQC: ∀ u, λ u ∈ {{ Y } , { Z } , { Y, Z }} . • Universal model of quantum computation ( Y, Z ) , α Z Robustly deterministic but no Pauli flow: 1 2 3 • { X, Z } -MBQC: ∀ u, λ u ∈ {{ 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. 19/22
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. 20/22
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. 20/22
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