Qudit versions of the π/8 gate: Applications in fault-tolerant QC and nonlocality

Transcription

1 Preliminaries Mathematical Structure Geometry Applications Qudit versions of the π/8 gate: Applications in fault-tolerant QC and nonlocality Mark Howard1 & Jiri Vala1,2 1 National 2 Dublin October 2012 University of Ireland, Maynooth, Ireland. Institute for Advanced Studies, Dublin, Ireland. IRC Empower Fellowship & SFI Grant No. 10/IN.1/I / 58

2 The U π/8 gate and its uses U π/8 UQC = Cliffords, U π/8 UQC Cliffords Preliminaries ( e i π 8 0 U π/8 = 0 e i π 8 ) ( e i π 4 Key References I ) P. O. Boykin, T. Mor, M. Pulver, V. Roychowdhury and F. Vatan. A new universal and fault-tolerant quantum basis Information Processing Letters 75, 3 pp , (2000). 2 / 58

3 The U π/8 gate and its uses U π/8 UQC = Cliffords, U π/8 UQC Cliffords Teleportation-based UQC Preliminaries ( e i π 8 0 U π/8 = 0 e i π 8 ) ( e i π 4 Key References II ) D. Gottesman and I. L. Chuang, Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations Nature 402, 6760 pp , (1999). 3 / 58

4 The U π/8 gate and its uses U π/8 UQC = Cliffords, U π/8 UQC Cliffords Teleportation-based UQC Transversal for R-M codes Preliminaries ( e i π 8 0 U π/8 = 0 e i π 8 ) ( e i π 4 Key References III ) B. Zeng, H. Chung, A. Cross and I. Chuang, Local unitary versus local Clifford equivalence of stabilizer and graph states, Phys. Rev. A 75, (2007). 4 / 58

5 The U π/8 gate and its uses Preliminaries U π/8 ( e i π 8 0 U π/8 = 0 e i π 8 ) ( e i π 4 Key References XII ) UQC = Cliffords, U π/8 UQC Cliffords Teleportation-based UQC Transversal for R-M codes Topological Protection (3D) S. Bravyi and R. Koenig Classification of topologically protected gates for local stabilizer codes, arxiv: (2012). 5 / 58

6 The U π/8 gate and its uses U π/8 Preliminaries U ( π/8 e i π ) ( 8 0 U π/8 Key = References e i π IV 8 0 e i π 4 ) UQC = Cliffords, U π/8 UQC Cliffords Teleportation-based UQC Transversal for R-M codes Topological Protection (3D) Secure assisted UQC A. M. Childs, Secure assisted quantum computation Quantum Info. Comput.5, pp. 456, (2005). 6 / 58

7 The U π/8 gate and its uses U π/8 Preliminaries U ( π/8 e i π ) ( 8 0 U π/8 Key = References e i π V 8 0 e i π 4 ) UQC = Cliffords, U π/8 UQC Cliffords Teleportation-based UQC Transversal for R-M codes Topological Protection (3D) Secure assisted UQC Measurement-based UQC with graph states M. Silva, V. Danos, E. Kashefi and H. Ollivier, A direct approach to fault-tolerance in measurement-based quantum computation via teleportation New Journal of Physics 9, 6 pp. 192, (2007). 7 / 58

8 The U π/8 gate and its uses U π/8 Preliminaries U ( π/8 e i π ) ( 8 0 U π/8 Key = References e i π VI 8 0 e i π 4 ) UQC = Cliffords, U π/8 UQC Cliffords Teleportation-based UQC Transversal for R-M codes Topological Protection (3D) Secure assisted UQC Measurement-based UQC with graph states Optimal CHSH game with ( )/ 2 M. Howard and J. Vala, Nonlocality as a benchmark for universal quantum computation in Ising anyon topological quantum computers, Phys. Rev. A 85, (2012). 8 / 58

9 The U π/8 gate and its uses U π/8 Preliminaries U ( π/8 e i π ) ( 8 0 U π/8 Key = References e i π VII 8 0 e i π 4 ) UQC = Cliffords, U π/8 UQC Cliffords Teleportation-based UQC Transversal for R-M codes Topological Protection (3D) Secure assisted UQC Measurement-based UQC with graph states Optimal CHSH game with ( )/ 2 Blind UQC A. Broadbent, J. Fitzsimons and E. Kashefi, Universal blind quantum computation, Annual IEEE Symposium on Foundations of Computer Science, pp , (2009). 9 / 58

10 Structure of Pauli/Clifford groups First 3 levels of the Clifford Hierarchy Preliminaries G = Pauli Group Key References II Clifford Group C = {C CGC = G} C 3 = { U UGU C } D. Gottesman and I. L. Chuang, Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations Nature 402, 6760 pp , (1999). 10 / 58

11 Structure of Pauli/Clifford groups First 3 levels of the Clifford Hierarchy U π/8 Preliminaries G = Pauli Group Key References II Clifford Group C = {C CGC = G} Toffoli C 3 = { U UGU C } We will focus on single, p-level particles Generalized σ x /σ z : X j = j + 1 mod p Z j = ω j j (ω = e 2πi/p ) Displacement operators, D (x z) = ω 2 1xz X x Z z, form Pauli group G D. Gottesman and I. L. Chuang, Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations Nature 402, 6760 pp , (1999). 11 / 58

12 Single-Qudit Clifford Gates Preliminaries C = {C (F χ) F SL(2, Z p ), χ Z 2 p}, F = ( α β ) ( γ δ with unit determinant. χ = xz ) Key References VIII is a vector of length 2. All elements of F, χ are from Z p = {0, 1,..., p 1} Explicit recipe for constructing a Clifford unitary (where τ = ω 2 1 ): C (F χ) = D (x z) V F { 1 p 1 p j,k=0 V F = τ β 1 (αk 2 2jk+δj 2 ) j k β 0 p 1 k=0 τ αγk2 αk k β = 0. D. M. Appleby, Properties of the extended Clifford group with applications to SIC-POVMs and MUBs, arxiv:quant-ph/ , (2009). 12 / 58

13 Single-Qudit Clifford Gates Preliminaries C = {C (F χ) F SL(2, Z p ), χ Z 2 p}, F = ( α β ) ( γ δ with unit determinant. χ = xz ) Key References VIII is a vector of length 2. All elements of F, χ are from Z p = {0, 1,..., p 1} Explicit recipe for constructing a Clifford unitary (where τ = ω 2 1 ): C (F χ) = D (x z) V F { 1 p 1 p j,k=0 V F = τ β 1 (αk 2 2jk+δj 2 ) j k β 0 p 1 k=0 τ αγk2 αk k β = 0. D. M. Appleby, Properties of the extended Clifford group with applications to SIC-POVMs and MUBs, arxiv:quant-ph/ , (2009). C ([ ] 1 0 [ ]) xz γ 1 13 / 58

14 Single-Qudit Clifford Gates Preliminaries C = {C (F χ) F SL(2, Z p ), χ Z 2 p}, F = ( α β ) ( γ δ with unit determinant. χ = xz ) Key References VIII is a vector of length 2. All elements of F, χ are from Z p = {0, 1,..., p 1} Explicit recipe for constructing a Clifford unitary (where τ = ω 2 1 ): C (F χ) = D (x z) V F { 1 p 1 p j,k=0 V F = τ β 1 (αk 2 2jk+δj 2 ) j k β 0 p 1 k=0 τ αγk2 αk k β = 0. D. M. Appleby, Properties of the extended Clifford group with applications to SIC-POVMs and MUBs, arxiv:quant-ph/ , (2009). C ([ ] 1 0 [ ]) xz SU(p) p > 3, γ 1 ( ) det C ([ ] 1 0 [ ]) xz = τ 2γ for p = 3, γ 1 14 / 58

15 U υ as qudit version of U π/8 (p > 3) p 1 Define U υ = U(υ 0, υ 1,...) = ω υ k k k (υ k Z p ) k=0 15 / 58

16 U υ as qudit version of U π/8 (p > 3) p 1 Define U υ = U(υ 0, υ 1,...) = ω υ k k k (υ k Z p ) k=0 Easy to show U υ D (x z) U υ = D (x z) ω (υ k+x υ k ) k k and in particular U υ D (1 0) U υ = D (1 0) ω (υ k+1 υ k ) k k * k k 16 / 58

17 U υ as qudit version of U π/8 (p > 3) p 1 Define U υ = U(υ 0, υ 1,...) = ω υ k k k (υ k Z p ) k=0 Easy to show U υ D (x z) U υ = D (x z) ω (υ k+x υ k ) k k and in particular U υ D (1 0) U υ = D (1 0) ω (υ k+1 υ k ) k k * Must have U υ D (1 0) U υ = ω ɛ C ([ ] 1 0 [ ]) 1z γ 1 p 1 = ω ɛ D (1 z ) τ γ k 2 k k ** k=0 k k 17 / 58

18 U υ as qudit version of U π/8 (p > 3) p 1 Define U υ = U(υ 0, υ 1,...) = ω υ k k k (υ k Z p ) k=0 Easy to show U υ D (x z) U υ = D (x z) ω (υ k+x υ k ) k k and in particular U υ D (1 0) U υ = D (1 0) ω (υ k+1 υ k ) k k * Must have U υ D (1 0) U υ = ω ɛ C ([ ] 1 0 [ ]) 1z γ 1 p 1 = ω ɛ D (1 z ) τ γ k 2 k k ** k=0 Equating * and ** : ω υ k+1 υ k = ω ɛ τ z ω kz τ k2 γ ( k Z p ), υ k = 12 1 k(γ + k(6z + (2k 3)γ ) + kɛ (υ 0 = 0) 18 / 58 k k

19 U υ as qudit version of U π/8 (p = 3) For p = 3 : ( ) det ζ 2γ C ([ ] 1 0 [ ]) 1z γ 1 = 1 (ζ = e 2πi 9 ) 2 U υ = ζ υ k k k (υ k Z 9 ) k=0 υ = (υ 0, υ 1, υ 2 ) =(0, 6z + 2γ + 3ɛ, 6z + γ + 6ɛ ) mod 9 19 / 58

20 U υ as qudit version of U π/8 (p = 3) For p = 3 : ( ) det ζ 2γ C ([ ] 1 0 [ ]) 1z γ 1 = 1 (ζ = e 2πi 9 ) 2 U υ = ζ υ k k k (υ k Z 9 ) k=0 υ = (υ 0, υ 1, υ 2 ) =(0, 6z + 2γ + 3ɛ, 6z + γ + 6ɛ ) mod 9 Examples: p = 3 : e 2πi e 2πi 9 [z =1,γ =2,ɛ =0] p = 5 : e 4πi e 2πi e 4πi e 2πi 5 [z =1,γ =4,ɛ =0] 20 / 58

21 Group Structure of U υ gates Can create p 3 gates U υ (z, γ, ɛ ) varying over z, γ, ɛ Z p. p 2 (p 1) non-clifford U υ varying over z, ɛ Z p, γ Z p. U υ D (1 0) U υ = ω ɛ C ([ ] 1 0 [ ]) 1z but C ([ ] γ 1 0 [ ]) 1z = D (1 z ) Easy to show p 3 gates {U υ } form a finite Abelian group. 21 / 58

22 Group Structure of U υ gates Can create p 3 gates U υ (z, γ, ɛ ) varying over z, γ, ɛ Z p. p 2 (p 1) non-clifford U υ varying over z, ɛ Z p, γ Z p. U υ D (1 0) U υ = ω ɛ C ([ ] 1 0 [ ]) 1z but C ([ ] γ 1 0 [ ]) 1z = D (1 z ) Easy to show p 3 gates {U υ } form a finite Abelian group. Use Fund. Thm. of finite Abelian groups to characterize {U υ } Group No. elements of order Min. no. of name 1 p p 2 p 3 generators p = 2 Z p = 3 Z 9 Z p > 3 Z 3 p 1 p / 58

23 All primes are odd except two, which is the oddest prime of all! Can create p 3 gates U υ (z, γ, ɛ ) varying over z, γ, ɛ Z p. p 2 (p 1) non-clifford U υ varying over z, ɛ Z p, γ Z p. Group Structure of U υ gates U υ D (1 0) U υ = ω ɛ C ([ ] 1 0 [ ]) 1z but C ([ ] γ 1 0 [ ]) 1z = D (1 z ) Easy to show p 3 gates {U υ } form a finite Abelian group. Use Fund. Thm. of finite Abelian groups to characterize {U υ } Group No. elements of order Min. no. of name 1 p p 2 p 3 generators p = 2 Z p = 3 Z 9 Z p > 3 Z 3 p 1 p / 58

24 All primes are odd except two, which is the oddest prime of all! Can create p 3 gates U υ (z, γ, ɛ ) varying over z, γ, ɛ Z p. p 2 (p 1) non-clifford U υ varying over z, ɛ Z p, γ Z p. Group Structure of U υ gates U υ D (1 0) U υ = ω ɛ C ([ ] 1 0 [ ]) 1z but C ([ ] γ 1 0 [ ]) 1z = D (1 z ) Easy to show p 3 Perhaps, in Quantum Information, gates {U υ } three form a is finite the second Abelian oddest group. prime! Use Fund. Thm. of finite Abelian groups to characterize {U υ } Group No. elements of order Min. no. of name 1 p p 2 p 3 generators p = 2 Z p = 3 Z 9 Z p > 3 Z 3 p 1 p / 58

25 Group Structure of U υ gates Can create p 3 gates U υ (z, γ, ɛ ) varying over z, γ, ɛ Z p. p 2 (p 1) non-clifford U υ varying over z, ɛ Z p, γ Z p. U υ D (1 0) U υ = ω ɛ C ([ ] 1 0 [ ]) 1z but C ([ ] γ 1 0 [ ]) 1z = D (1 z ) Easy to show p 3 gates {U υ } form a finite Abelian group. U υ (z 1, γ 1, ɛ 1 )U υ (z 2, γ 2, ɛ 2 ) = U υ (z 1 + z 2, γ 1 + γ 2, ɛ 1 + ɛ 2 ) Use Fund. Thm. of finite Abelian groups to characterize {U υ } Group No. elements of order Min. no. of name 1 p p 2 p 3 generators p = 2 Z p = 3 Z 9 Z p > 3 Z 3 p 1 p / 58

26 Qubit Geometry: Magic States and U π/ / 58

27 Qubit Geometry: Magic States and U π/ / 58

28 Qubit Geometry: Magic States and U π/ / 58

29 Qubit Geometry: Magic States and U π/8 0 + T H T T = 1 ( I + σ ) x + σ y + σ z 2 3 H H = 1 ( I + σ ) x + σ y Both T and H are eigenvectors of Clifford gates T is the most non-stabilizer qubit state H is the most non-stabilizer qubit state in the equatorial plane Note that H = U π/8 + = 1 2 diag(u π/8 ) 29 / 58

30 Qubit Geometry: Magic States and U π/8 0 + Preliminaries 1 T H Key References IX T T = 1 ( I + σ ) x + σ y + σ z 2 3 H H = 1 ( I + σ ) x + σ y 2 2 Both T and H are eigenvectors of Clifford gates T is the most non-stabilizer qubit state H is the most non-stabilizer qubit state in the equatorial plane Note that H = U π/8 + = 1 2 diag(u π/8 ) In [vdh 11], states were found to be maximally non-stabilizer, and Clifford eigenvectors, in all odd prime dimensions (similar to T?) We will argue that ψ Uυ U υ + is the qudit analogue of H. W. van Dam and M. Howard, Noise thresholds for higher-dimensional systems using the discrete Wigner function Phys. Rev. A. 83, , (2011). 30 / 58

31 Geometry: ψ Uυ as the qudit analogue of H STAB = Convex hull of qudit stabilizer states { p(p+1) = ρ ρ = q i ψ (i) ψ(i) ST AB i=1 i=1 { } = ρ min Tr [A(u)ρ] 0 u Z p+1 p ST AB, p(p+1) q i = 1 } 31 / 58

32 Geometry: ψ Uυ as the qudit analogue of H Define STAB = Convex hull of qudit stabilizer states { p(p+1) = ρ ρ = q i ψ (i) ψ(i) ST AB i=1 i=1 { } = ρ min Tr [A(u)ρ] 0 { Uθ = e iθ k k k (θ k R) ψ Uθ = eiθ k p k = U θ + u Z p+1 p so that ST AB, p(p+1) { {U υ } {U θ } { ψ Uυ } { ψ Uθ } q i = 1 } 32 / 58

33 Geometry: ψ Uυ as the qudit analogue of H Define STAB = Convex hull of qudit stabilizer states { p(p+1) = ρ ρ = q i ψ (i) ψ(i) ST AB i=1 i=1 { } = ρ min Tr [A(u)ρ] 0 { Uθ = e iθ k k k (θ k R) ψ Uθ = eiθ k p k = U θ + u Z p+1 p so that ST AB, p(p+1) { {U υ } {U θ } { ψ Uυ } { ψ Uθ } q i = 1 } ψ Uυ is farthest outside STAB of all ψ Uθ (for p = 2, 3, 5, 7 at least) ψ Uυ is also a Clifford eigenvector: C ([ ] 1 0 [ ]) 1 ψ U(z,γ,ɛ ) = ω ɛ ψ U(z,γ,ɛ ). γ 1 z 33 / 58

34 Geometry: U υ as the qudit analogue of U π/8 For { U U(p) E : ρ in ρ out Jamio lkowski state = J U (I U) p 1 ϱ E = [I E] ( j=0 p 1 j,k=0 jj p j,j k,k p ) CLIFF = Convex hull of qudit Clifford gates J C(F1 { j=p(p 2 1) } χ 1 ) J k=p 2 C(F2 χ 1 ) = ϱ E ϱ E = q j,k J J J C(Fj χ k ) C (Fj χ k ) C(Fj χ k ) j,k=1 { } = ϱ E min W W Tr [W ϱ E] 0 34 / 58

35 Geometry: U υ as the qudit analogue of U π/8 For { U U(p) E : ρ in ρ out Jamio lkowski state = J U (I U) p 1 ϱ E = [I E] ( j=0 p 1 j,k=0 jj p j,j k,k p ) Preliminaries J Uυ Key References IX{ CLIFF = Convex hull of qudit Clifford gates J C(Fj χ k ) = = { ϱ E ϱ E = j=p(p 2 1) k=p 2 j,k=1 ϱ E min W W Tr [W ϱ E] 0 q j,k J C(Fj χ k ) J C (Fj χ k ) } Seems that U υ is the most non-clifford U U(p) (for p = 2, 3, 5, 7,?) W. van Dam and M. Howard, Noise thresholds for higher-dimensional systems using the discrete Wigner function Phys. Rev. A. 83, , (2011). 35 / 58 }

36 Applications? U1 π/8 ( e i π 8 0 U π/8 = 0 e i π 8 ) ( e i π 4 ) UQC= Cliffords, U π/8 Teleportation-based UQC Transversal for some Reed-Muller codes Secure assisted UQC Measurement-based UQC with graph states Optimal CHSH game with ( )/ 2 Blind UQC 36 / 58

37 Application to Fault-tolerant QC We argued ψ Uυ was the qudit analogue of H = ψ Uπ/8. The key feature of H is that it is suitable for magic state distillation ρin φ Small Stabilizer Circuit U π/8 φ ρin. Magic State Distillation ρout H Trash ρin ρin 37 / 58

38 Application to Fault-tolerant QC We argued ψ Uυ was the qudit analogue of H = ψ Uπ/8. The key feature of H is that it is suitable for magic state distillation ρin φ Small Stabilizer Circuit U π/8 φ ρin. Magic State Distillation ρout H Trash ρin ρin 38 / 58

39 Application to Fault-tolerant QC We argued ψ Uυ was the qudit analogue of H = ψ Uπ/8. The key feature of H is that it is suitable for magic state distillation Preliminaries ρin φ Small Stabilizer Circuit U π/8 φ ρin Key ReferencesMagic X. State Distillation ρout H Trash ρin ρin Campbell, Anwar and Browne have shown ψ Uυ are qudit magic states. Nebe, Rains and Sloane have shown Cliffords,U υ enables UQC Geometrical features are good news! E. T. Campbell, H. Anwar and D. E. Browne, Magic state distillation in all prime dimensions using quantum Reed-Muller codes arxiv: v1, (2012). 39 / 58

40 UQC using perfect Cliffords + noisy U υ gates E ( ψuυ,ε)(ρ) (1 ε)u υ ρu υ + ε I p (ε depolarizing error rate) For what noise rates, ε, does E ( ψuυ,ε)+ Cliffords enable UQC? + E ( ψuυ, ε) (1 ε) ψ Uυ ψ Uυ + ε I p 40 / 58

41 UQC using perfect Cliffords + noisy U υ gates E ( ψuυ,ε)(ρ) (1 ε)u υ ρu υ + ε I p (ε depolarizing error rate) For what noise rates, ε, does E ( ψuυ,ε)+ Cliffords enable UQC? + E ( ψuυ, ε) (1 ε) ψ Uυ ψ Uυ + ε I p Postselection dilutes the noise: ε < ε. + 0 E ( ψuυ, ε) Π 0 (1 ε ) ψ Uυ ψ Uυ + ε I p Exact relationship: ε ε = p (p 1)ε Use this result, along with routines in [CAB 12], to find allowable ε 41 / 58

42 Noise thresholds for UQC Lower Bound Upper Bound p = % 45.32% p = % 78.63% p = % 95.20% p = % 97.63% Table: Bounds on threshold ε for UQC using noisy U υ + ideal Cliffords How are these values found? Lower bound: Postselction circuit & the best performing MSD routines given in [CAB 12]. Upper bound: ( [ ]) Explicit facets of CLIFF for which Tr W (1 ε) J U J U + ε I = 0 p 2 Note: U υ also maximally robust to phase damping noise (p = 2, 3, 5, 7,?) Open Question: Can the gaps be closed? 42 / 58

43 Applications? U1 π/8 ( e i π 8 0 U π/8 = 0 e i π 8 ) ( e i π 4 ) UQC= Cliffords, U π/8 Teleportation-based UQC Transversal for some Reed-Muller codes Secure assisted UQC Measurement-based UQC with graph states Optimal CHSH game with ( )/ 2 Blind UQC 43 / 58

44 CHSH Bell Inequality A 0 B 0 + A 0 B 1 + A 1 B 0 A 1 B 1 2 (λ(a j ), λ(b k ) = ±1) + 0 A A 1 2 B 0 B 1 44 / 58

45 CHSH Bell Inequality A 0 B 0 + A 0 B 1 + A 1 B 0 A 1 B 1 2 (λ(a j ), λ(b k ) = ±1) + 0 A 0 A 1 B 0 B 1 A 0 = X A 1 = Y B 0 = (X Y )/ 2 B 1 = (X + Y )/ / 58

46 CHSH Bell Inequality A 0 B 0 + A 0 B 1 + A 1 B 0 A 1 B 1 2 (λ(a j ), λ(b k ) = ±1) + 0 U π/8 A 0 A 1 B 0 B 1 A 0 = X A 1 = Y B 0 = X B 1 = Y / 58

47 CHSH Bell Inequality A 0 B 0 + A 0 B 1 + A 1 B 0 A 1 B 1 2 (λ(a j ), λ(b k ) = ±1) + 0 U π/8 A 0 A 1 J U π/8 B 0 B 1 A 0 = X A 1 = Y B 0 = X B 1 = Y / 58

48 CHSH Bell Inequality A 0 B 0 + A 0 B 1 + A 1 B 0 A 1 B 1 2 (λ(a j ), λ(b k ) = ±1) + 0 U π/8 A 0 A 1 B 0 B 1 A 0 = X A 1 = Y B 0 = X B 1 = Y B = Maximizing quantity B is related to maximizing p(a, b s, t) a+b=st mod 2 (a,b,s,t Z 2 ) settings where p(a, b s, t) is a conditional prob. outcomes 48 / 58

49 CHSH Bell Inequality B U π/8 B = ( 1) jk A j B k (λ(a j ), λ(b k ) = ±1) j,k Z 2 A 0 = X A 0 A 1 B 0 B 1 A 1 = Y B 0 = X B 1 = Y B = / 58

50 CHSH Bell Inequality + B 2 B = 0 U π/8 j,k Z 2 (e 2πi 2 ) jk A j B k (λ(a j ), λ(b k ) = ±1) A 0 A 1 B 0 B 1 A 0 = X A 1 = Y B 0 = X B 1 = Y B = / 58

51 CHSH Bell Inequality B U π/8 B = ω jk A j B k (λ(a j ), λ(b k ) = ±1) j,k Z 2 A 0 = X A 0 A 1 B 0 B 1 A 1 = Y B 0 = X B 1 = Y B = / 58

52 Generalized CHSH Bell Inequality for p = 3 B 4.5 B = n Z 3 j,k Z 3 ω njk A n j B n k (λ(a j ), λ(b k ) = {ω 0, ω 1, ω 2 }) 52 / 58

53 Generalized CHSH Bell Inequality for p = 3 B B = n Z 3 j,k Z 3 ω njk A n j B n k (λ(a j ), λ(b k ) = {ω 0, ω 1, ω 2 }) A 0 A 1 A 2 B 0 A j = ω j(j+1) XZ j B k = ω k(k+2) XZ 2k 0 U υ B 1 B 2 B = / 58

54 Generalized CHSH Bell Inequality for p = 3 B B = n Z 3 j,k Z 3 ω njk A n j B n k (λ(a j ), λ(b k ) = {ω 0, ω 1, ω 2 }) A 0 A 1 A 2 B 0 A j = ω j(j+1) XZ j B k = ω k(k+2) XZ 2k 0 U υ B 1 B 2 B = Maximizing quantity B is related to maximizing p(a, b s, t) a+b+st=0 mod 3 (a,b,s,t Z 3 ) settings where p(a, b s, t) is a conditional prob. outcomes 54 / 58

55 Generalized CHSH Bell Inequalities Originates with Ji et al., Reexamined by Liang et al. Se-Wan Ji, Jinhyoung Lee, James Lim, Koji Nagata, and Hai-Woong Lee Multisetting Bell inequality for qudits, Phys. Rev. A 78, (2008). Yeong-Cherng Liang, Chu-Wee Lim, Dong-Ling Deng Reexamination of a multisetting Bell inequality for qudits, Phys. Rev. A 80, (2009). Observation 1: For all the cases (p 17) that we have checked, U υ are optimal in the sense: Tr ( J Uυ J Uυ B) = λ max (B) 55 / 58

56 Generalized CHSH Bell Inequalities Originates with Ji et al., Reexamined by Liang et al. Se-Wan Ji, Jinhyoung Lee, James Lim, Koji Nagata, and Hai-Woong Lee Multisetting Bell inequality for qudits, Phys. Rev. A 78, (2008). Yeong-Cherng Liang, Chu-Wee Lim, Dong-Ling Deng Reexamination of a multisetting Bell inequality for qudits, Phys. Rev. A 80, (2009). Preliminaries Observation 1: For all the cases (p 17) that we have checked, U υ are optimal in the sense: Key References VI Tr ( J Uυ J Uυ B) = λ max (B) Observation 2: Suggests a natural generalization to p 3 of the qubit result relating capability of operation E + stabilizer ops Violation of CHSH inequality Universal QC (via MSD) M. Howard and J. Vala, Nonlocality as a benchmark for universal quantum computation in Ising anyon topological quantum computers, Phys. Rev. A 85, (2012). 56 / 58

57 Open Questions & Thanks Are qudits better in any way? How does one fairly compare qubits and qudits? Physical system that enables topologically protected qudit Cliffords? (cf. Ising anyons for p = 2) What does the rest of C 3 look like? What does the diagonal subset of C k look like? Can we close the gap between upper and lower bounds on noise thresholds? Are there applications in MUBs, SICs, Unitary designs? Can stronger statements be made relating nonlocality and UQC in the Clifford computer/magic state model of computation? Thanks to Earl Campbell for many helpful discussions & comments 57 / 58

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