Magic States and Contextuality Mark Howard (NUI Galway, Ireland) Outline of Talk 1. Universal Quantum Computing via Magic States 2. Quantum Contextuality 3. How 1 and 2 are related (esp. qudits) Quantum Computing 0 j0i (Classical) Bits Qubits (Quantum) Probabilistic 0=1 α 0 + β 1 j i j i 1 j1i Qudits::: X Are naturally occurring in many physical systems Are (probably) more difficult to experimentally prepare, control & × measure than qubits X Have nice symmetries, advantageous for fault-tolerance Qubits::: X Are what people actually want to use::: Qubits vs Qudits (Relevant Later) Let's define a qudit to be a p-level quantum system (p =odd prime) Qubit: α0 0 + α1 1 p−1 p−1j i j i 8 αk = 1; αk C >Qudit: αk k j j 2 <> j i kX=0 kX=0 > :> Mixed/impure states: ρ = p p = 1 ij iih ij i i i ! X X Qubits vs Qudits (Relevant Later) Let's define a qudit to be a p-level quantum system (p =odd prime) Qubit: α0 0 + α1 1 p−1 p−1j i j i 8 αk = 1; αk C >Qudit: αk k j j 2 <> j i kX=0 kX=0 > :> Mixed/impure states: ρ = p p = 1 ij iih ij i i i ! X X Qudits::: X Are naturally occurring in many physical systems Are (probably) more difficult to experimentally prepare, control & × measure than qubits X Have nice symmetries, advantageous for fault-tolerance Qubits::: X Are what people actually want to use::: Weyl-Heisenberg/Pauli Group: n a = [a1; a2; : : : ; an]; b = [b1; b2; : : : ; bn] F Binary Vectors 2 2 D(a; b) = Xa1 Zb1 Xa2 Zb2 Xan Zbn Pauli Operators ⊗ ⊗ · · · ⊗ ~ κ n WH2n = i D(a; b) a; b F ; κ Z4 Pauli/WH Group f j 2 2 2 g The Clifford Group: y Cliff2n = g U2n gD(a; b)g WH2n ; D(a; b) WH2n 2 j 2 8 2 It turns out that gD(a; b)gy = D([a; b]F ) where F is a 2n 2n binary matrix ± g g × Cliff2n =WH2n h Sp(2n; F2) Pauli Matrices: 0 1 1 0 0 i X = ;Z = ;Y = iXZ = − : 1 0 0 1 i 0 ! − ! ! The Clifford Group: y Cliff2n = g U2n gD(a; b)g WH2n ; D(a; b) WH2n 2 j 2 8 2 It turns out that gD(a; b)gy = D([a; b]F ) where F is a 2n 2n binary matrix ± g g × Cliff2n =WH2n h Sp(2n; F2) Pauli Matrices: 0 1 1 0 0 i X = ;Z = ;Y = iXZ = − : 1 0 0 1 i 0 ! − ! ! Weyl-Heisenberg/Pauli Group: n a = [a1; a2; : : : ; an]; b = [b1; b2; : : : ; bn] F Binary Vectors 2 2 D(a; b) = Xa1 Zb1 Xa2 Zb2 Xan Zbn Pauli Operators ⊗ ⊗ · · · ⊗ ~ κ n WH2n = i D(a; b) a; b F ; κ Z4 Pauli/WH Group f j 2 2 2 g It turns out that gD(a; b)gy = D([a; b]F ) where F is a 2n 2n binary matrix ± g g × Cliff2n =WH2n h Sp(2n; F2) Pauli Matrices: 0 1 1 0 0 i X = ;Z = ;Y = iXZ = − : 1 0 0 1 i 0 ! − ! ! Weyl-Heisenberg/Pauli Group: n a = [a1; a2; : : : ; an]; b = [b1; b2; : : : ; bn] F Binary Vectors 2 2 D(a; b) = Xa1 Zb1 Xa2 Zb2 Xan Zbn Pauli Operators ⊗ ⊗ · · · ⊗ ~ κ n WH2n = i D(a; b) a; b F ; κ Z4 Pauli/WH Group f j 2 2 2 g The Clifford Group: y Cliff2n = g U2n gD(a; b)g WH2n ; D(a; b) WH2n 2 j 2 8 2 Pauli Matrices: 0 1 1 0 0 i X = ;Z = ;Y = iXZ = − : 1 0 0 1 i 0 ! − ! ! Weyl-Heisenberg/Pauli Group: n a = [a1; a2; : : : ; an]; b = [b1; b2; : : : ; bn] F Binary Vectors 2 2 D(a; b) = Xa1 Zb1 Xa2 Zb2 Xan Zbn Pauli Operators ⊗ ⊗ · · · ⊗ ~ κ n WH2n = i D(a; b) a; b F ; κ Z4 Pauli/WH Group f j 2 2 2 g The Clifford Group: y Cliff2n = g U2n gD(a; b)g WH2n ; D(a; b) WH2n 2 j 2 8 2 It turns out that gD(a; b)gy = D([a; b]F ) where F is a 2n 2n binary matrix ± g g × Cliff2n =WH2n h Sp(2n; F2) Heisenberg Representation: Describe a state by its stabilizer instead::: n sj WH2n sj = ; 1 j 2 ( ) = f 2 j j i j i ≤ ≤ g 2n n S ( [a; b]j F sj = ; 1 j 2 2 2 j j i j i ≤ ≤ Updating description after Clifford g is easy, g j i 7! j i [a; b] [a; b]Fg Upshot: 7! Large class of states & operations efficiently/poly(n) simulable 1 0 Symmetry Symmetry = = • , group of =UQC = • , group of , T 0 eiπ/4 Cliff2n h i 6 h i Classical Simulation Rough Intuition: Simulating quantum theory is hard because 2n States' description exponentially long: C • j i 2 Evolution governed by 2n 2n Unitary Matrices. • × Updating description after Clifford g is easy, g j i 7! j i [a; b] [a; b]Fg Upshot: 7! Large class of states & operations efficiently/poly(n) simulable 1 0 Symmetry Symmetry = = • , group of =UQC = • , group of , T 0 eiπ/4 Cliff2n h i 6 h i Classical Simulation Rough Intuition: Simulating quantum theory is hard because 2n States' description exponentially long: C • j i 2 Evolution governed by 2n 2n Unitary Matrices. • × Heisenberg Representation: Describe a state by its stabilizer instead::: n sj WH2n sj = ; 1 j 2 ( ) = f 2 j j i j i ≤ ≤ g 2n n S ( [a; b]j F sj = ; 1 j 2 2 2 j j i j i ≤ ≤ 1 0 Symmetry Symmetry = = • , group of =UQC = • , group of , T 0 eiπ/4 Cliff2n h i 6 h i Classical Simulation Rough Intuition: Simulating quantum theory is hard because 2n States' description exponentially long: C • j i 2 Evolution governed by 2n 2n Unitary Matrices. • × Heisenberg Representation: Describe a state by its stabilizer instead::: n sj WH2n sj = ; 1 j 2 ( ) = f 2 j j i j i ≤ ≤ g 2n n S ( [a; b]j F sj = ; 1 j 2 2 2 j j i j i ≤ ≤ Updating description after Clifford g is easy, g j i 7! j i [a; b] [a; b]Fg Upshot: 7! Large class of states & operations efficiently/poly(n) simulable 1 0 Symmetry Symmetry = • , group of =UQC = • , group of , T 0 eiπ/4 h i 6 h i Classical Simulation Rough Intuition: Simulating quantum theory is hard because 2n States' description exponentially long: C • j i 2 Evolution governed by 2n 2n Unitary Matrices. • × Heisenberg Representation: Describe a state by its stabilizer instead::: n sj WH2n sj = ; 1 j 2 ( ) = f 2 j j i j i ≤ ≤ g 2n n S ( [a; b]j F sj = ; 1 j 2 2 2 j j i j i ≤ ≤ Updating description after Clifford g is easy, g j i 7! j i [a; b] [a; b]Fg Upshot: 7! Large class of states & operations efficiently/poly(n) simulable = Cliff2n Encode 1 logical qubit in 3: α 0 + β 1 α 000 + β 111 j Li j Li j i j i . 0 0 0 0 0 0 . 0 0 0 0 1 1 2 s 3 = 2 3 j 0 0 0 1 0 1 . 6 7 6 . 7 6 0 0 0 1 1 0 7 6 7 6 7 4 5 4 5 Unitary X⊗3 exchanges 000 111 j i $ j i Has the logical effect 0 1 j Li $ j Li In general UL \Normalizer of sj in U" 2 f g We would like code s.t. UL generates U2k (Universal) h i Alas, Eastin-Knill theorem prevents this Will show how to supplement Cliff2n with additional \ T " gate 1 0 Symmetry Symmetry = • , group of =UQC = • , group of , T 0 eiπ/4 h i 6 h i Fault-Tolerant Logic Use Quantum Error-Correcting to protect a 2k-dim subspace 2n n−k C sj = ; 1 j 2 j i 2 j i j i ≤ ≤ n o Unitary X⊗3 exchanges 000 111 j i $ j i Has the logical effect 0 1 j Li $ j Li In general UL \Normalizer of sj in U" 2 f g We would like code s.t. UL generates U2k (Universal) h i Alas, Eastin-Knill theorem prevents this Will show how to supplement Cliff2n with additional \ T " gate 1 0 Symmetry Symmetry = • , group of =UQC = • , group of , T 0 eiπ/4 h i 6 h i Fault-Tolerant Logic Use Quantum Error-Correcting to protect a 2k-dim subspace 2n n−k C sj = ; 1 j 2 j i 2 j i j i ≤ ≤ n o Encode 1 logical qubit in 3: α 0 + β 1 α 000 + β 111 j Li j Li j i j i . 0 0 0 0 0 0 . 0 0 0 0 1 1 2 s 3 = 2 3 j 0 0 0 1 0 1 . 6 7 6 . 7 6 0 0 0 1 1 0 7 6 7 6 7 4 5 4 5 We would like code s.t. UL generates U2k (Universal) h i Alas, Eastin-Knill theorem prevents this Will show how to supplement Cliff2n with additional \ T " gate 1 0 Symmetry Symmetry = • , group of =UQC = • , group of , T 0 eiπ/4 h i 6 h i Fault-Tolerant Logic Use Quantum Error-Correcting to protect a 2k-dim subspace 2n n−k C sj = ; 1 j 2 j i 2 j i j i ≤ ≤ n o Encode 1 logical qubit in 3: α 0 + β 1 α 000 + β 111 j Li j Li j i j i .