EarlMark Campbell Howard Sheffield APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield Stabilizer codes gates from finite Clifford group C Symmetry = • , group of = CNOT, H, S = √Z C h i h i Encoding/correction are Clifford. Typically UL gates too. Problem: Can't get transversal & universal (dense) set of gates UL (e.g. Toric) Solution: Supplement transversal gates with supply of \Magic States" EarlMark Campbell Howard Sheffield Need error-correcting code to protect our quantum computation ψ = α 0 + β 1 ··· | i | i |0i ··· |0i | i ··· 0 Uenc UL M1 M6 Ucorr | i ··· U ψ 0 L L ··· | i |0i | i ··· 0 | i ··· ψL = α 0L + β 1L | i | i | i APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield Symmetry = • , group of = CNOT, H, S = √Z C h i h i Solution: Supplement transversal gates with supply of \Magic States" EarlMark Campbell Howard Sheffield Need error-correcting code to protect our quantum computation ψ = α 0 + β 1 ··· | i | i |0i ··· |0i | i ··· 0 Uenc UL M1 M6 Ucorr | i ··· U ψ 0 L L ··· | i |0i | i ··· 0 | i ··· ψL = α 0L + β 1L | i | i | i Stabilizer codes gates from finite Clifford group C APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUMEncoding/correction COMPUTING are Clifford. Typically UL gates too. Problem: Can't get transversal & universal (dense) set of gates UL (e.g. Toric) Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield Symmetry = • , group of = CNOT, H, S = √Z C h i h i EarlMark Campbell Howard Sheffield Need error-correcting code to protect our quantum computation ψ = α 0 + β 1 ··· | i | i |0i ··· |0i | i ··· 0 Uenc UL M1 M6 Ucorr | i ··· U ψ 0 L L ··· | i |0i | i ··· 0 | i ··· ψL = α 0L + β 1L | i | i | i Stabilizer codes gates from finite Clifford group C APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUMEncoding/correction COMPUTING are Clifford. Typically UL gates too. Problem: Can't get transversal & universal (dense) set of gates UL (e.g. Toric) Howard & Campbell arXiv:1609.07488Solution: Supplement transversal gates with supply of \Magic States"Earl Campbell Sheffield It is not a Pauli eigenstate (stabilizer state) 0 | i 1 0 T = iπ=4 0 e m + HT T 1 | i | i = | i 1 T S | i S S T -type magic state T states (+Cliffords) enable T gates j i j i H m2 | i Adding the ability to do U promotes CliffordsH to Universal QC Y m3 62 C | i 1 0 Symmetry Symmetry = • , group of =UQC = • , group of , T 0 eiπ/4 h i 6 h i 1 EarlMark Campbell Howard Sheffield Q: What's a magic state? A: A state that enables a non-Clifford gate e.g., T = 1 ( 0 + eiπ=4 1 ) j i p2 j i j i APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield 0 | i m + HT T 1 | i | i = | i 1 T S | i S S H m2 | i Adding the ability to do U promotes CliffordsH to Universal QC Y m3 62 C | i 1 0 Symmetry Symmetry = • , group of =UQC = • , group of , T 0 eiπ/4 h i 6 h i 1 EarlMark Campbell Howard Sheffield Q: What's a magic state? A: A state that enables a non-Clifford gate e.g., T = 1 ( 0 + eiπ=4 1 ) j i p2 j i j i It is not a Pauli eigenstate (stabilizer state) 1 0 T = 0 eiπ=4 APPLICATION OF A RESOURCE THEORY FOR MAGICT -type STATES magic state T states (+Cliffords) enable T gates j i j i TO FAULT-TOLERANT QUANTUM COMPUTING Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield 0 | i m + HT T 1 | i | i = | i 1 T S | i S S H m2 | i H Y m3 | i 1 0 Symmetry Symmetry = • , group of =UQC = • , group of , T 0 eiπ/4 h i 6 h i 1 EarlMark Campbell Howard Sheffield Q: What's a magic state? A: A state that enables a non-Clifford gate e.g., T = 1 ( 0 + eiπ=4 1 ) j i p2 j i j i It is not a Pauli eigenstate (stabilizer state) 1 0 T = 0 eiπ=4 APPLICATION OF A RESOURCE THEORY FOR MAGICT -type STATES magic state T states (+Cliffords) enable T gates j i j i TO FAULT-TOLERANT QUANTUM COMPUTING Adding the ability to do U promotes Cliffords to Universal QC 62 C Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield Let's try to convert Billions Millions (i.e, reduce overhead) ! Overhead associated with MSD is polynomial in the number of T gates used Logical Cliffords (2% { 10% overhead) Logical Non-Clifford e.g. T gate (90% { 98% overhead including MSD) Build quality Connect Implement individual Billions Quantum qubits of them Algorithm 1 1 Clifford Z and S = Z 2 rotations \easy" but T = Z 4 gates hard since Encoded EarlMark Campbell Howard Sheffield ρHT HT HT n ··· ≈ | ih | Input: noisy ρin⊗ ρHT 0 ··· | i ρHT 0 ··· | i ρHT M M U 0 1 ··· 6 enc† | i ρHT 0 ··· | i ρHT 0 ··· | i ρHT 0 ··· | i Output: HT HT pure ρout ≈ | Lih L| ≈ Magic state distillation circuit MSD schematic APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield Build quality Connect Implement individual Billions Quantum qubits of them Algorithm Let's try to convert Billions Millions (i.e, reduce overhead) ! 1 1 Clifford Z and S = Z 2 rotations \easy" but T = Z 4 gates hard since Encoded EarlMark Campbell Howard Sheffield ρHT HT HT n ··· ≈ | ih | Input: noisy ρin⊗ ρHT 0 ··· | i ρHT 0 ··· | i ρHT M M U 0 1 ··· 6 enc† | i ρHT 0 ··· | i ρHT 0 ··· | i ρHT 0 ··· | i Output: HT HT pure ρout ≈ | Lih L| ≈ Magic state distillation circuit MSD schematic Overhead associated with MSD is polynomial in the number of T gates used Logical Cliffords (2% { 10% overhead) APPLICATIONLogical Non-Clifford OF A RESOURCE e.g. T gate THEORY (90% { 98% overhead including MSD) FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield 1 1 Clifford Z and S = Z 2 rotations \easy" but T = Z 4 gates hard since Encoded EarlMark Campbell Howard Sheffield ρHT HT HT n ··· ≈ | ih | Input: noisy ρin⊗ ρHT 0 ··· | i ρHT 0 ··· | i ρHT M M U 0 1 ··· 6 enc† | i ρHT 0 ··· | i ρHT 0 ··· | i ρHT 0 ··· | i Output: HT HT pure ρout ≈ | Lih L| ≈ Magic state distillation circuit MSD schematic Overhead associated with MSD is polynomial in the number of T gates used Logical Cliffords (2% { 10% overhead) APPLICATIONLogical Non-Clifford OF A RESOURCE e.g. T gate THEORY (90% { 98% overhead including MSD) FOR MAGIC STATES Build quality Connect Implement TO FAULT-TOLERANTindividual Billions Quantum QUANTUM COMPUTINGqubits of them Algorithm Howard & CampbellLet's try to convert Billions Millions (i.e, reduce overhead) ! arXiv:1609.07488 Earl Campbell Sheffield EarlMark Campbell Howard Sheffield ρHT HT HT n ··· ≈ | ih | Input: noisy ρin⊗ ρHT 0 ··· | i ρHT 0 ··· | i ρHT M M U 0 1 ··· 6 enc† | i ρHT 0 ··· | i ρHT 0 ··· | i ρHT 0 ··· | i Output: HT HT pure ρout ≈ | Lih L| ≈ Magic state distillation circuit MSD schematic Overhead associated with MSD is polynomial in the number of T gates used Logical Cliffords (2% { 10% overhead) APPLICATIONLogical Non-Clifford OF A RESOURCE e.g. T gate THEORY (90% { 98% overhead including MSD) FOR MAGIC STATES Build quality Connect Implement TO FAULT-TOLERANTindividual Billions Quantum QUANTUM COMPUTINGqubits of them Algorithm Howard & CampbellLet's try to convert Billions Millions (i.e, reduce overhead) ! arXiv:1609.07488 Earl Campbell 1 1 Clifford Z and S = Z 2 rotations \easy" but T = Z 4 gates hard since EncodedSheffield ::: or take log R log (ρ) 0; R ≥ log (ρ ρ ) log (ρ ) + log (ρ ) R 1 ⊗ 2 ≤ R 1 R 2 log ( (ρ)) log (ρ) R Estab ≤ R + (ρ) = min 2p + 1 ρ = (p + 1)ρ pρ− + R ρ ,ρ freef j − g −2P Robustness quantifies expensive stuff. Defn: How much free stuff must be mixed in to make your expensive stuff become free. + (ρ) = min 2p + 1 ρ = (p + 1)ρ pρ− + R ρ ,ρ stabf j − g −2P Resource Desiderata (ρ) 1; ( (ρ ) = 1) R ≥ R 2 Pstab (ρ ρ ) (ρ ) (ρ ) R 1 ⊗ 2 ≤ R 1 R 2 ( (ρ)) (ρ) R Estab ≤ R ::: Well-behaved quantifier ) EarlMark Campbell Howard Natural partition into easy/hard operations Resource Theory.Sheffield Expense quantifier must obey certain reasonable) properties to be useful! ρ Expensive Stuff ρ+ Free Stuff · ρ− APPLICATIONGeneric Resource OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield + (ρ) = min 2p + 1 ρ = (p + 1)ρ pρ− + R ρ ,ρ stabf j − g −2P Resource Desiderata ::: or take log R (ρ) 1; ( (ρ stab) = 1) log (ρ) 0; R ≥ R 2 P R ≥ (ρ1 ρ2) (ρ1) (ρ2) log (ρ ρ ) log (ρ ) + log (ρ ) R ⊗ ≤ R R R 1 ⊗ 2 ≤ R 1 R 2 ( stab(ρ)) (ρ) log ( (ρ)) log (ρ) R E ≤ R R Estab ≤ R ::: Well-behaved quantifier ) EarlMark Campbell Howard Natural partition into easy/hard operations Resource Theory.Sheffield Expense quantifier must obey certain reasonable) properties to be useful! ρ + Expensive Stuff (ρ) = min 2p + 1 ρ = (p + 1)ρ pρ− + R ρ ,ρ− freef j − g ρ+ 2P Free Stuff · Robustness quantifies expensive stuff.