Application of a Resource Theory for Magic States to Fault-Tolerant Quantum Computing

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 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 Symmetry FOR MAGIC STATES= • , group of = CNOT,H,S = √Z TO FAULT-TOLERANTC h i h i 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 EarlMark Campbell Howard Sheffield Need error-correcting code to protect our quantum computation

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 | i | i H m2 | i Adding the ability to do U promotes CliﬀordsH to Universal QC Y m3 6∈ C | i

1 0 Symmetry Symmetry = • , group of =UQC = • , group of , T 0 eiπ/4 h i 6 h i

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 ) | i √2 | i | i

APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield Adding the ability to do U promotes Cliﬀords to Universal QC 6∈ C

1 0 Symmetry Symmetry = • , group of =UQC = • , group of , T 0 eiπ/4 h i 6 h i

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 ) | i √2 | i | i It is not a Pauli eigenstate (stabilizer state)

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 ) | i √2 | i | i 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 APPLICATION OF A RESOURCE THEORY S S FOR MAGICT -type STATES magic state T states (+Cliﬀords) enable T gates | i | i H m2 TO FAULT-TOLERANT | i QUANTUM COMPUTING Adding the ability to do U promotes CliﬀordsH to Universal QC Y m3 6∈ C | i Howard & Campbell Symmetry Symmetry 1 0 arXiv:1609.07488 =Earl Campbell • , group of =UQC = • , group of , T 0 eiπ/4 h i 6 h Sheffieldi

1 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

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

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

Magic state distillation circuit MSD schematic

Magic state distillation circuit MSD schematic

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 free{ | − } −∈P 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 stab{ | − } −∈P

Resource Desiderata (ρ) 1, ( (ρ ) = 1) R ≥ R ∈ 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 stab{ | − } −∈P

Resource Desiderata ... or take log R (ρ) 1, ( (ρ stab) = 1) log (ρ) 0, R ≥ R ∈ 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− free{ | − } ρ+ ∈P

Free Stuff · Robustness quantiﬁes expensive stuﬀ.

ρ− Defn: How much free stuﬀ must be mixed in to make your expensive stuﬀ become free.

APPLICATIONGeneric Resource OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield EarlMark Campbell Howard Natural partition into easy/hard operations Resource Theory.Sheffield Expense quantiﬁer must obey certain reasonable⇒ properties to be useful!

+ Non-Stabilizer (ρ) = min 2p + 1 ρ = (p + 1)ρ pρ− + R ρ ,ρ− free{ | − } ρ+ ∈P

Stabilizer · Robustness quantiﬁes expensive stuﬀ.

ρ− 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− stab{ | − } APPLICATIONMulti-qubit QC OF A RESOURCE THEORY∈P ResourceFOR MAGIC Desiderata STATES ... or take log TO FAULT-TOLERANT R (ρ) 1, ( (ρ stab) = 1) log (ρ) 0, QUANTUMR ≥ COMPUTINGR ∈ P R ≥ (ρ1 ρ2) (ρ1) (ρ2) log (ρ ρ ) log (ρ ) + log (ρ ) R ⊗ ≤ R R R 1 ⊗ 2 ≤ R 1 R 2 Howard & Campbell ( stab(ρ)) (ρ) log ( stab(ρ)) log (ρ) arXiv:1609.07488R E ≤ R R E ≤ R Earl Campbell ... Well-behaved quantiﬁer Sheffield ⇒ Use CVX with MATLAB... Guaranteed to converge Syntax: variable x(n);minimize(norm(x,1));subject to A*x == b Only downside is n (vertices) grows rapidly 6, 60, 1080, 36720, 2423520,... { }

! 1 1 1 √2 Soln: ( T ) = x 1 = √2 : x = 0, 0, , 0, , − R | i || || √2 2 2

T Dual: min x 1 = max b y gives Witness Ax=b || || AT y 1 − || ||∞≤

EarlMark Campbell Howard To solve the geometrical problem for (ρ) rewrite as Linear Program:Sheffield R

(ρ) = min x 1 subject to Ax = b Non-Stabilizer R || ||

ρ+ where columns of A are vertices of PStab Stabilizer ·

ρ−

  I  1 1 1 1 1 1  1 h i  1  0 X  0 0 1 1 0 0  √2 | i Prob: Astab = h i  −  , b =   Y  0 0 0 0 1 1   1  h i −  √2  Z 1 1 0 0 0 0 0 APPLICATION OF A RESOURCE THEORYh i − + HT FOR MAGIC| i STATES| i

TO FAULT-TOLERANT1 | i QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield

1 Use CVX with MATLAB... Guaranteed to converge Syntax: variable x(n);minimize(norm(x,1));subject to A*x == b Only downside is n (vertices) grows rapidly 6, 60, 1080, 36720, 2423520,... { }

T Dual: min x 1 = max b y gives Witness Ax=b || || AT y 1 − || ||∞≤

EarlMark Campbell Howard To solve the geometrical problem for (ρ) rewrite as Linear Program:Sheffield R

(ρ) = min x 1 subject to Ax = b Non-Stabilizer R || ||

ρ+ where columns of A are vertices of PStab Stabilizer ·

ρ−

  I  1 1 1 1 1 1  1 h i  1  0 X  0 0 1 1 0 0  √2 | i Prob: Astab = h i  −  , b =   Y  0 0 0 0 1 1   1  h i −  √2  Z 1 1 0 0 0 0 0 h i − APPLICATION OF A RESOURCE THEORY ! √ + HT 1 1 1 2 | i | i Soln: ( T ) = x 1 = √2 : x = 0, 0, , 0, , − FOR MAGIC STATES R | i || || √2 2 2 TO FAULT-TOLERANT1 | i QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield

1 Use CVX with MATLAB... Guaranteed to converge Syntax: variable x(n);minimize(norm(x,1));subject to A*x == b Only downside is n (vertices) grows rapidly 6, 60, 1080, 36720, 2423520,... { }

EarlMark Campbell Howard To solve the geometrical problem for (ρ) rewrite as Linear Program:Sheffield R

(ρ) = min x 1 subject to Ax = b Non-Stabilizer R || ||

ρ+ where columns of A are vertices of PStab Stabilizer ·

ρ−

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield

1 EarlMark Campbell Howard To solve the geometrical problem for (ρ) rewrite as Linear Program:Sheffield R

(ρ) = min x 1 subject to Ax = b Non-Stabilizer R || ||

ρ+ where columns of A are vertices of PStab Stabilizer ·

ρ−

UseHoward CVX & Campbell with MATLAB... Guaranteed to converge Syntax:arXiv:1609.07488variable x(n);minimize(norm(x,1));subject toEarl A*x Campbell == b Sheffield Only downside is n (vertices) grows rapidly 6, 60, 1080, 36720, 2423520,... { }

1 2 Lower bounds on number of T gates and proving optimality Result: At least τ T gates are required to implement interesting non-Cliﬀord U.

3 Identify new circuit identities Result: Surprising Cliﬀord-equivalence of Magic states

EarlMark Campbell Howard Sheffield

Three immediate applications of Robustness of Magic

1 Simulation of quantum circuits Cliﬀord gates and T gates ⇒ Universal QC (shouldn’t be eﬃciently simulable) Result: Robustness gives an exponential simulation protocol with small exponent. (Not just T gates ... any third level hierarchy gate U works)

APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield 3 Identify new circuit identities Result: Surprising Cliﬀord-equivalence of Magic states

EarlMark Campbell Howard Sheffield

Three immediate applications of Robustness of Magic

2 Lower bounds on number of T gates and proving optimality Result: At least τ T gates are required to implement interesting non-Cliﬀord U. APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield EarlMark Campbell Howard Sheffield

Three immediate applications of Robustness of Magic

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield τ τ ... ( T ⊗ ) = 1.685 P R | i τ τ probability: qi = xi / xi [BSS‘16] χ( T ) 1.919 | | i | | | i⊗

Robustness has operational meaning as the classical simulation overhead ⇒

2 Adapt the eﬃcient classical simulation schemes for Cliﬀord circuits n τ τ to allow magic state inputs e.g., input= 0 ⊗ − T ⊗ | i | i Robustness gives a quasiprobability distribution over stabilizer states: ( ) X X X (ρ) = min xi ; ρ = xi (Stabilizer State) xi = 1 R x | | i i i i

Simulation takes longer to converge to desired accuracy (Chernoﬀ-Hoeﬀding) Require 2 (P x )2 ln 2 samples to get δ-close to real dist. with prob 1 δ2 i | i| −

EarlMark Campbell Howard Two most important steps toward simulation scheme: Sheffield

1 Realize that a quantum circuit with τ T gates is equivalent to a purely Cliﬀord circuit acting on τ magic states |T i T m1 | i T = S S S H m2 | i H Y m3 | i APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield P τ τ probability: qi = xi / xi [BSS‘16] χ( T ) 1.919 | | i | | | i⊗

Robustness has operational meaning as the classical simulation overhead ⇒

τ τ ... ( T ⊗ ) = 1.685 R | i Robustness gives a quasiprobability distribution over stabilizer states: ( ) X X X (ρ) = min xi ; ρ = xi (Stabilizer State) xi = 1 R x | | i i i i

Simulation takes longer to converge to desired accuracy (Chernoﬀ-Hoeﬀding) Require 2 (P x )2 ln 2 samples to get δ-close to real dist. with prob 1 δ2 i | i| −

EarlMark Campbell Howard Two most important steps toward simulation scheme: Sheffield

1 Realize that a quantum circuit with τ T gates is equivalent to a purely Clifford circuit acting on τ magic states |T i T m1 | i T = S S S 2 Adapt the efficient classical simulation schemes for Clifford circuits H m2 n τ τ to allow magic state| inputsi e.g., input= 0 ⊗ − T ⊗ H Y| im3 | i | i APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield P τ τ probability: qi = xi / xi [BSS‘16] χ( T ) 1.919 | | i | | | i⊗

Robustness has operational meaning as the classical simulation overhead ⇒

τ τ ... ( T ⊗ ) = 1.685 R | i

Simulation takes longer to converge to desired accuracy (Chernoﬀ-Hoeﬀding) Require 2 (P x )2 ln 2 samples to get δ-close to real dist. with prob 1 δ2 i | i| −

EarlMark Campbell Howard Two most important steps toward simulation scheme: Sheffield

1 Realize that a quantum circuit with τ T gates is equivalent to a purely Clifford circuit acting on τ magic states |T i T m1 | i T = S S S 2 Adapt the efficient classical simulation schemes for Clifford circuits H m2 n τ τ to allow magic state| inputsi e.g., input= 0 ⊗ − T ⊗ H Y| im3 | i | i Robustness gives a quasiprobability distribution over stabilizer states: APPLICATION OF A RESOURCE THEORY ( ) FOR MAGIC STATESX X X (ρ) = min xi ; ρ = xi (Stabilizer State) xi = 1 R x | | i TO FAULT-TOLERANTi i i QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield τ τ ... ( T ⊗ ) = 1.685 R | i [BSS‘16] χ( T τ ) 1.919τ | i⊗

Robustness has operational meaning as the classical simulation overhead ⇒

EarlMark Campbell Howard Two most important steps toward simulation scheme: Sheffield

1 Realize that a quantum circuit with τ T gates is equivalent to a purely Clifford circuit acting on τ magic states |T i T m1 | i T = S S S 2 Adapt the efficient classical simulation schemes for Clifford circuits H m2 n τ τ to allow magic state| inputsi e.g., input= 0 ⊗ − T ⊗ H YP| im3 | i probability:| i qi = xi / i xi Robustness gives a quasiprobability distribution| | | over| stabilizer states: APPLICATION OF A RESOURCE THEORY ( ) FOR MAGIC STATESX X X (ρ) = min xi ; ρ = xi (Stabilizer State) xi = 1 R x | | i TO FAULT-TOLERANTi i i QUANTUM COMPUTING Simulation takes longer to converge to desired accuracy (Chernoff-Hoeffding) Howard & 2CampbellP 2 2 Require δ2 ( i xi ) ln samples to get δ-close to real dist. with prob 1 arXiv:1609.07488| | Earl Campbell− Sheffield τ τ ... ( T ⊗ ) = 1.685 P R | i τ τ probability: qi = xi / xi [BSS‘16] χ( T ) 1.919 | | i | | | i⊗

EarlMark Campbell Howard Two most important steps toward simulation scheme: Sheffield

1 Realize that a quantum circuit with τ T gates is equivalent to a purely Clifford circuit acting on τ magic states |T i T m1 | i T = S S S 2 Adapt the efficient classical simulation schemes for Clifford circuits H m2 n τ τ to allow magic state| inputsi e.g., input= 0 ⊗ − T ⊗ H Y| im3 | i | i Robustness gives a quasiprobability distribution over stabilizer states: APPLICATION OF A RESOURCE THEORY ( ) FOR MAGIC STATESX X X (ρ) = min xi ; ρ = xi (Stabilizer State) xi = 1 R x | | i TO FAULT-TOLERANTi i i QUANTUM COMPUTING Simulation takes longer to converge to desired accuracy (Chernoff-Hoeffding) Howard & 2CampbellP 2 2 Require δ2 ( i xi ) ln samples to get δ-close to real dist. with prob 1 arXiv:1609.07488| | Earl Campbell− Robustness has operational meaning as the classical simulation overheadSheffield ⇒ P τ τ probability: qi = xi / xi [BSS‘16] χ( T ) 1.919 | | i | | | i⊗

EarlMark Campbell Howard Two most important steps toward simulation scheme: Sheffield

1 Realize that a quantum circuit with τ T gates is equivalent to a purely Clifford circuit acting on τ magic states |T i T m1 | i T = S S S 2 Adapt the efficient classical simulation schemes for Clifford circuits H m2 n τ τ τ τ to allow magic state| inputsi e.g., input= 0 ⊗ − T ⊗ ... ( T ⊗ ) = 1.685 H Y| im3 | i R | i | i Robustness gives a quasiprobability distribution over stabilizer states: APPLICATION OF A RESOURCE THEORY ( ) FOR MAGIC STATESX X X (ρ) = min xi ; ρ = xi (Stabilizer State) xi = 1 R x | | i TO FAULT-TOLERANTi i i QUANTUM COMPUTING Simulation takes longer to converge to desired accuracy (Chernoff-Hoeffding) Howard & 2CampbellP 2 2 Require δ2 ( i xi ) ln samples to get δ-close to real dist. with prob 1 arXiv:1609.07488| | Earl Campbell− Robustness has operational meaning as the classical simulation overheadSheffield ⇒ probability: q = x / P x i | i| i | i|

EarlMark Campbell Howard Two most important steps toward simulation scheme: Sheffield

1 Realize that a quantum circuit with τ T gates is equivalent to a purely Clifford circuit acting on τ magic states |T i T m1 | i T = S S S 2 Adapt the efficient classical simulation schemes for Clifford circuits H m2 n τ τ τ τ to allow magic state| inputsi e.g., input= 0 ⊗ − T ⊗ ... ( T ⊗ ) = 1.685 H Y| im3 | i R | i τ τ | i [BSS‘16] χ( T ⊗ ) 1.919 Robustness gives a quasiprobability distribution over stabilizer| states:i APPLICATION OF A RESOURCE THEORY ( ) FOR MAGIC STATESX X X (ρ) = min xi ; ρ = xi (Stabilizer State) xi = 1 R x | | i TO FAULT-TOLERANTi i i QUANTUM COMPUTING Simulation takes longer to converge to desired accuracy (Chernoff-Hoeffding) Howard & 2CampbellP 2 2 Require δ2 ( i xi ) ln samples to get δ-close to real dist. with prob 1 arXiv:1609.07488| | Earl Campbell− Robustness has operational meaning as the classical simulation overheadSheffield ⇒ EarlMark Campbell Howard Sheffield Circuit using n copies of resource state ρ has simulation cost (ρ)2n τ R Cost of simulating a circuit with T ⊗ ancilla? Submultiplicativity gives savings here:| i

( T )2τ 2τ R | i ∼ 2τ τ ( T,T ) 2 1.748 R | i ∼ 2τ τ ( T,T,T ) 3 1.701 R | i ∼ 2τ τ ( T,T,T,T ) 4 1.692 R | i ∼ 2τ τ APPLICATION OF( T,T,T,T,TA RESOURCE) 5 THEORY1.685 FOR MAGIC STATESR | i ∼ . . TO FAULT-TOLERANT . . 2τ n n τ τ QUANTUM COMPUTINGlim T ⊗ [1.457 , 1.685 ] n →∞ R | i ∼ ∈ Howard & Campbell 1+r +r +r Earl Campbellx y z Thm:arXiv:1609.07488Regularized robustness of ρ = (rx, ry, rz) lower-bounded by 2 Sheffield Idea: Compare robustness of target gate U := U + with ( T τ ) | i | i R | i⊗ Calculate and ﬁnd: ( T 2) < ( CS ) < ( T 3) R | i⊗ R | i R | i⊗ 1.747 < 2.2 . 2.219 meaning no possible scheme could implement CS using fewer than 3 T gates.

EarlMark Campbell Howard Sheffield A quantum algorithm will require a sequence of unitaries and measurements. These unitaries will not be Cliﬀord+T in general so we must compile. We must not waste hard-earned T gates

T m1 | i T S CS = = | i S ↔ T T † S S T m2 | i T Y m3 APPLICATION OF A RESOURCE THEORY| i FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield Calculate and ﬁnd: ( T 2) < ( CS ) < ( T 3) R | i⊗ R | i R | i⊗ 1.747 < 2.2 . 2.219 meaning no possible scheme could implement CS using fewer than 3 T gates.

T m1 | i T S CS = = | i S ↔ T T † S S T m2 | i T Y m3 APPLICATION OF A RESOURCE THEORY| i FOR MAGIC STATES TO FAULT-TOLERANT Idea:QUANTUMCompare COMPUTING robustness of target gate U := U + with ( T τ ) | i | i R | i⊗ Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield EarlMark Campbell Howard Sheffield A quantum algorithm will require a sequence of unitaries and measurements. These unitaries will not be Cliﬀord+T in general so we must compile. We must not waste hard-earned T gates

T m1 | i T S CS = = | i S ↔ T T † S S T m2 | i T Y m3 APPLICATION OF A RESOURCE THEORY| i FOR MAGIC STATES TO FAULT-TOLERANT Idea:QUANTUMCompare COMPUTING robustness of target gate U := U + with ( T τ ) | i | i R | i⊗ Howard & Campbell 2 3 Calculate and ﬁnd: ( T ⊗ ) < ( CS ) < ( T ⊗ ) arXiv:1609.07488 R | i R | i R | i Earl Campbell 1.747 < 2.2 . 2.219 Sheffield meaning no possible scheme could implement CS using fewer than 3 T gates. = Toﬀoli!

How do we know there is no way of doing Toﬀoli with 3 T gates (or 2 or 1)? Once again, our resource theory allows us to say

3 4 ( T ⊗ ) < ( Toﬀoli ) < ( T ⊗ ). R | i R | i R | i and so 4 T gates is the minimum possible. We have shown (non-)optimality of several important circuits

EarlMark Campbell Howard Sheffield Unitary synthesis is understood OK but ancilla-assisted is HARD

APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield How do we know there is no way of doing Toﬀoli with 3 T gates (or 2 or 1)? Once again, our resource theory allows us to say

3 4 ( T ⊗ ) < ( Toﬀoli ) < ( T ⊗ ). R | i R | i R | i and so 4 T gates is the minimum possible. We have shown (non-)optimality of several important circuits

EarlMark Campbell Howard Sheffield Unitary synthesis is understood OK but ancilla-assisted is HARD

= Toﬀoli! APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield Once again, our resource theory allows us to say

3 4 ( T ⊗ ) < ( Toﬀoli ) < ( T ⊗ ). R | i R | i R | i and so 4 T gates is the minimum possible. We have shown (non-)optimality of several important circuits

EarlMark Campbell Howard Sheffield Unitary synthesis is understood OK but ancilla-assisted is HARD

= Toﬀoli! APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES HowTO FAULT-TOLERANT do we know there is no way of doing Toﬀoli with 3 T gates (or 2 or 1)? QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield EarlMark Campbell Howard Sheffield Unitary synthesis is understood OK but ancilla-assisted is HARD

= Toffoli! APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES HowTO FAULT-TOLERANT do we know there is no way of doing Toffoli with 3 T gates (or 2 or 1)? OnceQUANTUM again, our COMPUTING resource theory allows us to say 3 4 ( T ⊗ ) < ( Toffoli ) < ( T ⊗ ). Howard & Campbell R | i R | i R | i arXiv:1609.07488 Earl Campbell and so 4 T gates is the minimum possible. Sheffield We have shown (non-)optimality of several important circuits EarlMark Campbell Howard Sheffield

+ + | i | i ¨¨ + ¨ + S | i ¨Cliﬀord¨∼ | i + + S | i | i T cost = 7 T cost = 4 APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield EarlMark Campbell Howard Sheffield

+ + | i | i + + S | i Cliﬀord∼ | i + + S | i | i T cost = 74 T cost = 4 APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT( CCZ ) = 2.555 = ( CS12,13 ) QUANTUMR | COMPUTINGi R | i

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield ( CCZ123,145 ) = 4.077 = ( CS12,13,14,15 ) R | i R | i

EarlMark Campbell Howard Sheffield

+ + | i | i + + S | i ¨¨ | i + ¨ + S | i ¨Cliﬀord¨∼ | i + + S | i | i + + S | i | i APPLICATIONT cost OF= A RESOURCE 11 THEORYT cost = 8 FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield EarlMark Campbell Howard Sheffield

+ + | i | i + + S | i | i + + S | i Cliﬀord∼ | i + + S | i | i + + S | i | i APPLICATIONT cost OF= A11 RESOURCE8 THEORYT cost = 8 FOR MAGIC STATES ( CCZ123,145 ) = 4.077 = ( CS12,13,14,15 ) TO FAULT-TOLERANTR | i R | i QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield ( T1,2,3CS12,13,23 )= 3.121 = ( T2,3CS12,13,23 ) R | i R | i

APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT QUANTUM COMPUTING

APPLICATION OF A RESOURCE THEORY ( T1,2,3CS12,13,23 )= 3.121 = ( T2,3CS12,13,23 ) FOR RMAGIC| STATES i R | i TO FAULT-TOLERANT QUANTUM COMPUTING

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield ( CCZ ) = 2.555 = ( CE ) R | i R | i

EarlMark Campbell Howard Sheffield + | i + ¨¨ | i + ¨ ¨Cliﬀord¨∼ + E | i | i + 0 | i | i T cost = 7 T cost = 4

APPLICATION OF A RESOURCE THEORY FOR MAGIC STATES TO FAULT-TOLERANT X QUANTUM COMPUTING Z E Howard & Campbell arXiv:1609.07488 Earl Campbell Y Sheffield EarlMark Campbell Howard Sheffield + + | i | i + + E | i Cliﬀord∼ | i + 0 | i | i T cost = 74 T cost = 4

APPLICATION( CCZ OF A RESOURCE) = 2.555 THEORY = ( CE ) FOR MAGICR | STATES i R | i TO FAULT-TOLERANT X QUANTUM COMPUTING Z E Howard & Campbell arXiv:1609.07488 Earl Campbell Y Sheffield Open Questions Q: Is there a scalable way of calculating a Magic Resource? Q: Is a measure that combines diﬀerent quantiﬁers possible/preferable? Q: Can we establish interconvertability results a la Entanglement? Q: Algorithms to calculate the T cost for ancilla states U ? | i

EarlMark Campbell Howard In Summary Sheffield Protecting qubits from errors incurs an overhead: While Cliﬀord gates (CNOT,H etc) are easily implementable, T gates are costly ...This suggests a resource theory picture We used Robustness of Magic as the resource quantiﬁer.

Howard & Campbell arXiv:1609.07488 Earl Campbell Sheffield EarlMark Campbell Howard In Summary Sheffield Protecting qubits from errors incurs an overhead: While Cliﬀord gates (CNOT,H etc) are easily implementable, T gates are costly ...This suggests a resource theory picture We used Robustness of Magic as the resource quantiﬁer.

1 Simulate quantum circuits with modestly exponential (in T gates) samples 2 Identify (non-)optimality of circuit synthesis (compilation)...prevent T wastage 3 Find Clifford-equivalent magic states Bonus: This norm-minimization approach (Ax = b) encompasses all similar known results (` norm, kets, Wigner polytope)...different quantifiers APPLICATION OF0 A RESOURCE THEORY OpenFOR MAGIC Questions STATES TO FAULT-TOLERANT Q: Is there a scalable way of calculating a Magic Resource? QUANTUM COMPUTING Q: Is a measure that combines different quantifiers possible/preferable? HowardQ: Can & Campbell we establish interconvertability results a la Entanglement? arXiv:1609.07488 Earl Campbell Q: Algorithms to calculate the T cost for ancilla states U ? Sheffield | i