Mathematics of efficient quantum compilers Adam Sawicki Center of - PowerPoint PPT Presentation

Mathematics of efficient quantum compilers Adam Sawicki Center of Theoretical Physics PAS, Warsaw, Poland

1. UNIVERSAL GATES 2. EFFICIENT UNIVERSAL GATES

Quantum circuit ◮ Quantum system consisting of n qubits: H = C 2 ⊗ . . . ⊗ C 2 ◮ Quantum Gates are unitary matrices from SU ( H ) ≃ SU ( 2 n ) U † U = I = UU † and det U = 1 ◮ 1 -qubit gates are unitary matrices belonging to SU ( 2 ) ⊂ SU ( 2 n ) � α β � | α | 2 + | β | 2 = 1 U = − β α ◮ k -qubit gate is are matrices from SU ( 2 k ) ⊂ SU ( 2 n )

Universal quantum gates ◮ S = { U 1 , . . . , U k } ⊂ SU ( H ) - a finite set of quantum gates ◮ S n = { U a 1 U a 2 · · · U a n : a i ∈ { 1 , . . . , k }} - the set of all words of the length n . ◮ S is universal or generates SU ( H ) , iff the set: ∞ � < S > := S n , n = 1 is dense in SU ( H ) . ◮ To check if < S > is dense in SU ( H ) we need a measure of distance: � Tr ( U − V )( U † − V † ) � U − V � = ◮ S is universal iff for every U ∈ SU ( H ) and ǫ > 0 there is n ∈ N such that for some w ∈ S n � U − w � < ǫ

Universal gates

Universal gates

Universal gates

Universal gates and ǫ -nets ◮ X -finite subset of SU ( H ) is an ǫ -net iff for every U ∈ SU ( H ) there is U n ∈ X such that || U − U n || < ǫ ◮ S is universal iff for every ǫ > 0 there is n such that S n is ǫ -net

Universal sets for n -qubit quantum computation ◮ Quantum system consisting of n qubits: H = C 2 ⊗ . . . ⊗ C 2 ◮ Theorem A universal set for n -qubit quantum computing consists of all 1 -qubit gates ( SU ( 2 ) ) and an additional 2 -qubit gate E that does not map simple tensors onto simple tensors (entangling gate). ◮ Typically we have access to a finite set S of 1 -qubit gates ◮ Fact : If S is universal for SU ( 2 ) then S ∪ { E } is universal for SU ( 2 n ) .

Properties of universal sets ◮ Theorem (Kuranishi ’49): Let S = { U 1 , . . . , U k } ⊂ SU ( 2 ) . Universal sets of cardinality k = |S| form an open and dense set in SU ( 2 ) × k . ◮ The probability that randomly chosen set of gates is universal is equal to 1 ! ◮ Universality checking algorithm – A.S., K. Karnas, Ann. Henri Poincar, 11, vol. 18, 3515-3552, (2017), A.S., K. Karnas, Phys. Rev. A 95, 062303 (2017)

Properties of universal sets ◮ How fast can we approximate gates? ◮ S = { U 1 , . . . , U k , U − 1 1 , . . . , U − 1 k } symmetric set of qubit gates ◮ Theorem (Solovay-Kitaev): Assume S is an universal set. For every U ∈ SU ( 2 ) , ǫ > 0 and � 1 � n > A log 3 ǫ there is U n ∈ S n such that � U − U n � < ǫ , where A depends on S . ◮ All universal sets are rather efficient.

◮ EFFICIENT UNIVERSAL GATES

Properties of universal sets ◮ S = { U 1 , . . . , U k , U − 1 1 , . . . , U − 1 k } symmetric set of qubit gates ◮ Theorem (Solovay-Kitaev): Assume S is an universal set. For every U ∈ SU ( 2 ) , ǫ > 0 and � 1 � n > A log 3 ǫ there is U n ∈ S n such that � U − U n � < ǫ , where A depends on S . ◮ All universal sets are roughly the same efficient. ◮ How A changes with S ? ◮ Is 3 in log 3 � 1 � optimal? ǫ

Properties of universal sets ◮ V B ǫ - the volume (wrt to the normalised Haar measure) of an ǫ -ball (wrt to H-S norm) in SU ( 2 ) a 1 ǫ 3 ≤ V ( B ǫ ) ≤ a 2 ǫ 3 ◮ The best case: < S > is free - |S n | = |S| ( |S| − 1 ) n − 1 |S n | a 2 ǫ 3 > 1 ⇓ � 1 � − log ( |S| a 2 ) 3 n > log ( |S| − 1 ) log log ( |S| − 1 ) + 1 ǫ ◮ c can’t be smaller than 1 .

Averaging operators ◮ T SU ( 2 ) : L 2 ( SU ( 2 )) → L 2 ( SU ( 2 )) � T SU ( 2 ) f ( h ) = fd µ SU ( 2 ) ◮ S = { U 1 , . . . , U k , U − 1 1 , . . . , U − 1 k } ⊂ SU ( 2 ) ◮ T S : L 2 ( SU ( 2 )) → L 2 ( SU ( 2 )) � k k � T S f ( h ) = 1 � � f ( U − 1 f ( U i h ) + h ) i |S| i = 1 i = 1 ◮ Powers T n S give averages over words of length n , S n ◮ Quantify efficiency of a universal set S by looking how fast T n S → T SU ( 2 )

Averaging operators � k k � T S f ( h ) = 1 � � f ( U − 1 f ( U i h ) + h ) i |S| i = 1 i = 1 ◮ || T S || = sup || f || = 1 || T S f || ◮ T S - bounded sefladjoint operator; a constant function is the eigenfunction with the eigenvalue 1 , � T S � = 1 hence the spectrum is in [ − 1 , 1 ] . ◮ Consider T S | L 2 0 ( SU ( 2 )) . If � T S | L 2 0 ( SU ( 2 )) � = λ 1 < 1 then and we have a spectral gap gap ( S ) = 1 − � T S | L 2 0 ( SU ( 2 )) �

Spectral gap S − T SU ( 2 ) � = � ( T S − T SU ( 2 ) ) n � = � T S − T SU ( 2 ) � n = � T n 0 ( SU ( 2 )) � n = ( 1 − gap ( S )) n ≤ e − n · gap ( S ) = � T S | L 2 ◮ The speed of convergence T n S → T SU ( 2 ) is determined by gap ( S ) ◮ (Bourgain, Gamburd ’11) Assume S is universal and matrices from S have algebraic entries. Then gap ( S ) > 0 . 2 √ |S|− 1 ◮ (Kesten ’59) gap ( S ) ≤ 1 − . |S| ◮ Conjecture (Sarnak): For any universal set T S has a spectral gap.

Spectral gap and efficient gates ◮ Theorem (Harrow et. al. ’02) Assume S is universal and T S has a spectral gap. For every U ∈ SU ( 2 ) , ǫ > 0 and � 1 � n > A log + B ǫ there is U n ∈ S n such that � U − U n � < ǫ , where B = log ( 8 / a 1 ) + 0 . 5 log ( 3 ) 3 A = log ( 1 / ( 1 − gap ( S ))) , log ( 1 / ( 1 − gap ( S ))) ◮ (Lubotzky, Phillips, Sarnak ’84): Using quaternion algebras constructed SU ( 2 ) -gates with the optimal spectral gap for |S| + 1 = p , where p = 1 mod 4 . ◮ Main challenge Construction of many qubit gates with the optimal spectral gap.

Spectral gap and efficient gates ◮ Calculation of gap ( S ) is in general a hard problem. ◮ Peter-Weyl theorem: L 2 ( SU ( 2 )) decomposes under the left regular representation as a direct sum of all irreducible representations of SU ( 2 ) ◮ SU ( 2 ) irreps are indexed by one nonnegative integer m . The dimension of m -irrep of S ( 2 ) is m + 1 . ◮ The restriction of T S to m -irrep ρ m : SU ( 2 ) → U ( m + 1 ) is the m + 1 × m + 1 matrix: k T S , m := 1 � ρ m ( U i ) + ρ m ( U i ) − 1 � � . 2 k i = 1 ◮ The spectral gap of T S , m is gap m ( S ) = 1 − � T S , m � op . ◮ The spectral gap of T S at the resolution r by gap ≤ r ( S ) = 0 < m ≤ r gap m ( S ) . inf

Spectral gap and efficient gates gap ( S ) = inf r gap ≤ r ( S ) ◮ (P . Varju ’13) Assume that S = { U 1 , . . . , U k } is universal. Then for every U ∈ SU ( 2 ) , ǫ > 0 and � 1 � n > A log , ǫ there is U n ∈ W n ( S ) such that � U − U n � < ǫ , where a A = gap ≤ b ǫ − c ( S ) , and a , b , c are some positive consts determined by SU ( 2 ) .

Spectral gap and efficient gates � 1 � a n > A log , A = ǫ gap ≤ b ǫ − c ( S ) ◮ Relation between the efficiency of ǫ -approxiamation of U ∈ SU ( 2 ) and gap ≤ r ( S ) at the scale r = b ǫ − c ◮ gap ≤ r ( S ) - can be calculated in finite time ◮ To do : Establishing values of a , b , c ◮ Question : How the spectral gap at resolution r is distributed for randomly chosen universal sets of the fixed cardinality 2 k ?

Spectral gap and efficient gates 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 0 0.05 0.1 0.15 0.2 0.25 Figure : (The distribution of gap 100 ( S ) made for a sample of 10 4 randomly chosen sets S = { U 1 , U 2 , U − 1 1 , U − 1 2 } . The optimal spectral gap has value √ 3 1 − 2

Spectral gap and efficient gates 0.03 0.025 0.02 0.015 0.01 0.005 mal 0 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 Figure : (a) The distribution of log ( gap ≤ 100 ( S )) made for a sample of 10 4 randomly chosen sets S = { U 1 , U 2 , U − 1 1 , U − 1 2 } . The optimal spectral gap has √ 3 value 1 − 2

Examples of gates with the optimal gap ( S ) � � � � � � 1 1 2 i 1 1 2 1 1 − 2 i 0 √ √ √ V 1 = V 2 = V 3 = 2 i 1 − 2 1 0 1 − 2 i 5 5 5 � exp � − i π � � � � i 1 1 0 H = √ , T = 8 . � i π � − 1 1 0 exp 2 8

Open problems ◮ Understand how the distributions of log ( gap ≤ r ( S )) and gap r ( S ) change when r → ∞ ◮ Contact: a.sawicki@cft.edu.pl