Overview of Quantum Algorithmic Tools András Gilyén Institute for Quantum Information and Matter Quantum Cryptanalysis of Post-Quantum Cryptography Berkeley, 22nd February 2020 Block-encodings and Quantum Singular Value Transformation One can efficiently construct block-encodings of I an efficiently implementable unitary U, I a sparse matrix with efficiently computable elements, I a matrix stored in a clever data-structure in a QRAM, I a density operator ρ given a unitary preparing its purification. I a POVM operator M given we can sample from the rand.var.: Tr(ρM), Implementing arithmetic operations on block-encoded matrices I Given block-encodings Aj we can implement convex combinations. I Given block-encodings A; B we can implement block-encoding of AB. Block-encoding A way to represent large matrices on a quantum computer efficiently " # A : U = () A = (h0ja ⊗ I)U j0ib ⊗ I : :: 1 / 15 I an efficiently implementable unitary U, I a sparse matrix with efficiently computable elements, I a matrix stored in a clever data-structure in a QRAM, I a density operator ρ given a unitary preparing its purification. I a POVM operator M given we can sample from the rand.var.: Tr(ρM), Implementing arithmetic operations on block-encoded matrices I Given block-encodings Aj we can implement convex combinations. I Given block-encodings A; B we can implement block-encoding of AB. Block-encoding A way to represent large matrices on a quantum computer efficiently " # A : U = () A = (h0ja ⊗ I)U j0ib ⊗ I : :: One can efficiently construct block-encodings of 1 / 15 I a sparse matrix with efficiently computable elements, I a matrix stored in a clever data-structure in a QRAM, I a density operator ρ given a unitary preparing its purification. I a POVM operator M given we can sample from the rand.var.: Tr(ρM), Implementing arithmetic operations on block-encoded matrices I Given block-encodings Aj we can implement convex combinations. I Given block-encodings A; B we can implement block-encoding of AB. Block-encoding A way to represent large matrices on a quantum computer efficiently " # A : U = () A = (h0ja ⊗ I)U j0ib ⊗ I : :: One can efficiently construct block-encodings of I an efficiently implementable unitary U, 1 / 15 I a matrix stored in a clever data-structure in a QRAM, I a density operator ρ given a unitary preparing its purification. I a POVM operator M given we can sample from the rand.var.: Tr(ρM), Implementing arithmetic operations on block-encoded matrices I Given block-encodings Aj we can implement convex combinations. I Given block-encodings A; B we can implement block-encoding of AB. Block-encoding A way to represent large matrices on a quantum computer efficiently " # A : U = () A = (h0ja ⊗ I)U j0ib ⊗ I : :: One can efficiently construct block-encodings of I an efficiently implementable unitary U, I a sparse matrix with efficiently computable elements, 1 / 15 I a density operator ρ given a unitary preparing its purification. I a POVM operator M given we can sample from the rand.var.: Tr(ρM), Implementing arithmetic operations on block-encoded matrices I Given block-encodings Aj we can implement convex combinations. I Given block-encodings A; B we can implement block-encoding of AB. Block-encoding A way to represent large matrices on a quantum computer efficiently " # A : U = () A = (h0ja ⊗ I)U j0ib ⊗ I : :: One can efficiently construct block-encodings of I an efficiently implementable unitary U, I a sparse matrix with efficiently computable elements, I a matrix stored in a clever data-structure in a QRAM, 1 / 15 I a POVM operator M given we can sample from the rand.var.: Tr(ρM), Implementing arithmetic operations on block-encoded matrices I Given block-encodings Aj we can implement convex combinations. I Given block-encodings A; B we can implement block-encoding of AB. Block-encoding A way to represent large matrices on a quantum computer efficiently " # A : U = () A = (h0ja ⊗ I)U j0ib ⊗ I : :: One can efficiently construct block-encodings of I an efficiently implementable unitary U, I a sparse matrix with efficiently computable elements, I a matrix stored in a clever data-structure in a QRAM, I a density operator ρ given a unitary preparing its purification. 1 / 15 Implementing arithmetic operations on block-encoded matrices I Given block-encodings Aj we can implement convex combinations. I Given block-encodings A; B we can implement block-encoding of AB. Block-encoding A way to represent large matrices on a quantum computer efficiently " # A : U = () A = (h0ja ⊗ I)U j0ib ⊗ I : :: One can efficiently construct block-encodings of I an efficiently implementable unitary U, I a sparse matrix with efficiently computable elements, I a matrix stored in a clever data-structure in a QRAM, I a density operator ρ given a unitary preparing its purification. I a POVM operator M given we can sample from the rand.var.: Tr(ρM), 1 / 15 Block-encoding A way to represent large matrices on a quantum computer efficiently " # A : U = () A = (h0ja ⊗ I)U j0ib ⊗ I : :: One can efficiently construct block-encodings of I an efficiently implementable unitary U, I a sparse matrix with efficiently computable elements, I a matrix stored in a clever data-structure in a QRAM, I a density operator ρ given a unitary preparing its purification. I a POVM operator M given we can sample from the rand.var.: Tr(ρM), Implementing arithmetic operations on block-encoded matrices I Given block-encodings Aj we can implement convex combinations. I Given block-encodings A; B we can implement block-encoding of AB. 1 / 15 Suppose that " # " # " # A : P & jw ihv j : P P(& )jw ihv j : U = = i i i i =) U = i i i i ; :: :: Φ :: d where Φ(P) 2 R is efficiently computable and UΦ is the following circuit: Alternating phase modulation sequence UΦ := H e−iφ1σz e−iφ2σz ··· e−iφd σz H 8 ··· j i⊗a < 0 : ··· U Uy ··· ··· ··· Simmilar result holds for even polynomials. Quantum Singular Value Transformation (QSVT) Main theorem about QSVT (G, Su, Low, Wiebe 2018) Let P :[−1; 1] ! [−1; 1] be a degree-d odd polynomial map. 2 / 15 " # P P(& )jw ihv j : =) U = i i i i ; Φ :: d where Φ(P) 2 R is efficiently computable and UΦ is the following circuit: Alternating phase modulation sequence UΦ := H e−iφ1σz e−iφ2σz ··· e−iφd σz H 8 ··· j i⊗a < 0 : ··· U Uy ··· ··· ··· Simmilar result holds for even polynomials. Quantum Singular Value Transformation (QSVT) Main theorem about QSVT (G, Su, Low, Wiebe 2018) Let P :[−1; 1] ! [−1; 1] be a degree-d odd polynomial map. Suppose that " # " # A : P & jw ihv j : U = = i i i i :: :: 2 / 15 d where Φ(P) 2 R is efficiently computable and UΦ is the following circuit: Alternating phase modulation sequence UΦ := H e−iφ1σz e−iφ2σz ··· e−iφd σz H 8 ··· j i⊗a < 0 : ··· U Uy ··· ··· ··· Simmilar result holds for even polynomials. Quantum Singular Value Transformation (QSVT) Main theorem about QSVT (G, Su, Low, Wiebe 2018) Let P :[−1; 1] ! [−1; 1] be a degree-d odd polynomial map. Suppose that " # " # " # A : P & jw ihv j : P P(& )jw ihv j : U = = i i i i =) U = i i i i ; :: :: Φ :: 2 / 15 Alternating phase modulation sequence UΦ := H e−iφ1σz e−iφ2σz ··· e−iφd σz H 8 ··· j i⊗a < 0 : ··· U Uy ··· ··· ··· Simmilar result holds for even polynomials. Quantum Singular Value Transformation (QSVT) Main theorem about QSVT (G, Su, Low, Wiebe 2018) Let P :[−1; 1] ! [−1; 1] be a degree-d odd polynomial map. Suppose that " # " # " # A : P & jw ihv j : P P(& )jw ihv j : U = = i i i i =) U = i i i i ; :: :: Φ :: d where Φ(P) 2 R is efficiently computable and UΦ is the following circuit: 2 / 15 Simmilar result holds for even polynomials. Quantum Singular Value Transformation (QSVT) Main theorem about QSVT (G, Su, Low, Wiebe 2018) Let P :[−1; 1] ! [−1; 1] be a degree-d odd polynomial map. Suppose that " # " # " # A : P & jw ihv j : P P(& )jw ihv j : U = = i i i i =) U = i i i i ; :: :: Φ :: d where Φ(P) 2 R is efficiently computable and UΦ is the following circuit: Alternating phase modulation sequence UΦ := H e−iφ1σz e−iφ2σz ··· e−iφd σz H 8 ··· j i⊗a < 0 : ··· U Uy ··· ··· ··· 2 / 15 Quantum Singular Value Transformation (QSVT) Main theorem about QSVT (G, Su, Low, Wiebe 2018) Let P :[−1; 1] ! [−1; 1] be a degree-d odd polynomial map. Suppose that " # " # " # A : P & jw ihv j : P P(& )jw ihv j : U = = i i i i =) U = i i i i ; :: :: Φ :: d where Φ(P) 2 R is efficiently computable and UΦ is the following circuit: Alternating phase modulation sequence UΦ := H e−iφ1σz e−iφ2σz ··· e−iφd σz H 8 ··· j i⊗a < 0 : ··· U Uy ··· ··· ··· Simmilar result holds for even polynomials. 2 / 15 Amplitude amplification and estimation Fixed-point amplitude ampl. (Yoder, Low, Chuang 2014) Amplitude amplification problem: Given U such that p ¯ p Uj0i = pj0ij goodi + 1 − pj1ij badi; prepare j goodi: 3 / 15 Amplitude amplification and estimation Fixed-point amplitude ampl. (Yoder, Low, Chuang 2014) Amplitude amplification problem: Given U such that p ¯ p Uj0i = pj0ij goodi + 1 − pj1ij badi; prepare j goodi: 2 ::::: 3 2 ::::: 3 6 7 6 7 6 good ::::: 7 6 ::::: 7 6 7 6 good 7 6 7 6 7 6 7 6 7 6 p ::::: 7 6 ::::: 7 6 p 7 =) 6 7 6 :::::: 7 6 0 ::::: 7 6 7 6 7 6 :::::: 7 6 0 ::::: 7 6 7 6 7 4 :::::: 5 4 0 ::::: 5 3 / 15 Amplitude amplification and estimation Fixed-point amplitude ampl.