QALGO workshop, Riga. 1 / 26. Quantum algorithms for linear algebra.
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1 QALGO workshop, Riga. 1 / 26 Quantum algorithms for linear algebra., Center for Quantum Technologies and Nanyang Technological University, Singapore. September 22, 2015
2 QALGO workshop, Riga. 2 / 26 Overview 1 Introduction 2 The augmented QRAM 3 Quantum Singular Value Estimation 4 Applications, Questions
3 QALGO workshop, Riga. > Introduction 3 / 26 Introduction Quantum algorithms for linear algebra: Linear systems [HHL09], preprints on quantum machine learning [LMR13...].
4 QALGO workshop, Riga. > Introduction 3 / 26 Introduction Quantum algorithms for linear algebra: Linear systems [HHL09], preprints on quantum machine learning [LMR13...]. Assumptions about data like sparsity or bounded l norm.
5 QALGO workshop, Riga. > Introduction 3 / 26 Introduction Quantum algorithms for linear algebra: Linear systems [HHL09], preprints on quantum machine learning [LMR13...]. Assumptions about data like sparsity or bounded l norm. Polynomial dependence on dimension of data for the general case.
6 QALGO workshop, Riga. > Introduction 3 / 26 Introduction Quantum algorithms for linear algebra: Linear systems [HHL09], preprints on quantum machine learning [LMR13...]. Assumptions about data like sparsity or bounded l norm. Polynomial dependence on dimension of data for the general case. Quantum algorithms that work for all data with poly-logarithmic dependence on dimension.
7 QALGO workshop, Riga. > Introduction 3 / 26 Introduction Quantum algorithms for linear algebra: Linear systems [HHL09], preprints on quantum machine learning [LMR13...]. Assumptions about data like sparsity or bounded l norm. Polynomial dependence on dimension of data for the general case. Quantum algorithms that work for all data with poly-logarithmic dependence on dimension. What linear algebra problems can we can solve with such guarantees?
8 QALGO workshop, Riga. > Introduction 4 / 26 Introduction How to design a quantum algorithm for linear algebra/machine learning?
9 QALGO workshop, Riga. > Introduction 4 / 26 Introduction How to design a quantum algorithm for linear algebra/machine learning? The encode-process-extract framework.
10 QALGO workshop, Riga. > Introduction 4 / 26 Introduction How to design a quantum algorithm for linear algebra/machine learning? The encode-process-extract framework. Encoding vectors and matrices into quantum states: the augmented QRAM.
11 QALGO workshop, Riga. > Introduction 4 / 26 Introduction How to design a quantum algorithm for linear algebra/machine learning? The encode-process-extract framework. Encoding vectors and matrices into quantum states: the augmented QRAM. Processing: Coherent operations on quantum encodings, quantum singular value estimation.
12 QALGO workshop, Riga. > Introduction 4 / 26 Introduction How to design a quantum algorithm for linear algebra/machine learning? The encode-process-extract framework. Encoding vectors and matrices into quantum states: the augmented QRAM. Processing: Coherent operations on quantum encodings, quantum singular value estimation. Extraction: Sampling quantum state to obtain classically useful information, low rank approximation [P14] and recommender systems [KP15].
13 QALGO workshop, Riga. > Introduction 5 / 26 Quantum memory models O( n) speedup for quantum search assumes that the data is stored in QRAM, i [n] φ i i QRAM i [n] φ i i x i (1)
14 QALGO workshop, Riga. > Introduction 5 / 26 Quantum memory models O( n) speedup for quantum search assumes that the data is stored in QRAM, i [n] φ i i QRAM i [n] φ i i x i (1) Storage time is discounted, pre-processing can be done at time of storage.
15 QALGO workshop, Riga. > Introduction 5 / 26 Quantum memory models O( n) speedup for quantum search assumes that the data is stored in QRAM, i [n] φ i i QRAM i [n] φ i i x i (1) Storage time is discounted, pre-processing can be done at time of storage. O(nnz(x)) overhead for pre-processing vectors is reasonable, all entries of vector must be read.
16 QALGO workshop, Riga. > Introduction 5 / 26 Quantum memory models O( n) speedup for quantum search assumes that the data is stored in QRAM, i [n] φ i i QRAM i [n] φ i i x i (1) Storage time is discounted, pre-processing can be done at time of storage. O(nnz(x)) overhead for pre-processing vectors is reasonable, all entries of vector must be read. Quantum big data model: Massive dataset stored in quantum memory, storage time discounted, find quantum speedups for linear algebra and machine learning tasks.
17 QALGO workshop, Riga. > Introduction 6 / 26 Encoding, augmented QRAM
18 QALGO workshop, Riga. > Introduction 6 / 26 Encoding, augmented QRAM Definition Vector state: Given x R n the vector state x is defined as i [n] x i i. 1 x
19 QALGO workshop, Riga. > Introduction 6 / 26 Encoding, augmented QRAM Definition Vector state: Given x R n the vector state x is defined as i [n] x i i. 1 x Vector state preparation using QRAM requires time Õ( n) in the worst case by the search lower bounds.
20 QALGO workshop, Riga. > Introduction 6 / 26 Encoding, augmented QRAM Definition Vector state: Given x R n the vector state x is defined as i [n] x i i. 1 x Vector state preparation using QRAM requires time Õ( n) in the worst case by the search lower bounds. But we can pre-process, change memory organization and add classical data structures once we have a QRAM.
21 QALGO workshop, Riga. > Introduction 6 / 26 Encoding, augmented QRAM Definition Vector state: Given x R n the vector state x is defined as i [n] x i i. 1 x Vector state preparation using QRAM requires time Õ( n) in the worst case by the search lower bounds. But we can pre-process, change memory organization and add classical data structures once we have a QRAM.
22 QALGO workshop, Riga. > Introduction 6 / 26 Encoding, augmented QRAM Definition Vector state: Given x R n the vector state x is defined as i [n] x i i. 1 x Vector state preparation using QRAM requires time Õ( n) in the worst case by the search lower bounds. But we can pre-process, change memory organization and add classical data structures once we have a QRAM. Theorem Vector state x can be prepared in time O(polylog(n)) if x R n is stored in the augmented QRAM.
23 QALGO workshop, Riga. > Introduction 7 / 26 Augmented QRAM A high level view of the augmented QRAM. Augmented RAM proposals GLM08 Augmented QRAM Data structures, memory organization. Data structures, memory organization. RAM proposals GLM08 QRAM
24 QALGO workshop, Riga. > Introduction 7 / 26 Augmented QRAM A high level view of the augmented QRAM. Augmented RAM proposals GLM08 Augmented QRAM Data structures, memory organization. Data structures, memory organization. RAM proposals GLM08 QRAM The bottleneck is the realization of the QRAM, augmentations are classical and can be implemented.
25 QALGO workshop, Riga. > Introduction 8 / 26 Quantum singular value estimation Obtain generic linear algebra algorithm with running time poly-logarithmic in matrix dimensions using augmented QRAM.
26 QALGO workshop, Riga. > Introduction 8 / 26 Quantum singular value estimation Obtain generic linear algebra algorithm with running time poly-logarithmic in matrix dimensions using augmented QRAM. Let M R m n have singular value decomposition M = i σ iu i v t i.
27 QALGO workshop, Riga. > Introduction 8 / 26 Quantum singular value estimation Obtain generic linear algebra algorithm with running time poly-logarithmic in matrix dimensions using augmented QRAM. Let M R m n have singular value decomposition M = i σ iu i v t i. Given a superposition over the singular vectors i α i v i (respectively u i ) we want to obtain i α i v i σ i where σ i σ i ± ɛ M F for all i.
28 QALGO workshop, Riga. > Introduction 8 / 26 Quantum singular value estimation Obtain generic linear algebra algorithm with running time poly-logarithmic in matrix dimensions using augmented QRAM. Let M R m n have singular value decomposition M = i σ iu i v t i. Given a superposition over the singular vectors i α i v i (respectively u i ) we want to obtain i α i v i σ i where σ i σ i ± ɛ M F for all i.
29 QALGO workshop, Riga. > Introduction 8 / 26 Quantum singular value estimation Obtain generic linear algebra algorithm with running time poly-logarithmic in matrix dimensions using augmented QRAM. Let M R m n have singular value decomposition M = i σ iu i v t i. Given a superposition over the singular vectors i α i v i (respectively u i ) we want to obtain i α i v i σ i where σ i σ i ± ɛ M F for all i. Theorem If M R m n stored in the augmented QRAM, there is an algorithm with running time O(polylog(mn)/ɛ) that performs quantum singular value estimation with probability at least 1 1/poly(n).
30 QALGO workshop, Riga. > Introduction 9 / 26 Quantum singular value estimation The self analysis approach from LMR13 requires time Õ(1/ɛ3 ), analysis of coherence is required [P14].
31 QALGO workshop, Riga. > Introduction 9 / 26 Quantum singular value estimation The self analysis approach from LMR13 requires time Õ(1/ɛ3 ), analysis of coherence is required [P14]. Reduce singular value estimation to estimating the principal angles between subspaces associated with M. [Jordan s lemma]
32 QALGO workshop, Riga. > Introduction 9 / 26 Quantum singular value estimation The self analysis approach from LMR13 requires time Õ(1/ɛ3 ), analysis of coherence is required [P14]. Reduce singular value estimation to estimating the principal angles between subspaces associated with M. [Jordan s lemma] Implement reflection in the subspaces as unitary operators using augmented QRAM.
33 QALGO workshop, Riga. > Introduction 9 / 26 Quantum singular value estimation The self analysis approach from LMR13 requires time Õ(1/ɛ3 ), analysis of coherence is required [P14]. Reduce singular value estimation to estimating the principal angles between subspaces associated with M. [Jordan s lemma] Implement reflection in the subspaces as unitary operators using augmented QRAM. Apply phase estimation to the product of the reflection operators to estimate the angles and thus the singular values.
34 QALGO workshop, Riga. > Introduction 9 / 26 Quantum singular value estimation The self analysis approach from LMR13 requires time Õ(1/ɛ3 ), analysis of coherence is required [P14]. Reduce singular value estimation to estimating the principal angles between subspaces associated with M. [Jordan s lemma] Implement reflection in the subspaces as unitary operators using augmented QRAM. Apply phase estimation to the product of the reflection operators to estimate the angles and thus the singular values. Applications: quantum projections/linear systems and low rank approximation by column selection/ recommender systems.
35 QALGO workshop, Riga. > Introduction 10 / 26 Overview Vector state preparation.
36 QALGO workshop, Riga. > Introduction 10 / 26 Overview Vector state preparation. Augmented QRAM.
37 QALGO workshop, Riga. > Introduction 10 / 26 Overview Vector state preparation. Augmented QRAM. Quantum singular value estimation.
38 QALGO workshop, Riga. > Introduction 10 / 26 Overview Vector state preparation. Augmented QRAM. Quantum singular value estimation. Applications and open questions.
39 QALGO workshop, Riga. > The augmented QRAM 11 / 26 Vector states using QRAM There is a unitary U 0 = φ where, φ = 1 n i [n] ( ) xi i 0 + β i 1 x (2)
40 QALGO workshop, Riga. > The augmented QRAM 11 / 26 Vector states using QRAM There is a unitary U 0 = φ where, φ = 1 n i [n] ( ) xi i 0 + β i 1 x (2) QRAM query, conditional rotation, post select on 0, erase 1 query register. Success probability. n x 2
41 QALGO workshop, Riga. > The augmented QRAM 11 / 26 Vector states using QRAM There is a unitary U 0 = φ where, φ = 1 n i [n] ( ) xi i 0 + β i 1 x (2) QRAM query, conditional rotation, post select on 0, erase 1 query register. Success probability. n x 2 φ = sin(θ) x, 0 + cos(θ) x, 1, reflection in S φ = US 0 U 1 and reflection in x, 0 is phase flip if ancilla is 1.
42 QALGO workshop, Riga. > The augmented QRAM 11 / 26 Vector states using QRAM There is a unitary U 0 = φ where, φ = 1 n i [n] ( ) xi i 0 + β i 1 x (2) QRAM query, conditional rotation, post select on 0, erase 1 query register. Success probability. n x 2 φ = sin(θ) x, 0 + cos(θ) x, 1, reflection in S φ = US 0 U 1 and reflection in x, 0 is phase flip if ancilla is 1. The product of reflections S φ S x is rotation by 2θ, after k iterations: ( S φ S x ) k φ = sin((2k + 1)θ) x, 0 + cos((2k + 1)θ) x, 1 (3)
43 QALGO workshop, Riga. > The augmented QRAM 12 / 26 Amplitude amplification As θ sin(θ) = 1 n x, for k = O( n x ) the success probability is a constant. x, 1 φ S φ S x φ θ 2θ x, 0 S φ S x φ S x φ
44 QALGO workshop, Riga. > The augmented QRAM 12 / 26 Amplitude amplification As θ sin(θ) = 1 n x, for k = O( n x ) the success probability is a constant. x, 1 φ S φ S x φ θ 2θ x, 0 S φ S x φ S x φ Amplitude amplification can be made exact if the success probability is known.
45 QALGO workshop, Riga. > The augmented QRAM 13 / 26 Coherent exact amplitude amplification There is a unitary operator U such that U i, 0 log n +1 = φ i = sin(θ i ) i, x i, 0 + cos(θ i ) i, x i, 1.
46 QALGO workshop, Riga. > The augmented QRAM 13 / 26 Coherent exact amplitude amplification There is a unitary operator U such that U i, 0 log n +1 = φ i = sin(θ i ) i, x i, 0 + cos(θ i ) i, x i, 1. x i R n and sin(θ i ) are stored in QRAM.
47 QALGO workshop, Riga. > The augmented QRAM 13 / 26 Coherent exact amplitude amplification There is a unitary operator U such that U i, 0 log n +1 = φ i = sin(θ i ) i, x i, 0 + cos(θ i ) i, x i, 1. x i R n and sin(θ i ) are stored in QRAM. T (U) Then i α i i, 0 l i α i i, x i requires time Õ( min i sin(θ i ) ).
48 QALGO workshop, Riga. > The augmented QRAM 13 / 26 Coherent exact amplitude amplification There is a unitary operator U such that U i, 0 log n +1 = φ i = sin(θ i ) i, x i, 0 + cos(θ i ) i, x i, 1. x i R n and sin(θ i ) are stored in QRAM. T (U) Then i α i i, 0 l i α i i, x i requires time Õ( min i sin(θ i ) ). Proof: Let t i be the exact number of rotations required, π/2 = 2t θ i + 1 choosing the largest θ i θ i. i Compute i α i i, 0 l, t i.
49 QALGO workshop, Riga. > The augmented QRAM 13 / 26 Coherent exact amplitude amplification There is a unitary operator U such that U i, 0 log n +1 = φ i = sin(θ i ) i, x i, 0 + cos(θ i ) i, x i, 1. x i R n and sin(θ i ) are stored in QRAM. T (U) Then i α i i, 0 l i α i i, x i requires time Õ( min i sin(θ i ) ). Proof: Let t i be the exact number of rotations required, π/2 = 2t θ i + 1 choosing the largest θ i θ i. i Compute i α i i, 0 l, t i. Apply unitary R( S φ S x ) where R 0 = 0, R t = t 1 if t > 1 and reflections S φ and S x are conditioned on auxiliary register being non zero.
50 QALGO workshop, Riga. > The augmented QRAM 14 / 26 Quantum key value map A quantum key value map with (K i, V i ) N for i [m] implements the following transformation in time Õ(1), α i V i (4) i [m] α i K i i [n]
51 QALGO workshop, Riga. > The augmented QRAM 14 / 26 Quantum key value map A quantum key value map with (K i, V i ) N for i [m] implements the following transformation in time Õ(1), α i V i (4) i [m] α i K i i [n] Store K i at address f(v i ), V i at address f(k i ), α i K i f(k),qram α i K i, f(k i ), V i i [n] i [n] f(k),f(v ) i [n] α i K i, f(v i ), V i QRAM,f(V ) i [n] α i f(v i ), V i (5)
52 QALGO workshop, Riga. > The augmented QRAM 14 / 26 Quantum key value map A quantum key value map with (K i, V i ) N for i [m] implements the following transformation in time Õ(1), α i V i (4) i [m] α i K i i [n] Store K i at address f(v i ), V i at address f(k i ), α i K i f(k),qram α i K i, f(k i ), V i i [n] i [n] f(k),f(v ) i [n] α i K i, f(v i ), V i QRAM,f(V ) i [n] α i f(v i ), V i (5) Collisions can be handled.
53 QALGO workshop, Riga. > The augmented QRAM 15 / 26 Sparse vector state preparation Set up a key value map X 0 + i t i between memory addresses and indices of v. v 1 0 v 3 0 v 5 0 v 7 v Sparse v R Address, content QRAM v 1 v 3 v 5 v 7 v 8 Address Index Quantum key value map Can prepare v in time Õ( nnz(v)). k 1 AA i [k] i 1 k i [k] v t i i + X 0 1 k i [k] v t i t i
54 QALGO workshop, Riga. > The augmented QRAM 16 / 26 Augmented QRAM components Vector state preparation time is Õ( n v ), this is constant if entries of v lie in [1/ n, 2/ n].
55 QALGO workshop, Riga. > The augmented QRAM 16 / 26 Augmented QRAM components Vector state preparation time is Õ( n v ), this is constant if entries of v lie in [1/ n, 2/ n]. Augmented QRAM is therefore organized into bins: v 1 v 2 v B 2 1, 8 2, 6 2, 7 3, 3 3, 4 Index Key value map 2, 1 2, 2 2, 3 2, 4 2, 5 Memory Address
56 QALGO workshop, Riga. > The augmented QRAM 16 / 26 Augmented QRAM components Vector state preparation time is Õ( n v ), this is constant if entries of v lie in [1/ n, 2/ n]. Augmented QRAM is therefore organized into bins: v 1 v 2 v B 2 1, 8 2, 6 2, 7 3, 3 3, 4 Index Key value map 2, 1 2, 2 2, 3 2, 4 2, 5 Memory Address In addition, we maintain counts and offsets as metadata in quantum memory.
57 QALGO workshop, Riga. > The augmented QRAM 17 / 26 Augmented QRAM architecture Augmented QRAM architecture: v 1, v 2, v 3 R 8 QRAM3 Metadata Controller Offsets v v v Key-value map (i, j) (k, t) QRAM2 B 1 B 2 B 3 B 4 B QRAM1
58 QALGO workshop, Riga. > The augmented QRAM 17 / 26 Augmented QRAM architecture Augmented QRAM architecture: v 1, v 2, v 3 R 8 QRAM3 Metadata Controller Offsets v v v Key-value map (i, j) (k, t) QRAM2 B 1 B 2 B 3 B 4 B QRAM1 Pre-processing can be done as the controller streams over the vector entries.
59 QALGO workshop, Riga. > The augmented QRAM 18 / 26 State preparation with augmented QRAM We describe i, 0 i, x, for query in superposition use coherent exact amplitude estimation.
60 QALGO workshop, Riga. > The augmented QRAM 18 / 26 State preparation with augmented QRAM We describe i, 0 i, x, for query in superposition use coherent exact amplitude estimation. The metadata: Counts c ik and offsets o ik for the elements of x in bin B k.
61 QALGO workshop, Riga. > The augmented QRAM 18 / 26 State preparation with augmented QRAM We describe i, 0 i, x, for query in superposition use coherent exact amplitude estimation. The metadata: Counts c ik and offsets o ik for the elements of x in bin B k. Let l(k) = 2 log(c(k)), smallest power of 2 more than c k.
62 QALGO workshop, Riga. > The augmented QRAM 18 / 26 State preparation with augmented QRAM We describe i, 0 i, x, for query in superposition use coherent exact amplitude estimation. The metadata: Counts c ik and offsets o ik for the elements of x in bin B k. Let l(k) = 2 log(c(k)), smallest power of 2 more than c k. Prepare state 1 n k, t + oik as follows, i 1 k [b] l(k) k [b] 1 i k [b] l(k) k [b] l(k) k, 0 log N t [c(k)] k, t, 0 + c(k)<t l(k) k, t, 1 (6)
63 QALGO workshop, Riga. > The augmented QRAM 19 / 26 State preparation with augmented QRAM Add ancilla and apply conditional rotation, 1 ( xj ) k, t + oik n 2 k 0 + β 1 (7)
64 QALGO workshop, Riga. > The augmented QRAM 19 / 26 State preparation with augmented QRAM Add ancilla and apply conditional rotation, 1 ( xj ) k, t + oik n 2 k 0 + β 1 (7) x j [2 k 1, 2 k ], success probability is at least 1/4.
65 QALGO workshop, Riga. > The augmented QRAM 19 / 26 State preparation with augmented QRAM Add ancilla and apply conditional rotation, 1 ( xj ) k, t + oik n 2 k 0 + β 1 (7) x j [2 k 1, 2 k ], success probability is at least 1/4. Exact amplitude amplification to get 1 n xj i, k, t + o ik.
66 QALGO workshop, Riga. > The augmented QRAM 19 / 26 State preparation with augmented QRAM Add ancilla and apply conditional rotation, 1 ( xj ) k, t + oik n 2 k 0 + β 1 (7) x j [2 k 1, 2 k ], success probability is at least 1/4. Exact amplitude amplification to get 1 n xj i, k, t + o ik. Key value map to obtain 1 n xi i, i, j = i, x.
67 QALGO workshop, Riga. > Quantum Singular Value Estimation 20 / 26 Jordan s lemma P, Q are projectors onto P, Q and P QP = λ i >0 λ iv i v t i.
68 QALGO workshop, Riga. > Quantum Singular Value Estimation 20 / 26 Jordan s lemma P, Q are projectors onto P, Q and P QP = λ i >0 λ iv i v t i. w i := Qv i / Qv i 2 is an eigenvector for QP Q with eigenvalue λ i and Span(v i, w i ) is invariant subspaces under the action of P, Q,
69 QALGO workshop, Riga. > Quantum Singular Value Estimation 20 / 26 Jordan s lemma P, Q are projectors onto P, Q and P QP = λ i >0 λ iv i v t i. w i := Qv i / Qv i 2 is an eigenvector for QP Q with eigenvalue λ i and Span(v i, w i ) is invariant subspaces under the action of P, Q, w i = Qw i QP v i σ 2 i = λ i = cos 2 (θ) QP Qw i θ P QP v i v i = P v i
70 QALGO workshop, Riga. > Quantum Singular Value Estimation 20 / 26 Jordan s lemma P, Q are projectors onto P, Q and P QP = λ i >0 λ iv i v t i. w i := Qv i / Qv i 2 is an eigenvector for QP Q with eigenvalue λ i and Span(v i, w i ) is invariant subspaces under the action of P, Q, w i = Qw i QP v i σ 2 i = λ i = cos 2 (θ) QP Qw i θ P QP v i v i = P v i Estimating singular values of P Q reduces to estimating the principal angles θ i.
71 QALGO workshop, Riga. > Quantum Singular Value Estimation 21 / 26 Singular values and principal angles If M F = 1 then M = A t B with A R mn m, B R mn n such that A t A = I m and B t B = I n.
72 QALGO workshop, Riga. > Quantum Singular Value Estimation 21 / 26 Singular values and principal angles If M F = 1 then M = A t B with A R mn m, B R mn n such that A t A = I m and B t B = I n. Proof: Select a i = i, m i and b j = p, j where p = i [m] m i i.
73 QALGO workshop, Riga. > Quantum Singular Value Estimation 21 / 26 Singular values and principal angles If M F = 1 then M = A t B with A R mn m, B R mn n such that A t A = I m and B t B = I n. Proof: Select a i = i, m i and b j = p, j where p = i [m] m i i. Orthogonality follows from definition and a i b j = m ij.
74 QALGO workshop, Riga. > Quantum Singular Value Estimation 21 / 26 Singular values and principal angles If M F = 1 then M = A t B with A R mn m, B R mn n such that A t A = I m and B t B = I n. Proof: Select a i = i, m i and b j = p, j where p = i [m] m i i. Orthogonality follows from definition and a i b j = m ij. Define projector P = AA t and Q = BB t, then P Q = AMB t is iso-spectral with M t M.
75 QALGO workshop, Riga. > Quantum Singular Value Estimation 21 / 26 Singular values and principal angles If M F = 1 then M = A t B with A R mn m, B R mn n such that A t A = I m and B t B = I n. Proof: Select a i = i, m i and b j = p, j where p = i [m] m i i. Orthogonality follows from definition and a i b j = m ij. Define projector P = AA t and Q = BB t, then P Q = AMB t is iso-spectral with M t M. It suffices to estimate the singular values of P Q, which are the principal angles between P and Q.
76 QALGO workshop, Riga. > Quantum Singular Value Estimation 22 / 26 Reflections What is the reflection in P = Col(A)?
77 QALGO workshop, Riga. > Quantum Singular Value Estimation 22 / 26 Reflections What is the reflection in P = Col(A)? Let U be the multiplication by A unitary, U x, 0 log n = Ax = i x i i, m i.
78 QALGO workshop, Riga. > Quantum Singular Value Estimation 22 / 26 Reflections What is the reflection in P = Col(A)? Let U be the multiplication by A unitary, U x, 0 log n = Ax = i x i i, m i. R A = UR 0 U 1 can be implemented as unitary using augmented QRAM.
79 QALGO workshop, Riga. > Quantum Singular Value Estimation 22 / 26 Reflections What is the reflection in P = Col(A)? Let U be the multiplication by A unitary, U x, 0 log n = Ax = i x i i, m i. R A = UR 0 U 1 can be implemented as unitary using augmented QRAM. Multiplication by B is simpler, V 0 log m, x = p, x.
80 QALGO workshop, Riga. > Quantum Singular Value Estimation 22 / 26 Reflections What is the reflection in P = Col(A)? Let U be the multiplication by A unitary, U x, 0 log n = Ax = i x i i, m i. R A = UR 0 U 1 can be implemented as unitary using augmented QRAM. Multiplication by B is simpler, V 0 log m, x = p, x. R B = V R 0 V 1 can be implemented as unitary using augmented QRAM.
81 QALGO workshop, Riga. > Quantum Singular Value Estimation 22 / 26 Reflections What is the reflection in P = Col(A)? Let U be the multiplication by A unitary, U x, 0 log n = Ax = i x i i, m i. R A = UR 0 U 1 can be implemented as unitary using augmented QRAM. Multiplication by B is simpler, V 0 log m, x = p, x. R B = V R 0 V 1 can be implemented as unitary using augmented QRAM. R A R B has eigenvalues e i2θ i, use phase estimation.
82 QALGO workshop, Riga. > Quantum Singular Value Estimation 23 / 26 Quantum singular value estimation
83 QALGO workshop, Riga. > Quantum Singular Value Estimation 23 / 26 Quantum singular value estimation Theorem If M R m n has SV D given by M = i σ iu i vi t and is stored in the augmented QRAM, there is a quantum algorithm that transforms i α i v i i α i v i σ i such that σ i σ i ± ɛ M F for all i with probability at least 1 1/poly(n) in time O(polylog(mn)/ɛ).
84 QALGO workshop, Riga. > Quantum Singular Value Estimation 23 / 26 Quantum singular value estimation Theorem If M R m n has SV D given by M = i σ iu i vi t and is stored in the augmented QRAM, there is a quantum algorithm that transforms i α i v i i α i v i σ i such that σ i σ i ± ɛ M F for all i with probability at least 1 1/poly(n) in time O(polylog(mn)/ɛ). Can be used for quantum projections.
85 QALGO workshop, Riga. > Applications, Questions 24 / 26 Applications Low rank approximation by column sampling.
86 QALGO workshop, Riga. > Applications, Questions 24 / 26 Applications Low rank approximation by column sampling. A comparison of quantum and classical CX decomposition for power law decay.
87 QALGO workshop, Riga. > Applications, Questions 25 / 26 Questions Quantum recommender systems.
88 QALGO workshop, Riga. > Applications, Questions 25 / 26 Questions Quantum recommender systems. Can quantum computing help with big data?
89 QALGO workshop, Riga. > Applications, Questions 25 / 26 Questions Quantum recommender systems. Can quantum computing help with big data? Quadratic speedups for Markov chain methods?
90 QALGO workshop, Riga. > Applications, Questions 25 / 26 Questions Quantum recommender systems. Can quantum computing help with big data? Quadratic speedups for Markov chain methods? Iterative quantum algorithms? Gradient descent? Optimization?
91 QALGO workshop, Riga. > Applications, Questions 25 / 26 Questions Quantum recommender systems. Can quantum computing help with big data? Quadratic speedups for Markov chain methods? Iterative quantum algorithms? Gradient descent? Optimization? Learning and inference on quantum max entropy/graphical models? Quantum exponential family?
92 QALGO workshop, Riga. > Applications, Questions 26 / 26 The Last Slide Thank You.
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