Quantum algorithms for searching, resampling, and...
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Quantum algorithms for searching,resampling, and hidden shift problems
Maris OzolsUniversity of Waterloo
IQC
November 7, 2011
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Outline
1. Quantum algorithms for searching
2. Quantum rejection sampling
3. Boolean hidden shift problem
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Previous work
I [arXiv:1004.2721] Adiabatic condition and the quantum hitting time ofMarkov chains
Hari Krovi, Maris Ozols, Jeremie RolandI Phys. Rev. A, vol. 82(2), pp. 022333 (2010)
I [arXiv:1002.2419] Finding is as easy as detecting for quantum walks
Hari Krovi, Frederic Magniez, Maris Ozols, Jeremie RolandI Lecture Notes in Computer Science, vol. 6198, pp. 540–551 (2010)I ICALP 2010I QIP 2011 (featured talk)
I [arXiv:1009.1195] Entanglement can increase asymptotic rates ofzero-error classical communication over classical channels
Debbie Leung, Laura Mancinska, William Matthews, Maris Ozols, Aidan RoyI Communications in Mathematical Physics (submitted)I QIP 2011 (featured talk)
I [arXiv:1103.2774] Quantum rejection sampling
Maris Ozols, Martin Roetteler, Jeremie RolandI QIP 2012 (invited talk)I ITCS 2012
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Quantum algorithms for searching
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Spatial search on a graph
Setup
I Graph with vertex set X
I Marked vertices: unknown M ⊆ XI Vertex register: current position
I Edges: legal moves
The problem
I Move the robot to amarked vertex x ∈M
I Complexity: # moves
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Spatial search on a graph
Setup
I Graph with vertex set X
I Marked vertices: unknown M ⊆ XI Vertex register: current position
I Edges: legal moves
The problem
I Move the robot to amarked vertex x ∈M
I Complexity: # moves
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Search via random walk
Markov chain on the graph
Stochastic matrix P = (pxy)
I pxy 6= 0 only if (x, y) is an edge
I stationary distribution: π = πP
Algorithm
I Start from random x ∼ πI Apply P until x is marked
Hitting time
HT(P,M) = expected # steps of P to reach any x ∈M
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Search via random walk
Markov chain on the graph
Stochastic matrix P = (pxy)
I pxy 6= 0 only if (x, y) is an edge
I stationary distribution: π = πP
Algorithm
I Start from random x ∼ πI Apply P until x is marked
Hitting time
HT(P,M) = expected # steps of P to reach any x ∈M
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Search via random walk
Markov chain on the graph
Stochastic matrix P = (pxy)
I pxy 6= 0 only if (x, y) is an edge
I stationary distribution: π = πP
Algorithm
I Start from random x ∼ πI Apply P until x is marked
Hitting time
HT(P,M) = expected # steps of P to reach any x ∈M
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Classical intuition
Absorbing walk
I Turn all outgoing transitions from marked vertices intoself-loops: P =
ÄPUU PUMPMU PMM
ä⇒ P ′ =
ÄPUU PUM0 I
äI Stationary distribution: πM = “π restricted to M”
Interpolation
I P (s) = (1− s)P + sP ′
I Stationary distribution: π(s) ∼Ä(1− s)πU πM
ä
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Classical intuition
Absorbing walk
I Turn all outgoing transitions from marked vertices intoself-loops: P =
ÄPUU PUMPMU PMM
ä⇒ P ′ =
ÄPUU PUM0 I
äI Stationary distribution: πM = “π restricted to M”
Interpolation
I P (s) = (1− s)P + sP ′
I Stationary distribution: π(s) ∼Ä(1− s)πU πM
ä
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The algorithm
Adiabatic version
I Define a Hamiltonian H(s)corresponding to P (s)
I Interpolate s from 0 to 1
Circuit version
I Use Szegedy’s method to define a unitary W (P (s))
I W (P (s)) has a unique 1-eigenvector |π(s)〉I Use phase estimation to measure in the eigenbasis of W (P (s))
Algorithm
1. Prepare |π〉
2. Project onto |π(s∗)〉 = |πU 〉+|πM 〉√2
3. Measure current vertex
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The algorithm
Adiabatic version
I Define a Hamiltonian H(s)corresponding to P (s)
I Interpolate s from 0 to 1
Circuit version
I Use Szegedy’s method to define a unitary W (P (s))
I W (P (s)) has a unique 1-eigenvector |π(s)〉I Use phase estimation to measure in the eigenbasis of W (P (s))
Algorithm
1. Prepare |π〉
2. Project onto |π(s∗)〉 = |πU 〉+|πM 〉√2
3. Measure current vertex
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The algorithm
Adiabatic version
I Define a Hamiltonian H(s)corresponding to P (s)
I Interpolate s from 0 to 1
Circuit version
I Use Szegedy’s method to define a unitary W (P (s))
I W (P (s)) has a unique 1-eigenvector |π(s)〉I Use phase estimation to measure in the eigenbasis of W (P (s))
Algorithm
1. Prepare |π〉
2. Project onto |π(s∗)〉 = |πU 〉+|πM 〉√2
3. Measure current vertex
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The main result
TheoremLet P be a reversible, ergodic Markov chain on a set X andM ⊆ X be a set of marked elements. A quantum algorithm canfind an element in M within
»HT(P,M) steps
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Quantum rejection sampling
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Classical rejection sampling
Classical resampling problem
I Given: Ability to sample from distribution p
I Task: Sample from distribution s
I Note: Distributions p and s are known
, but samples are pairs(k, ξ(k)) where ξ(k) is not accessible
Pξ(k)
k A
ξ(k)accept/reject
k
Classical algorithm
I Accept k with probability γskpk
I Complexity: Θ(1/γ) where 1/γ = maxkskpk
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Classical rejection sampling
Classical resampling problem
I Given: Ability to sample from distribution p
I Task: Sample from distribution s
I Note: Distributions p and s are known
, but samples are pairs(k, ξ(k)) where ξ(k) is not accessible
Pξ(k)
k A
ξ(k)accept/reject
k
Classical algorithm
I Accept k with probability γskpk
I Complexity: Θ(1/γ) where 1/γ = maxkskpk
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Classical rejection sampling
Classical resampling problem
I Given: Ability to sample from distribution p
I Task: Sample from distribution s
I Note: Distributions p and s are known, but samples are pairs(k, ξ(k)) where ξ(k) is not accessible
Pξ(k)
k A
ξ(k)accept/reject
k
Classical algorithm
I Accept k with probability γskpk
I Complexity: Θ(1/γ) where 1/γ = maxkskpk
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Classical rejection sampling
Classical resampling problem
I Given: Ability to sample from distribution p
I Task: Sample from distribution s
I Note: Distributions p and s are known, but samples are pairs(k, ξ(k)) where ξ(k) is not accessible
Pξ(k)
k A
ξ(k)accept/reject
k
Classical algorithm
I Accept k with probability γskpk
I Complexity: Θ(1/γ) where 1/γ = maxkskpk
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Classical rejection sampling
Classical resampling problem
I Given: Ability to sample from distribution p
I Task: Sample from distribution s
I Note: Distributions p and s are known, but samples are pairs(k, ξ(k)) where ξ(k) is not accessible
Pξ(k)
k A
ξ(k)accept/reject
k
Classical algorithm
I Accept k with probability γskpk
I Complexity: Θ(1/γ) where 1/γ = maxkskpk
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Classical rejection sampling
Classical resampling problem
I Given: Ability to sample from distribution p
I Task: Sample from distribution s
I Note: Distributions p and s are known, but samples are pairs(k, ξ(k)) where ξ(k) is not accessible
Pξ(k)
k A
ξ(k)accept/reject
k
Classical algorithm
I Accept k with probability γskpk
where ∀k : γskpk≤ 1
I Complexity: Θ(1/γ) where 1/γ = maxkskpk
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Classical rejection sampling
Classical resampling problem
I Given: Ability to sample from distribution p
I Task: Sample from distribution s
I Note: Distributions p and s are known, but samples are pairs(k, ξ(k)) where ξ(k) is not accessible
Pξ(k)
k A
ξ(k)accept/reject
k
Classical algorithm
I Accept k with probability γskpk
where γ = minkpksk
I Complexity: Θ(1/γ) where 1/γ = maxkskpk
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Classical rejection sampling
Classical resampling problem
I Given: Ability to sample from distribution p
I Task: Sample from distribution s
I Note: Distributions p and s are known, but samples are pairs(k, ξ(k)) where ξ(k) is not accessible
Pξ(k)
k A
ξ(k)accept/reject
k
Classical algorithm
I Accept k with probability γskpk
where γ = minkpksk
I Complexity: Θ(1/γ) where 1/γ = maxkskpk
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Quantum rejection sampling
Quantum resampling problem
I Given: Oracle O : |0〉 7→∑nk=1 πk|ξk〉|k〉
I Task: Perform transformation
n∑k=1
πk|ξk〉|k〉 7→n∑k=1
σk|ξk〉|k〉
I Note: Amplitudes πk and σk are known, but states |ξk〉 arenot known
TheoremThe quantum query complexity of the π → σ quantum resampling
problem is Θ(1/γ) where 1/γ = maxk
∣∣∣∣σkπk∣∣∣∣
Proof idea: Algorithm is based on amplitude amplification, but thelower bound is based on a hybrid argument
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Quantum rejection sampling
Quantum resampling problem
I Given: Oracle O : |0〉 7→∑nk=1 πk|ξk〉|k〉
I Task: Perform transformation
n∑k=1
πk|ξk〉|k〉 7→n∑k=1
σk|ξk〉|k〉
I Note: Amplitudes πk and σk are known, but states |ξk〉 arenot known
TheoremThe quantum query complexity of the π → σ quantum resampling
problem is Θ(1/γ) where 1/γ = maxk
∣∣∣∣σkπk∣∣∣∣
Proof idea: Algorithm is based on amplitude amplification, but thelower bound is based on a hybrid argument
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Quantum rejection sampling
Quantum resampling problem
I Given: Oracle O : |0〉 7→∑nk=1 πk|ξk〉|k〉
I Task: Perform transformation
n∑k=1
πk|ξk〉|k〉 7→n∑k=1
σk|ξk〉|k〉
I Note: Amplitudes πk and σk are known, but states |ξk〉 arenot known
TheoremThe quantum query complexity of the π → σ quantum resampling
problem is Θ(1/γ) where 1/γ = maxk
∣∣∣∣σkπk∣∣∣∣
Proof idea: Algorithm is based on amplitude amplification, but thelower bound is based on a hybrid argument
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Quantum rejection sampling
Quantum resampling problem
I Given: Oracle O : |0〉 7→∑nk=1 πk|ξk〉|k〉
I Task: Perform transformation
n∑k=1
πk|ξk〉|k〉 7→n∑k=1
σk|ξk〉|k〉
I Note: Amplitudes πk and σk are known, but states |ξk〉 arenot known
TheoremThe quantum query complexity of the π → σ quantum resampling
problem is Θ(1/γ) where 1/γ = maxk
∣∣∣∣σkπk∣∣∣∣
Proof idea: Algorithm is based on amplitude amplification, but thelower bound is based on a hybrid argument
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Quantum rejection sampling
Quantum resampling problem
I Given: Oracle O : |0〉 7→∑nk=1 πk|ξk〉|k〉
I Task: Perform transformation
n∑k=1
πk|ξk〉|k〉 7→n∑k=1
σk|ξk〉|k〉
I Note: Amplitudes πk and σk are known, but states |ξk〉 arenot known
TheoremThe quantum query complexity of the π → σ quantum resampling
problem is Θ(1/γ) where 1/γ = maxk
∣∣∣∣σkπk∣∣∣∣
Proof idea: Algorithm is based on amplitude amplification, but thelower bound is based on a hybrid argument
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Applications
Implicit use
I synthesis of quantum states [Grover, 2000]
I linear systems of equations [Harrow, Hassidim and Lloyd 2009]
I fast amplification of QMA [Nagaj, Wocjan, Zhang, 2009]
New applications
I speed up quantum Metropolis sampling algorithm by[Temme, Osborne, Vollbrecht, Poulin, Verstraete, 2011]
I new quantum algorithm for the hidden shift problem of anyBoolean function
New applications by others
I preparing PEPS states [Schwarz, Temme, Verstraete, 2011]
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Boolean hidden shift problem
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Motivation
Hidden shift and subgroup problems
Hiddenshift
problems
Hiddensubgroupproblems
Dihedralgroup
Symmetricgroup
QQk
Legendresymbol
[van Dam et al., 2003]
��
��
?
New algorithms ??
Attacks oncryptosystems
����3
Factoring[Shor, 1994]
���:Discretelogarithm
[Shor, 1994]
XXXz Pell’sequation
[Hallgren, 2002]ZZZ~
?
Latticeproblems[Regev, 2002]?
?
Graphisomorphism
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Boolean hidden shift problem (BHSP)
Problem
I Given: Complete knowledge of f : Zn2 → Z2 and access to ablack-box oracle for fs(x) := f(x+ s)
x⇒ ⇒ fs(x)
I Determine: The hidden shift s
Delta functions are hard
I f(x) := δx,x0
I Equivalent to Grover’s search: Θ(√
2n)
0
1
0n 1nx0
x0 + s
fs(x)
s
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Boolean hidden shift problem (BHSP)
Problem
I Given: Complete knowledge of f : Zn2 → Z2 and access to ablack-box oracle for fs(x) := f(x+ s)
x⇒ ⇒ fs(x)
I Determine: The hidden shift s
Delta functions are hard
I f(x) := δx,x0
I Equivalent to Grover’s search: Θ(√
2n)
0
1
0n 1n
f(x)
x0
x0 + s
fs(x)
s
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Boolean hidden shift problem (BHSP)
Problem
I Given: Complete knowledge of f : Zn2 → Z2 and access to ablack-box oracle for fs(x) := f(x+ s)
x⇒ ⇒ fs(x)
I Determine: The hidden shift s
Delta functions are hard
I f(x) := δx,x0
I Equivalent to Grover’s search: Θ(√
2n)
0
1
0n 1nx0 x0 + s
fs(x)
s
![Page 36: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/36.jpg)
Boolean hidden shift problem (BHSP)
Problem
I Given: Complete knowledge of f : Zn2 → Z2 and access to ablack-box oracle for fs(x) := f(x+ s)
x⇒ ⇒ fs(x)
I Determine: The hidden shift s
Delta functions are hard
I f(x) := δx,x0I Equivalent to Grover’s search: Θ(
√2n)
0
1
0n 1nx0 x0 + s
fs(x)
s
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Fourier transform of Boolean functions
The ±1-function (normalized)
I F (x) := 1√2n
(−1)f(x)
Fourier transform
I F (w) := 〈w|H⊗n|F 〉
Function f is bent if ∀w : |F (w)| = 1√2n
![Page 38: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/38.jpg)
Fourier transform of Boolean functions
The ±1-function (normalized)
I F (x) := 1√2n
(−1)f(x)
Fourier transform
I F (w) := 〈w|H⊗n|F 〉
Function f is bent if ∀w : |F (w)| = 1√2n
![Page 39: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/39.jpg)
Fourier transform of Boolean functions
The ±1-function (normalized)
I F (x) := 1√2n
(−1)f(x)
Fourier transform
I F (w) := 〈w|H⊗n|F 〉
Function f is bent if ∀w : |F (w)| = 1√2n
![Page 40: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/40.jpg)
Fourier transform of Boolean functions
The ±1-function (normalized)
I F (x) := 1√2n
(−1)f(x)
Fourier transform
I F (w) := 〈w|H⊗n|F 〉
Function f is bent if ∀w : |F (w)| = 1√2n
![Page 41: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/41.jpg)
Bent functions are easy
Preparing the “phase state”
I Phase oracle Ofs : |x〉 7→ (−1)fs(x)|x〉
|0〉⊗n |Φ(s)〉H⊗n H⊗nOfs
I |Φ(s)〉 :=∑w∈Zn2 (−1)s·wF (w)|w〉
Algorithm [Rotteler’10]
I If f is bent then ∀w : |F (w)| = 1√2n
and thus
H⊗n diag(|F (w)|F (w)
)|Φ(s)〉 = |s〉
I Complexity: Θ(1)
![Page 42: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/42.jpg)
Bent functions are easy
Preparing the “phase state”
I Phase oracle Ofs : |x〉 7→ (−1)fs(x)|x〉
|0〉⊗n |Φ(s)〉H⊗n H⊗nOfs
I |Φ(s)〉 :=∑w∈Zn2 (−1)s·wF (w)|w〉
Algorithm [Rotteler’10]
I If f is bent then ∀w : |F (w)| = 1√2n
and thus
H⊗n diag(|F (w)|F (w)
)|Φ(s)〉 = |s〉
I Complexity: Θ(1)
![Page 43: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/43.jpg)
Bent functions are easy
Preparing the “phase state”
I Phase oracle Ofs : |x〉 7→ (−1)fs(x)|x〉
|0〉⊗n |Φ(s)〉H⊗n H⊗nOfs
I |Φ(s)〉 :=∑w∈Zn2 (−1)s·wF (w)|w〉
Algorithm [Rotteler’10]
I If f is bent then ∀w : |F (w)| = 1√2n
and thus
H⊗n diag(|F (w)|F (w)
)|Φ(s)〉 = |s〉
I Complexity: Θ(1)
![Page 44: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/44.jpg)
Bent functions are easy
Preparing the “phase state”
I Phase oracle Ofs : |x〉 7→ (−1)fs(x)|x〉
|0〉⊗n |Φ(s)〉H⊗n H⊗nOfs
I |Φ(s)〉 :=∑w∈Zn2 (−1)s·wF (w)|w〉
Algorithm [Rotteler’10]
I If f is bent then ∀w : |F (w)| = 1√2n
and thus
H⊗n diag(|F (w)|F (w)
)|Φ(s)〉 = |s〉
I Complexity: Θ(1)
![Page 45: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/45.jpg)
Other Boolean functions?
Known
I delta functions are hard
I bent functions are easy
ProblemWhat is the quantum query complexity of the hidden shift problemfor an arbitrary Boolean function?
Three approaches
1. Grover-like [Grover’00] / quantum rejection sampling [ORR’11]
2. Pretty good measurement
3. Simon-like [Rotteler’10, GRR’11]
![Page 46: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/46.jpg)
Other Boolean functions?
Known
I delta functions are hard
I bent functions are easy
ProblemWhat is the quantum query complexity of the hidden shift problemfor an arbitrary Boolean function?
Three approaches
1. Grover-like [Grover’00] / quantum rejection sampling [ORR’11]
2. Pretty good measurement
3. Simon-like [Rotteler’10, GRR’11]
![Page 47: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/47.jpg)
Algorithm 1: Grover-like / quantum rejection sampling
Quantum resampling
∑w∈Zn2
(−1)s·wF (w)|w〉 7→∑w∈Zn2
(−1)s·w1√2n|w〉
Complexity: O(1/γ) where 1/γ = maxwσwπw
=1√2n· 1
Fmin
Performance
I Delta functions: O(√
2n)
I Bent functions: O(1)
Issues
I What if Fmin = 0?
I Undetectable anti-shifts: f(x+ s) = f(x) + 1
![Page 48: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/48.jpg)
Algorithm 1: Grover-like / quantum rejection sampling
Quantum resampling
∑w∈Zn2
(−1)s·wF (w)|w〉 7→∑w∈Zn2
(−1)s·w1√2n|w〉
Complexity: O(1/γ) where 1/γ = maxwσwπw
=1√2n· 1
Fmin
Performance
I Delta functions: O(√
2n)
I Bent functions: O(1)
Issues
I What if Fmin = 0?
I Undetectable anti-shifts: f(x+ s) = f(x) + 1
![Page 49: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/49.jpg)
Algorithm 1: Grover-like / quantum rejection sampling
Quantum resampling
∑w∈Zn2
(−1)s·wF (w)|w〉 7→∑w∈Zn2
(−1)s·w1√2n|w〉
Complexity: O(1/γ) where 1/γ = maxwσwπw
=1√2n· 1
Fmin
Performance
I Delta functions: O(√
2n)
I Bent functions: O(1)
Issues
I What if Fmin = 0?
I Undetectable anti-shifts: f(x+ s) = f(x) + 1
![Page 50: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/50.jpg)
Algorithm 1: Grover-like / quantum rejection sampling
Quantum resampling
∑w∈Zn2
(−1)s·wF (w)|w〉 7→∑w∈Zn2
(−1)s·w1√2n|w〉
Complexity: O(1/γ) where 1/γ = maxwσwπw
=1√2n· 1
Fmin
Performance
I Delta functions: O(√
2n)
I Bent functions: O(1)
Issues
I What if Fmin = 0?
I Undetectable anti-shifts: f(x+ s) = f(x) + 1
![Page 51: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/51.jpg)
Algorithm 1: Approximate version
I Aim for approximately flat state
I Fix success probability p
I Optimal target amplitudes are given by the “water filling”vector εp such that µT · εp
‖εp‖2 ≥√p where µw = 1√
2n
I Queries: O(1/‖εp‖2)
![Page 52: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/52.jpg)
Algorithm 1: Approximate version
I Aim for approximately flat state
I Fix success probability p
I Optimal target amplitudes are given by the “water filling”vector εp such that µT · εp
‖εp‖2 ≥√p where µw = 1√
2n
I Queries: O(1/‖εp‖2)
![Page 53: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/53.jpg)
Algorithm 1: Approximate version
I Aim for approximately flat state
I Fix success probability p
I Optimal target amplitudes are given by the “water filling”vector εp such that µT · εp
‖εp‖2 ≥√p where µw = 1√
2n
I Queries: O(1/‖εp‖2)
![Page 54: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/54.jpg)
Algorithm 1: Approximate version
I Aim for approximately flat state
I Fix success probability p
I Optimal target amplitudes are given by the “water filling”vector εp such that µT · εp
‖εp‖2 ≥√p where µw = 1√
2n
I Queries: O(1/‖εp‖2)
![Page 55: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/55.jpg)
Algorithm 1: Approximate version
I Aim for approximately flat state
I Fix success probability p
I Optimal target amplitudes are given by the “water filling”vector εp such that µT · εp
‖εp‖2 ≥√p where µw = 1√
2n
I Queries: O(1/‖εp‖2)
![Page 56: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/56.jpg)
Algorithm 1: Approximate version
I Aim for approximately flat state
I Fix success probability p
I Optimal target amplitudes are given by the “water filling”vector εp such that µT · εp
‖εp‖2 ≥√p where µw = 1√
2n
I Queries: O(1/‖εp‖2)
![Page 57: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/57.jpg)
Algorithm 1: Approximate version
I Aim for approximately flat state
I Fix success probability p
I Optimal target amplitudes are given by the “water filling”vector εp such that µT · εp
‖εp‖2 ≥√p where µw = 1√
2n
I Queries: O(1/‖εp‖2)
![Page 58: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/58.jpg)
Algorithm 1: Approximate version
I Aim for approximately flat state
I Fix success probability p
I Optimal target amplitudes are given by the “water filling”vector εp such that µT · εp
‖εp‖2 ≥√p where µw = 1√
2n
I Queries: O(1/‖εp‖2)
![Page 59: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/59.jpg)
Algorithm 2: Pretty good measurement
t
1st stage 2nd stage
|0〉⊗n
|0〉⊗n
|0〉⊗n
|0〉⊗n
H⊗n H⊗n
H⊗n H⊗n
H⊗n H⊗n
H⊗n H⊗n
.
.
.
.
.
.
.
.
.
Ofs
Ofs
Ofs
Ofs
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . .. . .
After stage 1: |Φ(s)〉⊗t =Ä∑
w∈Zn2 (−1)s·wF (w)|w〉ä⊗t
After stage 2: |Φt(s)〉 :=∑w∈Zn2 (−1)s·w|F tw〉|w〉
PGM: |Ets〉 := 1√2n
∑w∈Zn2 (−1)s·w |Ftw〉
‖|Ftw〉‖2|w〉
Success probability:∣∣∣〈Ets|Φt(s)〉∣∣∣2 =
1
2n
( ∑w∈Zn2
1√2n⁄�(F ∗ F )t (w)
)2
![Page 60: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/60.jpg)
Algorithm 2: Pretty good measurement
t
1st stage 2nd stage
|0〉⊗n
|0〉⊗n
|0〉⊗n
|0〉⊗n
H⊗n H⊗n
H⊗n H⊗n
H⊗n H⊗n
H⊗n H⊗n
.
.
.
.
.
.
.
.
.
Ofs
Ofs
Ofs
Ofs
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . .. . .
After stage 1: |Φ(s)〉⊗t =Ä∑
w∈Zn2 (−1)s·wF (w)|w〉ä⊗t
After stage 2: |Φt(s)〉 :=∑w∈Zn2 (−1)s·w|F tw〉|w〉
PGM: |Ets〉 := 1√2n
∑w∈Zn2 (−1)s·w |Ftw〉
‖|Ftw〉‖2|w〉
Success probability:∣∣∣〈Ets|Φt(s)〉∣∣∣2 =
1
2n
( ∑w∈Zn2
1√2n⁄�(F ∗ F )t (w)
)2
![Page 61: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/61.jpg)
Algorithm 2: Pretty good measurement
t
1st stage 2nd stage
|0〉⊗n
|0〉⊗n
|0〉⊗n
|0〉⊗n
H⊗n H⊗n
H⊗n H⊗n
H⊗n H⊗n
H⊗n H⊗n
.
.
.
.
.
.
.
.
.
Ofs
Ofs
Ofs
Ofs
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . .. . .
After stage 1: |Φ(s)〉⊗t =Ä∑
w∈Zn2 (−1)s·wF (w)|w〉ä⊗t
After stage 2: |Φt(s)〉 :=∑w∈Zn2 (−1)s·w|F tw〉|w〉
PGM: |Ets〉 := 1√2n
∑w∈Zn2 (−1)s·w |Ftw〉
‖|Ftw〉‖2|w〉
Success probability:∣∣∣〈Ets|Φt(s)〉∣∣∣2 =
1
2n
( ∑w∈Zn2
1√2n⁄�(F ∗ F )t (w)
)2
![Page 62: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/62.jpg)
Algorithm 2: Pretty good measurement
t
1st stage 2nd stage
|0〉⊗n
|0〉⊗n
|0〉⊗n
|0〉⊗n
H⊗n H⊗n
H⊗n H⊗n
H⊗n H⊗n
H⊗n H⊗n
.
.
.
.
.
.
.
.
.
Ofs
Ofs
Ofs
Ofs
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . .. . .
After stage 1: |Φ(s)〉⊗t =Ä∑
w∈Zn2 (−1)s·wF (w)|w〉ä⊗t
After stage 2: |Φt(s)〉 :=∑w∈Zn2 (−1)s·w|F tw〉|w〉
PGM: |Ets〉 := 1√2n
∑w∈Zn2 (−1)s·w |Ftw〉
‖|Ftw〉‖2|w〉
Success probability:∣∣∣〈Ets|Φt(s)〉∣∣∣2 =
1
2n
( ∑w∈Zn2
1√2n⁄�(F ∗ F )t (w)
)2
![Page 63: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/63.jpg)
Algorithm 2: Pretty good measurement
t
1st stage 2nd stage
|0〉⊗n
|0〉⊗n
|0〉⊗n
|0〉⊗n
H⊗n H⊗n
H⊗n H⊗n
H⊗n H⊗n
H⊗n H⊗n
.
.
.
.
.
.
.
.
.
Ofs
Ofs
Ofs
Ofs
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . .. . .
After stage 1: |Φ(s)〉⊗t =Ä∑
w∈Zn2 (−1)s·wF (w)|w〉ä⊗t
After stage 2: |Φt(s)〉 :=∑w∈Zn2 (−1)s·w|F tw〉|w〉
PGM: |Ets〉 := 1√2n
∑w∈Zn2 (−1)s·w |Ftw〉
‖|Ftw〉‖2|w〉
Success probability:∣∣∣〈Ets|Φt(s)〉∣∣∣2 =
1
2n
( ∑w∈Zn2
1√2n⁄�(F ∗ F )t (w)
)2
![Page 64: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/64.jpg)
Algorithm 2: Pros / cons
Performance
I Bent functions: O(1)
I Random functions: O(1)
I No issues with undetectable anti-shifts
Issues
I Delta functions: O(2n), no speedup
Note
I For some t ≤ n all amplitudes will be non-zero!
![Page 65: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/65.jpg)
Algorithm 3: Simon-like
I Oracle Ofks : |k〉|w〉 7→ (−1)f(x+ks)|k〉|w〉
|0〉
|0〉⊗n
H H
H⊗n H⊗nOfks
k
|Ψ(s)〉 :=∑w∈Zn2
F (w)|s · w〉|w〉
I Complexity: O(n/√If )
I Where If (w) is the influence of w ∈ Zn2 on f :
If (w) := Prx
îf(x) 6= f(x+ w)
óand If := minw If (w)
![Page 66: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/66.jpg)
Algorithm 3: Simon-like
I Oracle Ofks : |k〉|w〉 7→ (−1)f(x+ks)|k〉|w〉
|0〉
|0〉⊗n
H H
H⊗n H⊗nOfks
k
|Ψ(s)〉 :=∑w∈Zn2
F (w)|s · w〉|w〉
I Complexity: O(n/√If )
I Where If (w) is the influence of w ∈ Zn2 on f :
If (w) := Prx
îf(x) 6= f(x+ w)
óand If := minw If (w)
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Algorithm 3: Simon-like
I Oracle Ofks : |k〉|w〉 7→ (−1)f(x+ks)|k〉|w〉
|0〉
|0〉⊗n
H H
H⊗n H⊗nOfks
k
|Ψ(s)〉 :=∑w∈Zn2
F (w)|s · w〉|w〉
I Complexity: O(n/√If )
I Where If (w) is the influence of w ∈ Zn2 on f :
If (w) := Prx
îf(x) 6= f(x+ w)
óand If := minw If (w)
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Summary
Comparison
delta bent random F (w) = 0 issues
Grover-like O(√
2n) O(1) O(1) yesPGM O(2n) O(1) O(1) no
Simon-like O(n√
2n) O(n) O(n) no
Conclusions
I PGM and Simon-like are suboptimal in some cases
I the Grover-like algorithm fails when lots of Fourier coefficientsare equal to zero
![Page 69: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/69.jpg)
Open problems
The main goals
I Find an optimal quantum query algorithm for solving BHSP
I Prove a matching quantum query lower bound
Intermediate problems
I Find an intermediate class of functions as a new test case
I Decision trees?
I Related problems:
I Verification of s: O(1/√If)
I Extracting parity w · s: O(1/F (w)
)
I What is the classical query complexity of this problem?
I What can we say about the time complexity?
I Generalize everything from Z2 to ZdI Applications
I Breaking cryptosystems?
![Page 70: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/70.jpg)
Open problems
The main goals
I Find an optimal quantum query algorithm for solving BHSP
I Prove a matching quantum query lower bound
Intermediate problems
I Find an intermediate class of functions as a new test case
I Decision trees?
I Related problems:
I Verification of s: O(1/√If)
I Extracting parity w · s: O(1/F (w)
)
I What is the classical query complexity of this problem?
I What can we say about the time complexity?
I Generalize everything from Z2 to ZdI Applications
I Breaking cryptosystems?
![Page 71: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/71.jpg)
Open problems
The main goals
I Find an optimal quantum query algorithm for solving BHSP
I Prove a matching quantum query lower bound
Intermediate problems
I Find an intermediate class of functions as a new test case
I Decision trees?
I Related problems:
I Verification of s: O(1/√If)
I Extracting parity w · s: O(1/F (w)
)
I What is the classical query complexity of this problem?
I What can we say about the time complexity?
I Generalize everything from Z2 to ZdI Applications
I Breaking cryptosystems?
![Page 72: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/72.jpg)
Open problems
The main goals
I Find an optimal quantum query algorithm for solving BHSP
I Prove a matching quantum query lower bound
Intermediate problems
I Find an intermediate class of functions as a new test caseI Decision trees?
I Related problems:
I Verification of s: O(1/√If)
I Extracting parity w · s: O(1/F (w)
)
I What is the classical query complexity of this problem?
I What can we say about the time complexity?
I Generalize everything from Z2 to ZdI Applications
I Breaking cryptosystems?
![Page 73: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/73.jpg)
Open problems
The main goals
I Find an optimal quantum query algorithm for solving BHSP
I Prove a matching quantum query lower bound
Intermediate problems
I Find an intermediate class of functions as a new test caseI Decision trees?
I Related problems:
I Verification of s: O(1/√If)
I Extracting parity w · s: O(1/F (w)
)I What is the classical query complexity of this problem?
I What can we say about the time complexity?
I Generalize everything from Z2 to ZdI Applications
I Breaking cryptosystems?
![Page 74: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/74.jpg)
Open problems
The main goals
I Find an optimal quantum query algorithm for solving BHSP
I Prove a matching quantum query lower bound
Intermediate problems
I Find an intermediate class of functions as a new test caseI Decision trees?
I Related problems:I Verification of s: O
(1/√If)
I Extracting parity w · s: O(1/F (w)
)I What is the classical query complexity of this problem?
I What can we say about the time complexity?
I Generalize everything from Z2 to ZdI Applications
I Breaking cryptosystems?
![Page 75: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/75.jpg)
Open problems
The main goals
I Find an optimal quantum query algorithm for solving BHSP
I Prove a matching quantum query lower bound
Intermediate problems
I Find an intermediate class of functions as a new test caseI Decision trees?
I Related problems:I Verification of s: O
(1/√If)
I Extracting parity w · s: O(1/F (w)
)
I What is the classical query complexity of this problem?
I What can we say about the time complexity?
I Generalize everything from Z2 to ZdI Applications
I Breaking cryptosystems?
![Page 76: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/76.jpg)
Open problems
The main goals
I Find an optimal quantum query algorithm for solving BHSP
I Prove a matching quantum query lower bound
Intermediate problems
I Find an intermediate class of functions as a new test caseI Decision trees?
I Related problems:I Verification of s: O
(1/√If)
I Extracting parity w · s: O(1/F (w)
)I What is the classical query complexity of this problem?
I What can we say about the time complexity?
I Generalize everything from Z2 to ZdI Applications
I Breaking cryptosystems?
![Page 77: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/77.jpg)
Open problems
The main goals
I Find an optimal quantum query algorithm for solving BHSP
I Prove a matching quantum query lower bound
Intermediate problems
I Find an intermediate class of functions as a new test caseI Decision trees?
I Related problems:I Verification of s: O
(1/√If)
I Extracting parity w · s: O(1/F (w)
)I What is the classical query complexity of this problem?
I What can we say about the time complexity?
I Generalize everything from Z2 to ZdI Applications
I Breaking cryptosystems?
![Page 78: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/78.jpg)
Open problems
The main goals
I Find an optimal quantum query algorithm for solving BHSP
I Prove a matching quantum query lower bound
Intermediate problems
I Find an intermediate class of functions as a new test caseI Decision trees?
I Related problems:I Verification of s: O
(1/√If)
I Extracting parity w · s: O(1/F (w)
)I What is the classical query complexity of this problem?
I What can we say about the time complexity?
I Generalize everything from Z2 to Zd
I Applications
I Breaking cryptosystems?
![Page 79: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/79.jpg)
Open problems
The main goals
I Find an optimal quantum query algorithm for solving BHSP
I Prove a matching quantum query lower bound
Intermediate problems
I Find an intermediate class of functions as a new test caseI Decision trees?
I Related problems:I Verification of s: O
(1/√If)
I Extracting parity w · s: O(1/F (w)
)I What is the classical query complexity of this problem?
I What can we say about the time complexity?
I Generalize everything from Z2 to ZdI Applications
I Breaking cryptosystems?
![Page 80: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/80.jpg)
Open problems
The main goals
I Find an optimal quantum query algorithm for solving BHSP
I Prove a matching quantum query lower bound
Intermediate problems
I Find an intermediate class of functions as a new test caseI Decision trees?
I Related problems:I Verification of s: O
(1/√If)
I Extracting parity w · s: O(1/F (w)
)I What is the classical query complexity of this problem?
I What can we say about the time complexity?
I Generalize everything from Z2 to ZdI Applications
I Breaking cryptosystems?
![Page 81: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/81.jpg)
...any questions?
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Algorithm 2: Pretty good measurement
Why does it work?
I States: |Φt(s)〉 :=∑w∈Zn2 (−1)s·w|F tw〉|w〉
where ‖|F tw〉‖22 =îF 2ó∗t
(w) = 1√2n⁄�(F ∗ F )t (w)
I Convolution: (F ∗ F )(w) =∑x∈Zn2 F (x)F (w − x)
![Page 83: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/83.jpg)
Algorithm 2: Pretty good measurement
Why does it work?
I States: |Φt(s)〉 :=∑w∈Zn2 (−1)s·w|F tw〉|w〉
where ‖|F tw〉‖22 =îF 2ó∗t
(w) = 1√2n⁄�(F ∗ F )t (w)
I Convolution: (F ∗ F )(w) =∑x∈Zn2 F (x)F (w − x)
![Page 84: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/84.jpg)
Algorithm 2: Pretty good measurement
Why does it work?
I States: |Φt(s)〉 :=∑w∈Zn2 (−1)s·w|F tw〉|w〉
where ‖|F tw〉‖22 =îF 2ó∗t
(w) = 1√2n⁄�(F ∗ F )t (w)
I Convolution: (F ∗ F )(w) =∑x∈Zn2 F (x)F (w − x)
![Page 85: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/85.jpg)
Algorithm 2: Pretty good measurement
Why does it work?
I States: |Φt(s)〉 :=∑w∈Zn2 (−1)s·w|F tw〉|w〉
where ‖|F tw〉‖22 =îF 2ó∗t
(w) = 1√2n⁄�(F ∗ F )t (w)
I Convolution: (F ∗ F )(w) =∑x∈Zn2 F (x)F (w − x)
(F ∗ F )(w)
![Page 86: Quantum algorithms for searching, resampling, and …home.lu.lv/~sd20008/papers/slides/Comprehensive_slides.pdf · Quantum algorithms for searching, resampling, and hidden shift problems](https://reader031.fdocuments.us/reader031/viewer/2022021816/5a7741457f8b9aa3688dcaf2/html5/thumbnails/86.jpg)
Algorithm 2: Pretty good measurement
Why does it work?
I States: |Φt(s)〉 :=∑w∈Zn2 (−1)s·w|F tw〉|w〉
where ‖|F tw〉‖22 =îF 2ó∗t
(w) = 1√2n⁄�(F ∗ F )t (w)
I Convolution: (F ∗ F )(w) =∑x∈Zn2 F (x)F (w − x)
1√2n⁄�(F ∗ F )t (w)