A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs Robert ˇ Spalek ∗ joint work with Andris Ambainis † and Ronald de Wolf ∗ ∗ CWI, Amsterdam † University of Waterloo Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.1/14
Quantum algorithms • Grover search: find a given number in an unsorted database of n records in time O ( √ n ) • element distinctness: find a collision x i = x j in time O ( n 2 / 3 ) Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.2/14
Quantum algorithms • Grover search: find a given number in an unsorted database of n records in time O ( √ n ) • element distinctness: find a collision x i = x j in time O ( n 2 / 3 ) Quantum query complexity • allow quantum superposition, unitary evolution, and measurements • count the number of queries, one query maps i = queried bit | i, z � → ( − 1) x i | i, z � z = the rest of memory Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.2/14
Quantum query lower bounds Adversary method • [Bennett, Bernstein, Brassard & Vazirani, 1994] tight lower bound Ω( √ n ) for Grover search known 2 years before discovering the algorithm Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.3/14
Quantum query lower bounds Adversary method • [Bennett, Bernstein, Brassard & Vazirani, 1994] tight lower bound Ω( √ n ) for Grover search known 2 years before discovering the algorithm • [Ambainis, 2000 & 2003] generalized to all functions • easy to use, gives strong bounds Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.3/14
Quantum query lower bounds Adversary method • [Bennett, Bernstein, Brassard & Vazirani, 1994] tight lower bound Ω( √ n ) for Grover search known 2 years before discovering the algorithm • [Ambainis, 2000 & 2003] generalized to all functions • easy to use, gives strong bounds Polynomial method [Beals, Buhrman, Cleve, Mosca & de Wolf, 2000] • incomparable to the adversary method • hard to use for non-symmetric functions Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.3/14
Quantum query lower bounds Adversary method • [Bennett, Bernstein, Brassard & Vazirani, 1994] tight lower bound Ω( √ n ) for Grover search known 2 years before discovering the algorithm • [Ambainis, 2000 & 2003] generalized to all functions • easy to use, gives strong bounds Polynomial method [Beals, Buhrman, Cleve, Mosca & de Wolf, 2000] • incomparable to the adversary method • hard to use for non-symmetric functions • [Aaronson & Shi, 2002] tight lower bound Ω( n 2 / 3 ) for element distinctness Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.3/14
Adversary lower bounds • if an algorithm computes f , then it must distinguish between x, y such that f ( x ) � = f ( y ) Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.4/14
Adversary lower bounds • if an algorithm computes f , then it must distinguish between x, y such that f ( x ) � = f ( y ) computation starts in a fixed state and | ψ T x � | start � it has to diverge far enough after T • queries for each such x, y | ψ T y � Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.4/14
Adversary lower bounds • if an algorithm computes f , then it must distinguish between x, y such that f ( x ) � = f ( y ) computation starts in a fixed state and | ψ T x � | start � it has to diverge far enough after T • queries for each such x, y | ψ T y � • prove that one query cannot change the scalar product too much (for an average x, y ) = ⇒ lower bound on T Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.4/14
Adversary lower bounds • if an algorithm computes f , then it must distinguish between x, y such that f ( x ) � = f ( y ) computation starts in a fixed state and | ψ T x � | start � it has to diverge far enough after T • queries for each such x, y | ψ T y � • prove that one query cannot change the scalar product too much (for an average x, y ) = ⇒ lower bound on T Limitations 1. weak bounds for exponentially small success probability Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.4/14
Adversary lower bounds • if an algorithm computes f , then it must distinguish between x, y such that f ( x ) � = f ( y ) computation starts in a fixed state and | ψ T x � | start � it has to diverge far enough after T • queries for each such x, y | ψ T y � • prove that one query cannot change the scalar product too much (for an average x, y ) = ⇒ lower bound on T Limitations 1. weak bounds for exponentially small success probability 2. [Š & Szegedy, Zhang, 2004] bounds limited by √ C 0 C 1 for total functions C z is the z -certificate complexity of f Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.4/14
Our new lower-bound method • does not suffer from the 1 st limitation and possibly not even from the 2 nd Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.5/14
Our new lower-bound method • does not suffer from the 1 st limitation and possibly not even from the 2 nd • extends the adversary method above by taking into account the current knowledge of the algorithm at each step (the adversary method is oblivious to this and its bound is the same for each query) Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.5/14
Our new lower-bound method • does not suffer from the 1 st limitation and possibly not even from the 2 nd • extends the adversary method above by taking into account the current knowledge of the algorithm at each step (the adversary method is oblivious to this and its bound is the same for each query) • based on subspace analysis of the density matrix Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.5/14
Our new lower-bound method • does not suffer from the 1 st limitation and possibly not even from the 2 nd • extends the adversary method above by taking into account the current knowledge of the algorithm at each step (the adversary method is oblivious to this and its bound is the same for each query) • based on subspace analysis of the density matrix Applications • k -fold search (find k ones) • direct product theorems explained in a moment • time-space tradeoffs Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.5/14
Subspaces for k -fold search “know” at most j ones T j spanned by � | ψ i 1 ...i j � = | x � x : | x | = k T 0 T 1 T 2 T 3 T k x i 1 = ··· = x ij =1 . . . T 0 ⊆ T 1 ⊆ · · · ⊆ T k Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.6/14
Subspaces for k -fold search “know” at most j ones T j spanned by � | ψ i 1 ...i j � = | x � x : | x | = k T 0 T 1 T 2 T 3 T k x i 1 = ··· = x ij =1 . . . starting state T 0 T 0 ⊆ T 1 ⊆ · · · ⊆ T k Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.6/14
Subspaces for k -fold search “know” at most j ones T j spanned by � | ψ i 1 ...i j � = | x � x : | x | = k T 0 T 1 T 2 T 3 T k x i 1 = ··· = x ij =1 . . . starting state T 0 entire input space T k T 0 ⊆ T 1 ⊆ · · · ⊆ T k Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.6/14
Subspaces for k -fold search “know” at most j ones T j spanned by � | ψ i 1 ...i j � = | x � x : | x | = k T 0 T 1 T 2 S 3 T k x i 1 = ··· = x ij =1 . . . starting state T 0 entire input space T k “know” exactly j ones S j S j = T j ∩ T ⊥ T 0 ⊆ T 1 ⊆ · · · ⊆ T k j − 1 Robert ˇ Spalek, CWI – A New Quantum Lower-Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs – p.6/14
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