Quantum Time-Space Tradeoffs for Deciding Systems of Linear - PowerPoint PPT Presentation

Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities Robert ˇ Spalek sr@cwi.nl joint work with Andris Ambainis and Ronald de Wolf quant-ph/0511200 Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.1/17

Time-Space Tradeoffs • A relation between the running time and space complexity The more memory is available, the faster the algorithm can possibly run. Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.2/17

Time-Space Tradeoffs • A relation between the running time and space complexity The more memory is available, the faster the algorithm can possibly run. • Example: sorting of N numbers TS = N 2 Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.2/17

Systems of Linear Inequalities • Let A be a fixed N × N Boolean matrix Let x, b be integer input vectors of length N • The task is to output for each row whether Ax ≥ b Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.3/17

Systems of Linear Inequalities • Let A be a fixed N × N Boolean matrix Let x, b be integer input vectors of length N • The task is to output for each row whether Ax ≥ b • We study the query complexity with bounded error ◦ Classically TS = N 2 Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.3/17

Systems of Linear Inequalities • Let A be a fixed N × N Boolean matrix Let x, b be integer input vectors of length N • The task is to output for each row whether Ax ≥ b • We study the query complexity with bounded error ◦ Classically TS = N 2 ◦ Quantumly T 2 S = N 3 t, S ≤ N/t TS = N 2 , S > N/t if numbers in b are at most t • Omitting log-factors in the upper bounds Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.3/17

Upper Bound Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.4/17

Classical Algorithm • Split the matrix into ( N/S ) 2 blocks of size S × S A x b S S ≥ · Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.5/17

Classical Algorithm • Split the matrix into ( N/S ) 2 blocks of size S × S • Evaluate the output row-wise ◦ maintain S counters at the same time ◦ in each of the N/S blocks, read S inputs and update all counters using the fixed matrix A A x b S S ≥ · Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.5/17

Classical Algorithm • Split the matrix into ( N/S ) 2 blocks of size S × S • Evaluate the output row-wise ◦ maintain S counters at the same time ◦ in each of the N/S blocks, read S inputs and update all counters using the fixed matrix A • The query complexity is = N 2 T = N � N � A x b S · S · S S S S when the space is S ≥ · Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.5/17

Classical Algorithm • Split the matrix into ( N/S ) 2 blocks of size S × S • Evaluate the output row-wise ◦ maintain S counters at the same time ◦ in each of the N/S blocks, read S inputs and update all counters using the fixed matrix A • The query complexity is = N 2 T = N � N � A x b S · S · S S S S when the space is S ≥ · TS ≤ N 2 Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.5/17

Quantum Algorithm • Split the matrix into N/S row blocks of height S A x b S ≥ · N Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.6/17

Quantum Algorithm • Split the matrix into N/S row blocks of height S • Evaluate the output row-wise ◦ maintain S counters at the same time ◦ use quantum counting and Grover search to find non-zero inputs √ ◦ the speedup is N → NSt per row block A x b S ≥ · N Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.6/17

Quantum Algorithm • Split the matrix into N/S row blocks of height S • Evaluate the output row-wise ◦ maintain S counters at the same time ◦ use quantum counting and Grover search to find non-zero inputs √ ◦ the speedup is N → NSt per row block • The query complexity is A x b √ � T = N t NSt = N 3 / 2 S · S S ≥ · when the space is S N Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.6/17

Quantum Algorithm • Split the matrix into N/S row blocks of height S • Evaluate the output row-wise ◦ maintain S counters at the same time ◦ use quantum counting and Grover search to find non-zero inputs √ ◦ the speedup is N → NSt per row block • The query complexity is A x b √ � T = N t NSt = N 3 / 2 S · S S ≥ · when the space is S T 2 S ≤ N 3 t N Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.6/17

Quantum Algorithm (cont.) x A vector of counters y set of open rows with y i < b i U U S column sum of A over � = a j a the rows from U Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.7/17

Quantum Algorithm (cont.) x A vector of counters y set of open rows with y i < b i U U S column sum of A over � = a j a the rows from U Start at position p ← 1 and with U ← [1 , S ] . Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.7/17

Quantum Algorithm (cont.) x A vector of counters y set of open rows with y i < b i U U S column sum of A over a p k the rows from U ∈ [ S, 2 S ] While p ≤ N and U � = ∅ , do • Find by binary search some k such that p + k − 1 . . . quantum counting � S ≤ a j x j ≤ 2 S j = p Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.7/17

Quantum Algorithm (cont.) x A vector of counters y set of open rows with y i < b i U U S column sum of A over a p k the rows from U ∈ [ S, 2 S ] While p ≤ N and U � = ∅ , do • Find by binary search some k such that p + k − 1 . . . quantum counting � S ≤ a j x j ≤ 2 S j = p • Find all positions j inside [ p, p + k − 1] such that . . . quantum search a j x j > 0 Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.7/17

Quantum Algorithm (cont.) x A vector of counters y set of open rows with y i < b i U U S column sum of A over a p the rows from U While p ≤ N and U � = ∅ , do • Find by binary search some k such that p + k − 1 . . . quantum counting � S ≤ a j x j ≤ 2 S j = p • Find all positions j inside [ p, p + k − 1] such that . . . quantum search a j x j > 0 • Update the counters y , remove from U the rows that have been closed in this iteration, and set p ← p + k Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.7/17

Complexity of the Algorithm i : 1 2 3 4 5 In the i -th iteration of length k i , • cost of quantum counting with √ k i queries is negligible • quantum search costs √ k i r i t + √ k i s i , where ◦ r i is the number of closed rows ◦ s i is the total number added to counters in this iteration Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.8/17

Complexity of the Algorithm i : 1 2 3 4 5 In the i -th iteration of length k i , • cost of quantum counting with √ k i queries is negligible • quantum search costs √ k i r i t + √ k i s i , where ◦ r i is the number of closed rows ◦ s i is the total number added to counters in this iteration By Cauchy-Schwarz, �� � � � T = k i r i t + k i s i i �� � �� �� � ≤ k i t r i + k i s i √ ≤ NSt Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.8/17

Lower Bound Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.9/17

Direct Product Theorems • Suppose we need T ( f ) queries to compute f with small error. How hard is it to compute k independent instances f ( x 1 ) , . . . , f ( x k ) ? Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.10/17

Direct Product Theorems • Suppose we need T ( f ) queries to compute f with small error. How hard is it to compute k independent instances f ( x 1 ) , . . . , f ( x k ) ? • Relation between total number of queries T and overall success probability σ : T ≤ αk · T ( f ) ⇒ σ ≤ 2 − γk α, γ are small positive constants Robert ˇ Spalek, CWI – Quantum Time-Space Tradeoffs for Deciding Systems of Linear Inequalities – p.10/17