numbers from a polynomial size range in a general sorting algorithm (not necessarily based on comparisons): TS=Ω(n^{3/2}). Hence for small values of S the upper bound is almost tight. Classically the time-space tradeoff for sorting is TS=Θ(n^2).

DOI: 10.48550/arxiv.quant-ph/0211174

Related papers

Quantum time-space tradeoffs for sorting

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ACM Symposium on Theory of Computing (STOC)Jan 1, 20031 citation

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arXiv:2401.05321SIAM Journal on ComputingMay 20, 20261 citation

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arXiv:2301.05680ACM Transactions on Computation TheoryApr 23, 20252 citations

n

n

numbers from a polynomial size range in a general sorting algorithm (not necessarily based on comparisons): TS=\Omega(n^{3/2}). Hence for small values of S the upper bound is almost tight. Classically the time-space tradeoff for sorting is TS=\Theta(n^2).

Ω(n^{4/3} (\log \log n)/(S^{1/3} \log n))

lower bound on the number of queries that a fully general quantum algorithm with at most

S

qubits of memory requires to sort

n

numbers from

[n^2]

. Applying our noise operator argument involves purely classical reasoning, which makes it particularly simple to use. We also use it to prove that, for any strongly universal (pairwise independent) hash function family

H

from

n

bits to

m

bits, almost all hash functions in

H

require any algorithm with at most

S

qubits of memory to make

Ω(nm/S)

quantum queries to input

x

in order to compute

h(x)

, even with very small success probability. Previously, Mansour, Nisan, and Tiwari had shown a matching classical lower bound for computing

h(x)

with both

h

and

x

as inputs using their hash mixing lemma. Our noise operator method allows us to use a related property of hash functions to prove our quantum lower bounds.

\tilde{\Omega }(n^2)

, any classical matrix multiplication algorithm requires cumulative memory Ω ( n 6 / T ), any quantum sorting circuit requires cumulative memory Ω ( n 3 / T ), and any quantum circuit that finds k disjoint collisions in a random function requires cumulative memory Ω ( k 3 n / T 2 ).