Quantum Algorithms, Lower Bounds, and Time-Space Tradeoffs

Andris Ambainis, Robert Špalek, Ronald de Wolf

Hartmut Klauck

We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that in general a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We observe that for all storage bounds n/\log n\ge S\ge \log^3 n, one can devise a quantum algorithm that sorts n numbers (using comparisons only) in time T=O(n^{3/2}\log^{3/2} n/\sqrt S). We then show the following lower bound on the time-space tradeoff for sorting $n$ 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).

Hartmut Klauck

We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that in general a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We observe that for all storage bounds n/\log n\ge S\ge \log^3 n, one can devise a quantum algorithm that sorts n numbers (using comparisons only) in time T=O(n^{3/2}\log^{3/2} n/\sqrt S). We then show the following lower bound on the time-space tradeoff for sorting $n$ 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).

Dave Bacon, Wim van Dam

What quantum algorithms outperform classical computation and how do they do it?

Paul Beame, Blake Holman, Niels Kornerup

Time and space (memory) are two of the most important measures of cost in computation, even more so for quantum computation. Yet, our tools for proving unconditional quantum tradeoffs between time and space are surprisingly limited. The first quantum time-space tradeoff lower bounds were proven for sorting by Klauck, Špalek and de Wolf. Unfortunately, their method is limited to proving output-oblivious lower bounds (i.e. the lower bounds only apply to algorithms with a non-adaptive output schedule) and other methods have yielded nothing beyond output-oblivious lower bounds for sorting. Here, we prove the first fully general quantum time-space tradeoff lower bound for sorting. We do so by introducing a novel method based on the noise operator to add to the analysis toolkit for proving quantum query and time-space tradeoff lower bounds. By combining our resulting quantum noise stability bound with quantum recording query methods, we prove an $Ω(n^{4/3} (\log \log n)/(S^{1/3} \log n))$ 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.

Hartmut Klauck

We investigate the complexity of sorting in the model of sequential quantum circuits. While it is known that in general a quantum algorithm based on comparisons alone cannot outperform classical sorting algorithms by more than a constant factor in time complexity, this is wrong in a space bounded setting. We observe that for all storage bounds n/log ≥ S ≥ log3n, one can devise a quantum algorithm that sorts n numbers (using comparisons only) in time T=O(n3/2 log3/2 n/√S). We then show the following lower bound on the time-space tradeoff for sorting n numbers from a polynomial size range in a general sorting algorithm (not necessarily based on comparisons): TS=Ω(n3/2). Hence for small values of S the upper bound is almost tight. Classically the time-space tradeoff for sorting is TS=Θ(n2).