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Revision #2 to TR20-127 | 17th November 2020 16:12

$k$-Forrelation Optimally Separates Quantum and Classical Query Complexity

Revision #2
Authors: Nikhil Bansal, Makrand Sinha
Accepted on: 17th November 2020 16:12
Keywords:

Abstract:

Aaronson and Ambainis (SICOMP 18) showed that any partial function on $N$ bits that can be computed with an advantage $\delta$ over a random guess by making $q$ quantum queries, can also be computed classically with an advantage $\delta/2$ by a randomized decision tree making ${O}_q(N^{1-\frac{1}{2q}}\delta^{-2})$ queries. Moreover, they conjectured the $k$-Forrelation problem --- a partial function that can be computed with $q = \lceil k/2 \rceil$ quantum queries --- to be a suitable candidate for exhibiting such an extremal separation.

We prove their conjecture by showing a tight lower bound of $\widetilde{\Omega}(N^{1-1/k})$ for the randomized query complexity of $k$-Forrelation, where the advantage $\delta = 2^{-O(k)}$. By standard amplification arguments, this gives an explicit partial function that exhibits an $O_\epsilon(1)$ vs $\Omega(N^{1-\epsilon})$ separation between bounded-error quantum and randomized query complexities, where $\epsilon>0$ can be made arbitrarily small. Our proof also gives the same bound for the closely related but non-explicit $k$-Rorrelation function introduced by Tal (FOCS 20).

Our techniques rely on classical Gaussian tools, in particular, Gaussian interpolation and Gaussian integration by parts, and in fact, give a more general statement. We show that to prove lower bounds for $k$-Forrelation against a family of functions, it suffices to bound the $\ell_1$-weight of the Fourier coefficients between levels $k$ and $(k-1)k$. We also prove new interpolation and integration by parts identities that might be of independent interest in the context of rounding high-dimensional Gaussian vectors.

Changes to previous version:

Minor change to the figure format to fix an adobe acrobat specific issue.

Revision #1 to TR20-127 | 16th November 2020 23:44

$k$-Forrelation Optimally Separates Quantum and Classical Query Complexity

Revision #1
Authors: Nikhil Bansal, Makrand Sinha
Accepted on: 16th November 2020 23:44
Keywords:

Abstract:

Aaronson and Ambainis (SICOMP 18) showed that any partial function on $N$ bits that can be computed with an advantage $\delta$ over a random guess by making $q$ quantum queries, can also be computed classically with an advantage $\delta/2$ by a randomized decision tree making ${O}_q(N^{1-\frac{1}{2q}}\delta^{-2})$ queries. Moreover, they conjectured the $k$-Forrelation problem --- a partial function that can be computed with $q = \lceil k/2 \rceil$ quantum queries --- to be a suitable candidate for exhibiting such an extremal separation. \medskip

We prove their conjecture by showing a tight lower bound of $\widetilde{\Omega}(N^{1-1/k})$ for the randomized query complexity of $k$-Forrelation, where the advantage $\delta = 2^{-O(k)}$. By standard amplification arguments, this gives an explicit partial function that exhibits an $O_\epsilon(1)$ vs $\Omega(N^{1-\epsilon})$ separation between bounded-error quantum and randomized query complexities, where $\epsilon>0$ can be made arbitrarily small. Our proof also gives the same bound for the closely related but non-explicit $k$-Rorrelation function introduced by Tal (FOCS 20). \medskip

Our techniques rely on classical Gaussian tools, in particular, Gaussian interpolation and Gaussian integration by parts, and in fact, give a more general statement. We show that to prove lower bounds for $k$-Forrelation against a family of functions, it suffices to bound the $\ell_1$-weight of the Fourier coefficients between levels $k$ and $(k-1)k$. We also prove new interpolation and integration by parts identities that might be of independent interest in the context of rounding high-dimensional Gaussian vectors.

Changes to previous version:

Improved the advantage $\delta$ to $2^{-O(k)}$ compared to the previous version where $\delta$ was $1/\mathrm{polylog}^k(N)$ -- this strengthens the main conclusions. Added several figures and a reference to the independent work of Sherstov, Storozhenko and Wu who obtained a similar lower bound for the randomized query complexity of $k$-Rorrelation

Paper:

TR20-127 | 21st August 2020 21:46

$k$-Forrelation Optimally Separates Quantum and Classical Query Complexity

TR20-127
Authors: Nikhil Bansal, Makrand Sinha
Publication: 21st August 2020 22:02
Keywords:

Abstract:

Aaronson and Ambainis (SICOMP '18) showed that any partial function on $N$ bits that can be computed with an advantage $\delta$ over a random guess by making $q$ quantum queries, can also be computed classically with an advantage $\delta/2$ by a randomized decision tree making ${O}_q(N^{1-\frac{1}{2q}}\delta^{-2})$ queries. Moreover, they conjectured the $k$-Forrelation problem --- a partial function that can be computed with $q = \lceil k/2 \rceil$ quantum queries --- to be a suitable candidate for exhibiting such an extremal separation.

We prove their conjecture by showing a tight lower bound of $\widetilde{\Omega}_k(N^{1-1/k})$ for the randomized query complexity of $k$-Forrelation, where the advantage $\delta = 1/\mathrm{polylog}^k(N)$ and $\widetilde{\Omega}_k$ hides $\mathrm{polylog}^k(N)$ factors. Our proof relies on classical Gaussian tools, in particular, Gaussian interpolation and Gaussian integration by parts, and in fact, shows a more general statement, that to prove lower bounds for $k$-Forrelation against a family of functions, it suffices to bound the $\ell_1$-weight of the Fourier coefficients at levels $k, 2k, 3k, \ldots, (k-1)k$ for functions in the family.

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