Quantum query complexity, with advice

In the SEARCH WITH ADVICE problem, a single entry of interest within a database of N entries is to be found assuming that an ordering of the entries, from that with the highest probability of being the entry of interest (as determined by a so-called advice distribution) to that with the lowest, is provided. We present a quantum algorithm that, in the presence of significant levels of quantum noise, solves SEARCH WITH ADVICE for a power law advice distribution with average-case query complexity O(1) as N tends to infinity. Since as we also show the best classical algorithms for this problem exhibit average-case query complexity of order no better than log(N), our quantum algorithm provides a super-exponential reduction in query complexity.In the SEARCH WITH ADVICE problem, a single entry of interest within a database of N entries is to be found assuming that an ordering of the entries, from that with the highest probability of being the entry of interest (as determined by a so-called advice distribution) to that with the lowest, is provided. We present a quantum algorithm that, in the presence of significant levels of quantum noise, solves SEARCH WITH ADVICE for a power law advice distribution with average-case query complexity O(1) as N tends to infinity. Since as we also show the best classical algorithms for this problem exhibit average-case query complexity of order no better than log(N), our quantum algorithm provides a super-exponential reduction in query complexity.

arXiv:1710.10790Oct 30, 20170 citations

The Quantum Message Complexity of Distributed Wake-Up with Advice

Peter Robinson, Ming Ming Tan

We consider the distributed wake-up problem with advice, where nodes are equipped with initial knowledge about the network at large. After the adversary awakens a subset of nodes, an oracle computes a bit string (``the advice'') for each node, and the goal is to wake up all sleeping nodes efficiently. We present the first upper and lower bounds on the message complexity for wake-up in the quantum routing model, introduced by Dufoulon, Magniez, and Pandurangan (PODC 2025). In more detail, we give a distributed advising scheme that, given $α$ bits of advice per node, wakes up all nodes with a message complexity of $O( \sqrt{\frac{n^3}{2^{\max\{\lfloor (α-1)/2 \rfloor},0\}}}\cdot\log n )$ with high probability. Our result breaks the $Ω( \frac{n^2}{2^α} )$ barrier known for the classical port numbering model in sufficiently dense graphs. To complement our algorithm, we give a lower bound on the message complexity for distributed quantum algorithms: By leveraging a lower bound result for the single-bit descriptor problem in the query complexity model, we show that wake-up has a quantum message complexity of $Ω( n^{3/2} )$ without advice, which holds independently of how much time we allow. In the setting where an adversary decides which nodes start the algorithm, most graph problems of interest implicitly require solving wake-up, and thus the same lower bound also holds for other fundamental problems such as single-source broadcast and spanning tree construction.We consider the distributed wake-up problem with advice, where nodes are equipped with initial knowledge about the network at large. After the adversary awakens a subset of nodes, an oracle computes a bit string (``the advice'') for each node, and the goal is to wake up all sleeping nodes efficiently. We present the first upper and lower bounds on the message complexity for wake-up in the quantum routing model, introduced by Dufoulon, Magniez, and Pandurangan (PODC 2025). In more detail, we give a distributed advising scheme that, given $α$ bits of advice per node, wakes up all nodes with a message complexity of $O( \sqrt{\frac{n^3}{2^{\max\{\lfloor (α-1)/2 \rfloor},0\}}}\cdot\log n )$ with high probability. Our result breaks the $Ω( \frac{n^2}{2^α} )$ barrier known for the classical port numbering model in sufficiently dense graphs. To complement our algorithm, we give a lower bound on the message complexity for distributed quantum algorithms: By leveraging a lower bound result for the single-bit descriptor problem in the query complexity model, we show that wake-up has a quantum message complexity of $Ω( n^{3/2} )$ without advice, which holds independently of how much time we allow. In the setting where an adversary decides which nodes start the algorithm, most graph problems of interest implicitly require solving wake-up, and thus the same lower bound also holds for other fundamental problems such as single-source broadcast and spanning tree construction.

quant-pharXiv:2602.05801Feb 6, 20260 citations

On quantum versus classical query complexity.

Scott Aaronson, Andris Ambainis, Andrej Bogdanov +2 more

Electronic colloquium on computational complexityJan 1, 20212 citations

Improved Quantum Query Complexity on Easier Inputs

Noel T. Anderson, Jay-U Chung, Shelby Kimmel +2 more

Quantum span program algorithms for function evaluation sometimes have reduced query complexity when promised that the input has a certain structure. We design a modified span program algorithm to show these improvements persist even without a promise ahead of time, and we extend this approach to the more general problem of state conversion. As an application, we prove exponential and superpolynomial quantum advantages in average query complexity for several search problems, generalizing Montanaro's Search with Advice [Montanaro, TQC 2010].Quantum span program algorithms for function evaluation sometimes have reduced query complexity when promised that the input has a certain structure. We design a modified span program algorithm to show these improvements persist even without a promise ahead of time, and we extend this approach to the more general problem of state conversion. As an application, we prove exponential and superpolynomial quantum advantages in average query complexity for several search problems, generalizing Montanaro's Search with Advice [Montanaro, TQC 2010].

arXiv:2303.00217QuantumApr 8, 20240 citations

Quantum query algorithms and lower bounds

Andris Ambainis

Springer eBooksNov 2, 200726 citations

α

bits of advice per node, wakes up all nodes with a message complexity of

O( \sqrt{\frac{n^3}{2^{\max\{\lfloor (α-1)/2 \rfloor},0\}}}\cdot\log n )

with high probability. Our result breaks the

Ω( \frac{n^2}{2^α} )

barrier known for the classical port numbering model in sufficiently dense graphs. To complement our algorithm, we give a lower bound on the message complexity for distributed quantum algorithms: By leveraging a lower bound result for the single-bit descriptor problem in the query complexity model, we show that wake-up has a quantum message complexity of

Ω( n^{3/2} )

without advice, which holds independently of how much time we allow. In the setting where an adversary decides which nodes start the algorithm, most graph problems of interest implicitly require solving wake-up, and thus the same lower bound also holds for other fundamental problems such as single-source broadcast and spanning tree construction.