Ordered search is the problem of locating a target element in a sorted list of size $n$ using comparison queries. Classically, binary search requires $\lceil \log_2 n\rceil$ queries, which is optimal. Quantum algorithms offer a constant-factor speedup, but the precise constant has been a longstanding open question. We close this gap by exhibiting two new quantum algorithms for ordered search, each using the optimal $\frac{1}π\ln n+o(\log n)$ queries. The first, discovered by Claude Fable 5, is a simple zero-error algorithm derived from a continuum relaxation of the problem. The second, discovered by GPT-5.6-Sol (informed by Claude's zero-error algorithm), is an exact algorithm based on an analytic solution of the polynomial program of Farhi, Goldstone, Gutmann, and Sipser.Ordered search is the problem of locating a target element in a sorted list of size $n$ using comparison queries. Classically, binary search requires $\lceil \log_2 n\rceil$ queries, which is optimal. Quantum algorithms offer a constant-factor speedup, but the precise constant has been a longstanding open question. We close this gap by exhibiting two new quantum algorithms for ordered search, each using the optimal $\frac{1}π\ln n+o(\log n)$ queries. The first, discovered by Claude Fable 5, is a simple zero-error algorithm derived from a continuum relaxation of the problem. The second, discovered by GPT-5.6-Sol (informed by Claude's zero-error algorithm), is an exact algorithm based on an analytic solution of the polynomial program of Farhi, Goldstone, Gutmann, and Sipser.

quant-pharXiv:2609.34293Sep 29, 20260 citations

Quantum search of partially ordered sets

Ashley Montanaro

We investigate the generalisation of quantum search of unstructured and totally ordered sets to search of partially ordered sets (posets). Two models for poset search are considered. In both models, we show that quantum algorithms can achieve at most a quadratic improvement in query complexity over classical algorithms, up to logarithmic factors; we also give quantum algorithms that almost achieve this optimal reduction in complexity. In one model, we give an improved quantum algorithm for searching forest-like posets; in the other, we give an optimal $O(\sqrt{m})$ -query quantum algorithm for searching posets derived from $m \times m$ arrays sorted by rows and columns. This leads to a quantum algorithm that finds the intersection of two sorted lists of $n$ integers in $O(\sqrt{n})$ time, which is optimal.We investigate the generalisation of quantum search of unstructured and totally ordered sets to search of partially ordered sets (posets). Two models for poset search are considered. In both models, we show that quantum algorithms can achieve at most a quadratic improvement in query complexity over classical algorithms, up to logarithmic factors; we also give quantum algorithms that almost achieve this optimal reduction in complexity. In one model, we give an improved quantum algorithm for searching forest-like posets; in the other, we give an optimal $O(\sqrt{m})$-query quantum algorithm for searching posets derived from $m \times m$ arrays sorted by rows and columns. This leads to a quantum algorithm that finds the intersection of two sorted lists of $n$ integers in $O(\sqrt{n})$ time, which is optimal.

Quantum Information and ComputationJul 1, 20090 citations

Quantum bounds for ordered searching and sorting

Peter Høyer, Jan Neerbek, Yaoyun Shi

We consider the quantum complexities of searching an ordered list and sorting an un-ordered list. For searching an ordered list of N elements, we prove a lower bound of \\frac{1}{\\pi}(\\ln(N)-1) on the number of oracle queries that access the list elements. This improves the previously best lower bound of ({1/12}\\log_2(N) - O(1)) due to Ambainis. For sorting N numbers, we prove a lower bound of \\frac{N}{2\\pi}(\\ln(N)-1) on the number of binary comparisons. The previously best lower bound is \\Omega(N). Our proofs are based on a weighted all-pairs inner product argument, and our results generalize to bounded error quantum algorithms. Both results are proven in the so-called quantum black box model, a quantum analogue of classical decision trees. In addition to our lower bound results, we give an exact quantum algorithm for ordered searching using (\\log_3(N) + O(1)) queries, which is roughly 0.631 \\log_2(N). Although our algorithm is worse than that of Farhi, Goldstone, Gutmann and Sipser, which makes 0.526 \\log_2(N) queries, its philosophy is completely different. Our algorithm is a quantum version of the classical binary search algorithm, and it uses a quantum routine for traversing through a binary search tree faster than classically.We consider the quantum complexities of searching an ordered list and sorting an un-ordered list. For searching an ordered list of N elements, we prove a lower bound of \\frac{1}{\\pi}(\\ln(N)-1) on the number of oracle queries that access the list elements. This improves the previously best lower bound of ({1/12}\\log_2(N) - O(1)) due to Ambainis. For sorting N numbers, we prove a lower bound of \\frac{N}{2\\pi}(\\ln(N)-1) on the number of binary comparisons. The previously best lower bound is \\Omega(N). Our proofs are based on a weighted all-pairs inner product argument, and our results generalize to bounded error quantum algorithms. Both results are proven in the so-called quantum black box model, a quantum analogue of classical decision trees. In addition to our lower bound results, we give an exact quantum algorithm for ordered searching using (\\log_3(N) + O(1)) queries, which is roughly 0.631 \\log_2(N). Although our algorithm is worse than that of Farhi, Goldstone, Gutmann and Sipser, which makes 0.526 \\log_2(N) queries, its philosophy is completely different. Our algorithm is a quantum version of the classical binary search algorithm, and it uses a quantum routine for traversing through a binary search tree faster than classically.

Feb 15, 20013 citations

Quantum algorithms for the ordered search problem via semidefinite programming

Andrew M. Childs, Andrew J. Landahl, Pablo A. Parrilo

One of the most basic computational problems is the task of finding a desired item in an ordered list of $N$ items. While the best classical algorithm for this problem uses ${\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}N$ queries to the list, a quantum computer can solve the problem using a constant factor fewer queries. However, the precise value of this constant is unknown. By characterizing a class of quantum query algorithms for the ordered search problem in terms of a semidefinite program, we find quantum algorithms for small instances of the ordered search problem. Extending these algorithms to arbitrarily large instances using recursion, we show that there is an exact quantum ordered search algorithm using $4\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{605}\phantom{\rule{0.2em}{0ex}}N\ensuremath{\approx}0.433\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}N$ queries, which improves upon the previously best known exact algorithm.One of the most basic computational problems is the task of finding a desired item in an ordered list of $N$ items. While the best classical algorithm for this problem uses ${\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}N$ queries to the list, a quantum computer can solve the problem using a constant factor fewer queries. However, the precise value of this constant is unknown. By characterizing a class of quantum query algorithms for the ordered search problem in terms of a semidefinite program, we find quantum algorithms for small instances of the ordered search problem. Extending these algorithms to arbitrarily large instances using recursion, we show that there is an exact quantum ordered search algorithm using $4\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{605}\phantom{\rule{0.2em}{0ex}}N\ensuremath{\approx}0.433\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}N$ queries, which improves upon the previously best known exact algorithm.

arXiv:quant-ph/0608161Physical Review AMar 26, 200746 citations

Finding quantum algorithms via convex optimization

Andrew M. Childs, Andrew J. Landahl, Pablo A. Parrilo

In this paper we describe how to use convex optimization to design quantum algorithms for certain computational tasks. In particular, we consider the ordered search problem, where it is desired to find a specific item in an ordered list of N items. While the best classical algorithm for this problem uses log2N queries to the list, a quantum computer can solve this problem much faster. By characterizing a class of quantum query algorithms for ordered search in terms of a semidefinite program, we find quantum algorithms using 4log605N ≈ 0.433 log2N queries, which improves upon the previously best known exact algorithm.In this paper we describe how to use convex optimization to design quantum algorithms for certain computational tasks. In particular, we consider the ordered search problem, where it is desired to find a specific item in an ordered list of N items. While the best classical algorithm for this problem uses log2N queries to the list, a quantum computer can solve this problem much faster. By characterizing a class of quantum query algorithms for ordered search in terms of a semidefinite program, we find quantum algorithms using 4log605N ≈ 0.433 log2N queries, which improves upon the previously best known exact algorithm.

European Control Conference (ECC)Jul 1, 20070 citations

Information-theoretic lower bounds for quantum sorting

Jean Cardinal, Gwenaël Joret, Jérémie Roland

We analyze the quantum query complexity of sorting under partial information. In this problem, we are given a partially ordered set $P$ and are asked to identify a linear extension of $P$ using pairwise comparisons. For the standard sorting problem, in which $P$ is empty, it is known that the quantum query complexity is not asymptotically smaller than the classical information-theoretic lower bound. We prove that this holds for a wide class of partially ordered sets, thereby improving on a result from Yao (STOC'04).We analyze the quantum query complexity of sorting under partial information. In this problem, we are given a partially ordered set $P$ and are asked to identify a linear extension of $P$ using pairwise comparisons. For the standard sorting problem, in which $P$ is empty, it is known that the quantum query complexity is not asymptotically smaller than the classical information-theoretic lower bound. We prove that this holds for a wide class of partially ordered sets, thereby improving on a result from Yao (STOC'04).

arXiv:1902.06473Feb 18, 20191 citation

Ordered search is the problem of locating a target element in a sorted list of size

n

using comparison queries. Classically, binary search requires

\lceil \log_2 n\rceil

queries, which is optimal. Quantum algorithms offer a constant-factor speedup, but the precise constant has been a longstanding open question. We close this gap by exhibiting two new quantum algorithms for ordered search, each using the optimal

\frac{1}π\ln n+o(\log n)

queries. The first, discovered by Claude Fable 5, is a simple zero-error algorithm derived from a continuum relaxation of the problem. The second, discovered by GPT-5.6-Sol (informed by Claude's zero-error algorithm), is an exact algorithm based on an analytic solution of the polynomial program of Farhi, Goldstone, Gutmann, and Sipser.

O(\sqrt{m})

-query quantum algorithm for searching posets derived from

m \times m

arrays sorted by rows and columns. This leads to a quantum algorithm that finds the intersection of two sorted lists of

n

integers in

O(\sqrt{n})

time, which is optimal.

One of the most basic computational problems is the task of finding a desired item in an ordered list of

N

items. While the best classical algorithm for this problem uses

{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}N

queries to the list, a quantum computer can solve the problem using a constant factor fewer queries. However, the precise value of this constant is unknown. By characterizing a class of quantum query algorithms for the ordered search problem in terms of a semidefinite program, we find quantum algorithms for small instances of the ordered search problem. Extending these algorithms to arbitrarily large instances using recursion, we show that there is an exact quantum ordered search algorithm using $4\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{605}\phantom{\rule{0.2em}{0ex}}N\ensuremath{\approx}0.433\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}N$ queries, which improves upon the previously best known exact algorithm.

We analyze the quantum query complexity of sorting under partial information. In this problem, we are given a partially ordered set

P

and are asked to identify a linear extension of

P

using pairwise comparisons. For the standard sorting problem, in which

P

is empty, it is known that the quantum query complexity is not asymptotically smaller than the classical information-theoretic lower bound. We prove that this holds for a wide class of partially ordered sets, thereby improving on a result from Yao (STOC'04).