23 Search Results

Ordered search is the task of finding an item in an ordered list using comparison queries. The best exact classical algorithm for this fundamental problem uses [log₂n] queries for a list of length n. Quantum computers can achieve a constant-factor speedup, but the best possible coefficient of log₂n for exact quantum algorithms is only known to lie between (ln2)/π ≈ 0.221 and 4/log₂ 605 ≈ 0.4333. We consider a special class of translation-invariant algorithms with no workspace, introduced by Farhi, Goldstone, Gutmann, and Sipser, that has been used to find the best known upper bounds. First, we show that anymore » bounded-error, k-query quantum algorithm for ordered search can be implemented by a k-query algorithm in this special class. Second, we use linear programming to show that the best exact 5-query quantum algorithm can search a list of length 7265, giving an ordered search algorithm that asymptotically uses 5 log₇₂₆₅ n ≈ 0.390 log₂n quantum queries.« less

Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for nonnormal matrices. Here, we propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation method [D. An, J.-P. Liu, and L. Lin, Phys. Rev. Lett., 131 (2023), 150603; D. An, A. M. Childs,more » and L. Lin, Commun. Math. Phys. 407, 19 (2026)] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, 𝐴^−𝑘, and the exponential of the matrix inverse, 𝑒^−𝐴−1. The latter can be interpreted as the solution of a mass-matrix differential equation of the form form 𝐴⁢𝑢′⁡⁡(𝑡) =−𝑢⁡(𝑡). We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting 𝐴, thereby reducing the computational complexity.« less

We introduce a family of identities that express general linear non-unitary evolution operators as a linear combination of unitary evolution operators, each solving a Hamiltonian simulation problem. This formulation can exponentially enhance the accuracy of the recently introduced linear combination of Hamiltonian simulation (LCHS) method [An, Liu, and Lin, Physical Review Letters, 2023]. For the first time, this approach enables quantum algorithms to solve linear differential equations with both optimal state preparation cost and near-optimal scaling in matrix queries on all parameters.

To implement arbitrary quantum circuits in architectures with restricted interactions, one may effectively simulate all-to-all connectivity by routing quantum information. We consider the entanglement dynamics and routing between two regions only connected through an intermediate “bottleneck” region with few qubits. In such systems, where the entanglement rate is restricted by a vertex boundary rather than an edge boundary of the underlying interaction graph, existing results such as the small incremental entangling theorem give only a trivial constant lower bound on the routing time (the minimum time to perform an arbitrary permutation). We significantly improve the lower bound on the routingmore » time in systems with a vertex bottleneck. Specifically, for any system with two regions 𝐿,𝑅 with 𝑁_𝐿,𝑁_𝑅 qubits, respectively, coupled only through an intermediate region 𝐶 with 𝑁_𝐶 qubits, for any 𝛿 > 0 we show a lower bound of Ω⁢(𝑁$$^{1−𝛿}_{𝑅}$$/√𝑁_𝐿⁢𝑁_𝐶) on the Hamiltonian quantum routing time when using piecewise time-independent Hamiltonians, or time-dependent Hamiltonians subject to a smoothness condition. We also prove an upper bound on the average amount of bipartite entanglement between 𝐿 and 𝐶,𝑅 that can be generated in time 𝑡 by such architecture-respecting Hamiltonians in systems constrained by vertex bottlenecks, improving the scaling in the system size from 𝑂⁡(𝑁_𝐿⁢𝑡) to 𝑂⁡(√𝑁_𝐿⁢𝑡). As a special case, when applied to the star graph (i.e., one vertex connected to 𝑁 leaves), we obtain an Ω⁡(√𝑁^1−𝛿) lower bound on the routing time and on the time to prepare 𝑁/2 Bell pairs between the vertices. We also show that, in systems of free particles, we can route optimally on the star graph in time Θ⁡(√𝑁) using Hamiltonian quantum routing, obtaining a speedup over gate-based routing, which takes time Θ⁡(𝑁).« less

Here, we present an algorithm utilizing midcircuit measurement and feedback that prepares Dicke states with polylogarithmically many ancillae and polylogarithmic depth. Our algorithm uses only global midcircuit projective measurements and adaptively chosen global rotations. This improves over prior work that was only efficient for Dicke states of low weight or was not efficient in both depth and width. Our algorithm can also naturally be implemented in a cavity QED context using logarithmic time, zero ancillae, and atom-photon coupling scaling with the square root of the system size.

Existing schemes for demonstrating quantum computational advantage are subject to various practical restrictions, including the hardness of verification and challenges in experimental implementation. Meanwhile, analog quantum simulators have been realized in many experiments to study novel physics. In this work, we propose a quantum advantage protocol based on verification of an analog quantum simulation, in which the verifier need only run an $O\left({\lambda }^2\right)$ -time classical computation, and the prover need only prepare $O\left(1\right)$ samples of a history state and perform $O({\lambda }^{2}more »$ ) single-qubit measurements, for a security parameter $\lambda$ . We also propose a near-term feasible strategy for honest provers and discuss potential experimental realizations. Published by the American Physical Society 2025« less

Geometric locality is an important theoretical and practical factor for quantum low-density parity-check (qLDPC) codes that affects code performance and ease of physical realization. For device architectures restricted to two-dimensional (2D) local gates, naively implementing the high-rate codes suitable for low-overhead fault-tolerant quantum computing incurs prohibitive overhead. In this work, we present an error-correction protocol built on a bilayer architecture that aims to reduce operational overheads when restricted to 2D local gates by measuring some generators less frequently than others. We investigate the family of bivariate-bicycle qLDPC codes and show that they are well suited for a parallel syndrome-measurement schememore » using fast routing with local operations and classical communication (LOCC). Through circuit-level simulations, we find that in some parameter regimes, bivariate-bicycle codes implemented with this protocol have logical error rates comparable to the surface code while using fewer physical qubits. Published by the American Physical Society 2025« less

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We study the problem of implementing arbitrary permutations of qubits under interaction constraints in quantum systems that allow for arbitrarily fast local operations and classical communication (LOCC). In particular, we show examples of speedups over swap-based and more general unitary routing methods by distributing entanglement and using LOCC to perform quantum teleportation. We further describe an example of an interaction graph for which teleportation gives a logarithmic speedup in the worst-case routing time over swap-based routing. We also study limits on the speedup afforded by quantum teleportation—showing an $O\left(\sqrt{NlogN}\right)more »$ upper bound on the separation in routing time for any interaction graph—and give tighter bounds for some common classes of graphs. Published by the American Physical Society 2024« less