arXiv subjects

Peter Høyer

Publications and source records attributed to Peter Høyer.

6 recordsLinked to original sources

arXiv2026

The Role of Symmetry in Quantum Query-to-Communication Simulation

Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}^n to {-1,1} and G in {AND_2, XOR_2}, the bounded-error quantum communication complexity of the composed function f o G equals O(Q(f) log n), where Q(f) denotes the bounded-error quantum query complexity of f. This is achieved by Alice running the optimal quantum query algorithm for f, using a round of O(log n) qubits of communication to implement each query. This is in contrast with the classical setting, where it is easy to show that R^{cc}(f o G) is at most 2R(f), where R^{cc} and R denote bounded-error communication and query complexity, respectively. We show that the O(log n) overhead is required for some functions in the quantum setting, and thus the BCW simulation is tight. We note here that prior to our work, the possibility of Q^{cc}(f o G) = O(Q(f)), for all f and all G in {AND_2, XOR_2}, had not been ruled out. More specifically, we show the following. - We show that the log n overhead is *not* required when f is symmetric, generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05). - In order to prove the above, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. - In view of our first result above, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive, which is a weaker notion of symmetry. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2. - We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model.

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arXiv2026

Nonlocality distillation can outperform entanglement distillation

Given the goal of maximizing CHSH violation, we compare the optimal strategies of entanglement and nonlocality distillation. In the limit of the number of copies of the shared state, entanglement distillation is guaranteed to work by generating a Bell state. For a small number of copies of the state, we show that nonlocality distillation can achieve a higher CHSH value, even though optimal entanglement distillation requires communication. Nonlocality distillation not only outperforms entanglement distillation but also demonstrates superior resource efficiency across multiple metrics for quantum resource estimation.

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arXiv2022

Tight Bound for Estimating Expectation Values from a System of Linear Equations

The System of Linear Equations Problem (SLEP) is specified by a complex invertible matrix $A$, the condition number $κ$ of $A$, a vector $b$, a Hermitian matrix $M$ and an accuracy $ε$, and the task is to estimate $x^\dagger Mx$, where $x$ is the solution vector to the equation $Ax = b$. We aim to establish a lower bound on the complexity of the end-to-end quantum algorithms for SLEP with respect to $ε$, and devise a quantum algorithm that saturates this bound. To make lower bounds attainable, we consider query complexity in the setting in which a block encoding of $M$ is given, i.e., a unitary black box $U_M$ that contains $M/α$ as a block for some $α\in \mathbb R^+$. We show that the quantum query complexity for SLEP in this setting is $Θ(α/ε)$. Our lower bound is established by reducing the problem of estimating the mean of a black box function to SLEP. Our $Θ(α/ε)$ result tightens and proves the common assertion of polynomial accuracy dependence (poly$(1/ε)$) for SLEP, and shows that improvement beyond linear dependence on accuracy is not possible if $M$ is provided via block encoding.

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arXiv2022

Spatial Search via Memoryless Walk with Selfloop

The defining feature of memoryless quantum walks is that they operate on the vertex space of a graph, and therefore can be used to produce search algorithms with minimal memory. We present a memoryless walk that can find a unique marked vertex on a two-dimensional grid. Our walk is based on the construction proposed by Falk, which tessellates the grid with squares of size $2 \times 2$. Our walk uses minimal memory, $O(\sqrt{N \log N})$ applications of the walk operator, and outputs the marked vertex with vanishing error probability. To accomplish this, we apply a selfloop to the marked vertex - a technique we adapt from interpolated walks. We prove that with our explicit choice of selfloop weight, this forces the action of the walk asymptotically into a single rotational space. We characterize this space and as a result, show that our memoryless walk produces the marked vertex with a success probability asymptotically approaching one.

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arXiv2020

Analysis of Lackadaisical Quantum Walks

The lackadaisical quantum walk is a quantum analogue of the lazy random walk obtained by adding a self-loop to each vertex in the graph. We analytically prove that lackadaisical quantum walks can find a unique marked vertex on any regular locally arc-transitive graph with constant success probability quadratically faster than the hitting time. This result proves several speculations and numerical findings in previous work, including the conjectures that the lackadaisical quantum walk finds a unique marked vertex with constant success probability on the torus, cycle, Johnson graphs, and other classes of vertex-transitive graphs. Our proof establishes and uses a relationship between lackadaisical quantum walks and quantum interpolated walks for any locally arc-transitive graph.

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arXiv2012

Quantum Nonlocal Boxes Exhibit Stronger Distillability

The hypothetical nonlocal box (\textsf{NLB}) proposed by Popescu and Rohrlich allows two spatially separated parties, Alice and Bob, to exhibit stronger than quantum correlations. If the generated correlations are weak, they can sometimes be distilled into a stronger correlation by repeated applications of the \textsf{NLB}. Motivated by the limited distillability of \textsf{NLB}s, we initiate here a study of the distillation of correlations for nonlocal boxes that output quantum states rather than classical bits (\textsf{qNLB}s). We propose a new protocol for distillation and show that it asymptotically distills a class of correlated quantum nonlocal boxes to the value $1/2 (3\sqrt{3}+1) \approx 3.098076$, whereas in contrast, the optimal non-adaptive parity protocol for classical nonlocal boxes asymptotically distills only to the value 3.0. We show that our protocol is an optimal non-adaptive protocol for 1, 2 and 3 \textsf{qNLB} copies by constructing a matching dual solution for the associated primal semidefinite program (SDP). We conclude that \textsf{qNLB}s are a stronger resource for nonlocality than \textsf{NLB}s. The main premise that develops from this conclusion is that the \textsf{NLB} model is not the strongest resource to investigate the fundamental principles that limit quantum nonlocality. As such, our work provides strong motivation to reconsider the status quo of the principles that are known to limit nonlocal correlations under the framework of \textsf{qNLB}s rather than \textsf{NLB}s.

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