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Oracle problems as communication tasks and optimization of quantum algorithms

Amit Te'eni, Zohar Schwartzman-Nowik, Marcin Nowakowski, Paweł Horodecki, Eliahu Cohen

TL;DR

The paper reframes single-query quantum oracle problems as quantum communication tasks, using mutual information $I(J;Y)$ to quantify algorithm performance and drawing a precise analogy to Holevo-type bounds. A central theorem links optimal single-query performance to the discord between the oracle subsystem and the computer, with the post-query measurement basis chosen to minimize discord, thereby maximizing $I(J;Y)$; a complementary lower bound is tied to coherence. The framework is illustrated across canonical algorithms (Deutsch--Jozsa, Bernstein--Vazirani, Simon, phase estimation, and Shor--Kitaev/HSP), showing how the information-theoretic quantities evolve through pre-query, post-query, and final states, and highlighting when quantum resources provide a genuine advantage. The results offer physical intuition for quantum speedups, inform design of non-adaptive multi-query strategies, and suggest practical implications for hybrid quantum–classical learning tasks and Hamiltonian learning. Overall, the work connects information-theoretic quantities in quantum systems to the structure and performance of quantum oracle algorithms, with potential impact on both theory and applications in quantum information processing.

Abstract

Quantum query complexity studies the number of queries needed to learn some property of a black box. A closely related question is how well an algorithm can succeed with this learning task using only a fixed number of queries. In this work, we propose measuring an algorithm's performance using the mutual information between the output and the actual value. The task of optimizing this mutual information using a single query, is similar to a basic task of quantum communication, where one attempts to maximize the mutual information of the sender and receiver. We make this analogy precise by splitting the algorithm between two agents, obtaining a communication protocol. The oracle's target property plays the role of a message that Alice encodes into a quantum state, which is subsequently sent over to Bob. The first part of the algorithm performs this encoding, and the second part measures the state and aims to deduce the message from the outcome. Moreover, we formally consider the oracle as a separate subsystem, whose state records the unknown oracle identity. Within this construction, Bob's optimal measurement basis minimizes the quantum correlations between the two subsystems. We also find a lower bound on the mutual information, which is related to quantum coherence. These results extend to multiple-query algorithms. As a result, we describe the optimal non-adaptive algorithm that uses at most a fixed number of queries, for any oracle classification problem. We demonstrate our results by studying several well-known algorithms through the proposed framework. Finally, we discuss some practical implications of our results.

Oracle problems as communication tasks and optimization of quantum algorithms

TL;DR

The paper reframes single-query quantum oracle problems as quantum communication tasks, using mutual information to quantify algorithm performance and drawing a precise analogy to Holevo-type bounds. A central theorem links optimal single-query performance to the discord between the oracle subsystem and the computer, with the post-query measurement basis chosen to minimize discord, thereby maximizing ; a complementary lower bound is tied to coherence. The framework is illustrated across canonical algorithms (Deutsch--Jozsa, Bernstein--Vazirani, Simon, phase estimation, and Shor--Kitaev/HSP), showing how the information-theoretic quantities evolve through pre-query, post-query, and final states, and highlighting when quantum resources provide a genuine advantage. The results offer physical intuition for quantum speedups, inform design of non-adaptive multi-query strategies, and suggest practical implications for hybrid quantum–classical learning tasks and Hamiltonian learning. Overall, the work connects information-theoretic quantities in quantum systems to the structure and performance of quantum oracle algorithms, with potential impact on both theory and applications in quantum information processing.

Abstract

Quantum query complexity studies the number of queries needed to learn some property of a black box. A closely related question is how well an algorithm can succeed with this learning task using only a fixed number of queries. In this work, we propose measuring an algorithm's performance using the mutual information between the output and the actual value. The task of optimizing this mutual information using a single query, is similar to a basic task of quantum communication, where one attempts to maximize the mutual information of the sender and receiver. We make this analogy precise by splitting the algorithm between two agents, obtaining a communication protocol. The oracle's target property plays the role of a message that Alice encodes into a quantum state, which is subsequently sent over to Bob. The first part of the algorithm performs this encoding, and the second part measures the state and aims to deduce the message from the outcome. Moreover, we formally consider the oracle as a separate subsystem, whose state records the unknown oracle identity. Within this construction, Bob's optimal measurement basis minimizes the quantum correlations between the two subsystems. We also find a lower bound on the mutual information, which is related to quantum coherence. These results extend to multiple-query algorithms. As a result, we describe the optimal non-adaptive algorithm that uses at most a fixed number of queries, for any oracle classification problem. We demonstrate our results by studying several well-known algorithms through the proposed framework. Finally, we discuss some practical implications of our results.
Paper Structure (20 sections, 1 theorem, 39 equations, 1 figure, 5 tables)

This paper contains 20 sections, 1 theorem, 39 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Single-query oracle classification
  3. Single-query algorithms
  4. Remarks:
  5. The Deutsch--Jozsa algorithm - $QDJ \left( k \right)$
  6. Non-adaptive algorithms
  7. Main contributions
  8. Results
  9. Quantum discord and optimal quantum algorithms
  10. Coherence and a lower bound on the mutual information
  11. Simulation results for Deutsch--Jozsa
  12. Information-theoretic quantities for Bernstein--Vazirani
  13. The Shor--Kitaev algorithm
  14. Pre-query
  15. Post-query
  16. ...and 5 more sections

Key Result

Theorem 1

For any oracle problem of the form defined in oracle_problems and any fixed value of $n \geq m$, a single-query quantum algorithm of the form defined in algo that uses $n$ qubits obtains the maximal value of $I \left( J;Y \right)$ out of all such algorithms (i.e. is optimal) iff the following condit

Figures (1)

  • Figure 1: The analogy with quantum communication: first, Alice initializes the quantum computer to be in the state $\ket{\psi_0}$ and chooses an oracle $f$. Then she performs the first unitary gate $V$ on the computer, followed by the controlled gate $U$. As a result, the state of the computer is $U_f V \ket{\psi_0}$, which she subsequently sends to Bob. Bob is then tasked with choosing a gate $W$ to perform on the state, before measuring it in the computational basis, such that he would optimize the information he obtains on Alice's chosen oracle $f$ or its class $j$.

Theorems & Definitions (2)

  • Theorem
  • Remark