Optimal Quantum State Testing Even with Limited Entanglement

• URL: http://arxiv.org/abs/2604.07460v1
• Date: Wed, 08 Apr 2026 18:01:51 GMT
• Title: Optimal Quantum State Testing Even with Limited Entanglement
• Abstract summary: We consider the fundamental task of quantum state certification: given copies of an unknown quantum state $$, test whether it matches some target state $$ or is $$-far from it.<n>Our main result is a smooth copy complexity upper bound for state certification as a function of $t$, which achieves a near-optimal rate at $t = d2$.<n>We extend our techniques to develop new algorithms for the related tasks of mixedness testing and purity estimation, and show tradeoffs achieving the optimal rates for these problems at $t = d2$ as well.
• Score: 13.84229509611525
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this work, we consider the fundamental task of quantum state certification: given copies of an unknown quantum state $ρ$, test whether it matches some target state $σ$ or is $ε$-far from it. For certifying $d$-dimensional states, $Θ(d/ε^2)$ copies of $ρ$ are known to be necessary and sufficient. However, the algorithm achieving this complexity makes fully entangled measurements over all $O(d/ε^2)$ copies of $ρ$. Often, one is interested in certifying states to a high precision; this makes such joint measurements intractable even for low-dimensional states. Thus, we study whether one can obtain optimal rates for quantum state certification and related testing problems while only performing measurements on $t$ copies at once, for some $1 < t \ll d/ε^2$. While it is well-understood how to use intermediate entanglement to achieve optimal quantum state learning, the only protocol known to achieve optimal testing is the one using fully entangled measurements. Our main result is a smooth copy complexity upper bound for state certification as a function of $t$, which achieves a near-optimal rate at $t = d^2$. In the high-precision regime, i.e., for $ε< \frac{1}{\sqrt{d}}$, this is a strict improvement over the entanglement used by the aforementioned optimal protocol. We also extend our techniques to develop new algorithms for the related tasks of mixedness testing and purity estimation, and show tradeoffs achieving the optimal rates for these problems at $t = d^2$ as well. Our algorithms are based on novel reductions from testing to learning and leverage recent advances in quantum state tomography in a non-black-box fashion. We complement our upper bounds with smooth lower bounds that imply joint measurements on $t \geq d^{Ω(1)}$ copies are necessary to achieve optimal rates for certification in the high-precision regime.

Related papers

• Robust Average-Reward Markov Decision Processes: Minimax-Optimal Learning via Plug-in Reductions [51.50375419691955]
  Distributionally robust Markov decision processes provide a principled framework for sequential decision making under model uncertainty.<n>We study how many samples are necessary and sufficient to learn an $varepsilon$-optimal robust policy under the average-reward criterion.
  arXiv  Detail & Related papers  (2026-08-06T19:49:48Z)
• Adiabatic Quantum Phase Estimation [47.3103723337301]
  Quantum phase estimation (QPE) is a central algorithmic primitive that estimates eigenvalues of a Hamiltonian up to precision $$ in Heisenberg-limited time $T= (1/)$.<n>Standard gate-based implementations of QPE require deep controlled time-evolution circuits and are not native to analog hardware.<n>We present a simple adiabatic protocol for QPE that achieves the optimal Heisenberg-limited scaling $T = Oleft( frac1 logleft(-1right)right)$ in both the precision $$
  arXiv  Detail & Related papers  (2026-05-21T17:31:36Z)
• High-Probability Convergence in Decentralized Stochastic Optimization with Gradient Tracking [69.90407799170687]
  We study high-probability (HP) convergence guarantees in decentralized optimization.<n>Our results establish the same conditions on the horizon as in achieving comparable times.
  arXiv  Detail & Related papers  (2026-04-30T22:45:21Z)
• The Power of Two Bases: Robust and copy-optimal certification of nearly all quantum states with few-qubit measurements [5.266360532660872]
  We present robust certification protocols based on few-qubit measurements.<n>Our tests are based on a new uncertainty principle for conditional fidelities.
  arXiv  Detail & Related papers  (2026-02-12T06:08:04Z)
• Quantum Algorithms for Finite-horizon Markov Decision Processes [40.812944518646006]
  We design quantum algorithms that are more efficient than classical algorithms to solve time-dependent and finite-horizon Markov Decision Processes (MDPs)<n>In the exact dynamics setting, we prove that our $textbfQVI-1$ algorithm achieves a quadratic speedup in the size of the action space $(A)$.<n>In the generative model setting, we prove that our algorithms $textbfQVI-3$ and $textbfQVI-4$ achieve improvements in sample complexity over the state-of-the-art (SOTA) classical algorithm.
  arXiv  Detail & Related papers  (2025-08-07T09:00:23Z)
• Instance-Optimal Quantum State Certification with Entangled Measurements [1.8416014644193066]
  We prove nearly instance-optimal bounds for quantum state certification when the tester can perform fully entangled measurements.<n>We prove our lower bounds using a novel quantum analogue of the Ingster-Suslina method, which is likely to be of independent interest.
  arXiv  Detail & Related papers  (2025-07-08T14:15:46Z)
• Pauli measurements are not optimal for single-copy tomography [34.83118849281207]
  We prove a stronger upper bound of $O(frac10Nepsilon2)$ and a lower bound of $Omega(frac9.118Nepsilon2)$.<n>This demonstrates the first known separation between Pauli measurements and structured POVMs.
  arXiv  Detail & Related papers  (2025-02-25T13:03:45Z)
• An optimal tradeoff between entanglement and copy complexity for state tomography [24.737530909081915]
  We study tomography in the natural setting where one can make measurements of $t$ copies at a time. This is the first smooth entanglement-copy protocol known for any quantum learning task. A key insight is to use SchurilonWeyl sampling not to estimate the spectrum of $rho$, but to estimate the deviation of $rho$ from the maximally mixed state.
  arXiv  Detail & Related papers  (2024-02-26T07:18:57Z)
• The role of shared randomness in quantum state certification with unentangled measurements [36.19846254657676]
  We study quantum state certification using unentangled quantum measurements. $Theta(d2/varepsilon2)$ copies are necessary and sufficient for state certification. We develop a unified lower bound framework for both fixed and randomized measurements.
  arXiv  Detail & Related papers  (2024-01-17T23:44:52Z)
• Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints [62.72309460291971]
  We introduce a variational quantum algorithm for Goemans-Williamson algorithm that uses only $n+1$ qubits. Efficient optimization is achieved by encoding the objective matrix as a properly parameterized unitary conditioned on an auxilary qubit. We demonstrate the effectiveness of our protocol by devising an efficient quantum implementation of the Goemans-Williamson algorithm for various NP-hard problems.
  arXiv  Detail & Related papers  (2022-06-30T03:15:23Z)
• Tight Bounds for Quantum State Certification with Incoherent Measurements [18.566266990940374]
  When $sigma$ is the maximally mixed state $frac1d I_d$, this is known as mixedness testing. We focus on algorithms which use incoherent measurements, i.e. which only measure one copy of $rho$ at a time.
  arXiv  Detail & Related papers  (2022-04-14T17:59:31Z)
• Improved Sample Complexity for Incremental Autonomous Exploration in MDPs [132.88757893161699]
  We learn the set of $epsilon$-optimal goal-conditioned policies attaining all states that are incrementally reachable within $L$ steps. DisCo is the first algorithm that can return an $epsilon/c_min$-optimal policy for any cost-sensitive shortest-path problem.
  arXiv  Detail & Related papers  (2020-12-29T14:06:09Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.