Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer

• URL: http://arxiv.org/abs/2410.21201v1
• Date: Mon, 28 Oct 2024 16:48:21 GMT
• Title: Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer
• Authors: Qisheng Wang, Zhicheng Zhang,
• Abstract summary: Trace distance and infidelity, as basic measures of the closeness of quantum states, are commonly used in quantum state discrimination, certification, and tomography. We present a quantum algorithm that estimates the trace distance and square root fidelity between pure states to within additive error $varepsilon$, given sample access to their identical copies.
• Score: 7.319050391449301
• License:
• Abstract: Trace distance and infidelity (induced by square root fidelity), as basic measures of the closeness of quantum states, are commonly used in quantum state discrimination, certification, and tomography. However, the sample complexity for their estimation still remains open. In this paper, we solve this problem for pure states. We present a quantum algorithm that estimates the trace distance and square root fidelity between pure states to within additive error $\varepsilon$, given sample access to their identical copies. Our algorithm achieves the optimal sample complexity $\Theta(1/\varepsilon^2)$, improving the long-standing folklore $O(1/\varepsilon^4)$. Our algorithm is composed of a samplized phase estimation of the product of two Householder reflections. Notably, an improved (multi-)samplizer for pure states is used as an algorithmic tool in our construction, through which any quantum query algorithm using $Q$ queries to the reflection operator about a pure state $|\psi\rangle$ can be converted to a $\delta$-close (in the diamond norm) quantum sample algorithm using $\Theta(Q^2/\delta)$ samples of $|\psi\rangle$. This samplizer for pure states is shown to be optimal.

Related papers

• Near-Optimal Bounds for Learning Gaussian Halfspaces with Random Classification Noise [50.64137465792738]
  We show that any efficient SQ algorithm for the problem requires sample complexity at least $Omega(d1/2/(maxp, epsilon)2)$. Our lower bound suggests that this quadratic dependence on $1/epsilon$ is inherent for efficient algorithms.
  arXiv  Detail & Related papers  (2023-07-13T18:59:28Z)
• Quantum Metropolis-Hastings algorithm with the target distribution calculated by quantum Monte Carlo integration [0.0]
  Quantum algorithms for MCMC have been proposed, yielding the quadratic speedup with respect to the spectral gap $Delta$ compered to classical counterparts. We consider not only state generation but also finding a credible interval for a parameter, a common task in Bayesian inference. In the proposed method for credible interval calculation, the number of queries to the quantum circuit to compute $ell$ scales on $Delta$, the required accuracy $epsilon$ and the standard deviation $sigma$ of $ell$ as $tildeO(sigma/epsilon
  arXiv  Detail & Related papers  (2023-03-10T01:05:16Z)
• Fast Quantum Algorithms for Trace Distance Estimation [8.646488471216262]
  We propose efficient quantum algorithms for estimating the trace distance within additive error $varepsilon$ between mixed quantum states of rank $r$. We show that the decision version of low-rank trace distance estimation is $mathsfBQP$-complete.
  arXiv  Detail & Related papers  (2023-01-17T10:16:14Z)
• Quantum tomography using state-preparation unitaries [0.22940141855172028]
  We describe algorithms to obtain an approximate classical description of a $d$-dimensional quantum state when given access to a unitary. We show that it takes $widetildeTheta(d/varepsilon)$ applications of the unitary to obtain an $varepsilon$-$ell$-approximation of the state. We give an efficient algorithm for obtaining Schatten $q$-optimal estimates of a rank-$r$ mixed state.
  arXiv  Detail & Related papers  (2022-07-18T17:56:18Z)
• Quantum Approximation of Normalized Schatten Norms and Applications to Learning [0.0]
  This paper addresses the problem of defining a similarity measure for quantum operations that can be textitefficiently estimated We develop a quantum sampling circuit to estimate the normalized Schatten 2-norm of their difference and prove a Poly$(frac1epsilon)$ upper bound on the sample complexity. We then show that such a similarity metric is directly related to a functional definition of similarity of unitary operations using the conventional fidelity metric of quantum states.
  arXiv  Detail & Related papers  (2022-06-23T07:12:10Z)
• How to simulate quantum measurement without computing marginals [3.222802562733787]
  We describe and analyze algorithms for classically computation measurement of an $n$-qubit quantum state $psi$ in the standard basis. Our algorithms reduce the sampling task to computing poly(n)$ amplitudes of $n$-qubit states.
  arXiv  Detail & Related papers  (2021-12-15T21:44:05Z)
• Random quantum circuits transform local noise into global white noise [118.18170052022323]
  We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. For local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution $p_textnoisy$ of a generic noisy circuit instance shrink exponentially. If the noise is incoherent, the output distribution approaches the uniform distribution $p_textunif$ at precisely the same rate.
  arXiv  Detail & Related papers  (2021-11-29T19:26:28Z)
• 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)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)
• Quantum Algorithms for Simulating the Lattice Schwinger Model [63.18141027763459]
  We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In lattice units, we find a Schwinger model on $N/2$ physical sites with coupling constant $x-1/2$ and electric field cutoff $x-1/2Lambda$. We estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable---the mean pair density.
  arXiv  Detail & Related papers  (2020-02-25T19:18:36Z)
• Quantum Coupon Collector [62.58209964224025]
  We study how efficiently a $k$-element set $Ssubseteq[n]$ can be learned from a uniform superposition $|Srangle of its elements. We give tight bounds on the number of quantum samples needed for every $k$ and $n$, and we give efficient quantum learning algorithms.
  arXiv  Detail & Related papers  (2020-02-18T16:14:55Z)

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.