Simultaneous Estimation of Nonlinear Functionals of a Quantum State

• URL: http://arxiv.org/abs/2505.16715v1
• Date: Thu, 22 May 2025 14:23:48 GMT
• Title: Simultaneous Estimation of Nonlinear Functionals of a Quantum State
• Abstract summary: We consider a fundamental task in quantum information theory, estimating the values of $operatornametr(Orho)$, $operatornametr(Orho2)$,..., $operatornametr(Orhok)$ for an observable $O$ and a quantum state $rho$.<n>We show that $widetildeTheta(k)$ samples of $rho$ are sufficient to simultaneously estimate all the $k$ values.
• Score: 12.191929463091354
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider a fundamental task in quantum information theory, estimating the values of $\operatorname{tr}(O\rho)$, $\operatorname{tr}(O\rho^2)$, ..., $\operatorname{tr}(O\rho^k)$ for an observable $O$ and a quantum state $\rho$. We show that $\widetilde\Theta(k)$ samples of $\rho$ are sufficient and necessary to simultaneously estimate all the $k$ values. This means that estimating all the $k$ values is almost as easy as estimating only one of them, $\operatorname{tr}(O\rho^k)$. As an application, our approach advances the sample complexity of entanglement spectroscopy and the virtual cooling for quantum many-body systems. Moreover, we extend our approach to estimating general functionals by polynomial approximation.

Related papers

• Quantum Speedups for Stochastic Optimization with Heavy-Tailed Noise [49.730496294398726]
  We develop a novel quantum mean estimator for heavy-tailed random variables.<n>We derive stronger lower bounds for tail index>4/3$, showing that a nontrivial dependence on a dimension is unavoidable.<n>For a convex objective function, we propose a quantum projected gradient descent method.
  arXiv  Detail & Related papers  (2026-07-28T09:29:37Z)
• Faster quantum linear system solver beyond the condition number [50.84794327094274]
  We present two quantum algorithms that produce the normalized solution $|xrangle$ of linear system $Ax=| b rangle$ to accuracy $$ with complexity independent of the condition number $=lVert A-1rVert$.<n>Our filtering-based solver is extremely simple with a favorable runtime prefactor.
  arXiv  Detail & Related papers  (2026-07-08T17:49:40Z)
• Towards Minimax Estimation of High-Order Functionals by Quantum Arguments [13.491187998442596]
  We present a novel approach to the minimax estimation of high-order functionals from the perspective of quantum computing.<n>Our estimators are constructed under a unified framework using quantum primitives and run on a quantum computer.
  arXiv  Detail & Related papers  (2026-07-08T15:38:24Z)
• On estimating operator norm distance, with optimal trace distance estimation when one state is pure [14.360970343875588]
  We investigate the computational complexity of estimating the operator norm distance $rm T_infty(_infty), defined via the operator norm $|A|_infty = _max(A)$.<n>We provide efficient quantum estimators for the operator norm distance whose complexity is independent of the rank of the states.
  arXiv  Detail & Related papers  (2026-07-04T14:59:03Z)
• Optimal Scalar Quantization for Matrix Multiplication: Closed-Form Density and Phase Transition [50.36362492608702]
  We study entrywise scalar quantization of two matrices prior to multiplication.<n>We obtain a closed-form optimal point density [ star(u) propto exp!left(-fracu26right)bigl( (1-2)+2u22bigr), qquad u=fracx_X,, and prove a correlation-driven phase transition.
  arXiv  Detail & Related papers  (2026-03-20T01:53:44Z)
• Sublinear Time Quantum Sensitivity Sampling [57.356528942341534]
  We present a unified framework for quantum sensitivity sampling, extending the advantages of quantum computing to a broad class of classical approximation problems.<n>Our framework provides a streamlined approach for constructing coresets and offers significant runtime improvements in applications such as clustering, regression, and low-rank approximation.
  arXiv  Detail & Related papers  (2025-09-20T20:18:49Z)
• Improved Sample Upper and Lower Bounds for Trace Estimation of Quantum State Powers [9.136389487369117]
  We significantly improve the sample complexity of estimating $operatornametr(rhoq)$ in both the upper and lower bounds.<n>Our upper bounds are obtained by (non-plug-in) quantum estimators based on weak Schur sampling.
  arXiv  Detail & Related papers  (2025-05-14T17:06:33Z)
• Resource-efficient algorithm for estimating the trace of quantum state powers [1.5133368155322298]
  Estimating the trace of quantum state powers, $textTr(rhok)$, for $k$ identical quantum states is a fundamental task.<n>We introduce an algorithm that requires only $mathcalO(tilder)$ qubits and $mathcalO(tilder)$ multi-qubit gates.<n>We extend our algorithm to the estimation of $textTr(rhok)$ for arbitrary observables and $textTr(rhok)
  arXiv  Detail & Related papers  (2024-08-01T06:23:52Z)
• Measuring quantum relative entropy with finite-size effect [53.64687146666141]
  We study the estimation of relative entropy $D(rho|sigma)$ when $sigma$ is known.<n>Our estimator attains the Cram'er-Rao type bound when the dimension $d$ is fixed.
  arXiv  Detail & Related papers  (2024-06-25T06:07:20Z)
• Time-Efficient Quantum Entropy Estimator via Samplizer [7.319050391449301]
  Estimating the entropy of a quantum state is a basic problem in quantum information. We introduce a time-efficient quantum approach to estimating the von Neumann entropy $S(rho)$ and R'enyi entropy $S_alpha(rho)$ of an $N$-dimensional quantum state $rho.
  arXiv  Detail & Related papers  (2024-01-18T12:50:20Z)
• Asymptotically Optimal Quantum Amplitude Estimation by Generalized Qubitization [5.0755851789013535]
  We first show that the standard deviationally lower bounded by approximately $1.28 L-1$, where $L$ is the number of queries. Then we propose a generalized qubitization that can block-encode several functions simultaneously, and show how it can help estimating quantum amplitude to achieve the optimal accuracy.
  arXiv  Detail & Related papers  (2023-06-29T05:31:52Z)
• Enlarging the notion of additivity of resource quantifiers [62.997667081978825]
  Given a quantum state $varrho$ and a quantifier $cal E(varrho), it is a hard task to determine $cal E(varrhootimes N)$. We show that the one shot distillable entanglement of certain spherically symmetric states can be quantitatively approximated by such an augmented additivity.
  arXiv  Detail & Related papers  (2022-07-31T00:23:10Z)
• Quantum Resources Required to Block-Encode a Matrix of Classical Data [56.508135743727934]
  We provide circuit-level implementations and resource estimates for several methods of block-encoding a dense $Ntimes N$ matrix of classical data to precision $epsilon$. We examine resource tradeoffs between the different approaches and explore implementations of two separate models of quantum random access memory (QRAM) Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms.
  arXiv  Detail & Related papers  (2022-06-07T18:00:01Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• Simplest non-additive measures of quantum resources [77.34726150561087]
  We study measures that can be described by $cal E(rhootimes N) =E(e;N) ne Ne$.
  arXiv  Detail & Related papers  (2021-06-23T20:27:04Z)
• Improved quantum data analysis [1.8416014644193066]
  We give a quantum "Threshold Search" algorithm that requires only $O(log2 m)/epsilon2)$ samples of a $d$-dimensional state. We also give an alternative Hypothesis Selection method using $tildeO((log3 m)/epsilon2)$ samples.
  arXiv  Detail & Related papers  (2020-11-22T01:22:37Z)
• Hutch++: Optimal Stochastic Trace Estimation [75.45968495410048]
  We introduce a new randomized algorithm, Hutch++, which computes a $(1 pm epsilon)$ approximation to $tr(A)$ for any positive semidefinite (PSD) $A$. We show that it significantly outperforms Hutchinson's method in experiments.
  arXiv  Detail & Related papers  (2020-10-19T16:45:37Z)
• 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)
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits [0.755972004983746]
  Variational quantum algorithms (VQAs) optimize the parameters $vectheta$ of a parametrized quantum circuit. We prove two results, assuming $V(vectheta)$ is an alternating layered ansatz composed of blocks forming local 2-designs. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.
  arXiv  Detail & Related papers  (2020-01-02T18:18:25Z)

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.