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arXiv:2505.00698 · Physical Review A

Heisenberg-limited quantum algorithms for multiple observables estimation

Yuki Koizumi, Kaito Wada, Wataru Mizukami, Nobuyuki Yoshioka

Published Jan 15, 2026• 1 citation
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Abstract

In the accompanying paper of arXiv:2505.00697, we have presented a generalized scheme of adaptive quantum gradient estimation (QGE) algorithm, and further proposed two practical variants which not only achieve doubly quantum enhancement in query complexity regarding estimation precision and number of observables, but also enable minimal cost to estimate $k$-RDMs in fermionic systems among existing quantum algorithms. Here, we provide full descriptions on the algorithm, and provide theoretical guarantee for the estimation precision in terms of the root mean squared error. Furthermore, we analyze the performance of the quantum amplitude estimation algorithm, another variant of the Heisenberg-limited scaling algorithm, and show how the estimation error is minimized under the circuit structure that resembles the phase estimation algorithm. We finally describe the details for the numerical evaluation of the query complexity of the Heisenberg-limited algorithms and sampling-based methods to make a thorough comparison in the task of estimating fermionic $k$-RDMs.

DOI
10.1103/wrkp-qd33

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•Jul 9, 2021•0 citations
In the accompanying paper of arXiv:2505.00697, we have presented a generalized scheme of adaptive quantum gradient estimation (QGE) algorithm, and further proposed two practical variants which not only achieve doubly quantum enhancement in query complexity regarding estimation precision and number of observables, but also enable minimal cost to estimate kkk-RDMs in fermionic systems among existing quantum algorithms. Here, we provide full descriptions on the algorithm, and provide theoretical guarantee for the estimation precision in terms of the root mean squared error. Furthermore, we analyze the performance of the quantum amplitude estimation algorithm, another variant of the Heisenberg-limited scaling algorithm, and show how the estimation error is minimized under the circuit structure that resembles the phase estimation algorithm. We finally describe the details for the numerical evaluation of the query complexity of the Heisenberg-limited algorithms and sampling-based methods to make a thorough comparison in the task of estimating fermionic kkk-RDMs.
Achieving quantum advantage in efficiently estimating collective properties of quantum many-body systems remains a fundamental goal in quantum computing. While the quantum gradient estimation (QGE) algorithm has been shown to achieve doubly quantum enhancement in the precision and the number of observables, it remains unclear whether one benefits in practical applications. In this Letter, we present a generalized framework of the adaptive QGE algorithm and further propose two variants which enable us to estimate the collective properties of fermionic systems using the smallest cost among existing quantum algorithms. The first method utilizes the symmetry inherent in the target state, and the second method enables estimation in a single-shot manner using the parallel scheme. We show that our proposal offers a quadratic speedup compared with prior QGE algorithms in the task of fermionic partial tomography for systems with limited particle numbers. Furthermore, we provide numerical demonstrations showing that, for a problem of estimating 2-body fermionic reduced density matrices, our proposals improve the number of queries to the target state preparation oracle by a factor of 4.4 for the nitrogenase FeMo cofactor and by a factor of 7.8 for Fermi-Hubbard model of 200 sites in chemical accuracy.
In quantum mechanics, measuring the expectation value of a general observable has an inherent statistical uncertainty that is quantified by variance or mean squared error of measurement outcome. While the uncertainty can be reduced by averaging several samples, the number of samples should be minimized when each sample is very costly. This is especially the case for fault-tolerant quantum computing that involves measurement of multiple observables of nontrivial states in large quantum systems that exceed the capabilities of classical computers. In this work, we provide an adaptive quantum algorithm for estimating the expectation values of M general observables within root mean squared error ε simultaneously, using O ( ε − 1 M log  M ) queries to a state preparation oracle of a target state. This remarkably achieves the scaling of Heisenberg limit 1 / ε , a fundamental bound on the estimation precision in terms of mean squared error, together with the sublinear scaling of the number of observables M . The proposed method is an adaptive version of the quantum gradient-estimation algorithm and has a resource-efficient implementation due to its adaptiveness. Specifically, the space overhead in the proposed method is O ( M ) , which is independent from the estimation precision ε unlike noniterative algorithms. In addition, our method can avoid the numerical instability problem for constructing quantum circuits in a large-scale task (e.g., ε ≪ 1 in our case), which appears in the actual implementation of many algorithms relying on quantum signal processing techniques. Our method paves a new way to precisely understand and predict various physical properties in complicated quantum systems using quantum computers.
Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues of exponentially-large sparse matrices.The maximum rate at which these eigenvalues may be learned, –known as the Heisenberg limit–, is constrained by bounds on the circuit complexity required to simulate an arbitrary Hamiltonian. Single-control qubit variants of quantum phase estimation that do not require coherence between experiments have garnered interest in recent years due to lower circuit depth and minimal qubit overhead. In this work we show that these methods can achieve the Heisenberg limit, a l s o when one is unable to prepare eigenstates of the system. Given a quantum subroutine which provides samples of a `phase function' g ( k ) = ∑ j A j e i ϕ j k with unknown eigenphases ϕ j and overlaps A j at quantum cost O ( k ) , we show how to estimate the phases { ϕ j } with (root-mean-square) error δ for total quantum cost T = O ( δ − 1 ) . Our scheme combines the idea of Heisenberg-limited multi-order quantum phase estimation for a single eigenvalue phase [Higgins et al (2009) and Kimmel et al (2015)] with subroutines with so-called dense quantum phase estimation which uses classical processing via time-series analysis for the QEEP problem [Somma (2019)] or the matrix pencil method. For our algorithm which adaptively fixes the choice for k in g ( k ) we prove Heisenberg-limited scaling when we use the time-series/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberg-limited scaling as well.
The Non-Orthogonal Quantum Eigensolver (NOQE) provides an accurate framework for electronic-structure calculations, but the estimation of its Hamiltonian and overlap matrix elements relies on sampling and requires O(1/ε2)O(1/\varepsilon^2)O(1/ε2) circuit repetitions to achieve additive precision ε\varepsilonε. Here, we reformulate this matrix-element estimation step as a collection of amplitude-estimation tasks and integrate iterative quantum amplitude estimation into the NOQE workflow. The resulting protocol achieves near-Heisenberg query complexity O(1/ε)O(1/\varepsilon)O(1/ε) for these estimation tasks, by replacing incoherent statistical averaging with coherent amplitude amplification. We present explicit circuit constructions and the corresponding implementation procedure. Numerical simulations for the electronic states of the hydrogen molecule show that the proposed method reaches chemical accuracy with substantially fewer total queries than the original sampling-based protocol. Overall, this work provides a measurement-efficient route to high-precision energy estimation and illustrates how sampling-limited quantum algorithms can be systematically reformulated to leverage quantum coherence and achieve lower measurement costs.
Quantum kernel methods are among the leading candidates for achieving quantum advantage in supervised learning. A key bottleneck is the cost of inference: evaluating a trained model on new data requires estimating a weighted sum ∑i=1Nαik(x,xi)\sum_{i=1}^N α_i k(x,x_i)∑i=1N​αi​k(x,xi​) of NNN kernel values to additive precision ε\varepsilonε, where ααα is the vector of trained coefficients. The standard approach estimates each term independently via sampling, yielding a query complexity of O(N∥α∥22/ε2)O(N\lVertα\rVert_2^2/\varepsilon^2)O(N∥α∥22​/ε2). In this work, we identify two independent axes for improvement: (1) How individual kernel values are estimated (sampling versus quantum amplitude estimation), and (2) how the sum is approximated (term-by-term versus via a single observable), and systematically analyze all combinations thereof. The query-optimal combination, encoding the full inference sum as the expectation value of a single observable and applying quantum amplitude estimation, achieves a query complexity of O(∥α∥1/ε)O(\lVertα\rVert_1/\varepsilon)O(∥α∥1​/ε), removing the dependence on NNN from the query count and yielding a quadratic improvement in both ∥α∥1\lVertα\rVert_1∥α∥1​ and ε\varepsilonε. We prove a matching lower bound of Ω(∥α∥1/ε)Ω(\lVertα\rVert_1/\varepsilon)Ω(∥α∥1​/ε), establishing query-optimality of our approach up to logarithmic factors. Beyond query complexity, we also analyze how these improvements translate into gate costs and show that the query-optimal strategy is not always optimal in practice from the perspective of gate complexity. Our results provide both a query-optimal algorithm and a practically optimal choice of strategy depending on hardware capabilities, along with a complete landscape of intermediate methods to guide practitioners. All algorithms require only amplitude estimation as a subroutine and are thus natural candidates for early-fault-tolerant implementations.
Quantum phase estimation is a cornerstone in quantum algorithm design, allowing for the inference of eigenvalues of exponentially-large sparse matrices. The maximum rate at which these eigenvalues may be learned, --known as the Heisenberg limit--, is constrained by bounds on the circuit depth required to simulate an arbitrary Hamiltonian. Single-control qubit variants of quantum phase estimation have garnered interest in recent years due to lower circuit depth and minimal qubit overhead. In this work we show that these methods can achieve the Heisenberg limit, {\em also} when one is unable to prepare eigenstates of the system. Given a quantum subroutine which provides samples of a `phase function' g(k)=∑jAjeiϕjkg(k)=\sum_j A_j e^{i \phi_j k}g(k)=∑j​Aj​eiϕj​k with unknown eigenvalue phases ϕj\phi_jϕj​ and probabilities AjA_jAj​ at quantum cost O(k)O(k)O(k), we show how to estimate the phases {ϕj}\{\phi_j\}{ϕj​} with accuracy (root-mean-square) error δ\deltaδ for total quantum cost T=O(δ−1)T=O(\delta^{-1})T=O(δ−1). Our scheme combines the idea of Heisenberg-limited multi-order quantum phase estimation for a single eigenvalue phase \cite{Higgins09Demonstrating,Kimmel15Robust} with subroutines with so-called dense quantum phase estimation which uses classical processing via time-series analysis for the QEEP problem \cite{Somma19Quantum} or the matrix pencil method. For our algorithm which adaptively fixes the choice for kkk in g(k)g(k)g(k) we prove Heisenberg-limited scaling when we use the time-series/QEEP subroutine. We present numerical evidence that using the matrix pencil technique the algorithm can achieve Heisenberg-limited scaling as well.