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.
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.
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.
Quantum computers have great potential to uncover new and unexpected physical phenomena.They can simulate classically intractable many-particle quantum systems and probe them in a programmable, computational manner.They can also process experimental quantum data directly, enabling us to probe unseen aspects of physical systems that cannot be reached by any classical means.In both roles, refining quantum algorithms toward their fundamental limits expands and sharpens the reach of quantum computers.Among quantum algorithms, quantum estimation is of central importance because it elicits the physical properties of interest from quantum systems.Even in an ideal, noiseless setting, however, obtaining precise estimates of such properties is significantly resource-intensive, unless one leverages coherent quantum operations.While many quantum-enhanced (i.e., higher performance than any classical means enabled by quantum effects) estimation algorithms have been proposed under reasonable access assumptions, it is still an ongoing quest to design quantum algorithms achieving ultimate estimation efficiency allowed by quantum mechanics.In addition, rapid experimental progress makes us envision an early fault-tolerant regime with large-scale yet not fully error-corrected quantum computers.Toward this regime, it is also crucial to develop quantum estimation algorithms that elicit as much information of interest as possible from a quantum system subject to noise.In these contexts, this thesis advances quantum-enhanced algorithms for estimating physical properties in two respects: Heisenberg-limited multi-observable estimation and single-observable estimation for the early faulttolerant quantum computing (FTQC) regime.First, we provide an adaptive quantum algorithm for estimating expectation values of multiple observables simultaneously in an ideal (noiseless) setting.Given common access to block-encoded observables, a state-preparation unitary, and its inverse, this algorithm achieves a double quantum enhancement in query complexity to the state preparation and its inverse: the mean-squared-error (MSE) scaling at the Heisenberg limit and sublinear scaling in the number of observables.It also essentially matches the query lower bound in the small-additive-error regime (not in MSE), previously derived in quantum complexity theory.In addition, the proposed algorithm significantly reduces the number of ancilla qubits and the classical computation for circuit synthesis, compared to prior art for this task.The double enhancement and the circuit simplification come from a novel algorithm design that integrates key ideas from adaptive phase estimation, quantum amplification, and quantum gradient estimation.Next, we focus on expectation value estimation for a single observable in the early FTQC regime.In this regime, available coherent resources are mostly consumed for a state preparation unitary, and the dominant noise stems from that costly unitary and its inverse.Under a specific model reflecting these aspects, we derive the quantum-enhanced precision limit, that is, the minimal quantum Cramér-Rao bound over all possible estimation strategies in the model.Then, by designing an adaptive optimization based on Fisher information, we propose a quantum-enhanced estimation algorithm, with rigorous estimation guarantees, that asymptotically achieves the quantum Cramér-Rao bound, when the target system size is sufficiently large.Finally, we numerically demonstrate that the asymptotic behavior holds with only a few hundred measurements per step and a few tens of qubits.
Many quantum algorithms involve the evaluation of expectation values. Optimal strategies for estimating a single expectation value are known, requiring a number of state preparations that scales with the target error ϵ as O(1/ϵ). In this Letter, we address the task of estimating the expectation values of M different observables, each to within additive error ϵ, with the same 1/ϵ dependence. We describe an approach that leverages Gilyén et al.'s quantum gradient estimation algorithm to achieve O(sqrt[M]/ϵ) scaling up to logarithmic factors, regardless of the commutation properties of the M observables. We prove that this scaling is worst-case optimal in the high-precision regime if the state preparation is treated as a black box, even when the operators are mutually commuting. We highlight the flexibility of our approach by presenting several generalizations, including a strategy for accelerating the estimation of a collection of dynamic correlation functions.
Randomized quantum algorithms for the estimation of expectation values of molecular observables offer a promising route to early fault-tolerant quantum advantage by trading circuit depth for an increased number of samples. However, this begs the question whether the resulting measurement overhead remains feasible for industrially relevant molecular systems. In this work, we benchmark the randomized sampling algorithm of Wang et al. [1] through numerical simulations on a hydrogen molecule and resource estimates for various active spaces of the cytochrome P450, an industry-relevant system in the pharmaceutical industry. Our numerical simulations reveal that the analytical error bounds significantly overestimate the actual sampling error, whereas the analytical bound on the required imaginary evolution time is comparatively tight, exceeding a direct DMRG estimate by roughly 44% for the active space where the comparison is feasible. The measurement overhead nevertheless remains prohibitively high: even for the smallest active spaces, the expectation-value estimation alone requires around 1010 samples, scaling to beyond 1015 for the largest active spaces considered. These results suggest that substantial improvements in measurement strategies or algorithmic scaling are necessary before randomized early-fault-tolerant algorithms become competitive for industrial quantum chemistry applications.
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.
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)=∑jAjeiϕjk with unknown eigenvalue phases ϕj and probabilities Aj at quantum cost O(k), we show how to estimate the phases {ϕj} with accuracy (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 \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 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.