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arXiv:2503.10656

The Quantum Cramér-Rao lower bound (Why quantum computers won't work I)

Liam Paul McGuinness

Published Mar 8, 2025• 0 citations
arXivHTMLPDF

Abstract

Quantum information science currently poses a troubling contradiction. It can be summarized as: (1) To factor efficiently, quantum computers must perform exponentially precise energy estimation. (2) Exponentially precise energy estimation is impossible according to both the Heisenberg limit and the Cramér-Rao lower bound in quantum metrology. It is surprising that such a dramatic contradiction exists between two accepted predictions of quantum mechanics, and yet this contradiction it is not widely discussed. It is even more surprising when one notes it is not a minor discrepancy -- the two statements differ by an exponential margin. Not only that, whether (1) or (2) is correct is of fundamental importance to the realisation of an important class of quantum technologies. If (2) is correct, then quantum computers are much less powerful than expected. This work resolves the above contradiction by defining a computational model in which a wide range of computational problems are not solvable in polynomial time. We then show that this computational model applies to the majority of quantum algorithms, including Shor's algorithm.

DOI
10.48550/arxiv.2503.10656

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Quantum information science currently poses a troubling contradiction. It can be summarized as: (1) To factor efficiently, quantum computers must perform exponentially precise energy estimation. (2) Exponentially precise energy estimation is impossible according to both the Heisenberg limit and the Cramér-Rao lower bound in quantum metrology. It is surprising that such a dramatic contradiction exists between two accepted predictions of quantum mechanics, and yet this contradiction it is not widely discussed. It is even more surprising when one notes it is not a minor discrepancy -- the two statements differ by an exponential margin. Not only that, whether (1) or (2) is correct is of fundamental importance to the realisation of an important class of quantum technologies. If (2) is correct, then quantum computers are much less powerful than expected. This work resolves the above contradiction by defining a computational model in which a wide range of computational problems are not solvable in polynomial time. We then show that this computational model applies to the majority of quantum algorithms, including Shor's algorithm.
The quantum Fisher information constrains the achievable precision in parameter estimation via the quantum Cramér-Rao bound, which has attracted much attention in Hermitian systems since the 60s of the last century. However, less attention has been paid to non-Hermitian systems. In this Letter, working with different logarithmic operators, we derive two previously unknown expressions for quantum Fisher information, and two Cramér-Rao bounds lower than the well-known one are found for non-Hermitian systems. These lower bounds are due to the merit of non-Hermitian observable and it can be understood as a result of extended regimes of optimization. Two experimentally feasible examples are presented to illustrate the theory, saturation of these bounds and estimation precisions beyond the Heisenberg limit are predicted and discussed. A setup to measure non-Hermitian observable is also proposed.
Abstract Sensing and imaging are among the most important applications of quantum information science. To investigate their fundamental limits and the possibility of quantum enhancements, for decades researchers have relied on the quantum Cramér-Rao lower error bounds pioneered by Helstrom. Recent work, however, has called into question the tightness of those bounds for highly nonclassical states in the non-asymptotic regime, and better methods are now needed to assess the attainable quantum limits in reality. Here we propose a new class of quantum bounds called quantum Weiss-Weinstein bounds, which include Cramér-Rao-type inequalities as special cases but can also be significantly tighter to the attainable error. We demonstrate the superiority of our bounds through the derivation of a Heisenberg limit and phase estimation examples.
Quantum state estimation is a fundamental task in quantum information theory, where one estimates real parameters continuously embedded in a family of quantum states. In the theory of quantum state estimation, the widely used Cramér Rao approach which considers local estimation gives the ultimate precision bound of quantum state estimation in terms of the quantum Fisher information. However practical scenarios need not offer much prior information about the parameters to be estimated, and the local estimation setting need not apply. In general, it is unclear whether the Cramér-Rao approach is applicable for global estimation instead of local estimation. In this paper, we find situations where the Cramér-Rao approach does and does not work for quantum state estimation problems involving a family of bosonic states in a non-IID setting, where we only use one copy of the bosonic quantum state in the large number of bosons setting. Our result highlights the importance of caution when using the results of the Cramér-Rao approach to extrapolate to the global estimation setting.
Estimation of parameters is a pivotal task throughout science and technology. The quantum Cramér-Rao bound provides a fundamental limit of precision allowed to be achieved under quantum theory. For closed quantum systems, it has been shown how the estimation precision depends on the underlying dynamics. Here, we propose a general formulation for metrology scenarios in open quantum systems, aiming to relate the precision more directly to properties of the underlying dynamics. This feature may be employed to enhance an estimation precision, e.g., by quantum control techniques. Specifically, we derive a Cramér-Rao bound for a fairly large class of open system dynamics, which is governed by a (time-dependent) dynamical semigroup map. We illustrate the utility of this scenario through three examples.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.