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Fisher Information in Noisy Intermediate-Scale Quantum Applications

Johannes Jakob Meyer

arXiv:2103.15191v3quant-ph

TL;DR

Near-term quantum applications rely on parametrized quantum systems, but the broader usefulness of classical and quantum Fisher information beyond quantum sensing has only recently been explored. This article develops an intuitive tutorial, explains how both quantities can be computed and related in NISQ settings, and reviews applications in variational algorithms and quantum machine learning. It concludes that both quantities are useful because classical Fisher information describes measurable outputs while quantum Fisher information captures changes in the underlying quantum state, subject to measurement-compatibility and noise-related limitations.

  • Problem

    The broader use of classical and quantum Fisher information for understanding NISQ applications beyond quantum sensing remains insufficiently explored.

  • Method

    The article provides a tutorial on classical and quantum Fisher information, explains their NISQ computation and relationship, and reviews quantum-sensing results and applications in variational algorithms and quantum machine learning.

  • Results

    Classical Fisher information captures experimentally observable outputs, whereas quantum Fisher information captures information about the underlying quantum state and upper-bounds the classical Fisher information associated with measurements.

  • Takeaways & Limitations

    Classical and quantum Fisher information are useful complementary tools for studying near-term quantum applications beyond quantum sensing.

  • Takeaways & Limitations

    For multiple parameters, no single measurement necessarily achieves equality between the classical and quantum Fisher information matrices because individual optimal measurements may be incompatible.

Abstract

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The recent advent of noisy intermediate-scale quantum devices, especially near-term quantum computers, has sparked extensive research efforts concerned with their possible applications. At the forefront of the considered approaches are variational methods that use parametrized quantum circuits. The classical and quantum Fisher information are firmly rooted in the field of quantum sensing and have proven to be versatile tools to study such parametrized quantum systems. Their utility in the study of other applications of noisy intermediate-scale quantum devices, however, has only been discovered recently. Hoping to stimulate more such applications, this article aims to further popularize classical and quantum Fisher information as useful tools for near-term applications beyond quantum sensing. We start with a tutorial that builds an intuitive understanding of classical and quantum Fisher information and outlines how both quantities can be calculated on near-term devices. We also elucidate their relationship and how they are influenced by noise processes. Next, we give an overview of the core results of the quantum sensing literature and proceed to a comprehensive review of recent applications in variational quantum algorithms and quantum machine learning.

1 Parametrized Quantum States

Parametrized quantum states depend continuously on parameters, but Euclidean parameter distances need not reflect changes in the underlying states or measured output distributions. Pullbacks provide a way to measure parameter distances through quantum-state or probability-distribution distances, with monotonicity ensuring that quantum operations cannot increase distinguishability.

  • 1 Parametrized Quantum States: Parametrized quantum states are states that depend continuously on a parameter vector θ and arise in NISQ computing, optimal control, and quantum metrology.
  • 1 Parametrized Quantum States: Euclidean distances between parameters may poorly represent how strongly those parameters affect the associated quantum states.
  • 1 Parametrized Quantum States: Pullbacks measure parameter distances by evaluating distances between the corresponding quantum states or output probability distributions.
  • 1 Parametrized Quantum States: Monotonic distance measures cannot increase under a shared quantum operation, reflecting that operations cannot add information and noise reduces distinguishability.
  • 1 Parametrized Quantum States: Measurements collapse quantum states into classical outcome distributions, so distances may also be assessed in the space of measurement probabilities.

2 Information Matrices

Information matrices describe local changes in parametrized quantum states or measurement distributions by capturing the second-order response of a distance to a small parameter perturbation. Their metric structure incorporates the local information geometry of the underlying quantum or probabilistic objects.

  • 2 Information Matrices: A small perturbation θ + δ is analyzed through the distance d(θ, θ+δ), whose first nonzero Taylor contribution is second order.
  • 2 Information Matrices: The matrix M(θ) captures the local vicinity of θ when parameter-space distances are measured through distances between quantum states or output distributions.
  • 2 Information Matrices: Large entries of M(θ) indicate that changes in the corresponding parameters produce large changes in the underlying quantum state or output probability distribution.
  • 2 Information Matrices: The matrix M defines a metric that measures parameter-space lengths and angles while incorporating the local structure of the parametrized quantum state or probability distribution.
  • 2 Information Matrices: Information matrices are so named because they encode information about underlying quantum states and probability distributions.

3 The Classical Fisher Information

The classical Fisher information is derived from distances between parametrized measurement-outcome distributions and can be estimated on NISQ devices using data already collected for expectation values and their gradients. Its practical calculation is constrained by the sampling demands of estimating full distributions and their derivatives.

  • The classical Fisher information is defined for parametrized probability distributions of NISQ measurement outcomes, using the Kullback–Leibler divergence as a distance measure.The KL divergence induces the classical Fisher information matrix through a second-order expansion.
  • The KL divergence captures how quickly the false-negative error decreases when distinguishing two distributions under a false-positive constraint.
  • Monotonic distance measures yield a constant multiple of the classical Fisher information, establishing its uniqueness under this construction.The derivation assumes the relevant KL edge cases are excluded.
  • Calculation: Estimating the classical Fisher information requires measurement-outcome probabilities and their parameter derivatives, obtained from repeated NISQ experiments and derivative-estimation procedures.Bayesian and machine-learning methods can also estimate output distributions.
  • Calculation: In the worst case, estimating the full probability distribution requires samples proportional to the number of outcomes, usually exponential in the number of qubits.The burden is reduced when many output probabilities are very small.
  • Calculation: The processes used to compute expectation values and their gradients already provide the data needed for the full classical Fisher information matrix.Thus, its calculation is not harder than these routine NISQ tasks, although faithful full-distribution estimates generally require more samples.

4 The Quantum Fisher Information

The quantum Fisher information generalizes classical Fisher information to parametrized quantum states, using fidelity-based geometry and SLD operators. For NISQ circuits, generator variances and covariances provide practical calculations, although nonparallel gates are harder.

  • Foundations: Quantum Fisher information should reduce to classical Fisher information for classical states, but quantum monotone metrics are not unique.The fidelity-derived quantity is commonly called SLD quantum Fisher information.
  • Derivation: The quantum Fisher information is derived from fidelity-based distances between quantum states, with normalization chosen for consistency with classical Fisher information.The presentation focuses first on pure states and uses the fidelity-derived distance df.
  • Foundations: The SLD quantum Fisher information is the smallest monotone metric in a certain sense and remains defined for pure states.These properties distinguish it from many other quantum generalizations of classical Fisher information.
  • NISQ calculation: For a parameterized gate U(θ_i)=e^-iθ_iG_i, diagonal QFIM elements equal fourfold generator variances measurable from the state immediately before the gate.Parallel gates similarly yield off-diagonal elements through fourfold generator covariances.
  • NISQ calculation: QFIM elements for nonparallel gates are harder because the required observables depend on intermediary circuit elements.More sophisticated techniques, including parameter-shift rules, can still evaluate them.

5 Relation of Classical and Quantum Fisher Information

Monotonicity makes information matrices decrease under quantum operations, so the quantum Fisher information upper-bounds the classical Fisher information obtained after measurement. Equality is always attainable for one parameter but not necessarily for multiple parameters.

  • Monotonicity: Monotonicity of a distance measure implies that its information matrix decreases under quantum channels.The result follows by applying the distance inequality to infinitesimal parameter changes and retaining the second-order term.
  • Monotonicity: The matrix inequality A≥B means that A−B has only non-negative eigenvalues, equivalently δ^T Bδ≤δ^T Aδ for every vector δ.Figure 2 represents this relation by placing B's ellipse inside A's ellipse.
  • Quantum–classical relation: Because measurement is a quantum channel and produces a classical probability distribution, the quantum information matrix upper-bounds the corresponding classical Fisher information matrix.The quantum quantity measures state changes, while the classical quantity measures the portion visible after measurement.
  • Quantum–classical relation: The quantum Fisher information captures changes in the underlying quantum state, whereas classical Fisher information describes information available from measured outcomes.Both quantities therefore address different stages of extracting information from a parametrized quantum system.
  • Optimal measurements: For a single parameter, a measurement can always attain equality between classical and quantum Fisher information.For multiple parameters, individually optimal measurements may be incompatible, so simultaneous equality is not guaranteed.

6 The Role of Noise

Noise requires extending the pure-state construction to mixed states through Bures fidelity, but exact noisy-QFIM evaluation is resource-intensive. Several variational, truncation, and bound-based approaches seek more practical estimates for small or near-term systems.

  • Mixed states: Mixed-state quantum Fisher information uses Bures fidelity, which extends the pure-state fidelity to mixed quantum states.The associated Bures distance includes a prefactor chosen for consistency with classical Fisher information.
  • Mixed states: The noisy QFIM contains a classical contribution from changing eigenvalues and a quantum contribution from changing eigenstates.Eigenvalue changes correspond to the classical part, while eigenvector changes are non-classical.
  • Computational challenges: Exact noisy-QFIM evaluation usually requires full state tomography, whose sample count is exponential in the number of qubits.Nearly pure-state approximations can fail at certain noise levels.
  • Computational approaches: Variational quantum autoencoders, purification methods, and purity minimization have been proposed to reduce the resources needed to calculate Bures fidelity.The purification and purity-minimization approaches require multiple copies of input states and are therefore suitable only for small systems.
  • Bounds and approximations: Truncated QFIM approximations improve as more of the largest-eigenvalue eigenvectors are included.A separate hierarchy of randomized-measurement lower bounds becomes more accurate as its level increases, while computational complexity also increases.

7 NISQ Applications

This section presents quantum sensing as a framework for parameter estimation and explains how Fisher information quantifies attainable precision. It then connects these ideas to parametrized NISQ systems and quantum natural gradient optimization.

  • 7.1 Quantum Sensing: Quantum sensing imprints physical parameters onto a probe state, measures the resulting state, and estimates the parameters from the measurement distribution.The distribution depends on both the chosen measurement and the physical parameters.
  • 7.1 Quantum Sensing: The classical Fisher information enters the Cramér-Rao bound, which limits the covariance of unbiased parameter estimators.Large Fisher information indicates that parameter changes produce more distinguishable probability distributions.
  • 7.1 Quantum Sensing: In the infinite-sample limit, maximum likelihood estimation can saturate the scalar Cramér-Rao bound.This provides an asymptotically optimal estimator under the stated conditions.
  • 7.1 Quantum Sensing: The quantum Fisher information upper-bounds measurement-specific classical Fisher information and therefore yields the quantum Cramér-Rao bound.Optimal sensing involves choosing both a probe state and a measurement that extract information effectively.
  • 7.1 Quantum Sensing: A factor n signal enhancement produces a factor n^2 improvement in Fisher information, reaching Heisenberg scaling in idealized sensing.The enhancement arises because Fisher information contains the squared parameter derivative.
  • 7.1 Quantum Sensing: Noise can eliminate the advantage of generalized GHZ states, while less-entangled states or metrological error correction can preserve limited or Heisenberg scaling.For many noise models, only a constant-factor improvement over the standard quantum limit is possible without combining sensing with error correction.
  • 7.2 Quantum Natural Gradient Descent: Quantum natural gradient descent updates optimization using the inverse quantum Fisher information matrix and can improve optimization of large realistic quantum systems.The method follows optimization paths that differ from strategies unrelated to the underlying quantum states.

8 Beyond NISQ

The article also highlights Fisher-information methods beyond NISQ applications, including quantum programming, quantum error correction, resource theories, and other quantum-information settings.

  • 8 Beyond NISQ: Quantum metrology techniques can establish lower bounds on the size of quantum programs needed to implement unitaries with target precision.An optimal quantum program interpreter estimates the unitary to be performed from the quantum program.
  • 8 Beyond NISQ: Quantum Fisher information upper bounds provide a simple proof of the approximate Eastin-Knill theorem.The argument shows that certain quantum error-correcting codes with favorable properties cannot exist.
  • 8 Beyond NISQ: Fisher information can construct parameter-estimation tasks in which quantum resources provide an advantage, with a converse showing every resource is useful for some metrology task.The construction is presented as general within quantum resource theories.
  • 8 Beyond NISQ: Other applications of quantum Fisher information include quantum thermodynamics, quantum speed limits, and non-Markovianity.These examples illustrate the breadth of Fisher-information applications beyond sensing and NISQ devices.

9 Outlook

The outlook identifies broader applications for Fisher information in NISQ research while highlighting unresolved challenges in calculation, estimation, and theoretical understanding.

  • Computational challenges: More efficient techniques for calculating classical and quantum Fisher information on NISQ hardware are needed to increase their applicability.The quantum Fisher information is especially challenging for noisy quantum states because of its higher complexity.
  • Computational challenges: Rigorous analyses of classical Fisher information estimation strategies and sample complexity remain important open problems.Bayesian techniques may be promising for variational applications when circuit parameters are updated incrementally.
  • Future directions: Quantum Fisher information could support the analysis of quantum machine learning generalization bounds, including distinctions between data-input and trainable circuit parameters.The outlook calls for bounds that capture the influence of noise and limits of quantum-enhanced models.
  • Future directions: Comparing classical and quantum Fisher information may quantify information loss when variational algorithms estimate observables using measurements in many bases.Other quantum-sensing tools, such as measures of parameter incompatibility, may also inform near-term application analysis.
  • Future directions: Because both Fisher information quantities are susceptible to noise, they may be useful in quantum error correction and error mitigation.The proposed uses include both practical applications and theoretical tools for rigorous mathematical statements.
  • Future directions: Further applications of monotone metrics beyond quantum Fisher information could provide tighter bounds in contexts other than the quantum Cramér-Rao bound.Examples include Wigner-Yanase and Kubo-Mori information, alongside possible generalizations of quantum natural gradient descent.

A Derivation of the Classical Fisher Information

The classical Fisher information matrix is derived from the local second-order behavior of the Kullback-Leibler divergence under parameter perturbations.

  • Derivation: The derivation begins with the Kullback-Leibler divergence between a probability distribution and its perturbed counterpart.The perturbed expression is rewritten using log(a/b) = log a − log b.
  • Derivation: A second-order expansion around δ = 0 isolates the contribution relevant to the information metric.Only the second term contributes after taking second derivatives with respect to δ.
  • Derivation: Substituting ξ = θ + δ and transforming derivatives yields the Fisher-information expression in the original parameter coordinates.The derivation then renames ξ to θ and obtains a more familiar form.
  • Derivation: A further algebraic reformulation produces an equivalent expression for the information matrix.The reformulation uses an additional identity before taking the expectation over the probability distribution.
  • Derivation: The derivation concludes with equivalent formulas for the classical Fisher information matrix.The expectation step removes the second term in the intermediate expression.

B Properties of the Classical Fisher Information Matrix

The classical Fisher information matrix is characterized through its definition, structural properties, behavior under maps and reparametrization, and its connection to monotone distance measures.

  • Definition and properties: The classical Fisher information matrix is defined for a probability distribution p(θ) over d parameters θ ∈ R^d.It is presented as a matrix associated with the parameterized distribution.
  • Definition and properties: The matrix is real, symmetric, and positive semidefinite, so all its eigenvalues are non-negative.Positive semidefiniteness follows from the KL divergence attaining a minimum when the distributions coincide.
  • Definition and properties: The Fisher information matrix is convex and additive under direct sums of probability distributions.Convexity follows from joint convexity of the KL divergence, while additivity follows from the direct-sum structure.
  • Maps and transformations: It is non-increasing under stochastic maps, meaning stochastic post-processing cannot increase the matrix in the relevant matrix order.The proof derives the matrix inequality from monotonicity of the KL divergence under a stochastic map T.
  • Maps and transformations: Under parameter reparametrization, the Fisher information transforms according to the inverse Jacobian of the parameter mapping.The transformation rule is obtained by applying the Hessian coordinate-change result to the Fisher-information definition.
  • Monotone metrics: The second-order expansion of any monotonic distance measure yields a constant multiple of the classical Fisher information matrix.This establishes the matrix as unique up to scale among metrics generated by such expansions.

C Modified Distance Functions

Applying a scalar function to a distance changes its associated information matrix by a local scalar factor, subject to the distance being extremal at zero perturbation.

  • Modified distance functions: Post-processing a distance function changes its information matrix only by a scalar prefactor.The result follows from evaluating the second derivatives of h(g(δ)) at an extremal point.
  • Modified distance functions: At an extremal point δ*, the matrix associated with h(g(δ)) differs from that of g(δ) by f′(g(δ*)).The first-derivative terms vanish at the extremum, leaving the scalar factor multiplying the second derivative matrix.
  • Modified distance functions: For the distance functions considered, δ = 0 is extremal because d(θ, θ) = 0.A sensible transformed information matrix therefore requires f′(0) > 0.
  • Modified distance functions: The same reasoning applies to fidelity-based constructions of the quantum Fisher information matrix.The fidelity is extremal at δ = 0 with value 1, and alternative definitions may use its square root.
  • Modified distance functions: The two conventions for the information matrix associated with the modified distance differ by a factor of 1/2.This factor is reported for the conventions discussed in the section.

D Derivation of the Quantum Fisher Information

The quantum Fisher information is derived from the second-order change of fidelity between nearby parametrized quantum states. The derivation enforces consistency with classical Fisher information and yields the pure-state formula.

  • The derivation starts from the fidelity distance between two nearby pure states.
  • Expanding the state for a small parameter displacement reveals the second-order terms that determine the information metric.
  • The resulting information matrix is checked against a classical state measured in the computational basis to recover classical Fisher information.
  • Correcting the normalization factor makes the quantum Fisher information consistent with the classical limit.
  • For a pure state, the quantum Fisher information matrix is Fij = 4 Re[⟨∂iψ(θ)|∂jψ(θ)⟩−⟨∂iψ(θ)|ψ(θ)⟩⟨ψ(θ)|∂jψ(θ)⟩].

E Properties of the Quantum Fisher Information Matrix

The quantum Fisher information matrix is characterized through its relation to the Bures distance and has structural, invariance, monotonicity, and transformation properties. These properties establish how the matrix behaves under mixtures, quantum channels, tensor products, and parameter reparametrization.

  • For pure states, the mixed-state expression simplifies to the pure-state form derived earlier.
  • F is a real symmetric positive-semidefinite d × d matrix.
  • F is convex under mixing of quantum states, following the joint convexity of the Bures distance.
  • F is invariant under unitary transformations because the Bures distance is unitarily invariant.
  • F is additive under direct sums and tensor products, with tensorization following from the multiplicativity of Bures fidelity.
  • F is non-increasing under quantum channels, reflecting monotonicity of the Bures divergence.
  • Under parameter reparametrization, F transforms through the inverse Jacobian of the coordinate mapping.
  • The mixed-state quantum Fisher information matrix arises from the second-order expansion of the Bures distance.
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