Quantum Likelihood Estimation

• Quantum Likelihood Estimation is a hybrid quantum-classical algorithm that iteratively refines estimates of an unknown Hamiltonian using adaptive Bayesian inference.
• It employs a prepare–evolve–measure–update loop where experimental controls are optimized with simulated annealing to maximize mutual information.
• Benchmarks demonstrate QLE’s efficiency in reducing adaptive rounds, making it a powerful tool for parameter estimation and model validation in NISQ-era experiments.

Quantum Likelihood Estimation (QLE) is a hybrid quantum-classical method for Hamiltonian learning in which measurement data generated by a quantum system evolving under an unknown Hamiltonian are converted into likelihoods over a candidate set of Hamiltonians, and those likelihoods are incorporated through Bayesian inference into an iteratively refined classical weight distribution. In the formulation developed most explicitly in "Optimal Quantum Likelihood Estimation" (Levi et al., 31 Aug 2025), QLE is a prepare–evolve–measure–update loop whose efficiency depends strongly on experiment design at each iteration; the paper makes that design information-theoretically optimal, iteration by iteration, by choosing controls that maximize the mutual information between the hidden Hamiltonian label and the measurement outcome. Earlier work had already realized QLE experimentally as a component of quantum Hamiltonian learning, using a photonic quantum simulator to estimate likelihoods for an NV-center Hamiltonian and to drive sequential Bayesian updates (Wang et al., 2017).

1. Definition and historical placement

In its quantum-information usage, QLE addresses the problem of identifying which Hamiltonian in a candidate set $\{H_1,\dots,H_N\}$ generated observed dynamics under an unknown true Hamiltonian $H_{\text{true}}$. The method maintains a classical probability distribution over the candidate Hamiltonians and refines that distribution as additional quantum measurement data are acquired (Levi et al., 31 Aug 2025).

This places QLE squarely in the NISQ-era class of hybrid quantum-classical algorithms: a quantum device performs the expensive data-generating step, while a classical routine updates beliefs and chooses subsequent experiments. The original experimental realization in quantum Hamiltonian learning interfaced two quantum systems through a classical computer: a negatively charged nitrogen-vacancy center in diamond as the target system and a programmable silicon photonic quantum device as the likelihood engine. In that experiment, the likelihood for a candidate Hamiltonian parameter $\vec x$ was the return probability

$$\Pr(D|\vec x)=\left|\langle D|e^{-i\hat H(\vec x)t}|\psi\rangle\right|^2,$$

with $|D\rangle=|\psi\rangle=|m_s=0\rangle$, and Bayesian inference was implemented approximately by Sequential Monte Carlo (SMC) (Wang et al., 2017).

That experimental program established several points that remain central to QLE. Using 50 steps and 20 SMC particles, the protocol learned an NV-center Rabi frequency with final estimate

$f_{\text{QLE}}=(6.93\pm0.09)\,\text{MHz},$

consistent with the independently fitted value $f_0=6.90\,\text{MHz}$, and the paper reports a final quadratic loss of approximately $10^{-5}$. The same work also showed that saturation of posterior variance can diagnose model mismatch: in its simplest model the variance saturated at about $\sigma^2(\omega)\simeq 4.2\times 10^{-5}$, motivating an improved chirped Hamiltonian model with Bayes factor $K=560$. An interactive variant, IQLE, achieved

$H_{\text{true}}$0

with quadratic loss about $H_{\text{true}}$1 (Wang et al., 2017). These results established QLE not only as a parameter-estimation procedure but also as a model-validation method.

2. Core algorithmic structure

At its core, QLE is a prepare–evolve–measure–update protocol. In each iteration, the algorithm begins with an initial state $H_{\text{true}}$2, evolves it under the unknown unitary

$H_{\text{true}}$3

applies a final unitary $H_{\text{true}}$4 selecting the measurement basis, and measures in the computational basis. The outcome $H_{\text{true}}$5 is then used to compute likelihoods for each candidate Hamiltonian $H_{\text{true}}$6, after which Bayesian inference updates the weights $H_{\text{true}}$7 (Levi et al., 31 Aug 2025).

The Bayesian update rule is

$H_{\text{true}}$8

where $H_{\text{true}}$9 is the likelihood of observing outcome $\vec x$0 under hypothesis $\vec x$1. The posterior from one round becomes the prior for the next. In this sense, QLE is not a one-shot estimator but an adaptive sequential inference scheme.

For the optimized single-qubit setting analyzed in (Levi et al., 31 Aug 2025), the controls are the initial-state angles $\vec x$2,

$\vec x$3

the measurement-basis rotation angles $\vec x$4,

$\vec x$5

and the evolution time $\vec x$6, chosen in the range

$\vec x$7

where $\vec x$8 is the minimum energy gap among the candidate Hamiltonians.

The optimized algorithm, denoted QLE_optimization, dynamically selects the five-parameter control vector $\vec x$9 at every iteration. This is a substantive departure from earlier QLE practice, in which the initial state is often fixed and $$\Pr(D|\vec x)=\left|\langle D|e^{-i\hat H(\vec x)t}|\psi\rangle\right|^2,$$0 is chosen by the Particle Guess Heuristic (PGH). The point of the revised formulation is that discrimination power depends jointly on state preparation, measurement basis, and evolution time, so optimizing only one control leaves information on the table (Levi et al., 31 Aug 2025).

3. Information-theoretic formulation and adaptive control

The defining conceptual move in optimal QLE is to recast each iteration as a single-query oracle-identification problem. The hidden variable is the Hamiltonian label $$\Pr(D|\vec x)=\left|\langle D|e^{-i\hat H(\vec x)t}|\psi\rangle\right|^2,$$1, distributed according to the current candidate probabilities $$\Pr(D|\vec x)=\left|\langle D|e^{-i\hat H(\vec x)t}|\psi\rangle\right|^2,$$2, and the observable is the measurement outcome $$\Pr(D|\vec x)=\left|\langle D|e^{-i\hat H(\vec x)t}|\psi\rangle\right|^2,$$3. The joint classical-quantum state is written as

$$\Pr(D|\vec x)=\left|\langle D|e^{-i\hat H(\vec x)t}|\psi\rangle\right|^2,$$4

The experiment-design problem is then to choose the controls so as to maximize the mutual information $$\Pr(D|\vec x)=\left|\langle D|e^{-i\hat H(\vec x)t}|\psi\rangle\right|^2,$$5 (Levi et al., 31 Aug 2025).

The paper expresses the mutual information as

$$\Pr(D|\vec x)=\left|\langle D|e^{-i\hat H(\vec x)t}|\psi\rangle\right|^2,$$6

with the basis-dependent discord-like quantity

$$\Pr(D|\vec x)=\left|\langle D|e^{-i\hat H(\vec x)t}|\psi\rangle\right|^2,$$7

Because the candidate probabilities $$\Pr(D|\vec x)=\left|\langle D|e^{-i\hat H(\vec x)t}|\psi\rangle\right|^2,$$8 are fixed within a given iteration, $$\Pr(D|\vec x)=\left|\langle D|e^{-i\hat H(\vec x)t}|\psi\rangle\right|^2,$$9 is constant for that step. Maximizing $|D\rangle=|\psi\rangle=|m_s=0\rangle$0 is therefore equivalent to minimizing the conditional von Neumann entropy, which becomes the cost function actually optimized in the algorithm. Operationally, the selected controls are those that make the post-measurement state maximally informative about the true Hamiltonian (Levi et al., 31 Aug 2025).

The search over controls is performed by simulated annealing. Starting from a randomly initialized parameter vector

$|D\rangle=|\psi\rangle=|m_s=0\rangle$1

the algorithm generates neighbors according to

$|D\rangle=|\psi\rangle=|m_s=0\rangle$2

where $|D\rangle=|\psi\rangle=|m_s=0\rangle$3 denotes the Hadamard product. Candidates with lower cost are always accepted; higher-cost candidates are accepted with probability

$|D\rangle=|\psi\rangle=|m_s=0\rangle$4

The temperature is then reduced geometrically,

$|D\rangle=|\psi\rangle=|m_s=0\rangle$5

with $|D\rangle=|\psi\rangle=|m_s=0\rangle$6 in the reported implementation. The role of annealing is to explore broadly at high temperature and concentrate on low-entropy regions later, mitigating local-minimum trapping in a highly nonconvex control landscape (Levi et al., 31 Aug 2025).

A common simplification is to describe QLE as merely Bayesian Hamiltonian identification. That is accurate but incomplete. In the optimal formulation, the distinguishing feature is not Bayesian updating alone; it is adaptive quantum experiment design driven by an information-theoretic objective.

4. Convergence behavior and reported benchmarks

The most direct evidence for the revised formulation comes from the benchmark problems reported in (Levi et al., 31 Aug 2025). In a four-Hamiltonian test set,

$|D\rangle=|\psi\rangle=|m_s=0\rangle$7

the best static baseline required about 144 iterations on average to make all four cases converge, whereas the optimized QLE required only 9 iterations. When the success threshold was tightened from $|D\rangle=|\psi\rangle=|m_s=0\rangle$8 to $|D\rangle=|\psi\rangle=|m_s=0\rangle$9, the improvement became even more pronounced.

The same study also tested a control variant in which a discrete grid search was repeated at every iteration. That version converged in about 10 iterations, very close to the 9 iterations of the continuous simulated-annealing version. The explicit interpretation given in the paper is that the speedup comes mainly from the optimization strategy itself, not merely from the availability of a larger continuous parameter space (Levi et al., 31 Aug 2025).

A harder six-Hamiltonian benchmark, containing mixed Pauli terms and asymmetric shifts, sharpened the comparison. The original QLE failed to converge reliably, whereas the optimized version succeeded for all six Hamiltonians and required only about 4–5 iterations on average per Hamiltonian (Levi et al., 31 Aug 2025). The paper therefore frames the principal gain not as asymptotic complexity theory but as practical reduction in the number of adaptive experimental rounds needed to drive the posterior toward a single Hamiltonian hypothesis.

Two points follow from these benchmarks. First, QLE is highly parameter-sensitive if controls are chosen heuristically. Second, the large empirical improvement is consistent with the information-theoretic interpretation: the algorithm is faster when each query is selected to maximize expected information gain about the hidden Hamiltonian label.

5. Relation to other likelihood-based quantum inference methods

QLE belongs to a broader family of likelihood-based quantum inference procedures, but its target object, data model, and adaptation mechanism are specific. Related methods differ by whether they infer Hamiltonian parameters, amplitudes, density matrices, or latent states in continuously monitored or hidden-state models.

Setting	Likelihood object	Representative papers
Hamiltonian learning from thermal or projector data	Negative log-likelihood over Gibbs states	(Zhao et al., 2022)
Amplitude estimation	Binomial likelihood over amplification depths	(Tanaka et al., 2020, Callison et al., 2022)
Density-matrix inference and tomography	Quantum relative entropy or Gaussian likelihood with physical-state constraints	(Cranmer et al., 2019, Singh et al., 2016)
Continuous monitoring	$f_{\text{QLE}}=(6.93\pm0.09)\,\text{MHz},$0 from an unnormalized SME	(Gammelmark et al., 2012)
Quantum walks	Exact displacement/return likelihood $f_{\text{QLE}}=(6.93\pm0.09)\,\text{MHz},$1	(Ellinas et al., 2022)
Quantum hidden-state models	Sequence likelihood from Kraus-operator evolution	(Ghysels et al., 28 Jul 2025)

In maximum-likelihood Hamiltonian learning for many-body systems, the candidate model is updated directly in Hamiltonian space rather than by maintaining a discrete hypothesis distribution. The update

$f_{\text{QLE}}=(6.93\pm0.09)\,\text{MHz},$2

is constructed so that the negative log-likelihood decreases, at least to linear order in the step size, while the state remains Gibbsian throughout the optimization (Zhao et al., 2022). This is close in spirit to QLE, but it is not the same algorithmic object: the likelihood is defined from measurement frequencies on thermal or ground states, and the optimization variable is the Hamiltonian itself rather than a control policy over sequential experiments.

In amplitude estimation, likelihood-based quantum inference takes the form of maximum-likelihood fitting of measurement counts produced by repeated amplitude amplification. In the noisy formulation of (Tanaka et al., 2020), the model jointly estimates the target amplitude $f_{\text{QLE}}=(6.93\pm0.09)\,\text{MHz},$3 and a depolarization parameter $f_{\text{QLE}}=(6.93\pm0.09)\,\text{MHz},$4; in the improved MLQAE analysis of (Callison et al., 2022), the likelihood is built from counts gathered at different Grover depths, and exceptional amplitude values together with a depth-jittering fix are analyzed explicitly. These methods share QLE’s hybrid structure—quantum sampling plus classical likelihood optimization—but they estimate an amplitude parameter rather than a Hamiltonian label.

Density-matrix inference provides another neighboring lineage. "Inferring the quantum density matrix with machine learning" formulates Quantum Maximum Likelihood (QML) and Quantum Variational Inference (QVI) in terms of the quantum relative entropy, using a spectral variational family and optional quantum flows to learn a thermal density matrix and thereby recover eigenstates and eigenvalues (Cranmer et al., 2019). In experimental NMR tomography, maximum-likelihood estimation is used to enforce physicality by parameterizing the state as

$f_{\text{QLE}}=(6.93\pm0.09)\,\text{MHz},$5

thereby guaranteeing positivity and normalization throughout reconstruction (Singh et al., 2016). These approaches are likelihood-based quantum estimation, but their target is the density operator, not an unknown Hamiltonian chosen from a candidate set.

Likelihood generation from continuously monitored systems proceeds differently again. There, the stochastic master equation conditioned on the measurement record is simultaneously an equation for the conditional quantum state and for the likelihood of the record under candidate parameters. The unnormalized conditional state $f_{\text{QLE}}=(6.93\pm0.09)\,\text{MHz},$6 yields

$f_{\text{QLE}}=(6.93\pm0.09)\,\text{MHz},$7

which can be used directly in Bayesian updating or MCMC posterior sampling (Gammelmark et al., 2012). The shared structural feature with QLE is the centrality of likelihoods derived from quantum dynamics, but the data are full time-resolved records rather than discrete adaptive interrogation rounds.

6. Generalizations, scope, and nomenclature

The 2025 optimal-QLE formulation explicitly generalizes beyond Hamiltonian identification. The paper describes a class of quantum learning problems consisting of an initial state $f_{\text{QLE}}=(6.93\pm0.09)\,\text{MHz},$8, a CPTP map $f_{\text{QLE}}=(6.93\pm0.09)\,\text{MHz},$9, an attribute $f_0=6.90\,\text{MHz}$0, and hidden parameters $f_0=6.90\,\text{MHz}$1 drawn from a prior distribution. In that setting, the controlled parameters $f_0=6.90\,\text{MHz}$2 are chosen to maximize

$f_0=6.90\,\text{MHz}$3

with Bayesian or likelihood-based inference updating the hidden-parameter distribution after each query (Levi et al., 31 Aug 2025). This makes QLE a specific instance of a more general prepare–measure–update–optimize paradigm.

The same paper notes that the framework can be extended to noisy scenarios and to multi-qubit systems, while also observing that the parameter search becomes more difficult there and may benefit from quantum annealing in place of classical simulated annealing (Levi et al., 31 Aug 2025). A plausible implication is that the principal bottleneck in scaling QLE is not Bayesian updating itself but adaptive control optimization in larger experimental design spaces.

The term QLE is, however, acronymically overloaded. In mathematical physics and probability theory, it also denotes quantum Loewner evolution, especially QLE $f_0=6.90\,\text{MHz}$4. In that usage, QLE is a growth process on a doubly marked $f_0=6.90\,\text{MHz}$5-Liouville quantum gravity sphere, constructed by repeatedly growing small pieces of an $f_0=6.90\,\text{MHz}$6-type exploration, resampling the tip uniformly along the quantum boundary, and taking a subsequential limit as $f_0=6.90\,\text{MHz}$7. That process is used to define a metric on a dense subset of the surface and to prove symmetry and strict triangle inequality for the resulting distance on sampled points (Miller et al., 2015). This meaning is unrelated to Hamiltonian learning except for the shared initials.

There are also broader statistical uses of the language of quantum likelihood. One line of work develops a quantum framework for likelihood ratios using Hilbert-space encodings of contingency tables and overlap structure interpreted through non-orthonormal inner products (Bond et al., 2015). Another studies maximum-likelihood estimation in quantum probability on orthomodular posets and Greechie diagrams, where likelihoods

$f_0=6.90\,\text{MHz}$8

are optimized subject to contextual normalization constraints on operations (Navara et al., 2024). These usages widen the semantic field of “quantum likelihood estimation,” but they are conceptually distinct from the Hamiltonian-learning algorithm now commonly denoted QLE in quantum computing.

Taken together, the literature supports a narrow and a broad sense of the term. In the narrow and technically dominant quantum-computing sense, QLE is an adaptive Bayesian Hamiltonian-learning protocol accelerated by quantum-generated likelihood information and, in its current optimal form, by mutual-information-based experiment design (Levi et al., 31 Aug 2025). In the broader sense, it belongs to a wider class of likelihood-based quantum inference procedures that use quantum dynamics, quantum probability, or quantum-hidden-state representations to make latent physical parameters statistically identifiable.