Physical Likelihood Processor

• Physical Likelihood Processor is a framework that integrates physical data and hardware constraints to construct, optimize, and compose statistical likelihoods.
• It ensures physical admissibility in quantum state reconstructions and noise modeling by structurally enforcing criteria through maximum-likelihood methods.
• Applications span from processor design to analog front ends, reducing errors and improving execution fidelity in quantum and hybrid systems.

“Physical Likelihood Processor” is an Editor’s term for a family of systems in which physically generated data, physically constrained state spaces, or physical hardware are used to construct, optimize, or propagate likelihoods. In this usage, the processor may be a quantum-state reconstructor that searches only over valid density matrices, a processor-characterization pipeline that fits Lindblad generators directly to time-resolved tomographic data, a hardware architecture whose coupler graph is chosen to increase the likelihood of correct execution, or a physical front-end that evaluates statistical objectives through analog projections rather than full digital reconstruction (Singh et al., 2016, Samach et al., 2021, Lu et al., 2021, Zhang et al., 2022, Severin et al., 1 May 2026).

1. Formal interpretations of likelihood as a dynamical object

In one abstract formulation, a parameterized stochastic model is represented by a morphism

$$f:\Omega^n\times \mathbb R^p\times \mathbb R^a\to \mathbb R^b,$$

with conditional likelihood

$$L_f(x_p,x_a,x_b)=\frac{d\, f(-,x_p,x_a)_*\mu^n}{d\lambda^b}(x_b).$$

For compositional models, likelihoods combine by latent-variable integration: $$L_{f'\circ f}((x_q,x_p),x_a,x_c)=\int_{x_b\in \mathbb R^b} L_{f'}(x_q,x_b,x_c)L_f(x_p,x_a,x_b)\,d\lambda^b.$$ This establishes a semantics in which a processor can be understood as a device that composes stochastic mechanisms and then composes their likelihoods in parallel with them (Shiebler, 2020).

A related but distinct construction appears in mathematical finance, where positive discounted asset prices under a martingale measure are identified with filtered likelihood ratio processes. The key equivalence is

$$\frac{dQ_{i|\mathcal F_t}}{dQ_{|\mathcal F_t}}=\frac{X_t^i}{X_0^i},$$

so that a discounted price process becomes a filtered Radon–Nikodym derivative process, and conversely such filtered likelihood processes generate arbitrage-free discounted price models. In that setting, option payoffs are linked to tests, option prices become combinations of test level and power, and in special cases Delta and Gamma are derivatives of power functions (Janssen et al., 2013).

2. Likelihood-constrained reconstruction of quantum states

A canonical instance of a physical likelihood processor is maximum-likelihood quantum-state reconstruction on quantum hardware. In NMR quantum information processing, standard quantum state tomography expands the density operator in the Pauli basis,

$$\rho = \sum_{i=0}^{3}\sum_{j=0}^{3}\cdots\sum_{k=0}^{3} c_{ij\cdots k}\,\sigma_i\otimes\sigma_j\otimes\cdots\otimes\sigma_k,$$

and estimates the coefficients from expectation values extracted from free induction decay signals,

$$S(t)\propto \operatorname{Tr}\left\{\rho(t)\sum_k (I_{kx}-iI_{ky})\right\}.$$

Because linear inversion is applied to finite, imperfect data, the reconstructed matrix can be Hermitian and trace-normalized but not positive semidefinite, even though a physical density matrix must be Hermitian, have unit trace, and satisfy positivity. The maximum-likelihood remedy is to parameterize the state as

$$\rho=T^\dagger T,\qquad \operatorname{Tr}(T^\dagger T)=1,$$

which enforces physicality structurally rather than by external constraints. For two qubits, the fitted observables are

$$n_{jk}=\operatorname{Tr}\big[(\sigma_j\otimes\sigma_k)\rho\big],$$

and the minimized Gaussian negative log-likelihood is

$$\mathcal{L}(t_1,\dots,t_{16})=\sum_{j,k}\frac{(n_{jk}(t)-\bar n_{jk})^2}{2\bar\sigma_{jk}^{\,2}}.$$

The paper demonstrates the point on several prepared states. For $$\frac{1}{\sqrt{2}}(\lvert 00\rangle+\lvert 01\rangle)$$, standard QST returned eigenvalues $$L_f(x_p,x_a,x_b)=\frac{d\, f(-,x_p,x_a)_*\mu^n}{d\lambda^b}(x_b).$$0 and $$L_f(x_p,x_a,x_b)=\frac{d\, f(-,x_p,x_a)_*\mu^n}{d\lambda^b}(x_b).$$1, whereas MLE returned $$L_f(x_p,x_a,x_b)=\frac{d\, f(-,x_p,x_a)_*\mu^n}{d\lambda^b}(x_b).$$2. Analogous repairs of nonphysical reconstructions were shown for $$L_f(x_p,x_a,x_b)=\frac{d\, f(-,x_p,x_a)_*\mu^n}{d\lambda^b}(x_b).$$3, Bell states, a three-qubit $$L_f(x_p,x_a,x_b)=\frac{d\, f(-,x_p,x_a)_*\mu^n}{d\lambda^b}(x_b).$$4 state, and Bell-state entanglement decay, with the methodological conclusion that physical likelihood reconstruction prevents downstream corruption of fidelities, eigenvalue spectra, and partial-transpose entanglement indicators (Singh et al., 2016).

3. Quantum processors as objects of likelihood-based dynamical inference

The same logic extends from state estimation to processor-level noise modeling. Lindblad tomography reconstructs a time-independent Markovian generator from time-domain measurements by fitting the master equation

$$L_f(x_p,x_a,x_b)=\frac{d\, f(-,x_p,x_a)_*\mu^n}{d\lambda^b}(x_b).$$5

with $$L_f(x_p,x_a,x_b)=\frac{d\, f(-,x_p,x_a)_*\mu^n}{d\lambda^b}(x_b).$$6 Hermitian and positive semidefinite. The protocol first estimates SPAM from $$L_f(x_p,x_a,x_b)=\frac{d\, f(-,x_p,x_a)_*\mu^n}{d\lambda^b}(x_b).$$7 data and then maximizes a likelihood over all time points, using Cholesky parameterizations to enforce positivity of $$L_f(x_p,x_a,x_b)=\frac{d\, f(-,x_p,x_a)_*\mu^n}{d\lambda^b}(x_b).$$8, POVM elements, and the Lindblad matrix. On a superconducting processor with two characterized neighboring transmons, this recovered an always-on $$L_f(x_p,x_a,x_b)=\frac{d\, f(-,x_p,x_a)_*\mu^n}{d\lambda^b}(x_b).$$9 shift $$L_{f'\circ f}((x_q,x_p),x_a,x_c)=\int_{x_b\in \mathbb R^b} L_{f'}(x_q,x_b,x_c)L_f(x_p,x_a,x_b)\,d\lambda^b.$$0, in close agreement with an independent estimate of $$L_{f'\circ f}((x_q,x_p),x_a,x_c)=\int_{x_b\in \mathbb R^b} L_{f'}(x_q,x_b,x_c)L_f(x_p,x_a,x_b)\,d\lambda^b.$$1 kHz from device parameters. It also showed that single-qubit Lindblad fits fail when a neighboring qubit is prepared in $$L_{f'\circ f}((x_q,x_p),x_a,x_c)=\int_{x_b\in \mathbb R^b} L_{f'}(x_q,x_b,x_c)L_f(x_p,x_a,x_b)\,d\lambda^b.$$2, because the $$L_{f'\circ f}((x_q,x_p),x_a,x_c)=\int_{x_b\in \mathbb R^b} L_{f'}(x_q,x_b,x_c)L_f(x_p,x_a,x_b)\,d\lambda^b.$$3 term induces entanglement and apparent non-Markovianity in the reduced dynamics: the single-qubit fit then had average error $$L_{f'\circ f}((x_q,x_p),x_a,x_c)=\int_{x_b\in \mathbb R^b} L_{f'}(x_q,x_b,x_c)L_f(x_p,x_a,x_b)\,d\lambda^b.$$4, whereas the two-qubit Lindbladian fit reduced this to $$L_{f'\circ f}((x_q,x_p),x_a,x_c)=\int_{x_b\in \mathbb R^b} L_{f'}(x_q,x_b,x_c)L_f(x_p,x_a,x_b)\,d\lambda^b.$$5 (Samach et al., 2021).

A subsequent framework, LIMINAL, makes model selection itself likelihood-based. It fits nested Lindblad models to multinomial tomographic count data, parameterizes dissipators as

$$L_{f'\circ f}((x_q,x_p),x_a,x_c)=\int_{x_b\in \mathbb R^b} L_{f'}(x_q,x_b,x_c)L_f(x_p,x_a,x_b)\,d\lambda^b.$$6

and compares nested candidates with likelihood-ratio tests under Wilks-type asymptotics. In the five-qubit idling experiment, the selected model contained Hamiltonian terms up to 3-local and dissipator terms up to 2-local, with no statistically justified gain from adding 3-local dissipation. The supporting negative log-likelihoods for the most expressive models were tightly clustered—$$L_{f'\circ f}((x_q,x_p),x_a,x_c)=\int_{x_b\in \mathbb R^b} L_{f'}(x_q,x_b,x_c)L_f(x_p,x_a,x_b)\,d\lambda^b.$$7 for 3local H / 3local D, $$L_{f'\circ f}((x_q,x_p),x_a,x_c)=\int_{x_b\in \mathbb R^b} L_{f'}(x_q,x_b,x_c)L_f(x_p,x_a,x_b)\,d\lambda^b.$$8 for a2a H / 3local D, $$L_{f'\circ f}((x_q,x_p),x_a,x_c)=\int_{x_b\in \mathbb R^b} L_{f'}(x_q,x_b,x_c)L_f(x_p,x_a,x_b)\,d\lambda^b.$$9 for a2a H / a2a D, and $$\frac{dQ_{i|\mathcal F_t}}{dQ_{|\mathcal F_t}}=\frac{X_t^i}{X_0^i},$$0 for 3local H / a2a D—so the likelihood gain from 3-local dissipation was negligible relative to its added parameter count. The same framework was also used for driven single-qubit Hamiltonian recovery, shaped-pulse Hamiltonian reconstruction, and hidden-qubit tests in coupler-mediated dynamics (Severin et al., 1 May 2026).

4. Architecture as likelihood engineering

In another line of work, the processor itself is designed so that the physical architecture increases the likelihood of successful computation. Special-Purpose Quantum Processor Design treats the logical circuit as a weighted Circuit Coupling Graph with adjacency matrix

$$\frac{dQ_{i|\mathcal F_t}}{dQ_{|\mathcal F_t}}=\frac{X_t^i}{X_0^i},$$1

and synthesizes a planar Physical Coupling Graph subject to a maximum degree

$$\frac{dQ_{i|\mathcal F_t}}{dQ_{|\mathcal F_t}}=\frac{X_t^i}{X_0^i},$$2

The architecture search proceeds by profiling, pruning, high-degree handling, and recovering, while minimizing the communication surrogate

$$\frac{dQ_{i|\mathcal F_t}}{dQ_{|\mathcal F_t}}=\frac{X_t^i}{X_0^i},$$3

where $$\frac{dQ_{i|\mathcal F_t}}{dQ_{|\mathcal F_t}}=\frac{X_t^i}{X_0^i},$$4 is shortest-path distance on the candidate hardware graph. The primary routing-overhead metric is

$$\frac{dQ_{i|\mathcal F_t}}{dQ_{|\mathcal F_t}}=\frac{X_t^i}{X_0^i},$$5

the number of extra SWAP gates per original two-qubit gate. On 158 quantum programs collected from IBM Qiskit, the reported average values were $$\frac{dQ_{i|\mathcal F_t}}{dQ_{|\mathcal F_t}}=\frac{X_t^i}{X_0^i},$$6 for SPQPD, $$\frac{dQ_{i|\mathcal F_t}}{dQ_{|\mathcal F_t}}=\frac{X_t^i}{X_0^i},$$7 for a triangular lattice, $$\frac{dQ_{i|\mathcal F_t}}{dQ_{|\mathcal F_t}}=\frac{X_t^i}{X_0^i},$$8 for Li’s structure, and $$\frac{dQ_{i|\mathcal F_t}}{dQ_{|\mathcal F_t}}=\frac{X_t^i}{X_0^i},$$9 for a cross-square lattice. The method was therefore presented as a workload-specific planar processor synthesis technique that reduces SWAP-induced error accumulation and improves the probability of correct execution (Lu et al., 2021).

Energy-based quantum hardware provides a more literal physical realization of objective optimization. The D-Wave Two annealing processor was designed to minimize

$$\rho = \sum_{i=0}^{3}\sum_{j=0}^{3}\cdots\sum_{k=0}^{3} c_{ij\cdots k}\,\sigma_i\otimes\sigma_j\otimes\cdots\otimes\sigma_k,$$0

on a Chimera $$\rho = \sum_{i=0}^{3}\sum_{j=0}^{3}\cdots\sum_{k=0}^{3} c_{ij\cdots k}\,\sigma_i\otimes\sigma_j\otimes\cdots\otimes\sigma_k,$$1 graph with 512 qubits and 1472 programmable inter-qubit couplers. Its architecture combines tunably coupled rf-SQUID flux qubits with 4608 embedded flux DACs—512 single-stage, 3520 two-stage, and 512 three-stage—addressed with 56 wires and dissipating about $$\rho = \sum_{i=0}^{3}\sum_{j=0}^{3}\cdots\sum_{k=0}^{3} c_{ij\cdots k}\,\sigma_i\otimes\sigma_j\otimes\cdots\otimes\sigma_k,$$2 in worst-case full-chip reprogramming. Embedding studies for quantum-dot cellular automata networks on such sparse annealers then show that practical use depends on minor embedding, chain management, negotiated-congestion routing, and placement heuristics, because structured logical interactions must be compiled onto bounded-degree hardware graphs (Bunyk et al., 2014, Retallick et al., 2017).

5. Physical front-ends and accelerator architectures

Physical likelihood processing need not be purely digital. In broadband blind source separation, a silicon-photonic microring-resonator weight bank implements analog weighted projections

$$\rho = \sum_{i=0}^{3}\sum_{j=0}^{3}\cdots\sum_{k=0}^{3} c_{ij\cdots k}\,\sigma_i\otimes\sigma_j\otimes\cdots\otimes\sigma_k,$$3

while off-chip electronics optimize PCA and ICA objectives from low-order statistics of the projected output. The demonstrated system used a microring weight bank with dithering-based control, achieved effective weight precision of 9.0 bits rather than 6.7 bits, and performed blind source separation over carriers from 1 GHz to 19.2 GHz using a single photonic chip. Across 22 tested frequencies, the recovered-source SIR remained no less than 30 dB, and in the ill-conditioned mixing experiment the average SIR remained above 35 dB with dithering control, while without dithering the SIR was about 20 dB worse on average. This is not a full probabilistic likelihood engine, but it is a physical contrast evaluator: candidate demixing vectors are realized as analog optical weights, and the hardware returns statistical evidence in the form of variance and kurtosis objectives (Zhang et al., 2022).

A second family of accelerators replaces dense exact likelihood evaluation by sparse exact evaluation plus interpolation. In particle filtering, the Li-PDF method computes the true observation-model likelihood at a smaller set of fulcrums and then fits a numerical likelihood function $$\rho = \sum_{i=0}^{3}\sum_{j=0}^{3}\cdots\sum_{k=0}^{3} c_{ij\cdots k}\,\sigma_i\otimes\sigma_j\otimes\cdots\otimes\sigma_k,$$4 so that

$$\rho = \sum_{i=0}^{3}\sum_{j=0}^{3}\cdots\sum_{k=0}^{3} c_{ij\cdots k}\,\sigma_i\otimes\sigma_j\otimes\cdots\otimes\sigma_k,$$5

For multivariate fitting it uses local kernel averaging,

$$\rho = \sum_{i=0}^{3}\sum_{j=0}^{3}\cdots\sum_{k=0}^{3} c_{ij\cdots k}\,\sigma_i\otimes\sigma_j\otimes\cdots\otimes\sigma_k,$$6

thereby shifting runtime dependence from particle count toward fulcrum count. In wireless sensor networks, maximum-likelihood topology mapping similarly treats packet receptions as Bernoulli observations of latent geometry, estimating node coordinates by maximizing

$$\rho = \sum_{i=0}^{3}\sum_{j=0}^{3}\cdots\sum_{k=0}^{3} c_{ij\cdots k}\,\sigma_i\otimes\sigma_j\otimes\cdots\otimes\sigma_k,$$7

and in the mmWave extension multiplying this with sector-consistency constraints. In that setting, packet success and beam sector IDs become physical measurements from which a topology coordinate system is inferred (Li et al., 2013, Gunathillake, 2018).

At a larger numerical scale, profile-likelihood construction for global SMEFT fits has been recast as a GPU-native likelihood processor. The pipeline keeps the physics likelihood intact, tensorizes the prediction

$$\rho = \sum_{i=0}^{3}\sum_{j=0}^{3}\cdots\sum_{k=0}^{3} c_{ij\cdots k}\,\sigma_i\otimes\sigma_j\otimes\cdots\otimes\sigma_k,$$8

and couples it to a five-stage procedure of pre-scaling, pre-training, training, sampling, and maximizing. The resulting system evaluates the complete 42-dimensional combined SMEFT likelihood in about five hours on a single H100 GPU. For the combined analysis, the reported stages were 5.3 min of pre-scaling, 2.5 min of pre-training, 1.2 h of training, 17.6 min of sampling, and 3.7 h of profiling, replacing a CPU workflow that required 120 CPUs and about 20 h 50 min for sampling alone (Heimel et al., 2024).

6. Scope, applications, and limits

The expression “physical likelihood” is also used in a broader comparative sense, not only for hardware pipelines. In exoplanet habitability studies, for example, a relative Earth-normalized likelihood is defined as

$$\rho = \sum_{i=0}^{3}\sum_{j=0}^{3}\cdots\sum_{k=0}^{3} c_{ij\cdots k}\,\sigma_i\otimes\sigma_j\otimes\cdots\otimes\sigma_k,$$9

combining a temperature-dependent biospheric factor with an atmospheric-retention factor based on hydrodynamic escape and stellar-wind stripping. Under that metric, Proxima b was assigned $$S(t)\propto \operatorname{Tr}\left\{\rho(t)\sum_k (I_{kx}-iI_{ky})\right\}.$$0, $$S(t)\propto \operatorname{Tr}\left\{\rho(t)\sum_k (I_{kx}-iI_{ky})\right\}.$$1, $$S(t)\propto \operatorname{Tr}\left\{\rho(t)\sum_k (I_{kx}-iI_{ky})\right\}.$$2, yielding $$S(t)\propto \operatorname{Tr}\left\{\rho(t)\sum_k (I_{kx}-iI_{ky})\right\}.$$3 and $$S(t)\propto \operatorname{Tr}\left\{\rho(t)\sum_k (I_{kx}-iI_{ky})\right\}.$$4; TRAPPIST-1e was assigned $$S(t)\propto \operatorname{Tr}\left\{\rho(t)\sum_k (I_{kx}-iI_{ky})\right\}.$$5, $$S(t)\propto \operatorname{Tr}\left\{\rho(t)\sum_k (I_{kx}-iI_{ky})\right\}.$$6, $$S(t)\propto \operatorname{Tr}\left\{\rho(t)\sum_k (I_{kx}-iI_{ky})\right\}.$$7, yielding $$S(t)\propto \operatorname{Tr}\left\{\rho(t)\sum_k (I_{kx}-iI_{ky})\right\}.$$8 and $$S(t)\propto \operatorname{Tr}\left\{\rho(t)\sum_k (I_{kx}-iI_{ky})\right\}.$$9. This suggests that the underlying idea of a physical likelihood processor can extend from hardware inference engines to physically motivated comparative scoring systems (Lingam et al., 2017).

Across the literature, several limitations recur. Quantum dynamical reconstructions assume Markovian and often time-independent Lindblad evolution, while full tomography scales exponentially and remains practical mainly for small subsystems (Samach et al., 2021, Severin et al., 1 May 2026). Architecture-level designs such as SPQPD are algorithm-specific and do not model full hardware heterogeneity, calibration drift, or all chip-floorplanning constraints (Lu et al., 2021). Photonic and analog systems remain sensitive to drift, calibration precision, and limited model classes, as illustrated by the central role of 9-bit-equivalent weight control in broadband photonic BSS (Zhang et al., 2022). Sparse-hardware energy processors require embedding, chain management, and careful control infrastructure, so the intended logical objective is mediated by nontrivial compilation overhead (Bunyk et al., 2014, Retallick et al., 2017). The common implication is that a physical likelihood processor is strongest when physical admissibility, likelihood construction, and hardware constraints are co-designed rather than treated as separate stages.