Entropy Quantum Computing (EQC)

• Entropy Quantum Computing (EQC) is a dual-use paradigm that applies open-system photonic dynamics and entropy-focused methods for optimization and quantum information tasks.
• The Dirac-3 architecture implements time-bin photonic modes with gain, loss, measurement, and feedback to relax systems toward low-energy configurations.
• EQC techniques extend to machine learning and denoising applications, though current implementations are critiqued for their classical performance benchmarks.

Entropy Quantum Computing (EQC) is a term used in current literature in two overlapping but non-identical senses. In the narrow and most concrete sense, it denotes an open-system photonic optimization paradigm associated with Quantum Computing Inc.’s Dirac-3 platform, where gain, loss, measurement, and feedback are used to bias time-bin photonic dynamics toward low-cost configurations of a programmed Hamiltonian (Nguyen et al., 2024, Moosavian et al., 5 May 2026). In a broader and less standardized sense, EQC denotes entropy-centered approaches in which entropy, cross entropy, computational entropy, or entropy flow are treated as operational quantities for compression, optimization, denoising, thermodynamic control, or resource allocation in quantum technologies (Huang et al., 12 Feb 2025, Shangnan, 2021, Avidan et al., 16 Jun 2025). The term is therefore heterogeneous: the photonic optimization program is a specific hardware proposal, whereas the broader usage is closer to a research viewpoint.

1. Terminology, operative notion of entropy, and scope

In the QCi/Dirac-3 literature, EQC is presented as a computational paradigm that deliberately embraces environmental noise and decoherence in an open photonic system rather than suppressing them. The intended mechanism is an imaginary-time-like relaxation in which high-energy configurations are dissipatively suppressed while low-energy configurations are stabilized and thus preferentially observed (Nguyen et al., 2024, Moosavian et al., 5 May 2026). In the denoising formulation, the same paradigm is described as a hardware-native quantum optimization mechanism in an open quantum system that uses dissipation, frequent weak measurements, the quantum Zeno effect, and relaxation into a decoherence-free subspace to reach a least-loss configuration (Huang et al., 12 Feb 2025).

This usage does not identify EQC with direct minimization of Shannon entropy or von Neumann entropy as a primitive objective. Shannon entropy and von Neumann entropy are explicitly defined as

$$H(X)=-\sum_i p_i \log p_i,\qquad S(\rho)=-\mathrm{Tr}(\rho\log\rho),$$

but in the photonic optimization papers entropy is operationalized through open-system dynamics, differential loss, measurement, and feedback rather than by optimizing either formula directly (Huang et al., 12 Feb 2025).

A central point of dispute is the operative meaning of “entropy” in the current hardware. The critical comment argues that, in the present EQC realization, the relevant entropy is primarily classical shot noise arising from weak coherent states and photodetection, not a demonstrably nonclassical quantum resource (Moosavian et al., 5 May 2026). By contrast, the original experimental paper presents a forward-looking fully optical extension in which the relevant entropy functional would be the von Neumann entropy of an in-loop photonic state and the target principle would be free-energy minimization,

$F(\rho)=\mathrm{Tr}(\rho H)-T S(\rho),$

with a temperature-like parameter set by the entropy bath and non-unitary processes (Nguyen et al., 2024).

A broader, more interpretive usage extends the label EQC to entropy-centric quantum information tasks such as compression, computational entropy, and entropy-guided simulation. This suggests that EQC has become not only the name of a specific photonic optimizer, but also an umbrella term for research programs in which entropic quantities are elevated to algorithmic resources (Shangnan, 2021, Avidan et al., 16 Jun 2025).

2. Dirac-3 architecture and open-system photonic dynamics

The experimental EQC platform, Dirac-3, is a hybrid photonic–electronic machine based on temporal photonic modes. In the reported implementation, a continuous-wave C-band laser is attenuated by programmable variable optical attenuators, an electro-optic modulator driven by a DAC generates a train of time bins, and a periodically poled lithium niobate waveguide with an undepleted O-band pump implements sum-frequency generation (Nguyen et al., 2024). Upconverted photons are detected by a silicon single-photon detector, photon arrival times are histogrammed by time-correlated single-photon counting, and an FPGA computes feedback and reprograms the next optical waveform (Nguyen et al., 2024).

At the architectural level summarized by the critical comment, the loop contains optical gain, a mixer/encoder implementing linear transformations tied to the target cost function or Hamiltonian, a loss medium imparting linear and nonlinear losses, and TCSPC plus electro-optical feedback that updates losses and possibly gains based on measurements (Moosavian et al., 5 May 2026). A vector of variables

$v=(v_1,\ldots,v_n)$

is mapped onto photonic mode amplitudes or intensities associated with time bins. The mixer/encoder implements the problem’s couplings, including claimed all-to-all connectivity and up to fifth-order correlations, and the loop realizes an effective relaxation process targeting low-cost states (Moosavian et al., 5 May 2026).

The same hardware is described differently depending on the paper’s emphasis. In the original optimization paper, the state space is introduced through time-bin “qudits,” with weak coherent pulses populating the bins and single-photon-regime detection probabilities approximating modal probabilities (Nguyen et al., 2024). In the noise-reversal paper, Dirac-3 is described as a fiber-loop architecture with photon counting, weak coupling to the environment, rapid measurements, and quantum Zeno blockade guiding the system to a decoherence-free, least-loss solution state (Huang et al., 12 Feb 2025).

A crucial physical distinction is that the currently realized states are weak coherent states, not single-photon Fock states. The comment therefore identifies the dominant stochasticity as Poissonian shot noise rather than entanglement, non-Gaussianity, or other nonclassical resources (Moosavian et al., 5 May 2026). This is the main reason the current device is reinterpreted there as a classical, dissipative, analog stochastic optimizer rather than as a demonstrated nonclassical quantum computer.

3. Optimization formalism and statistical-mechanical models

The most general optimization form reported for Dirac-3 is a polynomial cost over nonnegative variables under a global sum constraint:

$$E=\sum_i C_i v_i+\sum_{i,j} J_{ij} v_i v_j+\sum_{i,j,k} T_{ijk} v_i v_j v_k +\sum_{i,j,k,l} Q_{ijkl} v_i v_j v_k v_l+\sum_{i,j,k,l,m} P_{ijklm} v_i v_j v_k v_l v_m,$$

with

$\sum_i v_i=R,\qquad v_i\ge 0.$

This supports dense couplings up to fifth order without quadratization and underlies both continuous and discrete encodings (Nguyen et al., 2024, Moosavian et al., 5 May 2026).

For MAX-CUT and related combinatorial tasks, the critical comment formalizes the device output as effective Gibbs sampling at inverse temperature $\beta$ determined by gains, losses, and feedback:

$$P_\beta(\sigma\mid G)=\frac{e^{\beta C(\sigma)}}{Z(\beta)},\qquad Z(\beta)=\sum_\sigma e^{\beta C(\sigma)}=\sum_k \Omega(k)e^{\beta k}.$$

The induced distribution on cut values is

$$P_\beta(C=k\mid G)=\frac{\Omega(k)e^{\beta k}}{\sum_{k'}\Omega(k')e^{\beta k'}}.$$

In this picture, low effective temperatures concentrate probability mass on high-cut or low-energy outcomes, while the density of states $\Omega(k)$ provides the entropic counterweight (Moosavian et al., 5 May 2026).

The original EQC paper motivates these dynamics through an imaginary-time analogy and a measurement–feedback map that suppresses high-loss configurations and amplifies low-loss ones (Nguyen et al., 2024). The critical comment, however, notes that no explicit Lindblad master equation with a specified Hamiltonian and jump operators is given, and no entropy-production bounds such as Spohn inequalities are provided. The formalism is therefore statistical-mechanical at the distribution level rather than a microscopic open-system derivation (Moosavian et al., 5 May 2026).

Specialized mappings illustrate the flexibility of the framework. In the noise-reversal formulation, the unknown noise counts $N_i$ are optimized under a known total-noise constraint,

$F(\rho)=\mathrm{Tr}(\rho H)-T S(\rho),$0

using a nearest-neighbor second-order spatial-correlation cost,

$F(\rho)=\mathrm{Tr}(\rho H)-T S(\rho),$1

which is converted into a quadratic Hamiltonian implemented as state-dependent optical loss (Huang et al., 12 Feb 2025). In the CVQBoost classifier, the training problem is reduced to a constrained quadratic form

$F(\rho)=\mathrm{Tr}(\rho H)-T S(\rho),$2

with $F(\rho)=\mathrm{Tr}(\rho H)-T S(\rho),$3 and $F(\rho)=\mathrm{Tr}(\rho H)-T S(\rho),$4, so that Dirac-3 can act as a continuous quadratic solver for ensemble weights (Emami et al., 14 Mar 2025).

4. Reported tasks, applications, and empirical results

The first experimental EQC paper reported three benchmark classes: a two-variable nonconvex polynomial minimization problem, QPLIB_0018 as a 50-variable continuous nonconvex quadratic program, and MAX-$F(\rho)=\mathrm{Tr}(\rho H)-T S(\rho),$5-CUT on a random graph with $F(\rho)=\mathrm{Tr}(\rho H)-T S(\rho),$6 and $F(\rho)=\mathrm{Tr}(\rho H)-T S(\rho),$7 edges for $F(\rho)=\mathrm{Tr}(\rho H)-T S(\rho),$8 (Nguyen et al., 2024, Moosavian et al., 5 May 2026). The same paper states that the machine supports up to $F(\rho)=\mathrm{Tr}(\rho H)-T S(\rho),$9 independent variables under a sum constraint and consumes below $v=(v_1,\ldots,v_n)$0 W of electrical power (Nguyen et al., 2024). For QPLIB_0018, it reports that over $v=(v_1,\ldots,v_n)$1 runs Dirac-3 returns the ground state in about $v=(v_1,\ldots,v_n)$2 of trials (Nguyen et al., 2024). For the MAX-$v=(v_1,\ldots,v_n)$3-CUT example, cut-size distributions are reported over $v=(v_1,\ldots,v_n)$4 runs and compared against an SDP relaxation baseline (Nguyen et al., 2024).

A subsequent paper uses EQC for noise reversal rather than generic optimization. There the device is treated as a quantum noise emulator that “observes and reproduces the quantum statistical properties of noise” and retrieves a concrete noise realization to subtract from measured data (Huang et al., 12 Feb 2025). Dirac-3 is reported to support $v=(v_1,\ldots,v_n)$5 modes, each holding up to $v=(v_1,\ldots,v_n)$6 effective photons, enabling exact processing up to $v=(v_1,\ldots,v_n)$7 pixels under the sum constraint (Huang et al., 12 Feb 2025). The paper reports qualitative recovery on 1D and 2D decaying sinusoids and on photon-counting LiDAR imagery. For a realistic space-LiDAR image, daytime-like Poisson noise with average amplitude approximately $v=(v_1,\ldots,v_n)$8 the average signal amplitude is reported to preserve the main features after column-wise EQC reversal, albeit with artificial features in low-signal regions; an extreme case with average noise approximately $v=(v_1,\ldots,v_n)$9 the signal is reported to remain recoverable in its main features but with many artificial features at image edges (Huang et al., 12 Feb 2025). The paper explicitly does not report SNR, PSNR, SSIM, or MSE.

A further application paper introduces CVQBoost, a boosting-style classifier co-designed for Dirac-3’s quadratic solver mode (Emami et al., 14 Mar 2025). On the Kaggle Credit Card Fraud Detection dataset, described there as roughly $$E=\sum_i C_i v_i+\sum_{i,j} J_{ij} v_i v_j+\sum_{i,j,k} T_{ijk} v_i v_j v_k +\sum_{i,j,k,l} Q_{ijkl} v_i v_j v_k v_l+\sum_{i,j,k,l,m} P_{ijklm} v_i v_j v_k v_l v_m,$$0 transactions with about $$E=\sum_i C_i v_i+\sum_{i,j} J_{ij} v_i v_j+\sum_{i,j,k} T_{ijk} v_i v_j v_k +\sum_{i,j,k,l} Q_{ijkl} v_i v_j v_k v_l+\sum_{i,j,k,l,m} P_{ijklm} v_i v_j v_k v_l v_m,$$1 fraud and $$E=\sum_i C_i v_i+\sum_{i,j} J_{ij} v_i v_j+\sum_{i,j,k} T_{ijk} v_i v_j v_k +\sum_{i,j,k,l} Q_{ijkl} v_i v_j v_k v_l+\sum_{i,j,k,l,m} P_{ijklm} v_i v_j v_k v_l v_m,$$2 features, CVQBoost is benchmarked against XGBoost under several balancing strategies and later extended to synthetic datasets ranging from $$E=\sum_i C_i v_i+\sum_{i,j} J_{ij} v_i v_j+\sum_{i,j,k} T_{ijk} v_i v_j v_k +\sum_{i,j,k,l} Q_{ijkl} v_i v_j v_k v_l+\sum_{i,j,k,l,m} P_{ijklm} v_i v_j v_k v_l v_m,$$3M to $$E=\sum_i C_i v_i+\sum_{i,j} J_{ij} v_i v_j+\sum_{i,j,k} T_{ijk} v_i v_j v_k +\sum_{i,j,k,l} Q_{ijkl} v_i v_j v_k v_l+\sum_{i,j,k,l,m} P_{ijklm} v_i v_j v_k v_l v_m,$$4M samples with $$E=\sum_i C_i v_i+\sum_{i,j} J_{ij} v_i v_j+\sum_{i,j,k} T_{ijk} v_i v_j v_k +\sum_{i,j,k,l} Q_{ijkl} v_i v_j v_k v_l+\sum_{i,j,k,l,m} P_{ijklm} v_i v_j v_k v_l v_m,$$5 to $$E=\sum_i C_i v_i+\sum_{i,j} J_{ij} v_i v_j+\sum_{i,j,k} T_{ijk} v_i v_j v_k +\sum_{i,j,k,l} Q_{ijkl} v_i v_j v_k v_l+\sum_{i,j,k,l,m} P_{ijklm} v_i v_j v_k v_l v_m,$$6 features (Emami et al., 14 Mar 2025). That study reports competitive AUC together with favorable training-time scaling as dataset size and feature dimension grow, and attributes the gain to the fact that the core optimization time on Dirac-3 does not blow up with the number of samples in the same way as the classical baseline (Emami et al., 14 Mar 2025). It also reports a known dynamic-range limitation around $$E=\sum_i C_i v_i+\sum_{i,j} J_{ij} v_i v_j+\sum_{i,j,k} T_{ijk} v_i v_j v_k +\sum_{i,j,k,l} Q_{ijkl} v_i v_j v_k v_l+\sum_{i,j,k,l,m} P_{ijklm} v_i v_j v_k v_l v_m,$$7 dB and higher run-to-run runtime variance due to analog hardware stochasticity (Emami et al., 14 Mar 2025).

These application results indicate that EQC has been used as more than a combinatorial optimizer. It has also been framed as a constrained quadratic solver for machine learning and as a hardware-native denoiser for photon-counting imaging. A plausible implication is that the most consistent near-term identity of Dirac-3 is not a universal quantum computer but a configurable analog optimizer specialized to sum-constrained polynomial objectives.

5. Critique, simulatability, and the status of quantum advantage

The most detailed critical assessment argues that the current EQC realization is well modeled as a canonical-ensemble Gibbs sampler with an effective temperature and that its stochasticity is consistent with classical Poissonian shot noise from weak coherent states and photodetection (Moosavian et al., 5 May 2026). The comment finds no evidence of entanglement, non-Gaussianity, or other nonclassical resources, and states that the observed behavior is consistent with “a classical dissipative sampler with measurement-and-feedback” (Moosavian et al., 5 May 2026).

On the reported benchmarks, the same comment concludes that current EQC does not outperform strong classical methods on commodity hardware. For the two-variable nonconvex test, it argues that plain gradient descent is a weak baseline and notes that Particle Swarm Optimization and Evolutionary Centers Algorithm converge robustly and rapidly to the global minimum in less than $$E=\sum_i C_i v_i+\sum_{i,j} J_{ij} v_i v_j+\sum_{i,j,k} T_{ijk} v_i v_j v_k +\sum_{i,j,k,l} Q_{ijkl} v_i v_j v_k v_l+\sum_{i,j,k,l,m} P_{ijklm} v_i v_j v_k v_l v_m,$$8 s on a CPU (Moosavian et al., 5 May 2026). For QPLIB_0018, it reports that Ipopt solves the instance to global optimality on a single CPU core in a fraction of a second, reliably and deterministically (Moosavian et al., 5 May 2026). For MAX-$$E=\sum_i C_i v_i+\sum_{i,j} J_{ij} v_i v_j+\sum_{i,j,k} T_{ijk} v_i v_j v_k +\sum_{i,j,k,l} Q_{ijkl} v_i v_j v_k v_l+\sum_{i,j,k,l,m} P_{ijklm} v_i v_j v_k v_l v_m,$$9-CUT on the $\sum_i v_i=R,\qquad v_i\ge 0.$0-node graph, exact branch-and-cut establishes the optimum, while simulated annealing and tabu search are reported to find optimal cuts with high success probabilities and sub-millisecond runtimes; for MAX-4-CUT, success rates of $\sum_i v_i=R,\qquad v_i\ge 0.$1 and $\sum_i v_i=R,\qquad v_i\ge 0.$2 are reported in approximately $\sum_i v_i=R,\qquad v_i\ge 0.$3–$\sum_i v_i=R,\qquad v_i\ge 0.$4 ms per run, whereas Dirac-3 distributions center below the optimum and device runtimes were not reported (Moosavian et al., 5 May 2026).

The critique also challenges one of the more prominent narrative points around approximation quality on dense random graphs. It proves that under fixed-$\sum_i v_i=R,\qquad v_i\ge 0.$5 Gibbs sampling on $\sum_i v_i=R,\qquad v_i\ge 0.$6, the probability of beating the Goemans–Williamson worst-case approximation threshold tends to $\sum_i v_i=R,\qquad v_i\ge 0.$7 rapidly in $\sum_i v_i=R,\qquad v_i\ge 0.$8, but then emphasizes that these instances are “average-case easy”: a random assignment already achieves cut value approximately $\sum_i v_i=R,\qquad v_i\ge 0.$9 and PTASs are known for dense instances (Moosavian et al., 5 May 2026). Surpassing the Goemans–Williamson threshold on such graphs is therefore not informative evidence of quantum or photonic advantage.

The resulting controversy is not whether the hardware is physically interesting, but what claims it can presently support. The comment explicitly states that its conclusions reflect the technology’s “current early stage of development” and are “not meant to discourage its pursuit” (Moosavian et al., 5 May 2026). Its recommendations are correspondingly concrete: provide a first-principles open-system model with explicit Hamiltonian and jump operators, derive conditions under which the steady state is Gibbsian, demonstrate nonclassical photonic states or non-Gaussianity indispensable to performance, benchmark against stronger classical baselines on harder instances, report time-to-solution and energy consumption, and quantify how modes, photons, pump power, and loop bandwidth scale with problem size (Moosavian et al., 5 May 2026).

6. Broader entropy-centric usages and related research programs

Outside the Dirac-3 literature, the label EQC is used more diffusely for quantum-computational programs organized around entropy as an operational resource. One strand concerns quantum compression and learning. A compression paper develops quantum cross entropy

$\beta$0

and proves

$\beta$1

so that mismatched variable-length quantum source coding has asymptotic rate given by quantum cross entropy and minimizing that rate recovers the von Neumann entropy (Shangnan, 2021). In this usage, EQC is not a photonic optimizer but an entropy-centered view of model mismatch, coding rate, and learning loss.

A second strand introduces fully quantum computational min- and max-entropies under explicit circuit-size restrictions. There the basic object is not physical dissipation but efficient distinguishability and efficient entanglement distillation. The computational min-entropy and max-entropy are defined through bounded-size channels, purified-distance smoothing, and a canonical “pretty good purification,” and the max-entropy is given an operational meaning as the efficiently achievable singlet fraction with the environment (Avidan et al., 16 Jun 2025). This is again an entropy-based computational theory rather than an implementation of Dirac-3.

A third strand develops quantum algorithms whose primitives are themselves entropic or distance-like quantities. Low-rank quantum algorithms have been proposed for computing the von Neumann entropy, Rényi entropy, trace distance, and fidelity by combining state block-encoding, quantum state eigenvalue transformation, and trace estimation (Wang et al., 2022). These results treat entropy as a first-class computational target and are sometimes presented as part of an entropy-centric quantum-computing toolkit.

Other work makes contact through thermodynamics and information flow. Replica-based analyses of entropy production emphasize that quantum entropy flow can contain coherent contributions with no analogue in ordinary physical observables, and propose a correspondence between Rényi-entropy flow and full counting statistics of energy transfer (Ansari et al., 2019). A separate transport analysis argues that the conventional heat formula is incomplete in the quantum limit and introduces a free-energy-flow correction,

$\beta$2

with the consequence that entropy current vanishes as $\beta$3 in accordance with the Third Law (Jimenez-Valencia et al., 31 Aug 2025). These connections to EQC are interpretive rather than terminologically settled, but they support a broader picture in which entropy flow is engineered rather than merely tolerated.

Quantum chemistry provides another extended usage. The AEGISS workflow combines single-orbital entropies from exploratory DMRG with atomic-orbital projections to construct compact active spaces, and the supplied exposition explicitly frames this as an EQC-style resource-allocation problem in which entropic information guides the construction of the computational subspace (Tarocco et al., 14 Aug 2025). On Ru(II) complexes, the reported output is typically a $\beta$4-orbital active space, corresponding to $\beta$5 spin-orbitals or qubits after mapping (Tarocco et al., 14 Aug 2025). This is far removed from Dirac-3 hardware, but it illustrates how the term EQC has broadened toward entropy-guided quantum workflow design.

Taken together, these usages indicate that “Entropy Quantum Computing” has not yet stabilized into a single canonical meaning. In the strict experimental sense, it refers to an open photonic optimization platform whose present realization is best understood through gain, loss, measurement, feedback, and effective Gibbs sampling (Nguyen et al., 2024, Moosavian et al., 5 May 2026). In the broader research sense, it denotes a family of approaches in which entropy-based quantities structure computation, learning, control, or simulation (Shangnan, 2021, Avidan et al., 16 Jun 2025). This suggests that EQC is presently both a specific contested hardware program and an emerging interdisciplinary vocabulary for entropy-centered quantum methods.