Equivalent Classical Operations (NEO)

• NEO is a framework that implements classical analogs of quantum gates using noise-based logic, accurately processing superpositions in high-dimensional spaces.
• It employs Instantaneous Noise-Based Logic (INBL) to construct XOR and XNOR gates that perform precise bitwise operations analogous to quantum controlled gates.
• The methodology scales polynomially in hardware complexity, offering a deterministic and error-free alternative to simulate select quantum operations.

Equivalent Classical Operations (NEO)

Equivalent classical operations within the Noise-Enabled Orthogonality (NEO) paradigm provide classical constructs that replicate the functional roles of key quantum gates, particularly within the superposition structure of high-dimensional Hilbert spaces. NEO-based methods, exemplified by the Instantaneous Noise-Based Logic (INBL) formalism, generate a full set of generalized logic gates (notably XOR and XNOR) that operate instantaneously and deterministically on superpositions of orthogonal basis signals, with hardware requirements scaling polynomially in the problem size. This framework enables entirely classical, hardware-efficient alternatives to specific classes of quantum operations, and thereby establishes a rigorous standard for equivalence between classical and quantum computational transformations (Kenarangui et al., 11 Nov 2025). The notion of “equivalent classical operation” applies broadly, not only in information-processing architectures but also in transport and entanglement settings, where classical operations can match or simulate certain quantum functionalities within well-defined limits (Liu et al., 2022, Sarkar et al., 2016).

1. Formal Definition and Operational Framework

In the NEO paradigm, equivalence refers to the implementation of a transformation or gate in a classical system that performs the same functional action—on individual basis vectors and on superpositions—as its quantum analogue. In the INBL setting, the Hilbert space is constructed from $2^M$ orthogonal stochastic product signals derived from $M$ noise-bits, with each bit realized as an independent, zero-mean orthogonal random telegraph wave (RTW). The mapping

$S_{b_1 \ldots b_M}(t) = \prod_{i=1}^M r_{i b_i}(t)$

encodes every bitstring to an orthogonal time-signal representative, forming a classical “Hilbert” hyperspace isomorphic to the computational basis of $M$-qubit systems.

Equivalent operations (e.g., XOR, XNOR) in this framework act pointwise on all basis vectors—analogously to how a CNOT or its complement (controlled-XNOR) would act on a quantum register—without requiring any access to or manipulation of the underlying reference noise system (Kenarangui et al., 11 Nov 2025). The requirement of universality over superpositions means that each classical gate must naturally extend its action to arbitrary linear combinations, ensuring exact correspondence at the superpositional level.

2. Construction of XOR and XNOR Gates in INBL

The implementation of pairwise operations such as XOR and XNOR in INBL relies on the multiplicative structure of the hyperspace signals:

• Each word $A(t)$ or $B(t)$ of length $M$ is encoded as the product of reference signals set by its bitpattern.
• The core bitwise operations are:
  • $$\mathrm{XOR}_i(t) = [G_i(t) \cdot G'_i(t)] \cdot r_{i0}(t)$$, yielding $r_{i1}(t)$ if the underlying bits differ, $r_{i0}(t)$ otherwise.
  • $M$0, yielding $M$1 if the bits are equal, $M$2 otherwise.
• The full gate operation for M-bit strings is:
  • $M$3,
  • $M$4.

Crucially, the action on superpositions is distributive, exactly mirroring the way quantum controlled gates act on a tensor product of superposed vectors. All processing is realized by combinational logic—multipliers—while the reference noise generator remains untouched, and the operations are fully general-purpose (no destructive reference re-initialization or limited-use gates as in earlier “reference-wiring” approaches) (Kenarangui et al., 11 Nov 2025).

3. Principles of Classical Replaceability and Channel Equivalence

The broader concept of classically equivalent operations has been addressed formally in the theory of classically replaceable operations (CRO). In this context, an operation is classically replaceable if, for a given input/output scenario, the quantum channel can be simulated—without loss of generality—by classical pre- and post-processing with a stochastic map in the relevant basis. Explicit criteria distinguish operations that are equivalent at the statistical or state level:

• In the “quantum-in/quantum-out” setting, $M$5 is a qqCRO if $M$6 with $M$7 stochastic, i.e., all action proceeds by measurement, classical processing, and re-preparation in the basis (Liu et al., 2022).
• Unitary operations qualify for classical equivalence (i.e., deterministic classical simulation) if the induced map is a permutation matrix in the measurement basis.
• Notably, the controlled-NOT (CNOT) gate does not belong to qqCRO, although it is dephasing-covariant (DIO). Thus, while some quantum gates have equivalent classical operations (e.g., Clifford gates under certain measurement settings), others cannot be fully captured except at the level of basis-state relabeling.

Resource-theoretic approaches quantitatively characterize the irreplaceability of quantum operations, with robustness measures capturing the minimal extent to which a quantum channel fails to be classically simulable (Liu et al., 2022).

4. Hardware Realization and Complexity in the NEO Approach

The INBL realization of equivalent operations achieves instantaneity and scalability, with hardware complexity linear in the number of encoded bits $M$8:

• $M$9 reference-noise lines (RTWs) supply the basis of the “Hilbert” hyperspace.
• An “Hilbert-space synthesizer” composed of trees of multipliers generates basis signals corresponding to the programmed bitstrings.
• XOR/XNOR units comprise banks of independent bitwise multipliers, aggregating results through final-stage multiplication without touching any reference-noise wires.

This configuration eliminates destructive reference manipulation and facilitates polynomial scaling of both gate-count and storage, in contrast to prior methods that required re-initialization or had non-combinational implementations. Simulations (up to $S_{b_1 \ldots b_M}(t) = \prod_{i=1}^M r_{i b_i}(t)$0) confirm zero error rates over $S_{b_1 \ldots b_M}(t) = \prod_{i=1}^M r_{i b_i}(t)$1 cycles and constant gate latency ($S_{b_1 \ldots b_M}(t) = \prod_{i=1}^M r_{i b_i}(t)$2 ns at 1 GHz clock), positioning INBL-based NEO operations as hardware-competitive realizations of superposition-based processing (Kenarangui et al., 11 Nov 2025).

5. Theoretical and Practical Criteria for Classical-Quantum Equivalence

The general criteria for establishing classical equivalence to quantum operations demand:

• The gate or channel must enact, for all basis and superposed inputs, the identical functional mapping as in the quantum computational model, including the preservation of global and local structure in the “Hilbert” space.
• For transport networks, the notion extends to the Transport Equivalent Network (TEN) formalism: two networks are transport-equivalent if, through unitary transformations that leave input/output ports invariant, they yield identical current-voltage characteristics globally (full $S_{b_1 \ldots b_M}(t) = \prod_{i=1}^M r_{i b_i}(t)$3–$S_{b_1 \ldots b_M}(t) = \prod_{i=1}^M r_{i b_i}(t)$4) and locally (all bond currents) across the full quantum-to-classical (high-dephasing) transition (Sarkar et al., 2016).
• In quantum information, operations are only classically equivalent if all statistics and output states (possibly after measurement and classical post-processing) are preserved under simulated action (Liu et al., 2022). Some channels (such as dephasing or entanglement-breaking channels) are fully replaceable, while others (including generic entangling gates) fundamentally resist classical simulation except in trivialized computational scenarios.

6. Unified Perspective and Impact on Quantum-Classical Boundaries

NEO and related frameworks reify a rigorous boundary between classical and quantum computational paradigms, providing explicit architectures in which classes of quantum operations—most notably those representable as classical basis permutations or measurable process–prepare channels—can be replaced fully by deterministic, scalable, and physically realizable classical circuits. Equivalence holds at the operational and statistical level for all superpositions in the constructed hyperspace. However, fundamental quantum resources such as context-sensitive coherence and certain types of entanglement (as made precise by SLOCC or Gaussian local unitary classification) remain intrinsically nonclassical outside of these equivalence categories (Liu et al., 2022, Giedke et al., 2013, Burchardt et al., 2020).

A plausible implication is that future classical architectures leveraging NEO principles may serve as universal simulators for large classes of quantum circuits—where classical equivalence is strictly provable—thus reducing quantum device requirements for error mitigation, variational, and certain simulation tasks. However, strict nonclassicality persists for operations outside deterministic classical equivalence, providing robust targets for the continued development of genuine quantum computational hardware.