Unoperation in Quantum Circuit Inversion

• Unoperation is a computational paradigm that generalizes traditional inversion by coherently enumerating all valid pre-images of an operation, including irreversible or nonunitary functions.
• Quantum circuit realizations, such as ripple-carry unadders and unmultipliers, enable the systematic and efficient inversion of classically hard-to-invert functions through superposition.
• The approach has significant implications for quantum cryptanalysis, error correction, and algorithm design by extending classical digital logic inversion into the quantum realm.

Unoperation refers to the construction, analysis, and application of computational devices—typically quantum circuits—that, given a valid output of an operation $\hat{O}$, coherently generate the set of all valid pre-images (inputs) $x$ such that $\hat{O}(x) = \tilde{x}$. This paradigm generalizes the notion of inversion from bijective/unitary to arbitrary (possibly surjective, non-invertible, or nonunitary) operations, with consequences for quantum algorithm design, complexity theory, and quantum cryptanalysis (Kohl, 9 Oct 2025). The implementation of unoperations, particularly for classically hard-to-invert functions such as addition or multiplication, leverages both circuit-level reversals of arithmetic processes and the unique ability of quantum computation to generate and manipulate superpositions of all valid pre-images in parallel.

1. Formal Definition and Conceptual Foundations

For a given operation $\mathcal{O}$ mapping an input set $X$ to an output set $Y$ via $\mathcal{O}(x) = \tilde{x}$, the unoperation $\mathfrak{Un}(\mathcal{O})$ is defined as the computation that, for a fixed output $\tilde{x}$, yields the set of all $x$ such that $\mathcal{O}(x) = \tilde{x}$: $$\mathfrak{Un}(\mathcal{O})(\tilde{x}) := \{ x \mid \mathcal{O}(x) = \tilde{x} \}$$ If $\mathcal{O}$ is bijective or unitary, the unoperation reduces to the usual inverse or quantum uncomputation. For non-injective or irreversible functions, the unoperation must enumerate (coherently, in the quantum case) the full pre-image set (Kohl, 9 Oct 2025).

The unoperation generalizes uncomputation by extending the domain from bijective/unitary functions to arbitrary classical or quantum operations, including those relevant for irreversible logic, error correction, or cryptanalytic inversion.

2. Quantum Circuit Realizations: Unaddition and Beyond

A canonical example is the “unaddition” circuit, which inverts binary addition. Given a sum $S$, unaddition generates all tuples $(a,b)$ such that $a+b=S$. Classically: $$\mathfrak{Un}(+)(3) = \{ (0,3), (1,2), (2,1), (3,0)\}$$ The paper implements this process via a ripple-carry-unadder (RCU), mirroring the classical ripple-carry adder (RCA), but with the computational flow reversed.

Full-Unadder Construction

The base unit is the full-unadder, which maps a single sum bit and carry ($c_\text{out}, \text{sum}$) to all valid $(a, b, c_\text{in})$:

$c_\text{out}$	sum	Output $(c_\text{in}, b, a)$	Probability amplitude/superposition
0	0	(0, 0, 0), (0, 1, 1), ...	uniform
1	1	...	...

The quantum circuit (see Fig. 2 in the source) acts on $$|c_\text{out}\rangle\,|\text{sum}\rangle\,|000\rangle$$ and uses universal gate primitives (H, X, $R_y$) to output $(c_\text{in}, b, a)$ as a uniform superposition of all valid solutions corresponding to the input bits (Kohl, 9 Oct 2025). An optimized version reduces the ancilla overhead, using a single work qubit.

RCU Assembly

The n-bit unadder is built by cascading n full-unadder units. Each stage receives as input the sum bit and the carry, outputting local solution bits and propagating carries downward. The global output is a superposition over all $(a,b)$ matching $a+b=S$ bitwise, with appropriate carry chain validation.

Extension: Unmultiplier

The unoperation for multiplication ("unmultiplier") is constructed by replacing the classical multiplier’s controlled-adders with unadders (RCUs). Inputs are arranged so that the quantum circuit generates the full pre-image set $(x, y)$ such that $x\cdot y = p$ for a given product $p$. Post-processing discards spurious outputs (invalid carry patterns), yielding the set of real integer factors (Kohl, 9 Oct 2025).

3. Resource Scaling and Complexity Analysis

The RCU and unmultiplier circuits operate with a qubit requirement

$\mathcal{O}((\log N)^2)$

for $N$-bit arithmetic, matching the leading order scaling of Shor's algorithm for integer factoring. Each output value leads, in general, to a superposition over the exponentially many valid pre-images (for surjective maps), leveraging quantum parallelism for full solution set enumeration. While classical post-processing is required to filter out invalid or spurious branches (such as duplicate factor orderings or forbidden carry chains), the construction maintains circuit depth and width in polynomial bounds (Kohl, 9 Oct 2025).

4. Application to Cryptanalysis and Trapdoor Functions

Many classical cryptographic primitives are based on trapdoor functions—easy to compute, hard to invert. The unoperation paradigm, implemented as a quantum circuit, provides an architectural template for constructing quantum devices that, for any efficiently computable function, generate all pre-images corresponding to a fixed output. In the specific application to integer factorization, the unmultiplier acts as an explicit quantum circuit for factorizing $N = x \cdot y$, with resource scaling that rivals Shor's algorithm and operates via a transparent extension of classical logic circuit principles (Kohl, 9 Oct 2025). This suggests a potentially broad impact on future quantum cryptanalytic techniques for functions beyond factoring.

5. Unoperation in Nonunitary and Irreversible Quantum Processes

The relation of unoperation to nonunitary evolution and irreversible processes is nontrivial. The engineered dissipation protocols of (Zapusek et al., 2022, Mourik et al., 2023) enable inherently irreversible logic gates (e.g., OR, NOR, XOR) in quantum systems, exploiting dissipative channels and additional excited state manifolds. Such “unoperation-like” processes drive the system from a single output back into all possible inputs or act as projectors that irreversibly map degenerate pre-images to a unique output. Although these protocols typically focus on implementing the direct irreversible map, the quantum unoperation is the systematic reverse: mapping a given output to its entire coherent pre-image set, not just an irreversible merger.

A plausible implication is that combining dissipation engineering with unoperation circuits may yield efficient schemes for probabilistic inversion, error filtering, or entropy extraction in quantum information processing.

6. Connections to Quantum Information Theory and Superselection Rules

In the context of quantum communications with unknown or partially characterized channels—modeled by “twirling” superoperators—the notion of unoperating a channel is limited by information loss. Only those operations that commute (or are consistent) with the twirling map can be unoperated or “undone” (Miatto, 2012). This physical limitation is reflected mathematically in the requirement that the unoperated map $\mathcal{O}'$ must satisfy

$$\mathcal{O}' \circ (\text{twirling}) = (\text{twirling}) \circ \mathcal{O}$$

where the partial inversion may be possible only for those subspaces preserved by twirling (often the diagonal). Thus, the fidelity of a physical unoperation is ultimately limited by channel knowledge and underlying superselection constraints.

This formalism demonstrates that the unoperation concept, though algorithmically general, respects deep quantum structural limitations imposed by physical decoherence, information loss, or symmetries.

7. Implications for Algorithm Design and Quantum Computing Architectures

The unoperation approach supports the systematic reversal of classical and quantum circuits for both reversible and irreversible functions. By treating unoperation as a first-class circuit-constructing primitive, entire families of inversion-based algorithms can be naturally described with explicit resource scaling and decomposition into elementary gate units. The methodology enables direct quantum circuit analogues of classical digital logic inversion, facilitates the design of hybrid quantum-classical routines, and underpins architectures requiring explicit pre-image enumeration.

A plausible implication is that further development of unoperation circuits may yield efficient alternatives to traditional Grover-style search for certain function inversion problems, especially where the underlying structure allows for compact circuit realization of full pre-image superpositions.

---

In summary, unoperation generalizes inversion and uncomputation to the domain of arbitrary (potentially non-invertible or nonunitary) processes, with concrete quantum circuit constructions for unaddition, unmultiplication, and beyond. The approach is both resource-efficient and aligned with classical digital circuit methodologies, potentially providing foundational tools for quantum cryptanalysis, trapdoor inversion, and a wide array of algorithmic applications (Kohl, 9 Oct 2025). The feasibility and limits of unoperation are intimately tied to the structure of the target operation, the availability of quantum parallelism, and the presence of physical constraints such as channel twirling or engineered dissipation.