Unaddition Quantum Circuit: Inverting Addition

• Unaddition quantum circuit is a quantum construct that uses coherent superposition to generate all valid input pairs for a given sum, inverting classical addition.
• It employs a quantum ripple-carry unadder design with optimized full-unadder gates to efficiently expand ambiguous input decompositions with reduced qubit requirements.
• The circuit has practical applications in quantum factoring and cryptanalysis by extracting valid multiplicative factors through a reversible inversion of surjective operations.

An unaddition quantum circuit is a quantum computational construct designed to generate, in coherent superposition, the complete set of inputs that could have led to a given output of a generally non-injective (surjective) operation—in particular, classical addition. Unlike uncomputation, which refers to the reversal of unitary evolution for bijective operations, unaddition is a specific instance of a broader quantum “unoperation” framework, purpose-built to invert surjective logic by coherently generating all valid input decompositions for a provided sum. This capability is fundamental for quantum algorithms parsing ambiguous inverses, such as in cryptanalysis or algebraic verification tasks. The unaddition circuit’s construction, efficiency, and broader significance have been recently delineated within the context of quantum factoring and general “unoperation” paradigm formulation (Kohl, 9 Oct 2025).

1. Foundational Principles and Formal Definition

In the formalism introduced by (Kohl, 9 Oct 2025), an “unoperation” $\mathfrak{Un}(\hat{O})$ is defined for any operation $\hat{O}: x \mapsto \tilde{y}$ as the mapping:

$$\mathfrak{Un}(\hat{O})(\tilde{y}) \triangleq \{ x \mid \hat{O}(x) = \tilde{y}\}$$

For the specific case of binary addition, this produces the set of all integer pairs that sum to a predetermined output: for $3 = a + b$, $$\mathfrak{Un}(+)(3) = \{(0,3), (1,2), (2,1), (3,0)\}$$.

This unaddition operation is inherently non-invertible, as multiple valid input pairs can correspond to a given sum. Thus, unaddition circuits must construct uniform (or algorithmically weighted) superpositions across all such inputs. This is distinguished from classical inverse computation, which either fails or is ill-defined for surjective operations.

2. Ripple-Carry Unadder Circuit Construction

The archetypal unaddition circuit is structured as a quantum analog of the classical ripple-carry adder (RCA), but in functional reversal. A classical RCA adds bits $a_i, b_i$ with carry-in $c_{\text{in},i}$ to produce the sum $s_i$ and carry-out $c_{\text{out},i}$ for all $i$:

$$a_i + b_i + c_{\text{in},i} = s_i + 2c_{\text{out},i}$$

The quantum ripple-carry unadder (RCU) accepts $(c_{\text{out}}, s)$ as input and coherently generates all valid $(a, b, c_{\text{in}})$ consistent with the bitwise arithmetic constraints. Its core block is the quantum full-unadder gate, which implements this input expansion in superposition.

Fundamentally, the unadder gate is defined to act as follows: for each $(c_{\text{out}}, s)$, it transforms initialized ancilla $|0\rangle$ into all valid $|(a, b, c_{\text{in}})\rangle$ such that $a + b + c_{\text{in}} = s + 2c_{\text{out}}$. Specific design ensures that, in the presence of ambiguity (multiple $a, b, c_{\text{in}}$ for fixed output), each configuration receives equal amplitude.

The circuit can be optimized from five to three qubits per gate by leveraging the unitary representation (see Eq. 2 of (Kohl, 9 Oct 2025)), with explicit matrix entries ensuring normalization and correct superposition weights. Cascading these gates across $n$ bits forms the full quantum RCU circuit.

3. Integration in Quantum Factoring via Unmultiplier

The unaddition quantum circuit is integral to the construction of an “unmultiplier” for quantum integer factoring. Classical multiplication is modelled as a sequence of controlled additions, often performed by aggregating partial products in a ripple-carry fashion.

Factoring a composite $N$ (i.e., given $p = x \cdot y$, recover $(x, y)$) thus reduces to inverting sequences of additions embedded in the multiplier circuit. By replacing each quantum adder with a ripple-carry unadder, the quantum circuit—provided with a product $p$—generates a superposition over all decompositions that could have yielded $p$ via classical multiplication.

A quantum feedback (QFB) stage is then applied to extract appropriate factors. Measuring suitable registers post-computation allows (with possible classical post-selection) isolation of correct $(x, y)$ pairs with $x \cdot y = p$.

4. Resource Requirements and Comparison with Established Schemes

Resource analysis in (Kohl, 9 Oct 2025) establishes that the unmultiplier approach with embedded unaddition requires a number of qubits scaling as $\mathcal{O}((\log N)^2)$ for factoring an integer $N$. This matches the asymptotic efficiency of known quantum factoring algorithms, including Shor’s algorithm, which also features polynomial-time resource estimates in the number of bits.

While the unaddition approach introduces quantum parallelism over candidate decompositions, it may incur overhead in classical post-processing. Invalid configurations (non-factors) produced in the superposition must be efficiently filtered, but this does not fundamentally alter the scaling of quantum resources.

5. Comparisons: Uncomputation, Unoptimization, and Related Paradigms

Unaddition is distinct from uncomputation, which presumes a forward circuit evolution that should be undone to erase ancillary information without measurement or collapse. The unaddition circuit, by contrast, reconstructs all preimages for a given output, leveraging the quantum ability to produce coherent superpositions with correct combinatorics for ambiguous logical inverses.

This is categorically different from “quantum circuit unoptimization” (Mori et al., 2023), which refers to intentionally adding redundant reversible gate operations—such as $(A, A^\dagger)$ pairs—to obscure circuit structure without altering its overall unitary action. While both “unaddition” and “unoptimization” increase circuit complexity, unaddition does so to reconstruct all input decompositions for a function, not to benchmark compiler performance or obfuscate gate structure.

6. Broader Implications and Theoretical Extensions

The introduction of the quantum “unoperation” as a primitive opens unexplored pathways in quantum algorithm design and cryptanalysis. Many cryptosystems rely on the one-wayness of function evaluation (e.g., multiplication or modular exponentiation); quantum unoperations generalize the approach of “compute all preimages” in superposition, potentially weakening the hardness assumptions underlying such protocols if efficient implementations are found.

Beyond factoring, if unaddition techniques can be modulated to construct quantum circuits for the “unoperation” of other non-bijective classical processes, it would pave the way for new classes of quantum algorithms exploiting non-injective symmetries. The methodology also suggests further hybridizations of classical circuit design—such as universal logic block translation—into the quantum domain.

7. Prospects, Limitations, and Implementation Considerations

The quantum ripple-carry unadder and related unaddition circuits, in theory, provide polynomial qubit usage for practical problem sizes. The optimized full-unadder gate is explicitly constructed, permitting direct realization on digital quantum architectures. Practical considerations include the fidelity of entangling gates, the need for ancillary registers initialized in well-defined states, and the management of amplitude normalization for large, degenerate preimage sets.

A potential limitation is the required classical post-processing or measurement postselection, as not all generated superposed candidates may be valid solutions to the specific task (e.g., not all decompositions generated by unaddition are consistent with multiplicative factoring constraints).

Experimental realization of large-scale unaddition circuits depends on the availability of high-fidelity multi-qubit gates and efficient state discrimination. Advances in quantum hardware may enable routine embedding of such unaddition modules as standard primitives for quantum algorithmics and potentially for cryptanalysis tasks.

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In summary, the unaddition quantum circuit operationalizes the inversion of surjective classical logic by generating—coherently and reversibly—all valid antecedents to a specified sum, with direct application to quantum factoring and the broader theory of quantum unoperations (Kohl, 9 Oct 2025). This conceptual and technical apparatus enables novel algorithmic frameworks and, if generalized, could provide systematic approaches for addressing other classically hard inversion problems using quantum superposition and parallelism.