A new quantum ripple-carry addition circuit

David Petrie Moulton; Samuel A. Kutin; Steven A. Cuccaro; Thomas G. Draper

REVIEW

A new quantum ripple-carry addition circuit

Steven A. Cuccaro, Thomas G. Draper, Samuel A. Kutin, David Petrie Moulton

Quantum Physics · Submitted 22 Oct 2004 · 4 references

Pith's verdict: The single-ancilla ripple-carry adder is real and checkable; the abstract's unqualified 'fewer gates' is the one claim that doesn't survive counting negations.

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CitationDavid Petrie Moulton, Samuel A. Kutin, Steven A. Cuccaro, Thomas G. Draper. "A new quantum ripple-carry addition circuit." arXiv:quant-ph/0410184 (2004). https://pith.science/paper/MK5HG3Y6

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Editor's note

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The thing worth your time here is the construction, not the framing. Cuccaro et al. build a ripple-carry adder that needs exactly one ancilla instead of the linear scratch space VBE used, and they do it with two small gadgets — an in-place majority and an 'unmajority and add' — that you can read off Figures 1 and 2 in about a minute. Figure 5 gives time-sliced pseudocode, Figure 6 a worked n = 6 case, and Table 1 the counts for the base adder plus four variants. I checked the 2n−1 Toffoli, 5n−3 CNOT, 2n−4 negation counts against the pseudocode and they hold; the depth 2n+4 is just the Toffoli slices plus five CNOT slices. Nothing is fitted, nothing is hidden, and the depth-reduction tricks in Section 3 are the kind of thing you can verify by staring at the diagram for two minutes. That is the contribution, and it is solid. It became a standard adder for a reason. The soft spots are small and mostly about framing. The abstract says 'fewer gates than previous ripple-carry adders,' but that only holds if you don't count negations — the paper's own Table 1 caption excludes them. Count them and you get 9n−8 against VBE's 8n−4, so CDKM is behind for n ≥ 5. Negations are cheap and absorb into neighbouring gates, so this is a scope qualifier rather than an error, but the abstract should have carried the qualifier. The depth win is sturdier: 2n+4 against 6n−2 in the same model, and Section 5 honestly reports the controlled-rotation depth of 6n−2, which is the number a hardware person will actually want. Second, the n-thresholds drift (n ≥ 2 in Section 3, n ≥ 3 in Table 1, n ≥ 4 in the pseudocode); cosmetic, but it makes the table harder to trust at a glance. Third, the counts are asserted rather than derived — trivially checkable, so I don't weight it. Reference [1] is an in-preparation self-citation, but it appears only in the conclusions about a no-ancilla variant and carries nothing load-bearing. Who should read it: anyone building arithmetic into a fault-tolerant stack, and anyone who wants a clean template for how to write a circuit paper — gadgets, pseudocode, worked instance, resource table. Recommendation: send it out. The construction is correct, the ancilla reduction is the real result, and the fixes are one sentence in the abstract and a consistent n.

Referee report

Significance

If the construction holds — and it does — constant-ancilla, linear-depth addition becomes a primitive available to any algorithm that needs arithmetic on a small register, which is a meaningful relaxation of the resource requirements of the standard VBE adder. The paper's virtues are the ones that make a construction paper easy to trust: the circuit is given at gate level rather than asserted, Figure 5 is pseudocode with one line per time-slice so that every entry of Table 1 can be checked by counting, Figure 6 supplies a fully worked n = 6 instance, and the variants in Section 4 are derived rather than merely listed. The authors are also candid about the costing model: the Table 1 caption states that negations are excluded from size and depth, and Section 5 discloses that under a controlled-rotation decomposition the depth becomes 6n − 2 rather than 2n + 4, with Figure 7 verifying the n = 5 case. That disclosure is exactly the kind of honesty that lets a reader recalibrate the headline numbers for their own hardware.

Major objections (1)

[Abstract (abstract:s2) and Section 1] The abstract's closing sentence, 'our circuit has lower depth and fewer gates than previous ripple-carry adders', is stated without the counting convention that the rest of the paper uses. The convention appears only in the Table 1 caption ('We do not include negations when counting size or depth'), and under a counting that includes negations the comparison reverses: the circuit of Figure 5 has (2n − 1) + (5n − 3) + (2n − 4) = 9n − 8 gates against the VBE adder's (4n − 2) + (4n − 2) = 8n − 4, so CDKM has more raw gates for n ≥ 5 (at n = 6, 46 versus 44). This is a scope qualifier rather than an error, since the exclusion is disclosed and negations are legitimately cheap or absorbable on many architectures, but the abstract is the one place a reader will look for the headline and it does not carry the qualifier. I would ask for one clause in the abstract, e.g. 'fewer gates when negations are not counted', or a parenthetical pointing to Table 1. The depth comparison (2n + 4 versus 6n − 2) is in the same Toffoli-equals-one-time-slice model and is fine as stated, since Section 5 gives the controlled-rotation alternative explicitly.

Minor objections (7)

[Abstract (abstract:s1)] 'Previous addition circuits required linearly many ancillary qubits' is broader than what the paper establishes. The comparison actually made in Section 1 is against VBE, which needs n − O(1) ancillae, and Section 5 itself cites [1] as a proposed no-ancilla version. Suggest narrowing to 'previous ripple-carry adders' or 'the standard constructions' so the sentence matches the evidence presented.
[Section 3, Figure 5 caption, Table 1 caption] The range of validity is stated three different ways: the text says 'Assuming n ≥ 2, the circuit size is ...', the Figure 5 caption says the pseudocode 'is valid only for n ≥ 4', and the Table 1 caption says 'for n ≥ 3'. These may all be correct for their respective objects, but the reader cannot tell which statement governs which claim. Please state once, explicitly, the range for the pseudocode, the range for the gate-count formulas, and the range for each row of Table 1.
[Section 5, reference [1]] The statement that 'A version of our ripple-carry adder has been proposed that uses no ancillae [1]' rests on a reference listed as 'in preparation'. Either sketch the construction in a sentence or two so the claim is checkable, or mark it explicitly as unpublished work whose details are not available.
[Section 4.3 and Table 1, 'Compare' rows] The comparator is introduced in a single sentence ('we compute the high bit of a − b, which is 1 if and only if a < b'), yet Table 1 reports 2n − 1 Toffolis, 4n − 3 CNOTs and depth 2n + 3 for it. Unlike the adder rows, these numbers cannot be verified from the text, because the two complementation time-slices mentioned in Section 4.3 are not accounted for in the stated depth. A short derivation or a figure for the comparator would make the row checkable in the same way the adder rows are.
[Figures 1, 2, 3] The gate diagrams are rendered in a character-based style in which the control and target of each CNOT and Toffoli are not unambiguously marked, and the wire labels (which qubit is A, which is B, which is the carry) are only inferable from the surrounding text. Redrawing these three figures with standard symbols and explicit wire labels would substantially reduce the effort needed to check the MAJ and UMA definitions.
[Section 5] 'It is well-known that a Toffoli gate can be built from five controlled rotations' would benefit from a citation, since the subsequent 6n − 2 depth claim depends on that decomposition.
[Throughout] The manuscript contains numerous broken words and ligature artifacts (e.g. 'additio n', 'pseudoco de', 'lab eled', 'Toﬀoli', 'ancillae' versus 'ancilla'), presumably from typesetting. These should be cleaned up in the final version, and the singular/plural of 'ancilla' regularized.

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The paper

The paper claims

A quantum adder that needs one scratch qubit instead of a whole register, and still runs in linear depth.

The reading

The paper sets out to build a reversible circuit that adds two $n$-bit numbers in place, writing the sum back into one of the inputs, and to do it with a single ancillary qubit rather than the $n-O(1)$ scratch qubits the standard ripple-carry construction requires. The authors care because addition is the primitive underneath essentially every quantum arithmetic routine, and scratch qubits are expensive: they must be initialized, kept coherent, and error-corrected. Their circuit keeps the classical ripple-carry shape, sweeping carries up from the low-order bit and then sweeping back down to erase them, but fuses the two sweeps into a matched pair of gates. For $n$-bit inputs the result uses $2n-1$ Toffoli gates, $5n-3$ CNOTs and $2n-4$ negations, at depth $2n+4$, against $4n+O(1)$ Toffolis, $4n+O(1)$ CNOTs and linearly many ancillae for the earlier construction. The same machinery is then trimmed into a modulo-$2^n$ adder, an adder that accepts an incoming carry, a circuit that computes only the high bit, and, by complementing one input, a comparator.

What carries the argument

The MAJ/UMA pair is the load-bearing device. MAJ is an in-place majority gate: two CNOTs followed by a Toffoli, computing $c_{i+1} = a_i b_i$ XOR $a_i c_i$ XOR $b_i c_i$ into the $a_i$ wire. UMA is its exact inverse composed with the sum step, so that applying it returns $a_i$ to its original value, passes $c_i$ back down the chain, and writes $s_i = a_i$ XOR $b_i$ XOR $c_i$ into $b_i$. Because MAJ writes the carry into a wire that already holds data, no scratch register is needed, and because UMA both uncomputes the carry and delivers the sum in one gate, the downward sweep costs no more than the upward one. The paper offers two UMA implementations, a two-CNOT version that is conceptually simpler and a three-CNOT version that admits more parallelism; only the three-CNOT version supports the depth optimizations.

Extended reading notes

Core claim

The discovery is a matched pair of reversible gates that makes a carry bit usable and then disposable without ever storing it in a dedicated wire. The first, called MAJ, computes the majority of three bits in place using two CNOTs and one Toffoli: applied to $(a_i, b_i, c_i)$ it leaves the majority $c_{i+1}$ written into the $a_i$ slot. The second, called UMA (UnMajority and Add), undoes that computation while simultaneously emitting the sum bit, restoring $a_i$ and routing $c_i$ onward. Stringing MAJ gates up the bit positions and UMA gates back down produces a complete in-place ripple-carry adder whose only extra wire is one ancilla holding $c_0$. The authors then shave the depth by commuting gates across neighbouring time-slices, by exploiting $c_0 = 0$ to replace the first MAJ with a single Toffoli, and by writing the final carry straight to the output rather than parking it in a register and copying it out. The resulting circuit is given as pseudocode (valid for $n$ at least 4) and as a worked six-bit diagram.

Load-bearing premise

The claim of lower depth and fewer gates than the earlier adder rests on a costing convention the paper adopts but does not argue for: one Toffoli equals one time-slice, negations are free, and any two qubits can interact directly.

Editorial extensions

If this is right

Any quantum arithmetic routine built by repeated addition, such as modular exponentiation, can be compiled with a constant rather than linear workspace per adder, which reduces the number of qubits that must be kept coherent and error-corrected.
The same circuit family, with small edits, delivers addition modulo $2^n$, addition with an incoming carry, a high-bit-only circuit, and a comparator, each with its own tabulated gate count and depth (Table 1).
Because the adder is its own inverse when run backwards, subtraction follows from complementing one input at the start and complementing the result at the end, and comparison follows from reading the high bit of the difference.
Rewriting the Toffolis as controlled rotations changes the depth accounting: the authors report a controlled-rotation depth of $6n-2$ rather than $2n+4$, because adjacent Toffolis can be overlapped.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's central resource comparison is only as strong as its cost model. Counting a Toffoli as one time-slice, excluding negations, and ignoring qubit connectivity all favour this design; under a fault-tolerant cost model where each Toffoli expands into several expensive non-Clifford gates, the relevant metric becomes T-depth, and the ordering against the earlier adder would need to be recompu
The one-ancilla property matters most in architectures where qubits are the binding constraint rather than gate time, such as trapped ions or small superconducting devices; the paper does not work out which hardware regimes actually prefer this trade.
The authors' own open question, whether constant-ancilla addition forces linear depth, is a clean separation problem: a proof that it does would explain why the logarithmic-depth constructions all pay with linearly many ancillae.
The comparator variant is a single-ancilla primitive for the comparison step inside amplitude-estimation-style algorithms, and it would be worth checking whether it lowers the qubit budget of those routines in practice.

Original abstract

We present a new linear-depth ripple-carry quantum addition circuit. Previous addition circuits required linearly many ancillary qubits; our new adder uses only a single ancillary qubit. Also, our circuit has lower depth and fewer gates than previous ripple-carry adders.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,825 input tokens.

What the abstract itself states, as probabilities: claims first 0.12 · states limitation 0.06 · human participants 0.02 · controlled comparison 0.42 · randomized assignment 0.01 · states quantitative result 0.90 · compares with prior results 0.98 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present a new linear-depth ripple-carry quantum addition circuit. Previous addition circuits required linearly many ancillary qubits; our new adder uses only a single ancillary qubit. Also, our circuit has lower depth and fewer gates than previous ripple-carry adders.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a new linear-depth ripple-carry quantum addition circuit.	contribution 1.00	0.94	0.61	asserted 1.00
s1	Previous addition circuits required linearly many ancillary qubits; our new adder uses only a single ancillary qubit.	contribution 0.88	0.82	0.32	asserted 1.00
s2	Also, our circuit has lower depth and fewer gates than previous ripple-carry adders.	result 0.70	0.86	0.24	asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from proof or derivation 0.70.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The in-place majority gate MAJ The second component, depicted in

**Figure 2.** Two implementations of the UMA gate The effect of using these two gates together is shown in

**Figure 3.** Combining the MAJ and UMA gates 0 = c0 M c0 ⊕ a0 U 0 b0 A b0 ⊕ a0 M s0 a0 J M c1 ⊕ a1 U A a0 b1 A b1 ⊕ a1 M s1 a1 J M c2 ⊕ a2 U A a1 b2 A b2 ⊕ a2 M s2 a2 J M c3 ⊕ a3 U A a2 b3 A b3 ⊕ a3 M s3 a3 J M c4 ⊕ a4 U A a3 b4 A b4 ⊕ a4 M s4 a4 J M c5 ⊕ a5 U A a4 b5 A b5 ⊕ a5 M s5 a5 J c6 r A a5 z z ❡ z ⊕ s6

**Figure 4.** A simple ripple-carry adder for n = 6. 3 Improving the circuit We can reduce the depth of the basic circuit of

**Figure 5.** The ripple-carry adder for n ≥ 4. Each line of pseudocode corresponds to a single time-slice. b0 r r ❡ s0 a0 r r r a0 0 ❡ ❡ r r r ❡ ❡ 0 b1 ❡ r ❡ ❡ r ❡ ❡ s1 a1 r r ❡ ❡ r r r ❡ ❡ r r a1 b2 ❡ r ❡ ❡ r ❡ ❡ s2 a2 r r ❡ ❡ r r r ❡ ❡ r r a2 b3 ❡ r ❡ ❡ r ❡ ❡ s3 a3 r r ❡ ❡ r r r ❡ ❡ r r a3 b4 ❡ r ❡ ❡ r ❡ ❡ s4 a4 r r ❡ ❡ r r ❡ ❡ r r a4 b5 ❡ r ❡ ❡ s5 a5 r r r r r a5 z ❡ ❡ z ⊕ s6

**Figure 6.** The ripple-carry adder for n = 6. 4

**Figure 7.** 5-bit ripple carry adder written in terms of contro

Figures reproduced from arXiv:quant-ph/0410184 by the authors.

References

Works cited (4)

[1]

Richard J. Dore and Samuel A. Kutin, A logarithmic-depth quantum comparison circuit with one ancilla , in preparation. 7 b0 r r r r ❡ r s0 a0 r ❡ r ❡ r r a0 0 ♥i ♥i ❡ ♥−i r r r r r ♥i ♥i ❡ ♥−i 0 b1 ❡ r ❡ r ❡ ❡ r ❡ r ❡ ❡ s1 a1 r r ♥i ♥i ❡ ♥−i r r r r r ♥−i ❡ ♥i ♥i r r a1 b2 ❡ r ❡ r ❡ ❡ r ❡ r ❡ ❡ s2 a2 r r ♥i ♥i ❡ ♥−i r r r r r ♥−i ❡ ♥i ♥i r r a2 b3 ❡ r ❡ r...

View on Pith
[2]

Thomas G. Draper, Addition on a quantum computer , quant-ph/0008033

View on Pith Pith review arXiv
[3]

Thomas G. Draper, Samuel A. Kutin, Eric M. Rains, and Krys ta M. Svore, A logarithmic- depth quantum carry-lookahead adder , EQIS, 2004, quant-ph/0406142

View on Pith Pith review arXiv 2004
[4]

Vlatko Vedral, Adriano Barenco, and Artur Ekert, Quantum networks for elementary arithmetic operations, quant-ph/9511018. 9

View on Pith Pith review arXiv

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quant-ph 2026-09 conditional novelty 6.0 of 10

A variational quantum algorithm reconstructs 1D Burgers and Kuramoto-Sivashinsky velocity fields from sparse time-resolved sensors by encoding the full spacetime solution in one quantum state and minimizing a combined...
Factorized Boolean representations for efficient quantum synthesis
quant-ph 2026-08 conditional novelty 6.0 of 10

Factorizing containment and complementary-polarity structure out of minimized ESOP expressions reduces quantum T-count and qubit count, with a provable representation-level monotonicity guarantee.
QSCI-CMP: Quantum-Selected Configuration Interaction with Chemically Motivated Preselection
quant-ph 2026-08 accept novelty 6.0 of 10

QSCI-CMP cuts the quantum sampling cost of SQD-AA by classically preselecting low-excitation, low-seniority determinants and using a compact oracle that checks those properties in superposition.
Hybrid quantum floating-point method for sharp arithmetic
quant-ph 2026-07 conditional novelty 6.0 of 10

A classically-enriched quantum floating-point variable stores offset and scaling classically, reducing approximation error by up to 89% after repeated additions compared to prior mono-quantum coding.
Quantum Kravchuk Transform using $\mathfrak{su}(2)$ fast-forwarding
quant-ph 2026-06 unverdicted novelty 6.0 of 10

Quantum algorithm implements Kravchuk transform via su(2) fast-forwarding with logarithmic scaling in dimension and error.
Optimized Point Addition Circuits for Elliptic Curve Discrete Logarithms
quant-ph 2026-06 unverdicted novelty 6.0 of 10

Explicit quantum circuits for elliptic-curve point addition achieve 6.5-10% fewer Toffoli gates and 1.5% more qubits than Babbush et al. for secp256k1, plus a generic prime-field version.
Towards Deploying Optimistic Quantum Fourier Transforms: An Architecture-Algorithm Co-Design Study
quant-ph 2026-05 unverdicted novelty 6.0 of 10

A hot-zone architecture for OQFT on reconfigurable neutral-atom hardware yields tunable latency via 2-4 zones, converging to roughly 500 extra logical ancillae and 128-qubit peak parallelism for half-time performance ...
Space-Efficient Quantum Algorithm for Elliptic Curve Discrete Logarithms with Resource Estimation
quant-ph 2026-04 conditional novelty 6.0 of 10

A space-efficient quantum ECDLP algorithm uses 5n + 4⌊log₂n⌋ + O(1) logical qubits and O(n³) Toffoli gates, lowering the 256-bit estimate from 2124 to 1333 qubits.
Quantum Elastic Network Models and their Application to Graphene
quant-ph 2026-01 conditional novelty 6.0 of 10

A quantum algorithm for coupled oscillators is adapted to elastic network models, with an efficient connectivity oracle for graphene and applications to heat transfer and rippling — at the cost of a coarse two-bucket ...
Benchmarking Quantum Simulation of Chemical Hamiltonians using the Sorted-List Encoding
quant-ph 2025-10 conditional novelty 6.0 of 10

Sorted-list encoding gives O(N log M) qubits and gate scalings for fermionic simulation, matching first-quantized in plane-wave trotterization and beating Jordan-Wigner only at low electron filling, while first-quanti...
Efficient Quantum Control via Automatic Control Skips
quant-ph 2025-05 conditional novelty 6.0 of 10

A generic algorithm automatically identifies conjugation-pair subcircuits that can skip control in quantum circuits, with an NP-hardness proof and a dynamic-programming approximation showing large practical reductions.
Fast correlated decoding of transversal logical algorithms
quant-ph 2025-05 conditional novelty 6.0 of 10

Decoding only back-propagated reliable logical Pauli products turns transversal-circuit decoding into a matchable graph, so fast minimum-weight perfect matching works with memory-like thresholds.
Decentralized Framework for Teleportation in Quantum Core Interconnects
quant-ph 2025-05 conditional novelty 6.0 of 10

A two-way teleportation strategy, where both source and destination qubits move toward an intermediate core, reduces end-to-end communication latency by 30-40% and circuit depth by up to 24% in simulations of multi-co...
S-SYNC: Shuttle and Swap Co-Optimization in Quantum Charge-Coupled Devices
quant-ph 2025-05 conditional novelty 6.0 of 10

S-SYNC unifies shuttling and SWAP operations into a single 'generic swap' on a static graph, and a greedy heuristic co-optimizes them to cut shuttling by 3.69x and raise success rate by 1.73x on average in simulation.
TrapSIMD: SIMD-Aware Compiler Optimization for 2D Trapped-Ion Quantum Machines
quant-ph 2025-04 conditional novelty 6.0 of 10

A SIMD-style abstraction and two scheduling passes for trapped-ion QCCD machines are reported to cut simulated execution time by up to 3.82x and improve simulated fidelity.
BOSS: Blocking algorithm for optimizing shuttling scheduling in Ion Trap
quant-ph 2024-12 conditional novelty 6.0 of 10

BOSS uses union-find circuit blocking and a half-block tape schedule to cut shuttle operations in linear-tape trapped-ion compilation, achieving up to 96.1% fewer shuttles on tested benchmarks.
On the practicality of quantum sieving algorithms for the shortest vector problem
quant-ph 2024-10 unverdicted novelty 6.0 of 10

Quantum sieving for SVP in dimension 400 needs ~10^13 physical qubits and ~10^31 years under optimistic assumptions, offering no practical speedup over classical methods.
Quantum Monte Carlo algorithm for option pricing and its complexity analysis
quant-ph 2023-01 unverdicted novelty 6.0 of 10

A quantum Monte Carlo algorithm solves multidimensional Black-Scholes PDEs for option pricing with polynomial complexity in dimension d and accuracy 1/ε, with rigorous error bounds and a claimed speedup over classical...
Quantum computing enhanced computational catalysis
quant-ph 2020-07 conditional novelty 6.0 of 10

A new double-factorized qubitization algorithm reduces the Toffoli gate count for quantum chemistry resource estimates on a real CO2-fixation catalyst to about 10^10 gates and 4000 logical qubits.
Solving Gauss's Law on Digital Quantum Computers with Loop-String-Hadron Digitization
hep-lat 2018-12 accept novelty 6.0 of 10

Using the loop-string-hadron basis turns SU(2) Gauss's law into Abelian flux-conservation checks that can be implemented with explicit, simple quantum circuits.
Open Quantum Assembly Language
quant-ph 2017-07 conditional novelty 6.0 of 10

OpenQASM 2.0 defines a compact assembly-like language for universal quantum circuits with measurement, reset, conditional gates, and user-defined gate subroutines.
Constant-Depth Quantum Barrier Option Pricing via a Static Bivariate Oracle Framework
quant-ph 2026-10 conditional novelty 5.0 of 10

A bivariate quantum oracle based on the Brownian reflection principle reduces barrier-option monitoring from O(d) comparators to two comparators, giving constant-depth scaling in the number of monitoring dates.
When More Becomes Less: Topology-Reversed Three-Qubit Gate Performance on IBM Quantum Processors
quant-ph 2026-08 conditional novelty 5.0 of 10

On linear IBM processors, the eight-CZ CCZ decomposition outperforms the routed six-CX one, which becomes twelve native CZ gates.
Position: Quantum Program Generation Must Prioritize Validity Over Probabilistic Scaling
cs.LG 2026-07 conditional novelty 5.0 of 10

The paper argues that probabilistic scaling alone cannot fix the validity gap in quantum circuit generation, so quantum code assistants must build verification into generation rather than filter outputs after the fact.

See all 73 Pith citations

How later papers used it

119 later papers cite it

Evidence: substantial. 119 of the citing papers share no author with it, and 55 of those carry a confident label from Pith's citation classifier.

Pith read 28 of the 188 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

11 later papers by other authors are listed below: 10 used its method or tool and 1 compared its own results with it.

Uses its method or tool 10 of 48 papers with this label

By $m<n$ and the compilation given in Ref. [ref] , add can be implemented by one ancilla qubit and $𝒪(m)$ not, cnot and toffoli gates.

Efficient Quantum Algorithm for All Quantum Wavelet Transforms (2023) · Complexity of key subroutines · 0.99
Utilized four adders, including this paper's linear-depth majority ripple adder, to generate circuits for each multiplier design.

High Performance Quantum Modular Multipliers (2018) · 1.00 · paraphrased: the licence does not allow quoting
This performs the MAJ sweep of Cuccaro's ripple carry adder [ref] .

How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits (2019) · Spacetime layout · 1.00
Used an adder described in another work and optimized from this paper for use with T gates.

Factoring 2048-bit RSA Integers in 177 Days with 13436 Qubits and a Multimode Memory (2021) · 0.98 · paraphrased: the licence does not allow quoting
Based its quantum adder on the ripple carry adder design in this paper.

T-count Optimized Quantum Circuits for Bilinear Interpolation (2018) · 1.00 · paraphrased: the licence does not allow quoting
Implemented each direction-controlled update via multi-controlled add/subtract circuits from this paper using Toffoli and CNOT gates.

A Grover-Based Quantum Algorithm for Solving Perfect Mazes via Fitness-Guided Search (2025) · 1.00 · paraphrased: the licence does not allow quoting
We will be using the Quantum Ripple Carry Adder, proposed by Cuccaro et al. [ref] .

A Fully Quantum Algorithm for Image Edge Detection (2026) · Addition · 1.00
Selected this paper's CDKM adder for the adder circuit each round requires.

Logic Synthesis for Fault-Tolerant Quantum Computers (2013) · 1.00 · paraphrased: the licence does not allow quoting
Using Cuccaro ripple-carry arithmetic [ref] for the additions and subtractions, each such operation costs $2b_μ+𝒪(1)$ Toffoli gates and has depth $2b_μ+𝒪(1)$.

Efficient Quantum Circuits for Coherent Conversion Between General First- and Second-Quantized Many-Body Representations (2026) · Implementation of the canonical reconstruction map $C$: lower canonical GT rows from occupation vectors · 1.00
Made use of this paper's implementations adapting the quantum ripple-adder for other arithmetic tasks for its oracle definitions.

Function Maximization with Dynamic Quantum Search (2019) · 1.00 · paraphrased: the licence does not allow quoting

Compares with it 1 of 7 papers with this label

We compare our circuit with several known in-place quantum adders: Cuccaro Ripple-Carry Adder (RCA) [ref] , Draper Carry-Lookahead Adder (CLA) [ref] , Takahashi RCA [ref] and Gidney RCA [ref] .

New Circuit for Quantum Adder by Constant (2025) · Complexity analysis · 1.00

How the literature uses it

116 arXiv papers cite this in 189 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 94 sentences without a confident label

Builds on it 1 paper

The two adders are generalized from their binary counterparts [ref] , but the generalizations are somewhat nontrivial, as seen later.

Improved Quantum Ternary Arithmetics · Quantum Ternary Adders · 0.97

Uses its method or tool 49 papers

The circuit is adapted from the construction introduced in Ref. [ref] , and is conceptually equivalent to a reversible subtraction $a-b$, where the sign of the result is extracted without explicitly storing the full difference [ref] .

Demonstration of a quantum comparator on an ion-trap quantum device · Quantum Comparator · 0.96
By $m<n$ and the compilation given in Ref. [ref] , add can be implemented by one ancilla qubit and $𝒪(m)$ not, cnot and toffoli gates.

Efficient Quantum Algorithm for All Quantum Wavelet Transforms · Complexity of key subroutines · 0.99
We utilize the four adders discussed in the appendix: the Fourier basis adder [ref] , the logarithmic-depth prefix (or carry-lookahead) adder [ref] , the linear-depth majority ripple adder [ref] , and the rippled prefix adder defined in circuits:prefix_adders.

High Performance Quantum Modular Multipliers · Resource Evaluation · 1.00

Compares with it 6 papers

We compare our circuit with several known in-place quantum adders: Cuccaro Ripple-Carry Adder (RCA) [ref] , Draper Carry-Lookahead Adder (CLA) [ref] , Takahashi RCA [ref] and Gidney RCA [ref] .

New Circuit for Quantum Adder by Constant · Complexity analysis · 1.00
This compares to a KQ for a 32-bit CDKM ripple-carry adder [ref] of 12,474.

A Space-Efficient Design for Reversible Floating Point Adder in Quantum Computing · Metrics for Fault-Tolerant Quantum Circuit · 0.98
Note the additional number of qubits is $2n-√(n)$, which is less than twice the minimum $2n$ qubits [ref] .

An $\Theta(\sqrt{n})$-depth Quantum Adder on a 2D NTC Quantum Computer Architecture · Required Space · 0.96

Cites it as background 19 papers

Many implementations of addition circuits are known [ref] , but the fault-tolerant preparation of Fourier states has received less attention.

Distillation protocols for Fourier states in quantum computing · Introduction · 0.89
This operator is a building block in quantum arithmetic within ripple-carry adders [ref] , which are in turn essential components of numerous quantum algorithms [ref] .

Ancilla-free Quantum Adder with Sublinear Depth · MCX ladders · 0.91
The first method uses Toffoli gates that mirror classical binary compositions [ref] .

Truncated phase-based quantum arithmetic: error propagation and resource reduction · Introduction · 0.80

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Richard J. Dore and Samuel A. Kutin, A logarithmic-depth quantum comparison circuit with one ancilla , in preparation. 7 b0 r r r r ❡ r s0 a0 r ❡ r ❡ r r a0 0 ♥i ♥i ❡ ♥−i r r r r r ♥i ♥i ❡ ♥−i 0 b1 ❡ r ❡ r ❡ ❡ r ❡ r ❡ ❡ s1 a1 r r ♥i ♥i ❡ ♥−i r r r r r ♥−i ❡ ♥i ♥i r r a1 b2 ❡ r ❡ r ❡ ❡ r ❡ r ❡ ❡ s2 a2 r r ♥i ♥i ❡ ♥−i r r r r r ♥−i ❡ ♥i ♥i r r a2 b3 ❡ r ❡ r...

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[2] [2]

Thomas G. Draper, Addition on a quantum computer , quant-ph/0008033

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[3] [3]

Thomas G. Draper, Samuel A. Kutin, Eric M. Rains, and Krys ta M. Svore, A logarithmic- depth quantum carry-lookahead adder , EQIS, 2004, quant-ph/0406142

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[4] [4]

Vlatko Vedral, Adriano Barenco, and Artur Ekert, Quantum networks for elementary arithmetic operations, quant-ph/9511018. 9

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