Addition on a Quantum Computer

Thomas G. Draper

REVIEW

Addition on a Quantum Computer

Thomas G. Draper

Quantum Physics · Submitted 7 Aug 2000 · 10 references

Pith's verdict: The QFT-basis adder is real and correct; the Shor qubit count in Section 6 is asserted, not derived.

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arXiv IDquant-ph/0008033 (version 1)

Pith IDBUZYCIW4, Pith's permanent record of this paper and its review

CitationThomas G. Draper. "Addition on a Quantum Computer." arXiv:quant-ph/0008033 (2000). https://pith.science/paper/BUZYCIW4

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Editor's note

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This is the paper that put the Fourier-basis adder into the literature, and the core of it holds up. Every adder Draper cites works in the computational basis with explicit carry propagation and 3n qubits. His move — do the addition as conditional phase accumulation on a Fourier-transformed register — is genuinely different, and the derivation in Section 5 is short enough to check line by line. The QFT factorization, the |phi_k(a)> definition, and the single-qubit phase trace all come out right. The commutativity claim is true for the boring reason: every rotation is diagonal in the computational basis, so they commute, and the parallelization argument (n+1 slices, log2 n with truncation) follows. The classical-constant trick is the sleeper contribution — you can add a known number to a superposition without ever putting it in a register, which is why this thing shows up in simulation oracles and phase estimation. Credit earned, no caveats needed on the core. Two soft spots, and I'd rank them differently than the reader did. The reader's weakest_assumption — that the carry-out is silently dropped — is half wrong. Section 3 opens by setting a in Z_{2^n}, the integers mod 2^n, and the QFT in Eq. (1) is defined on that group. The modular arithmetic is declared. What's true is that the scope statement never makes it into Section 5 or the abstract, and the Section 2 diagram draws n+1 output lines against Section 5's n. That's a clarity defect worth a paragraph, not a correctness problem. Where the mod-2^n restriction actually bites is Section 6. Shor's modular exponentiation needs arithmetic mod N, the semiprime, not mod 2^n. Getting from one to the other costs a comparison and conditional subtraction, which costs at least a qubit and more gates. The paper gives no construction and no register count for that step, so '2n qubits' is asserted rather than derived. That's the one quantitative claim the Section 5 derivation doesn't support. Minor: the approximate adder's n log2 n cost is transferred from the AQFT by analogy with no error bound attached. Fine for a 1998 note, worth flagging. Who should read it: anyone building quantum arithmetic, and anyone doing low-qubit factoring register accounting. It deserves a serious referee — the central claim is correct and the construction is standard now, and the fixes are all scope statements.

Referee report

Significance

If the construction holds, it is a genuine conceptual shift: addition is performed by phase accumulation in the Fourier basis rather than by carry propagation, and the adder needs no temporary carry qubits, so the two operand registers suffice. Two consequences are notable and, as far as I can check, correct. First, because the control bits enter only as classical rotation angles, a classical constant can be added to a quantum superposition without ever being written into a quantum register, which is a real practical advantage for oracle-style and arithmetic-heavy algorithms. Second, the mutual commutativity of the rotation sequence is not an approximation but an exact property of diagonal gates, so the parallelization argument is structurally sound rather than heuristic. The paper is also unusually checkable for its genre: the central claim is a single-qubit phase identity that a reader can verify line by line, and the derivation is parameter-free, with no fitted constants or hidden ancilla assumptions. These are real strengths and should be weighed alongside the concerns below.

Major objections (2)

[§6, Implications] The sentence 'Replacing the addition used by Zalka with quantum addition allows us to perform Shor's algorithm using only 2n qubits' is the one quantitative claim in the paper that the Section 5 derivation does not support. The adder of Section 5 computes addition modulo 2^n, as Section 3 declares when it sets a in Z_{2^n}; Shor's modular exponentiation requires arithmetic modulo N, the semiprime, not modulo 2^n. Passing from one to the other requires a comparison and conditional subtraction (or an equivalent modular-reduction step), which costs additional gates and, in the usual constructions, at least one additional qubit. The manuscript supplies no such construction and no register-by-register count for the 2n figure, so the number is asserted rather than derived. This does not affect the correctness of the adder itself, and the required fix is bounded: either give the reduction circuit and an explicit qubit accounting, or qualify the claim to say that the 2n count applies to the modular-2^n adder and that the mod-N reduction is not costed here. As written, the sentence is likely to be quoted as a derived result, so it should be made precise before publication.
[§5, Transform Addition diagram, versus §2, Reversible Adder diagram] The two adders being compared do not produce the same output. The reversible adder of Section 2 is drawn with an explicit fourth output line |(a+b)_4> for a 3-bit addition, i.e. n+1 sum bits, whereas the QFT adder diagram in Section 5 has only n output lines and the phase trace terminates at 0.a_n…a_1 + 0.b_n…b_1, which is (a+b)/2^n mod 1. The modular scope is declared in Section 3 ('Let a ∈ Z_{2^n}'), so this is not a hidden error, but the declaration is not carried into Section 5 or into the abstract, and a reader moving from Section 2 to Section 5 will reasonably expect the carry-out to appear. Since what the adder computes is the paper's central claim, Section 5 should state explicitly that the output is a+b mod 2^n, that the carry-out is not produced, and how a caller who needs the full n+1 bit sum should recover it. The abstract's phrasing about reducing qubit count should likewise be scoped to modular addition.

Minor objections (8)

[§5, final paragraph] The approximate-addition claim rests on the sentence 'Using analogous reasoning, it is quite clear that the approximate quantum addition can be computed efficiently using only n log2 n operations as well.' The analogy is plausible, since the discarded rotations are the same ones discarded in the AQFT, but the error bound is not given. A one- or two-line argument bounding the accumulated phase error in terms of the truncation threshold, in the style of the Barenco et al. result invoked in Section 4, would make the claim citable rather than asserted.
[§5, parallelization paragraph] The n+1 time-slice figure is explicitly conditional on a machine able to execute n^2 independent two-qubit gates simultaneously, and the log2 n figure on the truncated schedule. The conditionality is stated, but it would help to repeat it in the abstract and in Section 6, since the parallelization claim is one of the paper's three headline contributions and is architecture-dependent in a way the other two are not.
[§5, Transform Addition] The full unitary implemented by the circuit is never written out. Since the b register is used only as control and is left unchanged, the map is |a>|b> → |a+b mod 2^n>|b>, and stating this explicitly next to the diagram would remove any ambiguity about whether b is consumed, preserved, or overwritten.
[§2, Classical Addition] The sentence 'All of the implementations use at least 3n qubits to add two n-bit numbers' attributes a single qubit count to references [2], [5], [9] and [10], whose constructions differ in their ancilla accounting. The counts should be attributed per reference, or the sentence softened to 'of order 3n'.
[References] Reference [4] (Coppersmith, 'An approximate Fourier transform useful in quantum factoring') appears in the bibliography but is never cited in the body, even though it is directly relevant to the approximate-QFT discussion of Section 4. Either cite it there or remove it.
[§3, Eq. (1) and the factorization that follows] The factorization sum_k e(ak/2^n)|k> = |φ_n(a)> ⊗ … ⊗ |φ_1(a)> omits the 1/2^{n/2} normalization carried by Eq. (1). The missing factor is harmless but should be restored for consistency.
[§1, Introduction] The notation F(a) is introduced in the second paragraph without definition; the reader must infer from context that it denotes the quantum Fourier transform. Defining it at first use, or writing QFT(a), would help.
[Figures (Sections 2 and 5)] The wire diagrams are rendered as character art and the control connections in the 'Transform Addition' diagram are difficult to follow, particularly where the b-register control lines cross the φ-register rotation lines. Redrawing these as proper figures with labelled gate symbols would materially improve readability, since the diagrams carry the construction.

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The paper

The paper claims

A quantum adder that skips the carry bits entirely, by doing the addition as phase rotations in the Fourier basis.

The reading

The paper proposes computing $a+b$ on a quantum computer without ever building a carry chain. The trick is to Fourier-transform the register holding $a$ first, so that the state becomes a product of phases $|0> + e(a/2^k)|1>$ on each qubit; adding $b$ then amounts to nudging each of those phases by the corresponding bits of $b$, and the inverse transform returns $a+b$. Because the phase nudges are conditional rotations rather than reversible logic gates, no temporary carry qubits are needed, so an $n$-bit sum costs only the $2n$ qubits that hold the two operands instead of the $3n$ that standard reversible adders require. A second consequence is that if $b$ is an ordinary classical number, the rotations are known in advance and can simply be programmed in, letting a classical constant be added to a quantum superposition without ever writing that constant into a quantum register. A third is that all the rotations commute, so the circuit can be run in roughly $n+1$ parallel time slices, or about $log_2 n$ if small rotations are truncated.

What carries the argument

The load-bearing object is the factored form of the Fourier-transformed register, $|a> ightarrow |varphi_n(a)> otimes ... otimes |varphi_1(a)>$, together with the conditional rotation $R_k$ that multiplies the $|11>$ component of two qubits by $e(1/2^k)$. Because the transform turns the integer $a$ into a set of binary fractions $0.a_k...a_1$, addition of $b$ becomes bitwise phase addition with no interaction between qubits beyond the controlled phase, and the inverse transform converts the accumulated phases back into an integer. The same structure gives the paper's two side results: a classical $b$ means the controls are known constants, so the rotations reduce to fixed phase gates; and since controlled phase rotations commute, the gate sequence has no ordering constraint, unlike the Hadamard-and-rotation ordering required by the QFT itself.

Extended reading notes

Core claim

The central claim is that addition is diagonal in the Fourier basis. Writing $e(t) = e^{2pi i t}$ and letting $a$ have binary digits $a_n ... a_1$, the quantum Fourier transform maps $|a>$ to the product state $|varphi_n(a)> otimes ... otimes |varphi_1(a)>$ with $|varphi_k(a)> = (1/sqrt2)(|0> + e(a/2^k)|1>)$, so each qubit carries the lower $k$ digits of $a$ as a binary fraction. The author then shows that a sequence of conditional rotation gates $R_k$, controlled on the bits of $b$, carries $|varphi_n(a)>$ through the chain $0.a_n...a_1 ightarrow 0.a_n...a_1 + 0.b_n ightarrow ... ightarrow 0.a_n...a_1 + 0.b_n...b_1$, which is exactly $|varphi_n(a+b)>$, and similarly for every other qubit. Since these rotations are mutually commuting and are conditioned on bits that may be purely classical, the paper concludes that addition needs no carry workspace, costs about $n log_2 n$ operations when truncated in the manner of the approximate QFT, and parallelizes to $n+1$ time slices or $log_2 n$ with truncation.

Load-bearing premise

The adder only ever produces the sum modulo $2^n$: the paper's own classical comparison circuit carries an explicit $(n+1)$-st sum bit, while the Fourier adder has just $n$ output lines and its phase trace ends at $0.a_n...a_1 + 0.b_n...b_1$, a fraction taken mod 1, so the carry-out is silently dropped and the paper never says so or explains how a caller could recover it.

Editorial extensions

If this is right

An $n$-bit adder can be built from $2n$ qubits rather than the $3n$ that reversible carry-based adders need, since no temporary carry register is allocated.
A classical constant can be added to a quantum superposition by programming fixed rotations, without encoding the constant in a quantum register.
Because the rotation sequence is order-independent, a machine able to apply $n^2$ independent two-qubit gates at once can perform the addition in about $n+1$ time slices, or $log_2 n$ slices with truncated rotations.
Substituting this adder into Shor's algorithm lowers the register count from the $3n$ qubits reported in prior work to $2n$, while keeping the operation count at $O(n^3)$.
The approximate-QFT truncation argument carries over directly, giving an approximate adder that also runs in $n log_2 n$ operations.

Reading between the lines

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

The construction suggests a general principle worth stating separately: any arithmetic that is diagonal in the Fourier basis - addition, and by extension controlled accumulation - is cheap in that basis, so the interesting design question becomes which other operations can be conjugated into commuting phase gates.
The classical-constant mode points toward a natural architecture in which a quantum accumulator is repeatedly nudged by program-supplied constants, which is the shape of the modular-exponentiation inner loop in factoring algorithms.
The paper's own comparison is against a classical reversible adder drawn with an explicit fourth output line for the carry, while the Fourier adder has only $n$ output lines; a reader extending this work should check whether the missing $(n+1)$-st bit is needed by the intended application before adopting the $2n$-qubit accounting.
The truncation argument borrowed from the approximate QFT is asserted by analogy rather than bounded here, so an explicit error analysis for the truncated adder - how many rotations may be dropped for a target success probability - is a concrete open calculation.

Original abstract

A new method for computing sums on a quantum computer is introduced. This technique uses the quantum Fourier transform and reduces the number of qubits necessary for addition by removing the need for temporary carry bits. This approach also allows the addition of a classical number to a quantum superposition without encoding the classical number in the quantum register. This method also allows for massive parallelization in its execution.

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, 3,466 input tokens.

What the abstract itself states, as probabilities: claims first 0.12 · states limitation 0.06 · human participants 0.02 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.33 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

A new method for computing sums on a quantum computer is introduced. This technique uses the quantum Fourier transform and reduces the number of qubits necessary for addition by removing the need for temporary carry bits. This approach also allows the addition of a classical number to a quantum superposition without encoding the classical number in the quantum register. This method also allows for massive parallelization in its execution.

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	A new method for computing sums on a quantum computer is introduced.	contribution 1.00	0.89	0.42	asserted 1.00
s1	This technique uses the quantum Fourier transform and reduces the number of qubits necessary for addition by removing the need for temporary carry bits.	method 0.94	0.82	0.30	asserted 1.00
s2	This approach also allows the addition of a classical number to a quantum superposition without encoding the classical number in the quantum register.	contribution 0.89	0.78	0.13	asserted 0.92
s3	This method also allows for massive parallelization in its execution.	contribution 0.75	0.73	0.13	asserted 0.92

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from proof or derivation 0.68.

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

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S sections
T body sentences
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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

References

Works cited (10)

[1]

A. Barenco, A. Ekert, K. Suominen, P. T¨ orm¨ a, “Approxim ate quantum Fourier transform and decoherence”, quant-ph/9601018

View on Pith Pith review arXiv
[2]

D. Beckman, A. N. Chari, S. Devabhaktuni, J. Preskill, “E ﬃcient networks for quantum factoring”, Phys. Rev. A 54, 1034

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[3]

R. Cleve, A. Ekert, C. Macchiavello, M. Mosca, “Quantum a lgorithms re- visited”, quant-ph/9708016

View on Pith Pith review arXiv
[4]

D. Coppersmith, 1994, “An approximate Fourier transfor m useful in quan- tum factoring”, IBM Research Report No. RC19642

View on Pith 1994
[5]

P. Gossett, “Quantum carry-save arithmetic”, quant-ph /9808061

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[6]

C. Miquel, J. P. Paz, R. Perazzo, “Factoring in a dissipat ive quantum computer”, Phys. Rev. A 54, 2605

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[7]

A. Pittenger, “An introduction to quantum computing alg orithms”, Birkh¨ auser: 2000

View on Pith 2000
[8]

A. Steane, “Quantum computing”, quant-ph/9709022

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[9]

V. Vedral, A. Barenco, A. Ekert, “Quantum networks for el ementary arith- metic operations”, Phys. Rev. A 54, 147

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[10]

C. Zalka, “Fast Versions of Shor’s quantum factoring al gorithm”, quant- ph/9806084. 8

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How later papers used it

91 later papers cite it

Evidence: substantial. 91 of the citing papers share no author with it, and 30 of those carry a confident label from Pith's citation classifier.

Pith read 20 of the 150 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

7 later papers by other authors are listed below: 7 used its method or tool.

Uses its method or tool 7 of 26 papers with this label

This leads to a complexity of $𝒪 ( n_g_max^2 )$ CNOT gates when using the Draper adder [ref] to implement the comparison operations.

Efficient and fail-safe quantum algorithm for the transport equation (2022) · Complexity of the specular reflection step · 0.89
Utilized this paper's Fourier basis adder among four adders discussed in the appendix.

High Performance Quantum Modular Multipliers (2018) · 1.00 · paraphrased: the licence does not allow quoting
The circuit proceeds in two ancilla-free stages: * STEP$(w)$: prepare the uniform superposition over $[0,w)$ using the Shukla–Vedula step circuit [ref] at $𝒪(m)$ cost. * ADD$(k_s)$: shift the register by $k_s$ in place using the Draper QFT-based constant adder [ref] : apply QFT, accumulate $𝒪(m)$ phase rotations, then apply QFT$^†$.

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We use circuits like the one in <ref> [ref] to encode the number of units produced per shift.

QISS: Quantum Industrial Shift Scheduling Algorithm (2024) · Data encoding · 1.00
Quantum arithmetic based on the Fourier transform [ref] enables the implementation of the operator $U_ℛ^(ind_s)_L,1D^⊤$ [equation]

Quantum Domain Decomposition for Preconditioning the Finite Element Method (2026) · Implementation · 1.00
Included this paper's QFT adder in its in-house library as an in-place adder using no extra auxiliary qubits with an O(M^2) CX-count.

Design and synthesis of scalable quantum programs (2024) · 0.97 · paraphrased: the licence does not allow quoting
The basic idea is to use a modular adder [ref] and the linear combinations of unitaries (LCU) method originally introduced in [ref] .

A quantum advection-diffusion solver using the quantum singular value transform (2025) · Block encoding of finite-difference operators · 0.96

How the literature uses it

86 arXiv papers cite this in 136 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 · 72 sentences without a confident label

Builds on it 1 paper

Later this work was extended to multiplication and modular arithmetic [ref] .

Distributed Quantum Machine Learning · Distributed Adder · 0.85

Uses its method or tool 26 papers

The `multiples of' oracle is built reusing two pieces of existing quantum software, the linear multi-controlled gate [ref] and the modulo addition [ref] .

Operating with Quantum Integers: an Efficient 'Multiples of' Oracle · Background · 0.99
In this section we will introduce a novel streaming operation based on the Quantum Draper Adder [ref] , which we specialize to an efficient quantum incrementation (decrementation) procedure that is cheaper than methods currently in use [ref] ; see Section [ref].

Efficient and fail-safe quantum algorithm for the transport equation · Efficient quantum streaming operation · 0.89
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 4 papers

The total two-qubit gate count of the multiplier is, [equation] parallelized to a depth of, [equation] Though a significant speedup over Fourier modular multiplication via modular addition [ref] , the overall depth scales as nlogn, offering no speedup over the binary-basis multiplier constructed with logarithmic-depth adders.

High Performance Quantum Modular Multipliers · Analysis · 0.92
Previous published incrementers (not counting an unpublished version of our construction being cited by [ref] ) required either $O(n^2)$ gates or $ω(1)$ ancillae [ref] .

Factoring with n+2 clean qubits and n-1 dirty qubits · Overview and Improvements · 0.93
One of the improvements we propose in this paper with respect to other known methods [ref] is the use of the Quantum Draper Adder (QDA) [ref] inspired approach to implement the quantum incrementation operation, leading to a cheaper quantum primitive for the streaming operation.

Efficient and fail-safe quantum algorithm for the transport equation · Complexity comparison of incrementation operations · 0.94

Cites it as background 22 papers

Later, Draper [ref] proposed a quantum algorithm for summing integers stored in two quantum registers based on the quantum Fourier transform.

Quantum Algorithm based on Quantum Fourier Transform for Register-by-Constant Addition · Introduction · 0.83
Known as the Draper adder [ref] , this involves the application of a quantum Fourier transform (QFT), a sequence of structured $Z-$rotations to each qubit, followed by an inverse QFT (IQFT).

Truncated phase-based quantum arithmetic: error propagation and resource reduction · Introduction · 0.86
Quantum adders that operate in the quantum Fourier transform basis [ref] have also been used to implement modular addition, multiplication, and exponentiation.

High Performance Quantum Modular Multipliers · Prior Modular Multiplication Implementations · 0.83

Discussion

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