Quantum Summation: Concepts & Applications

• Quantum summation is a collection of methods where the central operation of addition is encoded into quantum phases, amplitudes, correlations, or diagrammatic representations.
• It underpins techniques in secure multiparty computation, Hamiltonian simulation with ZXW calculus, and closed-form evaluation of spectral and perturbative series.
• Approaches range from oracle-based modular arithmetic and amplitude estimation to QFT-based adders, highlighting both query-optimized and architecture-oriented strategies.

Quantum summation is not a single problem but a cluster of quantum and quantum-inspired constructions in which addition is the central operation. In the literature considered here, the summed object may be an oracle string $f:Z_n\to Z_k$, a family of private integers held by multiple parties, a Hamiltonian decomposition $H=\sum_i H_i$, or coefficients extracted from Hilbert-space expansions and divergent perturbation series. Accordingly, quantum summation may mean exact modulo-$k$ evaluation of $\sum_x f(x)$, privacy-preserving distributed aggregation, diagrammatic manipulation of $e^{-iHt}$, or closed-form recovery of infinite series from spectral data (Meyer et al., 2011, Yang et al., 2022, Shaikh et al., 2022, Mali et al., 2023, Zinn-Justin, 2010).

1. Principal meanings of the term

The cited literature uses “quantum summation” in several technically distinct senses. At the oracle level, the task is to compute

$S=\sum_{x=0}^{n-1} f(x)\pmod{k},$

given black-box access to $U_f$ (Meyer et al., 2011). In secure computation, the task is to recover a global sum such as

$S=\sum_{i=1}^m x^{(i)} \pmod{2^n}$

or a component-wise sum modulo $d$, while revealing minimal information about the inputs (Otero et al., 2023, Yang et al., 2022). In diagrammatic quantum dynamics, the task is to represent $H=\sum_i H_i$ and exponentials such as $H=\sum_i H_i$0 entirely inside ZXW calculus (Shaikh et al., 2022). In mathematical physics, “summation” can denote closed-form evaluation of convergent or divergent series by quantum-mechanical constructions, either from Hilbert-space completeness or from resummation of perturbative expansions (Mali et al., 2023, Zinn-Justin, 2010).

Setting	Summed object	Output form
Oracle/query model	$H=\sum_i H_i$1	Quantum algorithm output
Secure multiparty computation	$H=\sum_i H_i$2 or $H=\sum_i H_i$3	Distributed or server-recovered sum
ZXW calculus	$H=\sum_i H_i$4, $H=\sum_i H_i$5	Diagrammatic representation
Hilbert-space series generation	$H=\sum_i H_i$6 or related series	Closed-form identity
Divergent-series resummation	$H=\sum_i H_i$7	Convergent approximant

A plausible implication is that “quantum summation” functions less as a single algorithmic primitive than as a recurring structural motif: linear aggregation is encoded into phases, amplitudes, correlations, or diagrammatic composition, and then read out by measurement, rewriting, or analytic continuation.

2. Hamiltonian summation and exponentiation in ZXW calculus

In ZXW calculus, a wide class of sums of linear operators, including arbitrary qubits Hamiltonians, can be represented directly by controlled diagrams. If $H=\sum_i H_i$8 denotes the controlled diagram for $H=\sum_i H_i$9, Proposition 3.2 gives a controlled representation of

$k$0

using a top W-spider with one control wire and branches weighted by the coefficients $k$1. Feeding $k$2 or $k$3 into the control yields the weighted sum $k$4. The relevant addition mechanism is supported by W-spider fusion together with associativity and commutativity, so nested and reordered sums behave diagrammatically as expected (Shaikh et al., 2022).

The same paper uses this summation formalism to express the linearity of the time-dependent Schrödinger equation,

$k$5

States are interpreted in ZXW normal form, differentiation is implemented by a ZXW differentiation gadget, and a weighted 2-leg W-spider shows that if $k$6 and $k$7 solve the equation, then $k$8 also solves it. The framework is also used to give a diagrammatic representation of the Hamiltonian in Greene-Diniz et al., described there as the first paper that models carbon capture using quantum computing (Shaikh et al., 2022).

Exponentiation is obtained in principle from the Cayley–Hamilton theorem. For $k$9 with minimal polynomial of degree $\sum_x f(x)$0,

$\sum_x f(x)$1

Each power $\sum_x f(x)$2 is encoded by iterated serial composition of the controlled-$\sum_x f(x)$3 diagram, and a W-spider forms the linear combination. The same formalism yields a truncated Taylor expansion,

$\sum_x f(x)$4

and first-order Trotterization for $\sum_x f(x)$5,

$\sum_x f(x)$6

The paper’s conclusion states that ZXW calculus thereby admits a compact diagrammatic sum, a direct diagrammatic exponentiation via Cayley–Hamilton, a diagrammatic Taylor expansion, and diagrammatic Trotterization, with intended use in quantum chemistry and condensed-matter physics (Shaikh et al., 2022).

3. Spectral summation formulas and divergent-series resummation

One line of work derives nontrivial infinite sums from elementary quantum eigenproblems. For a particle in a one-dimensional infinite potential well of width $\sum_x f(x)$7, with $\sum_x f(x)$8 and $\sum_x f(x)$9, the time-independent Schrödinger equation

$e^{-iHt}$0

with boundary conditions $e^{-iHt}$1, has eigenvalues

$e^{-iHt}$2

and eigenfunctions

$e^{-iHt}$3

Completeness and orthonormality imply that any normalized $e^{-iHt}$4 with zero boundary conditions has coefficients

$e^{-iHt}$5

By choosing explicit trial states, the paper obtains closed-form series. For $e^{-iHt}$6, one finds

$e^{-iHt}$7

hence

$e^{-iHt}$8

For $e^{-iHt}$9, normalization yields

$S=\sum_{x=0}^{n-1} f(x)\pmod{k},$0

A further example is

$S=\sum_{x=0}^{n-1} f(x)\pmod{k},$1

The general framework starts from any exactly solvable quantum system whose eigenfunctions form an orthonormal basis in $S=\sum_{x=0}^{n-1} f(x)\pmod{k},$2, computes overlaps in closed form, and uses $S=\sum_{x=0}^{n-1} f(x)\pmod{k},$3 to obtain summation formulas; convergence is secured because $S=\sum_{x=0}^{n-1} f(x)\pmod{k},$4 and the basis is complete (Mali et al., 2023).

A distinct use of summation arises in quantum field theory, where perturbative series

$S=\sum_{x=0}^{n-1} f(x)\pmod{k},$5

often have coefficients with factorial growth

$S=\sum_{x=0}^{n-1} f(x)\pmod{k},$6

and hence zero radius of convergence. Order-Dependent Mapping (ODM) introduces, at truncation order $S=\sum_{x=0}^{n-1} f(x)\pmod{k},$7, a mapping

$S=\sum_{x=0}^{n-1} f(x)\pmod{k},$8

followed by a re-expansion

$S=\sum_{x=0}^{n-1} f(x)\pmod{k},$9

and an order-dependent parameter $U_f$0 fixed by either

$U_f$1

or

$U_f$2

Heuristic steep-descent analysis attributed in the paper to Seznec–Zinn-Justin yields

$U_f$3

so that $U_f$4 geometrically fast in a sector of the complex $U_f$5-plane of opening angle $U_f$6. In the zero-dimensional $U_f$7 integral, one obtains

$U_f$8

with $U_f$9; numerically, at $S=\sum_{i=1}^m x^{(i)} \pmod{2^n}$0 the error is $S=\sum_{i=1}^m x^{(i)} \pmod{2^n}$1, and at $S=\sum_{i=1}^m x^{(i)} \pmod{2^n}$2 it is $S=\sum_{i=1}^m x^{(i)} \pmod{2^n}$3. The paper also notes rigorous convergence proofs, including full convergence on the entire Riemann surface in the special $S=\sum_{i=1}^m x^{(i)} \pmod{2^n}$4 case by Guida–Konishi–Suzuki (Zinn-Justin, 2010).

4. Query models, amplitude estimation, and exact circuit adders

In the oracle model, quantum summation generalizes parity. For $S=\sum_{i=1}^m x^{(i)} \pmod{2^n}$5 with oracle

$S=\sum_{i=1}^m x^{(i)} \pmod{2^n}$6

the goal is to compute

$S=\sum_{i=1}^m x^{(i)} \pmod{2^n}$7

Meyer and Pommersheim show that for integers $S=\sum_{i=1}^m x^{(i)} \pmod{2^n}$8, there is an explicit algorithm using exactly $S=\sum_{i=1}^m x^{(i)} \pmod{2^n}$9 adaptive queries and succeeding with worst-case probability

$d$0

If $d$1, the algorithm succeeds with certainty. Even when the output is incorrect, it is within $d$2 of the true sum with probability at least $d$3. In the special case $d$4, this recovers the exact quantum parity result that $d$5 queries suffice (Meyer et al., 2011).

A later line of work formulates secure summation through amplitude estimation. There, a quantum summation oracle

$d$6

encodes a Boolean function into the amplitude of a single ancilla qubit. In the bitwise protocol, an index register is prepared in the uniform state

$d$7

and for each bit position $d$8 the parties sequentially encode their bits so that the probability of measuring $d$9 is

$H=\sum_i H_i$0

Standard quantum amplitude estimation, described there via Brassard–Høyer–Mosca–Tapp ’02, yields an estimate $H=\sum_i H_i$1, from which a classical bit-sum $H=\sum_i H_i$2 is reconstructed and carry-propagated to the full integer sum. The complexity formulas stated in that work are

$H=\sum_i H_i$3

The same source also describes an alternate QFT-based adder: an accumulator register of $H=\sum_i H_i$4 qubits is Fourier-transformed, each party applies a controlled adder $H=\sum_i H_i$5, and inverse QFT yields $H=\sum_i H_i$6; those adders decompose into $H=\sum_i H_i$7 controlled-phase gates (Sandhu et al., 26 Jun 2026).

These constructions show two complementary regimes. The multi-query algorithm targets query minimization for an abstract oracle problem, whereas the amplitude-estimation and QFT-adder constructions encode addition directly into amplitudes or phases of explicit circuits. This suggests a division between query-optimal and architecture-oriented formulations rather than a single universal protocol.

5. Secure multiparty quantum summation

Secure multiparty quantum summation is the most extensive use of the term in the cited corpus. Here the standard goal is to compute a modulo sum while preventing disclosure of individual inputs, subject to assumptions about outside eavesdroppers, dishonest participants, or a semi-trusted server or third party.

Protocol family	Main quantum resource	Stated feature
Tree-type QFT summation	$H=\sum_i H_i$8-level GHZ-type states	Secret-by-secret modulo-$H=\sum_i H_i$9 addition (Yang et al., 2022)
Improved GHZ-like summation	GHZ-like states plus genuineness checks	No participant QFT or $H=\sum_i H_i$00 (Zhang et al., 2019)
Teleportation-based summation	Bell pairs on a cycle	Congenitally free from Trojan horse attacks (Zhang et al., 2019)
Entanglement-swapping summation	$H=\sum_i H_i$01-level cat states and Bell states	Semi-trusted TP reconstructs $H=\sum_i H_i$02 (Chang et al., 2021)
No-TP Bell-state summation	$H=\sum_i H_i$03-dimensional Bell states	Improved collusion resistance (Li et al., 2023)
Lightweight three-user summation	Single-particle states	No entanglement swapping, Pauli operations, CNOT, Hadamard gate, or pre-shared key (Ye et al., 2022)
Three-party semiquantum summation	Single qubits	One quantum user and two classical users (Jia-Li et al., 2022)

In the tree-type QFT protocol, $H=\sum_i H_i$04 prepares $H=\sum_i H_i$05 copies of an $H=\sum_i H_i$06-qudit GHZ-type state,

$H=\sum_i H_i$07

each party applies $H=\sum_i H_i$08 followed by the cyclic shift $H=\sum_i H_i$09, measures in the computational basis, and $H=\sum_i H_i$10 computes

$H=\sum_i H_i$11

The paper states that the protocol resists outside attacks and participant attacks, that one party cannot obtain other parties’ private integer strings, that it is secure against colluding attacks performed by at most $H=\sum_i H_i$12 parties, and that it computes addition in a secret-by-secret way rather than a bit-by-bit way (Yang et al., 2022).

Several papers expose weaknesses in earlier QFT/GHZ-like designs. One shows that the participant who prepares the initial quantum states can replace genuine entanglement by single-particle states $H=\sum_i H_i$13 and then recover every honest party’s secret by subtracting the known $H=\sum_i H_i$14 from publicly announced outcomes; the proposed fix is an extra entanglement-check round using random measurement bases $H=\sum_i H_i$15 and $H=\sum_i H_i$16 on $H=\sum_i H_i$17 shared states (Gu et al., 2019). Another identifies two state-generator attacks: a measure–reprepare attack in the Fourier basis and a partial QFT-entanglement attack. Its improved protocol introduces a random entanglement-genuineness test, then replaces participant-side $H=\sum_i H_i$18 operations by Fourier-basis measurements plus classical masks $H=\sum_i H_i$19, so that participant gate complexity is reduced from $H=\sum_i H_i$20 to $H=\sum_i H_i$21 classical operations (Zhang et al., 2019).

Other protocols avoid that particular architecture. In the teleportation-based scheme, a malicious but non-collusive third party prepares Bell pairs on the cycle $H=\sum_i H_i$22–$H=\sum_i H_i$23–$H=\sum_i H_i$24–$H=\sum_i H_i$25–$H=\sum_i H_i$26, participants encode bits with either $H=\sum_i H_i$27 or the identity, Bell-state measurements propagate the parity, and the final output is

$H=\sum_i H_i$28

The authors state that because encoded qubits never leave their owners, the protocol is congenitally free from Trojan horse attacks (Zhang et al., 2019). In the entanglement-swapping scheme, each secret $H=\sum_i H_i$29 is split as $H=\sum_i H_i$30, embedded into Bell states by $H=\sum_i H_i$31, and combined with $H=\sum_i H_i$32-level cat states; the semi-trusted TP reconstructs

$H=\sum_i H_i$33

from collapsed cat-state parameters and public $H=\sum_i H_i$34 values (Chang et al., 2021).

Third-party-free variants also appear. The improved $H=\sum_i H_i$35-party Bell-state protocol for $H=\sum_i H_i$36 doubles each component as $H=\sum_i H_i$37, inserts a dedicated checking phase with random checking qudits, and derives

$H=\sum_i H_i$38

Its qubit-efficiency estimate is

$H=\sum_i H_i$39

for small $H=\sum_i H_i$40, and the paper states that the detection phase is designed to block the collusion loophole in Wu et al.’s original circle-type scheme (Li et al., 2023). The lightweight three-user protocol uses only single-particle states chosen from $H=\sum_i H_i$41, Bell measurements by Alice, and public one-time-pad strings

$H=\sum_i H_i$42

with correctness following from

$H=\sum_i H_i$43

Its qubit-efficiency tends to $H=\sum_i H_i$44 for large $H=\sum_i H_i$45 (Ye et al., 2022). The semiquantum variant replaces two of the users by classical participants restricted to CTRL or SIFT operations on single qubits, again using Bell-grouping and masked announcements to recover $H=\sum_i H_i$46 (Jia-Li et al., 2022).

6. Noise, experiments, and structural relations to other tasks

Network noise strongly constrains distributed summation. In the Distributed QFT-Based Adder (DQA), each of $H=\sum_i H_i$47 parties and a server share one $H=\sum_i H_i$48 pair for every logical qubit. The server applies $H=\sum_i H_i$49 to its output register, remote controlled-$H=\sum_i H_i$50 rotations accumulate the phases corresponding to each input, and $H=\sum_i H_i$51 returns the output register to $H=\sum_i H_i$52, where

$H=\sum_i H_i$53

Under dephasing or depolarising noise on the shared entanglement, each server qubit has density operator

$H=\sum_i H_i$54

with

$H=\sum_i H_i$55

After inverse QFT, the measurement probabilities are

$H=\sum_i H_i$56

so for the correct outcome $H=\sum_i H_i$57,

$H=\sum_i H_i$58

The paper concludes that for any fixed $H=\sum_i H_i$59, $H=\sum_i H_i$60 decreases exponentially in $H=\sum_i H_i$61, that no sharp threshold in $H=\sum_i H_i$62 exists, and that the most likely wrong outcome flips the most significant bit first. It also removes the trusted server assumption by using Shamir secret sharing, at the cost of $H=\sum_i H_i$63 DQA rounds and $H=\sum_i H_i$64 Bell-pair uses overall (Otero et al., 2023).

Experimental realization has so far been limited in scale. On the IBM ibmqx2 five-qubit processor, secure multiparty quantum summation for one-qubit secret states was implemented with 8192 shots in three representative cases: $H=\sum_i H_i$65 with $H=\sum_i H_i$66; $H=\sum_i H_i$67 with $H=\sum_i H_i$68; and $H=\sum_i H_i$69 with $H=\sum_i H_i$70. Reported state-tomography fidelities were $H=\sum_i H_i$71, $H=\sum_i H_i$72, and $H=\sum_i H_i$73, respectively. The same work proposed square and cubic summation by locally replacing $H=\sum_i H_i$74 with $H=\sum_i H_i$75 or $H=\sum_i H_i$76, and on IBM’s Custom Topology simulator obtained unit-probability outputs for the examples $H=\sum_i H_i$77 modulo $H=\sum_i H_i$78: square sum $H=\sum_i H_i$79 and cubic sum $H=\sum_i H_i$80 (Majumder et al., 2017).

A broader structural claim appears in recent work on auction protocols. There, revenue estimation, threshold testing, maximum-bid identification, and winner determination are reduced to repeated invocations of a quantum summation oracle acting on indicator functions. Conversely, summation protocols are embedded as auxiliary subroutines within auction frameworks. The analysis is stated to be protocol-agnostic and applicable across gate-based and photonic implementations, and a proof-of-concept numerical validation of a two-bidder sealed-bid auction is reported on IBM optical quantum hardware (Sandhu et al., 26 Jun 2026).

Taken together, these results show that quantum summation is a recurrent primitive rather than a narrow protocol class. It appears in query complexity, secure distributed computation, Hamiltonian simulation, spectral analysis, and perturbative resummation. What changes across these settings is not the centrality of addition, but the representation in which the sum becomes accessible: oracle phases, ancilla amplitudes, multipartite correlations, ZXW diagrams, or normalized expansion coefficients.