Generalized Shor's Algorithm

• Generalized Shor’s Algorithm is a family of quantum period-finding techniques that extend the classic modular exponentiation method using compiled circuits and flexible register architectures.
• It employs iterative phase estimation, Fourier transforms, and tailored arithmetic operations to optimize resource usage and enhance scalability.
• The approach broadens applications in factoring and discrete logarithms by integrating robust post-processing methods and noise-resilient implementations.

Searching arXiv for recent and foundational papers on generalized variants and implementations of Shor’s algorithm. Generalized Shor’s algorithm denotes a family of extensions, reformulations, and implementation strategies that preserve Shor’s central reduction of factoring to order finding while modifying one or more of its structural layers: the modular-exponentiation oracle, the Fourier-processing stage, the register architecture, the initialization model, the classical post-processing, or the target algebraic problem. In its standard form, Shor’s algorithm factors an odd composite integer $N$ by finding the order $r$ of a coprime base $a$, where $a^r \equiv 1 \pmod N$, using a quantum routine built from superposition over exponents, modular exponentiation, and the quantum Fourier transform (QFT). Generalizations retain that period-finding backbone but broaden its operational scope: they may compile the modular arithmetic to a fixed instance, replace full control registers by iterative or distributed phase estimation, express modular multiplication through generalized discrete Fourier transforms, exploit divisors of the recovered period rather than only the half-period, extend the framework to discrete logarithms and short discrete logarithms, or analyze performance under arbitrary pure, pseudo-pure, and noisy initial states (Gamel et al., 2013, Patoary et al., 13 Mar 2025, Martin-Lopez et al., 2011, Liao et al., 11 Dec 2025, Ekerå et al., 2017, Ye et al., 16 Aug 2025).

1. Canonical order-finding structure

In the standard formulation, Shor’s algorithm factors an odd composite integer $N$ by reducing factoring to the problem of finding the multiplicative order of an integer $a$ coprime to $N$. The relevant periodic function is

$f_{a,N}(x) = a^x \bmod N,$

and the order $r$ is the smallest positive integer such that

$a^r \equiv 1 \pmod N.$

When $r$0 is even and $r$1, nontrivial factors can be obtained from

$r$2

The quantum subroutine prepares a superposition over exponents, applies modular exponentiation, performs an inverse QFT or an equivalent phase-estimation routine on the exponent register, and measures an outcome from which $r$3 is recovered by classical post-processing such as continued fractions (Gamel et al., 2013, Martin-Lopez et al., 2011).

A general implementation therefore consists of three universal blocks: a superposition over an exponent register, a modular-exponentiation unitary, and a Fourier-processing stage. One common formulation uses the modular multiplication operator

$r$4

whose eigenvalues are

$r$5

Quantum phase estimation then infers $r$6 from controlled powers of $r$7, and the order $r$8 is reconstructed classically (Patoary et al., 13 Mar 2025).

This canonical structure remains the reference point for most generalizations. Some variants preserve the mathematical reduction but alter the architecture of QFT and control registers; others preserve the oracle–QFT pattern but change how modular multiplication is synthesized; still others leave the quantum subroutine unchanged and generalize only the classical factor-recovery stage (Martin-Lopez et al., 2011, Patoary et al., 13 Mar 2025, Liao et al., 11 Dec 2025).

2. Compiled and instance-specific generalizations

A major practical generalization replaces uniform, instance-independent modular exponentiation by compiled circuits specialized to a fixed pair $r$9. Such compiled versions preserve the global logic of superposition, modular-exponentiation-type evolution, QFT, and measurement, but hard-code the actual truth table of $a$0 on the relevant domain, often only over one period. The principal motivation is that general modular exponentiation is the dominant bottleneck in Shor’s algorithm, whereas compiled circuits can drastically reduce both qubit count and gate count on small instances (Gamel et al., 2013).

For $a$1, the compiled mappings for $a$2 and $a$3 admit fully compiled forms in which the modular-exponentiation output is relabeled by a classical logarithm map, so that the quantum circuit reduces essentially to copying input bits to output bits. For $a$4 with $a$5, the outputs $a$6 can be relabeled to $a$7, reducing the number of output qubits and yielding a partially compiled circuit with 2 Toffolis and 6 CNOTs, and a fully compiled circuit that uses only two input qubits and two output qubits. For $a$8 with $a$9, the classical map $a^r \equiv 1 \pmod N$0 similarly compresses the output alphabet and reduces the gate count to 3 Toffolis and 7 CNOTs (Gamel et al., 2013).

The same line of work emphasizes that allowed periods are constrained by the Carmichael function

$a^r \equiv 1 \pmod N$1

for semiprimes $a^r \equiv 1 \pmod N$2, and that the order $a^r \equiv 1 \pmod N$3 must divide $a^r \equiv 1 \pmod N$4. This provides a systematic method for selecting small semiprimes and bases with desired periodic structure, thereby turning compiled Shor circuits into controlled benchmarks for specific order-finding instances rather than merely ad hoc demonstrations (Gamel et al., 2013).

An extreme form of compilation shows that, for any $a^r \equiv 1 \pmod N$5 with distinct odd primes $a^r \equiv 1 \pmod N$6, one can choose a base $a^r \equiv 1 \pmod N$7 such that $a^r \equiv 1 \pmod N$8, giving period $a^r \equiv 1 \pmod N$9. In this case the compiled circuit can be reduced to two coherent qubits and constant depth, or even mimicked by a coin toss at the level of output statistics. This does not constitute a scalable factoring procedure because the special base is constructed using the factorization of $N$0, but it sharply illustrates that experimental difficulty is governed by the period length rather than the numerical size of $N$1 (Smolin et al., 2013).

Compiled generalizations therefore do not alter the mathematical content of Shor’s algorithm; instead they interpolate between the abstract uniform algorithm and experimentally realizable resource-optimized subfamilies. This suggests a continuum from simple monoperiodic functions through compiled modular exponentiation for modest $N$2 to fully generic modular arithmetic (Gamel et al., 2013).

3. Resource-architectural generalizations

A second major axis of generalization concerns the implementation architecture of the order-finding circuit. One important example is iterative phase estimation with qubit recycling, where the $N$3-qubit control register is replaced by a single qubit that is recycled $N$4 times. This semi-classical QFT scheme uses a sequence of measurements and feed-forward operations, with each round contributing one bit of phase precision. For full-scale factoring, this reduces the total qubit count from approximately $N$5 to $N$6, at the cost of a polynomial increase in time while retaining the exponential speedup (Martin-Lopez et al., 2011).

The same work also generalizes the work register beyond qubits by encoding the active modular-exponentiation values in a qutrit. In the $N$7, $N$8, $N$9 experiment, only the values $a$0 occur, so the work register is implemented as a single qutrit with labels $a$1. This demonstrates that compiled instances can be represented in higher-dimensional systems when the reachable subset of computational states is much smaller than the full binary encoding (Martin-Lopez et al., 2011).

A different architectural generalization distributes phase estimation across two quantum computers. In the distributed order-finding algorithm, one machine estimates coarse bits of $a$2 and another estimates finer bits of the same quantity by applying phase estimation to a powered modular-multiplication unitary. Quantum teleportation of the work register ensures that both machines act on the same eigenstate and hence the same eigenphase. Classical post-processing then merges the partial estimates into a single rational approximation to $a$3. This reduces nearly $a$4 qubits per node relative to the traditional $a$5-bit order-finding algorithm and lowers the circuit depth of each computer, while preserving polynomial-time complexity and requiring $a$6 communication via teleportation (Xiao et al., 2022).

A further architectural variant is an $a$7-bit general-purpose implementation of period finding using explicit gate-level reversible arithmetic rather than compiled circuits. This design decomposes modular exponentiation into ripple-carry adders, comparators, restricted modulo blocks, modular addition, modular doubling, and modular multiplication, yielding $a$8 gate complexity for modular exponentiation. For classical $a$9 and $N$0, the design uses $N$1 qubits; for quantum $N$2, the estimate is $N$3 qubits. The significance of this construction is that it makes “generalized Shor” operationally precise: the architecture depends only on bit-length $N$4, not on a fixed demonstration instance (Davies et al., 2016).

These resource-architectural generalizations reveal that the Shor framework is not tied to the textbook “$N$5-qubit control plus $N$6-qubit work” layout. The same order-finding logic can be realized by recycling, by distributed phase estimation, by qudit encodings, or by scalable arithmetic blueprints, with different qubit–depth–communication trade-offs (Martin-Lopez et al., 2011, Xiao et al., 2022, Davies et al., 2016).

4. Fourier-centric and arithmetic generalizations

Another line of generalization recasts modular multiplication itself as a Fourier-structured operation. In a discrete-Fourier-transform-based construction, the modular multiplication operator

$N$7

is expressed as

$N$8

where $N$9 is the discrete Fourier transform over $f_{a,N}(x) = a^x \bmod N,$0 and $f_{a,N}(x) = a^x \bmod N,$1 is a phase-rescaled Fourier transform with matrix elements proportional to $f_{a,N}(x) = a^x \bmod N,$2. The key two-register phase operator is

$f_{a,N}(x) = a^x \bmod N,$3

which, combined with an ancilla equal superposition, realizes the generalized DFT blocks needed for modular multiplication (Patoary et al., 13 Mar 2025).

In this formulation, modular multiplication uses two $f_{a,N}(x) = a^x \bmod N,$4-qubit registers, equal-superposition preparation $f_{a,N}(x) = a^x \bmod N,$5, phase gates $f_{a,N}(x) = a^x \bmod N,$6, and a QFT-based reset routine. The resulting modular multiplication has gate complexity

$f_{a,N}(x) = a^x \bmod N,$7

and modular exponentiation built from $f_{a,N}(x) = a^x \bmod N,$8 such blocks has complexity $f_{a,N}(x) = a^x \bmod N,$9. The proposal is not asymptotically optimal relative to advanced multiplication schemes, but it is structurally attractive because the entire construction is expressed in terms of QFT-type modules and phase gates rather than bespoke reversible arithmetic (Patoary et al., 13 Mar 2025).

A related hardware-oriented proposal uses QFT-based Draper adders to build multiplier/accumulator blocks and modular dividers, yielding a modular exponentiation circuit with depth about $r$0, $r$1 qubits, and total quantum cost $r$2. The depth can be decreased by more than three times if approximate QFT is used for each adder. This is again a generalization at the level of arithmetic synthesis rather than algorithmic reduction: the order-finding problem is unchanged, but the arithmetic oracle is re-engineered for low depth (Pavlidis et al., 2012).

A more radical Fourier reinterpretation replaces multiplicative order finding in $r$3 by phase order finding in an additive phase group modulo $r$4. In that formulation, modular exponentiation is replaced by a phase-based unitary

$r$5

with $r$6 defined from $r$7 and $r$8. Controlled phase gates and inverse QFT then recover the relevant period information. This suggests a generalized Shor template in which the arithmetic oracle is encoded directly in a phase group and the algorithm becomes a phase-estimation routine over that representation (Fleury et al., 2022).

Taken together, these Fourier-centric variants suggest that the distinction between the “arithmetic” and “Fourier” parts of Shor’s algorithm is not fundamental. Modular multiplication itself can be decomposed into DFTs, rescaled phases, and diagonal operators, so generalized Shor can be viewed as a family of Fourier-based order-finding constructions rather than a single arithmetic circuit design (Patoary et al., 13 Mar 2025, Fleury et al., 2022).

5. Post-processing and algebraic generalizations

Generalizations need not change the quantum routine at all; they may instead broaden the classical use of the recovered period. A notable example is generalized period decomposition, in which the order $r$9 returned by quantum phase estimation is not discarded when the standard half-period condition fails. Rather than requiring $a^r \equiv 1 \pmod N.$0 to be even and $a^r \equiv 1 \pmod N.$1, the method factors $a^r \equiv 1 \pmod N.$2 and tests arbitrary prime divisors $a^r \equiv 1 \pmod N.$3 through

$a^r \equiv 1 \pmod N.$4

The corresponding “All-$a^r \equiv 1 \pmod N.$5” algorithm leaves the quantum subroutine unchanged and modifies only the post-processing (Liao et al., 11 Dec 2025).

The stated motivation is that canonical Shor discards many useful quantum runs because it checks only the half-period. The generalized method instead traverses all distinct prime divisors of $a^r \equiv 1 \pmod N.$6, and when the base is a perfect square $a^r \equiv 1 \pmod N.$7, also tests the fallback

$a^r \equiv 1 \pmod N.$8

Across more than 1,000,000 simulated test cases on 2–8 digit semiprimes, the reported success rates exceed $a^r \equiv 1 \pmod N.$9 for 7-digit numbers and $r$00 for 8-digit numbers, while preserving polynomial-time complexity and requiring only modest additional classical gcd computations (Liao et al., 11 Dec 2025).

A different algebraic generalization embeds Shor’s approach within a quantum version of Pollard’s Rho. In that framework, factoring proceeds by analyzing cycle lengths of an iterated function $r$01 modulo the prime factors of $r$02. If the cycle lengths modulo $r$03 and $r$04 differ, there exists $r$05 such that

$r$06

Choosing the linear family $r$07 yields $r$08, exactly the periodic function used in Shor’s algorithm. On this reading, Shor is a particular case of a broader quantum Pollard framework in which quantum period finding is applied to structured iterates $r$09 (Bastos et al., 2020).

The discrete-logarithm setting provides another natural generalization. In cyclic groups of known order $r$10, one considers the state

$r$11

and applies QFTs to registers encoding $r$12 and $r$13. A semi-classical-QFT-compatible modification replaces ranges $r$14 by powers of two, enabling qubit recycling while preserving the discrete-logarithm information. The resulting expected single-run success probability is heuristically about $r$15 to $r$16; with modest extra group operations or a limited classical search, it can exceed $r$17, and tends to one as $r$18 (Ekerå, 2019).

Short discrete logarithms further specialize this structure by assuming $r$19. A multi-run lattice-based generalization then reduces the exponent length in the modular exponentiation stage. In the RSA-factoring application of that framework, Shor’s $r$20-bit exponent length is replaced by one only slightly more than $r$21 bits. The quantum core remains QFT-based period extraction, but the problem is recast using side information into a short-discrete-logarithm instance, trading quantum register size for multiple runs and classical lattice post-processing (Ekerå et al., 2017).

These post-processing and algebraic generalizations show that “generalized Shor” is not limited to modifying the period-finding circuit itself. The same quantum subroutine can support richer classical extraction strategies and broader algebraic targets, including discrete logarithms, short discrete logarithms, order-finding with side information, and Pollard-style cycle frameworks (Liao et al., 11 Dec 2025, Ekerå, 2019, Ekerå et al., 2017, Bastos et al., 2020).

6. Generalized initialization, noise, and resource-theoretic perspectives

A further notion of generalized Shor’s algorithm concerns the initial state and noise model rather than the arithmetic task. One such formulation allows the control register $r$22 to begin in an arbitrary pure state

$r$23

instead of the uniform Hadamard state, while keeping modular exponentiation and inverse QFT unchanged. The resulting probability of measuring $r$24 is

$r$25

A lower bound on the total success probability is

$r$26

where $r$27, showing that efficiency is retained provided the amplitude distribution is not too sparse (Ye et al., 16 Aug 2025).

The same work generalizes the input to pseudo-pure states on the joint register $r$28,

$r$29

and derives the exact relation

$r$30

Thus pseudo-pure Shor behaves as an attenuated version of the pure-state algorithm, with success probability approximately scaled by $r$31 when $r$32 is large (Ye et al., 16 Aug 2025).

Under a depolarizing channel, the final noisy success probability obeys the lower bound

$r$33

so the useful part of the success probability scales with $r$34. This quantifies how generalized initialization and decoherence degrade order finding without changing the algorithmic structure (Ye et al., 16 Aug 2025).

A complementary resource-theoretic study analyzes coherence and entanglement dynamics across the standard steps of Shor’s algorithm. After Hadamards on the first register, geometric coherence is

$r$35

with zero geometric entanglement. Modular exponentiation preserves the three coherence measures studied but generates entanglement. The inverse QFT reduces coherence to

$r$36

while entanglement further increases or changes depending on Hamming-weight structure. The overall effect is that Shor’s algorithm depletes coherence and produces entanglement, with coherence scaling governed initially by the register dimension $r$37 and finally by the order $r$38 (Ye et al., 8 Apr 2026).

These results imply that generalized Shor algorithms can also be understood as variants in coherence budget, entanglement production, and tolerance to initialization and noise. This suggests a broader interpretation in which order finding is a structured conversion of basis coherence into useful entanglement and phase information, rather than only a number-theoretic routine (Ye et al., 16 Aug 2025, Ye et al., 8 Apr 2026).

7. Significance and scope of the generalized framework

The accumulated literature indicates that generalized Shor’s algorithm is not a single algorithm but a stratified research program. At the most conservative end, it denotes generic scalable realizations of the original order-finding circuit for arbitrary $r$39, with explicit arithmetic and QFT resources (Davies et al., 2016). At the experimental end, it denotes compiled families of instance-specific circuits that preserve period-finding structure while sacrificing uniformity (Gamel et al., 2013, Smolin et al., 2013). At the architectural end, it includes iterative, qudit-based, hybrid, and distributed implementations that preserve the reduction while changing qubit count, depth, or communication (Martin-Lopez et al., 2011, Xiao et al., 2022).

At the algebraic end, generalized Shor encompasses discrete logarithms over arbitrary cyclic groups, short discrete logarithms, order finding with side information, and Pollard-style iterated-function frameworks where Shor’s original factoring algorithm appears as a special linear case (Ekerå et al., 2017, Ekerå, 2019, Bastos et al., 2020). At the classical-wrapper end, it includes post-processing schemes such as generalized period decomposition that dramatically raise practical success probabilities without altering quantum phase estimation (Liao et al., 11 Dec 2025). At the physical level, it includes generalized initialization, pseudo-pure-state variants, and noisy implementations whose performance can be bounded analytically (Ye et al., 16 Aug 2025).

A plausible implication is that the enduring core of Shor’s algorithm is narrower than its historical implementations suggest: a quantum routine that transforms periodicity or eigenphase structure into classical information about an order. Everything else—modular arithmetic synthesis, register organization, classical factor-recovery logic, initialization model, and target algebraic problem—admits meaningful generalization. On that reading, “generalized Shor’s algorithm” names a family of Fourier-based, period-finding quantum procedures unified by their exploitation of hidden cyclic structure, rather than a single fixed factoring circuit (Patoary et al., 13 Mar 2025, Liao et al., 11 Dec 2025, Ekerå et al., 2017).