Abelian Hidden Shift Problem

• AHShP is the problem of recovering an unknown group element that translates one function to another on a finite abelian group.
• It leverages Fourier analysis to extract spectral properties like flatness and influence, which are critical in determining algorithm efficiency.
• The problem spans diverse formulations—including Boolean and complex-valued cases—leading to varied quantum and classical recovery approaches.

The Abelian Hidden Shift Problem (AHShP) is the problem of recovering an unknown group element $s$ that relates two functions on an abelian group by translation. In its standard form, one is given black-box access to $f_0,f_1:G\to S$ on a finite abelian group $G$, with the promise that $f_1(x)=f_0(x+s)$ for all $x\in G$, and the task is to determine $s$. Across the literature, this basic formulation appears in several closely related oracle models: Boolean functions on $(\mathbb{Z}_2)^n$, injective hidden shifts over general abelian groups, complex scalar- and vector-valued functions with oracle access to $\widehat f$, and state-based variants where the shift is encoded in relative translation symmetry. The common technical core is Fourier analysis over $G$, but the algorithmic behavior depends sharply on spectral flatness, injectivity, and the available access model (Gavinsky et al., 2011, Roetteler, 2016, Adonsou et al., 25 Jul 2025).

1. Formal statement and oracle models

In the standard finite-group version, $G$ is a finite abelian group written additively, $f_0,f_1:G\to S$0 is a finite set, and one is given oracle access to two functions $f_0,f_1:G\to S$1 such that there exists an unknown shift $f_0,f_1:G\to S$2 with

$f_0,f_1:G\to S$3

The objective is to recover $f_0,f_1:G\to S$4. When $f_0,f_1:G\to S$5 is injective, the hidden shift is uniquely determined, and for injective hidden shifts over cyclic groups one obtains the familiar connection to hidden subgroup formulations over semidirect products (Roetteler, 2016).

A particularly important specialization is the Boolean Hidden Shift Problem (BHSP) on $f_0,f_1:G\to S$6. There one is given oracle access to Boolean functions $f_0,f_1:G\to S$7 satisfying $f_0,f_1:G\to S$8 for a unique $f_0,f_1:G\to S$9, together with the promise that $G$0 has no nontrivial self-shift: if $G$1, then $G$2. The standard query oracle is

$G$3

and similarly for $G$4; a single-qubit $G$5 gate on the ancilla converts this into an effective phase oracle by phase kickback (Gavinsky et al., 2011).

Other formulations enlarge the function class or alter the available access. For finite abelian $G$6, complex-valued variants take $G$7 with

$G$8

and assume quantum access not only to $G$9 but also to the Fourier transform $f_1(x)=f_0(x+s)$0 of the unshifted function. In the bent scalar case, phase oracles

$f_1(x)=f_0(x+s)$1

are available via additive-oracle constructions (Adonsou et al., 25 Jul 2025).

A state-based formulation replaces function oracles by the states

$f_1(x)=f_0(x+s)$2

Under $f_1(x)=f_0(x+s)$3, one has $f_1(x)=f_0(x+s)$4, so the hidden shift becomes a relative translation between two states (Hinsche et al., 21 May 2025).

2. Fourier structure and identifiability

Fourier analysis is the organizing language of AHShP. For a finite abelian group $f_1(x)=f_0(x+s)$5 with character group $f_1(x)=f_0(x+s)$6, the unitary quantum Fourier transform acts by

$f_1(x)=f_0(x+s)$7

and the shift relation implies

$f_1(x)=f_0(x+s)$8

in the convention $f_1(x)=f_0(x+s)$9. Hidden-shift algorithms exploit this multiplicative character factor to isolate $x\in G$0 through interference or Fourier sampling (Adonsou et al., 25 Jul 2025).

In the Boolean case, one encodes $x\in G$1 by the $x\in G$2-valued function

$x\in G$3

with Fourier transform

$x\in G$4

Parseval’s identity gives $x\in G$5. Two further quantities are central: the autocorrelation

$x\in G$6

and the directional influence

$x\in G$7

The key identity proved for BHSP is

$x\in G$8

which identifies the probability mass of the Fourier spectrum on a codimension-one coset with the influence of direction $x\in G$9. The minimum influence

$s$0

then controls how rapidly Fourier samples yield linearly independent constraints on the hidden shift (Gavinsky et al., 2011).

Flat spectra are the extremal favorable case. For Boolean bent functions on $s$1, $s$2 for all $s$3, so Fourier labels are sampled uniformly. For abelian difference sets $s$4, Turyn’s theorem gives

$s$5

for every nontrivial character $s$6, producing the same constant-magnitude phenomenon on the nontrivial spectrum. In the complex-valued setting, bentness is expressed by

$s$7

in the scalar case, or by corresponding norm equalities in the vector-valued case (Roetteler, 2016, Adonsou et al., 25 Jul 2025).

These spectral conditions are not merely technical. They determine whether the Fourier-domain amplitudes can be “flattened” exactly, whether a direct reduction to an abelian hidden subgroup problem is available, and whether a constant number of samples can suffice.

3. Structured efficient regimes

Several important AHShP families admit especially direct quantum algorithms because their spectra are flat or nearly flat. For quadratic Boolean functions on $s$8, a bent quadratic $s$9 has a dual bent function $(\mathbb{Z}_2)^n$0, and the hidden shift can be recovered exactly by preparing a uniform superposition, querying the shifted function in phase, applying the Walsh–Hadamard transform, multiplying by the dual phase $(\mathbb{Z}_2)^n$1, and applying a final Walsh–Hadamard transform. In the full-rank quadratic case this outputs $(\mathbb{Z}_2)^n$2 with probability $(\mathbb{Z}_2)^n$3; in the rank-deficient case it recovers the shift up to the kernel. The same work gives an $(\mathbb{Z}_2)^n$4-query quantum algorithm for learning an unknown quadratic form, contrasting with the stated classical $(\mathbb{Z}_2)^n$5-query requirement, and extends the hidden-shift method to functions close to quadratics in the sense of large Gowers $(\mathbb{Z}_2)^n$6 norm (0911.4724).

Difference sets furnish a broader abelian class with flat nontrivial Fourier magnitude. If $(\mathbb{Z}_2)^n$7 is a $(\mathbb{Z}_2)^n$8-difference set and one is given the hidden shift of its characteristic function, a single-query correlation algorithm prepares the uniform superposition, imprints the membership function of $(\mathbb{Z}_2)^n$9 in phase, applies the QFT, normalizes the nontrivial Fourier amplitudes by a diagonal unitary

$\widehat f$0

and then applies the inverse QFT. The resulting state contains a spike at $\widehat f$1 with success probability

$\widehat f$2

Special cases include Paley difference sets, where the single-run success probability is $\widehat f$3; Hadamard difference sets, which reproduce shifted bent-function algorithms with success probability exactly $\widehat f$4; and Singer difference sets, for which

$\widehat f$5

and efficient implementations are available for $\widehat f$6 (Roetteler, 2016).

A different structured regime is $\widehat f$7. There the shift appears after Fourier sampling as a one-qubit phase state

$\widehat f$8

with $\widehat f$9 uniformly random. A Kuperberg-style phase-addition gadget combines such states so that the label becomes even in every coordinate, permitting division by $G$0 and a descent from $G$1 to $G$2. After $G$3 levels, the remaining phases are $G$4, and measurement in the $G$5 basis yields linear equations modulo $G$6. The resulting algorithm has time

$G$7

uses $G$8 oracle calls, classical space $G$9, and quantum space $G$0, with success probability at least $G$1 after amplification (Csáji, 2021).

These structured families show that AHShP is not a single algorithmic phenomenon. Bentness, flat nontrivial spectrum, and 2-adic phase structure each support different exact or near-exact mechanisms for recovering the shift.

4. Boolean hidden shifts beyond the bent case

The BHSP on $G$2 is notable because the efficient quantum procedure extends far beyond the bent case. Starting from $G$3, one prepares a uniform superposition over $G$4, queries $G$5 into an ancilla, applies a $G$6 gate to induce the phase $G$7, queries $G$8, and then applies $G$9 to the data register. The resulting state can be written as

$f_0,f_1:G\to S$00

where $f_0,f_1:G\to S$01. Measuring yields a pair $f_0,f_1:G\to S$02 satisfying the linear equation

$f_0,f_1:G\to S$03

over $f_0,f_1:G\to S$04, and the marginal distribution is

$f_0,f_1:G\to S$05

Repeated sampling therefore produces a linear system whose unique solution is the hidden shift once the sampled $f_0,f_1:G\to S$06-vectors span $f_0,f_1:G\to S$07 (Gavinsky et al., 2011).

The rate at which the span grows is governed by the minimum influence $f_0,f_1:G\to S$08. If the current span has codimension $f_0,f_1:G\to S$09 and is orthogonal to a nonzero $f_0,f_1:G\to S$10, then the probability that the next sample lies outside that span is exactly

$f_0,f_1:G\to S$11

This implies expected sample complexity $f_0,f_1:G\to S$12, and with amplitude amplification one obtains the main query bound: $f_0,f_1:G\to S$13 expected oracle queries, with success probability $f_0,f_1:G\to S$14 in the ideal noiseless model. Under a promise $f_0,f_1:G\to S$15, the query bound becomes

$f_0,f_1:G\to S$16

with success probability at least $f_0,f_1:G\to S$17 (Gavinsky et al., 2011).

For random Boolean functions, the same analysis yields an average-case quantum–classical separation. For any fixed $f_0,f_1:G\to S$18, the events $f_0,f_1:G\to S$19 are mutually independent with probability $f_0,f_1:G\to S$20, and a Chernoff bound shows that with probability at least $f_0,f_1:G\to S$21, every $f_0,f_1:G\to S$22 satisfies $f_0,f_1:G\to S$23. Hence $f_0,f_1:G\to S$24 with overwhelming probability, so the quantum algorithm runs in $f_0,f_1:G\to S$25 queries and $f_0,f_1:G\to S$26 time with bounded error, whereas any classical algorithm that solves a uniformly random BHSP instance with success probability at least $f_0,f_1:G\to S$27 needs $f_0,f_1:G\to S$28 oracle queries (Gavinsky et al., 2011).

The bent case sits at one extreme of this picture. When the spectrum is flat, the sampled equations are unbiased random linear constraints, $f_0,f_1:G\to S$29 samples suffice by Gaussian elimination, and with access to the dual bent function there is a one-query quantum algorithm. The general Boolean case, however, does not rely on subgroup periodicity; the paper explicitly notes that unlike the bent extremal case, arbitrary Boolean hidden shifts do not seem to allow a direct reduction to an abelian hidden subgroup problem (Gavinsky et al., 2011).

5. Generalizations with enhanced access: complex functions and state-based formulations

A distinct 2025 line of work studies hidden shifts for complex scalar- and vector-valued functions on finite abelian groups under the explicit assumption that the Fourier transform of the unshifted function is available as a quantum oracle. In the scalar bent case, the exact circuit

$f_0,f_1:G\to S$30

maps the uniform superposition to $f_0,f_1:G\to S$31, using the identity $f_0,f_1:G\to S$32; success probability is $f_0,f_1:G\to S$33, and each phase oracle can be implemented using two calls to additive oracles. For non-bent functions, the paper introduces boundedness conditions such as $f_0,f_1:G\to S$34-boundedness, with

$f_0,f_1:G\to S$35

and proves that the corresponding approximate algorithm succeeds with probability

$f_0,f_1:G\to S$36

More flexible $f_0,f_1:G\to S$37-bounded models use subset post-selection, and vector-valued extensions replace magnitudes by norms. A one-register refinement introduces tunable phases $f_0,f_1:G\to S$38 and can achieve $f_0,f_1:G\to S$39 under specific conditions on $f_0,f_1:G\to S$40 and the flatness of $f_0,f_1:G\to S$41, even for some non-bent functions (Adonsou et al., 25 Jul 2025).

This model changes the hidden-shift landscape. The crucial resource is access to $f_0,f_1:G\to S$42: by interfering $f_0,f_1:G\to S$43 with $f_0,f_1:G\to S$44, the algorithm removes dependence on the amplitude structure of $f_0,f_1:G\to S$45 and isolates the shift with a constant number of queries. The paper presents this as a model difference from standard AHShP, where one typically has black-box access only to $f_0,f_1:G\to S$46 and $f_0,f_1:G\to S$47 and must rely on subexponential methods or special-case structure (Adonsou et al., 25 Jul 2025).

A related but conceptually different reformulation appears in the abelian StateHSP framework. Here one fixes a finite abelian group $f_0,f_1:G\to S$48, a unitary representation $f_0,f_1:G\to S$49, and an unknown state $f_0,f_1:G\to S$50. The hidden symmetry subgroup $f_0,f_1:G\to S$51 is defined by the conditions $f_0,f_1:G\to S$52 for $f_0,f_1:G\to S$53 and $f_0,f_1:G\to S$54 for $f_0,f_1:G\to S$55. The character POVM

$f_0,f_1:G\to S$56

produces outcomes supported on the annihilator

$f_0,f_1:G\to S$57

and $f_0,f_1:G\to S$58 copies suffice to generate $f_0,f_1:G\to S$59 and reconstruct $f_0,f_1:G\to S$60 (Hinsche et al., 21 May 2025).

For hidden shifts, the same paper interprets $f_0,f_1:G\to S$61 and $f_0,f_1:G\to S$62 as a pair of translated states and uses the covariance identity

$f_0,f_1:G\to S$63

Conditioned on a Fourier outcome $f_0,f_1:G\to S$64, interference between the $f_0,f_1:G\to S$65 and $f_0,f_1:G\to S$66 branches leaves a control qubit in

$f_0,f_1:G\to S$67

so $f_0,f_1:G\to S$68- and $f_0,f_1:G\to S$69-basis measurements estimate $f_0,f_1:G\to S$70 and $f_0,f_1:G\to S$71. Under injective $f_0,f_1:G\to S$72, the Fourier label $f_0,f_1:G\to S$73 is uniform over $f_0,f_1:G\to S$74, and the shift can then be reconstructed from character evaluations by linear algebra over the invariant-factor decomposition of $f_0,f_1:G\to S$75 (Hinsche et al., 21 May 2025).

These generalizations emphasize that “AHShP” encompasses materially different computational models. Constant-query exact recovery, polylogarithmic postprocessing, and state-based symmetry learning all become possible once one strengthens the form of access beyond the standard pair of black-box function oracles.

6. Relation to hidden subgroup problems, classical algorithms, and current boundaries

AHShP is closely related to hidden subgroup problems, but the relation is model- and structure-dependent. In abelian groups, a hidden shift problem can be reduced to a hidden subgroup problem over $f_0,f_1:G\to S$76 by defining a function on the semidirect product; in the cyclic case, the hidden shift over $f_0,f_1:G\to S$77 is equivalent to the dihedral HSP. Difference-set algorithms exploit precisely this interface in injective or effectively injectivized settings, and the 2025 infinite-group work states the same reduction for abelian groups and extends it to a wreath-product reduction in general groups (Roetteler, 2016, Kuperberg, 24 Jul 2025).

At the same time, some of the most informative AHShP results arise where direct HSP reduction is absent or unhelpful. The Boolean hidden shift algorithm for arbitrary $f_0,f_1:G\to S$78 works with highly non-injective functions and is governed by Fourier mass and influence rather than subgroup periodicity. The paper explicitly contrasts this with the bent case, where reduction to an abelian HSP with a quantum hiding function is available (Gavinsky et al., 2011).

The present boundary between efficient and subexponential algorithms is visible in the $f_0,f_1:G\to S$79 setting. For AHShP in $f_0,f_1:G\to S$80 with visible subgroup $f_0,f_1:G\to S$81 and a bound on the bit complexity of a representative of the shift, a 2025 paper outlines a stretched-exponential-time quantum algorithm based on approximate phase qubits, a collimation sieve, and final Fourier measurement over a finite subgroup $f_0,f_1:G\to S$82. The stated complexity is

$f_0,f_1:G\to S$83

quantum time, polynomial quantum space, $f_0,f_1:G\to S$84 classical space, and success probability at least $f_0,f_1:G\to S$85, with a corresponding corollary for HSP in finitely generated virtually abelian groups. The same work notes that some statistical estimates needed for a rigorous proof are omitted, and it frames optimization of the $f_0,f_1:G\to S$86 bound and sharper acceptance analyses as open directions (Kuperberg, 24 Jul 2025).

On the classical side, highly structured AHShP instances can nevertheless be tractable. The hidden shifted power problem over $f_0,f_1:G\to S$87, with oracle

$f_0,f_1:G\to S$88

is explicitly identified as an AHShP instance with template $f_0,f_1:G\to S$89, or equivalently with a multiplicative character oracle. For $f_0,f_1:G\to S$90 and $f_0,f_1:G\to S$91, the paper gives deterministic recovery algorithms using $f_0,f_1:G\to S$92 oracle calls and time

$f_0,f_1:G\to S$93

under assumptions on nonresidues, as well as a randomized algorithm with expected $f_0,f_1:G\to S$94 calls and expected time

$f_0,f_1:G\to S$95

with success probability at least $f_0,f_1:G\to S$96. These results do not contradict quantum speedups for generic AHShP; rather, they show that additive-combinatorial and analytic structure can make specific classical hidden-shift instances much easier than the general black-box problem (Bourgain et al., 2011).

Across these works, several limitations recur. Many algorithms assume exact oracle access and do not analyze noisy queries; the BHSP paper is explicit that oracle errors or noise are not analyzed. Efficient QFT implementations are typically assumed, as is coherent access to spectral data such as $f_0,f_1:G\to S$97 or the diagonal normalization $f_0,f_1:G\to S$98. In the difference-set framework, efficiency of $f_0,f_1:G\to S$99 is family-specific; in the complex-valued framework, access to $G$00 is the decisive extra assumption. This suggests a broad taxonomy of AHShP instances: cases with flat or controllably bounded spectra, cases with enriched spectral access, and generic cases where existing methods remain subexponential or depend on average-case structure (Gavinsky et al., 2011, Roetteler, 2016, Adonsou et al., 25 Jul 2025, Kuperberg, 24 Jul 2025).

In that sense, AHShP functions as a unifying problem class rather than a single algorithmic template. It interpolates between exact constant-query quantum recovery for bent or Fourier-oracle models, influence-governed average-case quantum speedups for Boolean functions, low-space polynomial-time algorithms in 2-adic product groups, specialized classical algorithms for multiplicative templates over finite fields, and stretched-exponential sieve methods in visible quotients of $G$01. A plausible implication is that the decisive invariant is not “abelianity” alone, but the extent to which the Fourier data of the hiding function can be accessed, flattened, or shown to anti-concentrate.