REVIEW
Is Grover's Algorithm a Quantum Hidden Subgroup Algorithm ?
Quantum Physics · Submitted 15 Mar 2006 · 32 references
Pith's verdict: A neat conceptual observation—Grover's search as an HSP on S_N—with a solid negative claim about Fourier sampling, but a real gap in the push construction that is patchable.
T0 · AI review
Why Pith says this
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Referee report
The paper
The paper claims
Grover's unstructured search can be rephrased as a non-abelian hidden subgroup problem on the symmetric group, and the standard hidden-subgroup algorithm provably cannot solve it.
The reading
What carries the argument
The carrying object is the hidden subgroup structure of Grover's oracle: a function $\phi:S_N\to S$ whose fibers are cosets of the stabilizer $\operatorname{Stab}_{j_0}$, together with the transversal $\tau((0\,\ell)\operatorname{Stab}_0)=(0\,\ell)$ used to push the HSP through the coset space $S_N/\operatorname{Stab}_0$. The comparison also relies on the generic Fourier-sampling subroutine QRand, the standard QHS algorithm whose failure is diagnosed through the normal-subgroup obstruction and the conjugacy of the stabilizers.
Extended reading notes
Core claim
The paper's central claim is that Grover's oracle can be viewed as a map $\phi:S_N\to\{0,\dots,N-1\}$ with hidden subgroup structure, where the hidden subgroup is the stabilizer $\operatorname{Stab}_{j_0}$ of the unknown marked label $j_0$. The cosets $(\ell\,j_0)\operatorname{Stab}_{j_0}$ for $\ell=0,\dots,N-1$ form a complete set of coset representatives, so recovering $\operatorname{Stab}_{j_0}$ is equivalent to recovering $j_0$. Using the pushing construction originally developed for infinite-group HSPs, the authors compare Grover's oracle with Shor's oracle and conclude both are pushed HSP oracles. They then show that the standard non-abelian QHS algorithm cannot solve the Grover HSP: Fourier sampling on $S_N$ would at best recover the largest normal subgroup inside $\operatorname{Stab}_{j_0}$, which is trivial, and the mutually conjugate stabilizers are information-theoretically indistinguishable to this method. Thus Grover's algorithm solves an HSP that the generic QHS method provably fails on.
Load-bearing premise
The whole comparison leans on treating the set of cosets $S_N/\operatorname{Stab}_0$ as a valid setting for pushing the HSP, even though the stabilizer is not a normal subgroup and the quotient is not a group; if that step lacks justification, Grover's oracle and the pushed HSP are not known to be the same problem.
Editorial extensions
If this is right
- If the central claim is right, Grover's algorithm is not a stand-alone search heuristic but an instance of the hidden subgroup framework, bringing it under the same conceptual umbrella as Shor's algorithm.
- The standard Fourier-sampling QHS algorithm cannot serve as a general-purpose replacement for Grover, because it provably fails on the stabilizer problem that encodes unstructured search.
- Any modified QHS algorithm that does solve the Grover HSP cannot be asymptotically faster than Grover's algorithm, by the optimality of Grover's search.
- The same hidden-subgroup viewpoint may extend to Grover's algorithm for multiple marked labels, as the paper notes, and may be applicable to more general oracles that carry transitive symmetry actions.
- The failure mode identified here sharpens the question of what non-standard techniques are needed for non-abelian HSPs beyond Fourier sampling.
Reading between the lines
- A possible way to salvage the push comparison is to replace the non-normal stabilizer by its normal core (the largest normal subgroup contained in it); the resulting HSP would be a genuinely group-theoretic quotient, but it would no longer encode the full search item and would likely require a different oracle construction.
- The paper's diagnosis suggests a general principle: any oracle whose hidden subgroup is non-normal and whose conjugate versions are symmetric under a transitive group action will foil the standard QHS algorithm, so quantum speedups for such problems must come from non-Fourier methods.
- One could test the framework computationally for small $N$ by implementing the pushed oracle on cosets of $S_N$ and running a brute-force classical simulation of QRand; the simulation should show no $j_0$-dependent measurement statistics if the paper's claim is correct.
- The push relationship between Shor and Grover may be more than analogy: defining HSPs on quotient sets rather than quotient groups might supply a common language for other quantum algorithms whose oracles admit partial symmetry, such as amplitude amplification variants.
Original abstract
The arguments given in this paper suggest that Grover's and Shor's algorithms are more closely related than one might at first expect. Specifically, we show that Grover's algorithm can be viewed as a quantum algorithm which solves a non-abelian hidden subgroup problem (HSP). But we then go on to show that the standard non-abelian quantum hidden subgroup (QHS) algorithm can not find a solution to this particular HSP. This leaves open the question as to whether or not there is some modification of the standard non-abelian QHS algorithm which is equivalent to Grover's algorithm.
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.
The abstract, sentence by sentence
The arguments given in this paper suggest that Grover's and Shor's algorithms are more closely related than one might at first expect. Specifically, we show that Grover's algorithm can be viewed as a quantum algorithm which solves a non-abelian hidden subgroup problem (HSP). But we then go on to show that the standard non-abelian quantum hidden subgroup (QHS) algorithm can not find a solution to this particular HSP. This leaves open the question as to whether or not there is some modification of the standard non-abelian QHS algorithm which is equivalent to Grover's algorithm.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The arguments given in this paper suggest that Grover's and Shor's algorithms are more closely related than one might at first expect. | interpretation 0.41 | 0.55 | 0.28 | hedged possibility 0.99 |
| s1 | Specifically, we show that Grover's algorithm can be viewed as a quantum algorithm which solves a non-abelian hidden subgroup problem (HSP). | result 0.72 | 0.92 | 0.71 | asserted 1.00 |
| s2 | But we then go on to show that the standard non-abelian quantum hidden subgroup (QHS) algorithm can not find a solution to this particular HSP. | result 0.94 | 0.90 | 0.24 | asserted 1.00 |
| s3 | This leaves open the question as to whether or not there is some modification of the standard non-abelian QHS algorithm which is equivalent to Grover's algorithm. | future work 0.57 | 0.21 | 0.15 | hedged possibility 0.48 |
What was examined, and what was not
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References
Discussion
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